UNIVERSITÀ DEGLI STUDI DI BARI
Dottorato di Ricerca in Matematica
XXIII Ciclo – A.A. 2009/2010
Settore Scientifico-Disciplinare:
MAT/06 – Probabilità
Tesi di Dottorato
On a new family of products for
smooth random variables
and their applications:
heat equation, stochastic integrals,
SDE’s and Hölder-Young
inequalities
Candidato:
Paolo DA PELO
Supervisori della tesi:
Prof. N. CUFARO PETRONI,
Dott. A. LANCONELLI
Coordinatore del Dottorato di Ricerca:
Prof. L. LOPEZ
iii
A mia moglie
Contents
Introduction
vii
1 Stochastic tools
1.1 Malliavin calculus . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Wick product and the space Gλ . . . . . . . . . . . . . . . . . .
1.3 Heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
4
6
11
2 Anti-Wick Product and heat equation
15
2.1 A new product for smooth random variables . . . . . . . . . . . 15
2.2 Application to the heat equation . . . . . . . . . . . . . . . . 26
3 Newton-Leibniz rule for stochastic integrals and
tions of SDE’s
3.1 α-products . . . . . . . . . . . . . . . . . . . . . .
3.2 A new representation for the Wick product . . . .
3.3 Stochastic integrals with generic evaluating point .
3.4 Newton-Leibniz rule . . . . . . . . . . . . . . . . .
3.4.1 Case 1 < α ≤ 2 . . . . . . . . . . . . . . . .
3.4.2 Case 0 < α < 1 . . . . . . . . . . . . . . . .
3.5 Wong-Zakai-type theorems . . . . . . . . . . . . . .
approxima.
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4 Sharp inequalities for Gaussian Wick products
55
4.1 A connection between the Gaussian Wick product and Lebesgue
convolution product . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2 Hölder-Young-Lieb inequality . . . . . . . . . . . . . . . . . . . 58
A Hermite polynomials
83
Introduction
In the present work we study a new product between random variables, the
anti-Wick product, a generalization of this product, the α-products, and their
connections with the heat equation and stochastic integration. We will show
some useful properties of these products which are at some extents very similar to some peculiar features of the Wick product. In particular we will show
a new representation for the heat equation, and we will prove, with several
results, an intrinsic connection between these products and the α-integrals,
i.e. stochastic integrals in which we fix the evaluating point of the Riemann
sums in tαi = ti + α2 (ti+1 − ti ). This link, at some extents, generalizes the
connection between Wick product and Itô integral to every α-integral.
The Wick product ”” was first introduced by Wick [34], in quantum field
theory, in order to renormalize some infinite quantitites, and to obtain useful
information. In stochastic analysis, according to [10], the Wick product was
first introduced by Hida and Ikeda [8]. Meyer and Yan extended to cover Wick
products of Hida distributions [26].
In the last decades Wick product has been applied to stochastic differential
equations, stochastic partial differential equations and many other fields. In
stochastic analysis, most of the research work on Wick product are on Hida distribution spaces or other spaces of generalized random variables [9, 10, 18, 29].
The Wick product provides a way to define a multiplication for singular white
noise, in fact the pointwise one does not make sense in Hida space.
The most important feature of the Wick product is its deep and natural connection with the Itô integral. In fact, the Wick product restores the fundamental theorem of calculus for the Itô integral [10]; moreover, stochastic
differential equations in which products are interpreted as Wick products,
possess a powerful solution technique via the S-transform. Hence the Wick
calculus with ordinary calculus rules is equivalent to Itô calculus governed by
the Itô formula.
viii
Introduction
It is well known (see for instance [10]) that for any t ∈ [0, T ] and n ≥ 1,
Btn := Bt · · · Bt
|
{z
}
n−times
=
hn,t (Bt ),
where hn,t denotes the n-th order Hermite polynomial with parameter t and
leading coefficient one. More generally, if g : R → R is a real analytic function
with expansion
X
an xn , x ∈ R,
g(x) =
n≥0
then
u(t, x) := g (Bt )|Bt =x
X
n an Bt =
n≥0
Bt =x
solves the backward heat equation
1
∂t u(t, x) + ∂xx u(t, x) = 0.
2
See [20] and the references quoted there.
Let us now briefly describe the contents of the present work, organized into
three chapters.
In the first chapter we give some stochastic tools that we will use along the
thesis. In particular we give some preliminary results about Malliavin calculus that are needed in the sequel, we introduce the spaces Gλ that will be our
framework, then we present the Wick product and we illustrate some of its
properties.
In the second chapter we introduce a new product for smooth random variables, defined in terms of Malliavin derivatives and named anti-Wick product.
We show some useful properties which are to some extents opposite to some
peculiar features of the Wick product and then we give some examples of antiWick products between random variables. Finally we show that the unique
solution of the heat equation
1
∂t u(t, x) − ∂xx u(t, x) = 0.
2
(0.0.1)
Introduction
ix
with initial condition f can be explicitly represented in terms of this multiplication and the data f . We also show that the solution of (0.0.1) with initial
data f · g can be written as the anti-Wick product of the two solutions with
data f and g, respectively.
In the third chapter, we introduce the α-products, a generalization of the
anti-Wick product. We show that the limit of the α-products for α → 0 is the
Wick product. Then we prove the fundamental theorem of calculus for the
α-integrals and the α-versions of real functions. At the end we prove a version
of the Wong-Zakai theorem for the α-products.
In the last chapter we first show an important connection between the finite dimensional Gaussian Wick product and Lebesgue convolution product.
Then this connection will be used to prove an important Hölder inequality for
the norms of Gaussian Wick products. We also obtain an alternative proof
of Nelson hypercontractivity inequality, and prove a more general inequality
whose marginal cases are the Hölder and Nelson inequalities mentioned before.
Since the Gaussian probability measure exists even in the infinite dimensional
case, the above three inequalities can be extended, via a classic Fatou’s lemma
argument, to the infinite dimensional framework.
Acknowledgements
My deepest gratitude is for Dott. Lanconelli. He has been a great supervisor
and without him nothing of this thesis could have been written. I am a
physicist and he had the patience to bring me in the ordered mathematical
way of thinking. Most of this work comes from his intuition, his patience and
his great passion for his work. He’s a good friend, a good researcher and a
good man.
A great thanks is for Prof. Stan from Ohio State University. Thanks for
the ample hospitality he and his family gave me during my period of research
in Ohio in U.S.A. He’s a great mathematician, with a deep mathematical
knowledge. The last chapter of this work is the fruit of our collaboration and
his geniality.
I want to express my thanks to Prof. Cufaro for accepting me in this Ph.D.
and for his useful advices.
A special thank to my best mathematical friends, Marco and Lucio. Thanks
for your precious friendship, for our funny conversations, for sharing your life
with me, the work was not so heavy with you. And thanks to all my colleagues
for their frindly company.
At last, but not at least, my greatest thanks to my wife Mariangela, she
encouraged me so many times...she’s by my side.
Chapter 1
Stochastic tools
In this section we set the necessary tools for our work. For more information
we refer the reader to one of the books [14, 17, 30, 32, 33].
Let (Ω, F, P) be a complete probability space.
Definition 1.0.1. A one dimensional Brownian motion is a family of random
variables {Bt }0≤t≤T (·) on Ω with values in R such that:
1. B0 = 0
a.e.;
2. for almost all, ω ∈ Ω, Bt has continuous paths;
3. for all 0 ≤ t1 ≤ t2 ≤ · · · ≤ tn ≤ T , the increments Bt2 − Bt1 , . . . , Btn −
Btn−1 , are independent;
4. for all 0 ≤ s ≤ t ≤ T , Bt − Bs has normal distribution N (0, t − s)
The Brownian motion Bt induces a filtration on Ω, Ft , which is the σalgebra generated by the random variables Bs for every s ∈ [0, t] and assume
that FT = F.
In the same way a Brownian motion can be introduced as a Wiener process
in the probability space (C0 ([0, T ]), B(C0 ), µ) where C0 ([0, T ]) is called Wiener
space and it is the space of all continuous functions ω : [0, T ] → R such that
ω(0) = 0, B(C0 ) is the Borel σ-algebra on C0 and µ is the Wiener measure. It
is possible to show that the coordinate process Bt (ω) := ω(t) is a Brownian
motion with respect to the measure µ.
It is known that it is not possible to use the Brownian motion as integrator of
a Riemann-Stieltjes integral. In fact we have the following theorem [32]
2
Chapter 1. Stochastic tools
Theorem 1.0.2. Let g : [0, T ] → R have finite variation and let f : [0, T ] → R
be continuous. Denote {∆n }n∈N a sequence of partitions of the real interval
[a, b] and |∆n | the norm of the partition. Then the limit
Z
T
f (t)dg(t) := lim
X
|∆n |→0
0
f (τi )(g(ti ) − g(ti−1 ))
(1.0.1)
i=1
exists and is independent of the choice of the partitions and of the choice of
the evaluating points τi ∈ [ti−1 , ti ]. The (1.0.1) defines the Riemann-Stieltjes
integral of f with respect to g. Moreover if g ∈ C 1 ([a, b]) then
Z
T
Z
f (t)dg(t) =
0
T
f (t)g 0 (t)dt
0
Given a Brownian motion Bt (ω), we could fix ω and define in this way a
Riemann-Stieljes integral with B as integrator. But this is not possible since
B has not finite variation.
Thanks to Wiener and Itô we can still build a definition of stochastic integral
for deterministic functions (Wiener integral) and stochastic processes (Itô integral), by using B as integrator, but we can’t do that for each ω. We can do
that just as a convergence in the Hilbert space
Z
2
2
L (Ω) := {X : Ω → R with E[|X| ] :=
|X(ω)|2 dP (ω) < +∞},
Ω
or at most in probability.
In general this definition depends on the evaluating point τi ∈ [ti−1 , ti ]. In
fact it is known that the first notion of stochastic integral, the Itô integral of
an integrable stochastic process ut can be written in Riemann sums
Z
T
ut dBt = lim
0
|∆n |→0
X
uti−1 (Bti − Bti−1 )
inL2 (Ω)
i=1
and it corresponds to a choice of the evaluating point in the left end point
τi = ti−1 .
A different choice of the evaluating point produces a definition of a new different integral. For example, if τi = ti−12+ti , so we choose the mid point, we
obtain the Stratonovich integral, if τi = ti we find the backward integral and
so on.
3
Example 1.0.3. It is possible to show that (see for example [30, 17])
Rt
Bt2
t
• Itô integral
0 Bs dBs = 2 − 2
Rt
• Stratonovich integral
0
Rt
• Backward integral
0
Bs dBs1 =
Bt2
2
Bt2
2
+
Bs dBs2 =
t
2
The following theorem is an important result about density in L2 (Ω).
Theorem 1.0.4. The vector space generated by the set of linear combinations
of the random variables
n
Z
E(f ) := exp
T
0
1
f (t)dBt −
2
Z
T
f 2 (t)dt ,
o
f ∈ L2 ([0, T ])
0
is dense in L2 (Ω).
E(f ) are called stochastic exponentials and are very important random
variables, since, thanks to the good property of density in L2 (Ω), they are
very useful for proving several properties of random variables. Stochastic exponentials also belong to Lp for every p ≥ 1. Now we shall show a fundamental
theorem in stochastic analysis that provides an useful representation for random variables, called Wiener-Itô decomposition or Chaos decomposition .
Theorem 1.0.5 (Wiener-Itô theorem). Let X ∈ L2 (Ω). Then there exists
a unique sequence {hn }n≥0 where for every n ≥ 1, hn ∈ L2 ([0, T ]n ) are
symmetric functions, and h0 ∈ R, such that
X
X=
In (hn )
inL2 (Ω)
n≥0
where
Z
T
Z
tn
Z
t2
···
In (hn ) := n!
0
0
hn (t1 , t2 , . . . , tn )dBt1 . . . dBtn
0
are called multiple Wiener-Itô integrals or homogeneous chaos of order n.
Moreover
X
E[X 2 ] =
n!|hn |2L2 ([0,T ]n ) .
n≥0
4
Chapter 1. Stochastic tools
Proposition 1.0.6. Multiple Wiener-Itô integrals satisfy the following properties:
1.
E[In (hn )Im (gm )] =
n 0 n 6= m
n!hhn , gn iL2 ([0,T ]n )
n=m
2.
E[(In (hn ))2 ] = n!|hn |2L2 ([0,T ]n )
3.
In (fn⊗n ) = hn,|fn |2 2
L ([0,T ])
where h·, ·i denotes the inner product in L2 ([0, T ]) and hn,σ2 are the Hermite
polynomials of degree n and parameter σ 2 .
For a brief introduction to Hermite polynomials see the Appendix A.
Multiple Wiener-Itô integrals form an orthogonal basis of L2 (Ω).
Example 1.0.7. Let us show the chaos decomposition of some random variables:
Bt = I1 (1[0,T [ ), where 1 is the indicator function;
Bt2 = t + I2 (1⊗2
[0,T [ );
X f ⊗n E(f ) =
In
n!
n≥0
1.1
Malliavin calculus
This section contains preliminary results about Malliavin derivatives that are
needed in the sequel. We refer the reader to the book of Nualart [27].
Let I be the set of random variables F of the form
F = φ I1 (h1 ), . . . , I1 (hn )
where φ : Rn → R are functions belonging to Cp∞ , i.e. the space of of all
infinitely differentiable functions with derivatives |Dα φ| ≤ C(1 + kxkk ) for
any order α, h1 , . . . , hn ∈ L2 ([0, T ]).
5
1.1. Malliavin calculus
Definition 1.1.1. For F ∈ I, the Malliavin derivative DF of F is the stochastic process {Dt F }t∈[0,T ] defined by
Dt F :=
n
X
∂ φ I1 (h1 ), . . . , I1 (hn ) hi (t)
∂xi
t ∈ [0, T ] a.s.
(1.1.1)
i=1
D· F is in L2 (Ω × [0, T ]). For example the Malliavin derivative of chaos of
RT
first order I1 (h) is Dt I1 (h) = Dt 0 h(s)dBs = h(t). We also endow I with
the norm:
i 1
hZ T
2
2 21
kF k1,2 = (E[F ]) + E
|Dt F |2 dt
0
D1,2
Then
denotes the Banach space which is the completion of I with respect
to the norm k · k1,2 .
P
Theorem 1.1.2. Let F ∈ L2 (Ω) with Wiener-Itô decomposition F = n≥0 In (hn ).
Then
X
n · n!|hn |2L2 ([0,T ]) < +∞ converges.
F ∈ D1,2 if and only if
n≥1
Moreover, if this is the case we have
X
Dt F =
nIn−1 (hn (·, t))
n≥1
and
E
hZ
0
T
i X
|Dt F |2 dt =
n · n!|hn |2L2 ([0,T ]n )
n≥1
Example 1.1.3. Let us consider the stochastic exponential E(f ). The chaos
⊗n
P
decomposition is E(f ) = n≥0 In (fn! ) . It is easy to prove that E(f ) ∈ D1,2 .
We have that
Dt E(f ) =
X
nIn−1
f ⊗n (·, t) n!
n≥1
=
X
n≥1
In−1
f ⊗(n−1) (n − 1!)
= E(f ) · f (t)
· f (t)
6
Chapter 1. Stochastic tools
The following proposition provides a chain rule for the Malliavin derivative and then a Leibniz rule of differentiation for a product between random
variables.
Proposition 1.1.4. Let F ∈ D1,2 and ϕ : R → R with bounded derivative,
then ϕ(F ) ∈ D1,2 and Dt ϕ(F ) = ϕ0 (F )Dt F .
Moreover, if G ∈ D1,2 is such that F G ∈ D1,2 , then Dt (F G) = GDt F + F Dt G.
In the same way, we can define, for F ∈ I, and k ≥ 1
Dtk1 ,...,tk := Dt1 Dt2 . . . Dtk F.
Then Dk,2 denotes the completion of I with respect to the norm
1
kF kk,2 := (E[F 2 ]) 2 +
k
X
hZ
E
[0,T ]j
j=1
|Dtj1 ,...,tn F |2 dt1 · · · dtj
i 1
2
.
Stochastic exponentials belong to Dk,2 .
1.2
Wick product and the space Gλ
Let X and Y be elements of L2 (Ω) with chaos decomposition
X
X
X=
In (hn )
Y =
In (gn );
n≥0
n≥0
we define a new random variable, named the Wick product of X and Y and
denoted by X Y , as
X
X Y =
In (kn ),
n≥0
where
kn :=
n
X
ˆ n−j , n ≥ 0;
hj ⊗g
j=0
ˆ stands for the symmetric tensor product.
here the symbol ⊗
It is clear that F, G ∈ L2 (Ω) does not imply that F G is a well-defined object
in L2 , in general it does not happen.
Example 1.2.1.
Z
I1 (f ) I1 (g) = I1 (f ) · I1 (g) −
T
f (t)g(t)dt
0
7
1.2. Wick product and the space Gλ
then, given a Brownian motion B
Bt Bt = I1 (1[0,t[ ) I1 (1[0,t[ ) = Bt2 − t.
We can calculate the n-th Wick power of I1 (f ), since I1 (f ) I1 (f ) is still in
L2 (Ω), and we obtain
(I1 (f ))n = hn,|f |2 2
L ([0,T ])
(I1 (f )) = In (f ⊗n )
Another example is
E(f ) E(g) = E(f + g)
We list below some useful properties of the Wick product.
For arbitrary X, Y, Z ∈ L2 (Ω) we have
• X Y = Y X (commutative law)
• X (Y Z) = (X Y ) Z (associative law)
• X (Y + Z) = (X Y ) + (X Z) (distributive law)
For more details on the Wick product we refer to several reviews of white
noise [7, 12, 15] or the books [10, 16]. One of the most striking features of the
Wick product is its relation to Itô-Skorohod integration. This relation can be
expressed as
Z T
Z T
Yt dBt =
Yt Wt dt
(1.2.1)
0
0
where Wt is the singular white noise, it is defined in the Hida space and it
is the time derivative of Bt . Here, the left hand side denotes the Skorohod
integral of the stochastic process Yt , which coincides with the Itô integral if Y
is adapted, while the right hand side is to be interpreted as an integral with
values in the Hida distribution space S ∗ .
If f is a real analytic function with Taylor expansion
X
f (x) =
an xn , with an ∈ R,
n≥0
then the Wick version f (X) of a random variable X ∈ S ∗ is defined by
X
f (X) =
an X n ∈ S ∗ .
n≥0
8
Chapter 1. Stochastic tools
We can take the Wick version of the exponential function as example. It is
defined by
exp {I1 (f )} =
X I n (f )
1
n!
n≥0
=
X In (f ⊗n )
n≥0
n!
= E(f ),
so the Wick version of the exponential function is the stochastic exponential.
By applying the (1.2.1) to the Wick version of an analytic function, we restore
the rules of ordinary calculus, in particular we have
Z
T
f (Bt )dBt = F (Bt )
0
where f is the derivative of F . This relation explain why the Wick product
is so natural and important in stochastic calculus. It is also the key to the
fact that Itô calculus with ordinary multiplication is equivalent to ordinary
calculus with Wick multiplication. Let us remember that the same connection
is known between the pointwise multiplication and Stratonovich integral.
Another result that confirms the deep connection between Itô integral and the
Wick product is the Wong-Zakai theorem [35] adapted to the Wick product
by Hu and Øksendal [11]. Let us remember the first result due to Wong and
Zakai.
Consider an Itô stochastic differential equation of the form
dXt = b(t, Xt )dt + σ(t, Xt )dBt ,
t > 0 X0 = x ∈ R
(1.2.2)
where b and σ are Lipshitz continuous functions of at most linear growth.
Then we know that a unique, strong solution Xt exists. We can approximate
this equation and its solution. According to [10] let ρ ≥ 0 be a smooth function
C ∞ -differentiable on R with compact support on [0, T ] and such that
Z
ρ(t)dt = 1.
R
For k ∈ N define
φk (t) = kρ(kt)
for t ∈ R
and let
Wtk :=
Z
T
φk (s − t)dBs
0
9
1.2. Wick product and the space Gλ
for t ∈ R be a smoothed white noise process. As an approximation of the SDE
above we can solve the equation
dYtk = b(t, Ytk ) + σ(t, Ytk ) · Wtk
t > 0; Y0k = x
(1.2.3)
as an ordinary differential equation in t for each ω. Then, by the Wong-Zakai
theorem, we know that Ytk → Yt as k → ∞, uniformly on bounded t-intervals
for each ω, where Yt is the solution of the Stratonovich equation
dYt = b(t, Yt ) + σ(t, Yt ) ◦ dBt
t > 0; Y0 = x.
So the approximated equation (1.2.3) is not the right choice. Hu and Øksendal
[11] conjectured and proved that, in the quasilinear case (σ(t, Xt ) = σ(t)Xt ),
we recover the solution Xt of (1.2.2), if we replace the ordinary product in the
(1.2.3) by the Wick product.
Let us introduce our framework, the following family of Hilbert spaces of
smooth random variables that was defined by Potthoff and Timpel [31] and
further studied by Benth and Potthoff [3] ant others [2, 28].
For λ ≥ 1, let
n
o
X
X
Gλ :=
X=
In (hn ) ∈ L2 (Ω) :
n!λ2n |hn |2L2 ([0,T ]n ) < +∞ .
n≥0
n≥0
Note that G1 = L2 (Ω) and for 1 ≤ λ < µ, Gµ ⊂ Gλ ⊂ L2 (Ω). We also denote
\
G :=
Gλ .
λ≥1
The most representative element of G is E(f ) with f ∈ L2 ([0, T ]). In fact,
since
X f ⊗n E(f ) =
In
,
n!
n≥0
for any λ ≥ 1 one has
X λ2n
n≥0
n!
|f |2n
L2 ([0,T ]) < +∞.
It is known that
forX, Y ∈ G,
X Y ∈ G.
10
Chapter 1. Stochastic tools
If A : L2 ([0, T ]) → L2 ([0, T ]) is a bounded linear operator then its second
quantization operator Γ(A) : G → G is defined as
X
In (hn )
Γ(A)X = Γ(A)
n≥0
:=
X
In (A⊗n hn ).
n≥0
With this notation the space Gλ previously defined can be described as
Gλ = {X ∈ L2 (Ω) : kXkGλ := kΓ(λI)Xk2 < +∞},
where I stands for the identity operator on L2 ([0, T ]). In the sequel the operator Γ(λI) will be denoted simply by Γ(λ). More properties of the second
quantization operator in [13, 21]
Observe that
Γ(A)E(f ) = E(Af ).
In particular it is easy to see that for any k ≥ 1, and any λ > 1, Gλ ⊂ Dk,2 .
This consideration permits to write two important formulas that relate Wick
and ordinary products via Malliavin derivatives:
X (−1)n Z
X Y =
Dtn1 ,...,tn X · Dtn1 ,...,tn Y dt1 ...dtn ,
n!
n
[0,T ]
n≥0
X 1 Z
X ·Y =
Dn
X Dtn1 ,...,tn Y dt1 ...dtn .
n! [0,T ]n t1 ,...,tn
n≥0
These two formulas are present in the literature, formulas (3.8) and (3.9) in
Nualart and Zakai [28] and formulas (2.3) and (2.4) in Hu and Øksendal [11].
Hu and Yan [12] proved these formulas for X and Y having all Malliavin
derivatives, and with a convergence in L1 , while in Lanconelli [21] they were
proved for X, Y ∈ G with a convergence in G.
Now, a sufficient condition for X Y to be square integrable is provided
by the next theorem which was proved among other things in Kuo, Saito and
Stan [19], Theorem 9.
Theorem 1.2.2 (Kuo, Saito, Stan). If X ∈ G√u , Y ∈ G√v , where (1/u) +
(1/v) = 1, then X Y ∈ L2 (Ω). More precisely,
√
√
kX Y k ≤ kΓ( u)Xk · kΓ( v)Y k,
11
1.3. Heat equation
or equivalently,
kX Y k ≤ kXk√2 · kY k√2
.
In Lanconelli and Stan [22], a sufficient condition was proved for L1 and
L∞ :
Theorem 1.2.3 (Lanconelli, Stan). Let u and v be two positive numbers such
that (1/u) + (1/v) = 1. If X ∈ G√u , Y ∈ G√v , then
•
•
√
√
kX Y k1 ≤ kΓ( u)Xk1 · kΓ( v)Y k1 ,
√
√
kX Y k∞ ≤ kΓ( u)Xk∞ · kΓ( v)Y k∞ .
In the last chapter of this thesis we will show a stronger result, i.e. sharp
Lp -inequalities for Wick products for every p ≥ 1.
1.3
Heat equation
Now we shall give a brief introduction to the Heat Equation. In order to do
that we will follow entirely the approach given in the book of Prof. Pascucci
[32]. Let us introduce the following partial differential equation in [0, T ] × R
1
Lu := c ∂xx u(t, x) + b ∂x u(t, x) − au(t, x) − ∂t u(t, x) = 0
2
(1.3.1)
where c, b, a are real constants. This equation is a generic parabolic equation
with constant coefficients. For c = 1 and b = a = 0 we obtain the Heat
Equation
1
∂t u(t, x) = ∂xx u(t, x).
(1.3.2)
2
Let us consider the classical Cauchy problem for the operator L in (1.3.1)
n Lu = 0, x ∈ R, t ∈]0, T ]
u(0, x) = f (x), x ∈ R
(1.3.3)
where f ∈ Cb (R) is a continuous and bounded function on R, called initial
data of the problem.
Let us denote by C 1,2 the class of functions that have continuous second order
12
Chapter 1. Stochastic tools
derivatives in x and continue first order derivative in t.
A classical solution of the Cauchy problem is a function
u ∈ C 1,2 (]0, T ] × R) ∩ C([0, T ] × R)
that satisfies (1.3.3).
Definition 1.3.1. We call fundamental solution for L a function Γ(t, x), defined on ]0, T ] × R, such that, for every f ∈ Cb (R), the function defined by
n R Γ(t, x − y)f (y)dy, t ∈]0, T ], x ∈ R
R
u(t, x) =
f (x), t = 0, x ∈ R
is a classical solution of the Cauchy problem (1.3.3).
The heat equation has a fundamental solution and it is given by the Gaussian function
1 − x2
Γ(t, x) = √
e 2t , t > 0, x ∈ R.
(1.3.4)
2πt
Then
Z
Γ(t, x − y)f (y)dy
(1.3.5)
u(t, x) =
R
Z
(x−y)2
1
√
=
e− 2t f (y)dy, x ∈ R, t > 0,
2πt R
is a solution for the Cauchy problem associated to the heat equation.
Let us observe that we can enlarge the condition of boundedness on f in order
to keep the convergence of the integral in (1.3.5). In fact the condition
γ
|f (y)| ≤ c1 ec2 |y| ,
c1 , c2 ∈ R, γ < 2
(1.3.6)
is enough to achieve the convergence for every (t, x) ∈]0, T ] × R.
Now we shall give an important result about the uniqueness of the solution of
(1.3.3).
Theorem 1.3.2. There exists at most one classical solution u ∈ C 1,2 (]0, T ] ×
R) ∩ C([0, T ] × R) of the problem
n Lu = 0, x ∈ R, t ∈]0, T ]
(1.3.7)
u(0, x) = f (x), x ∈ R
such that
2
|u(t, x)| ≤ CeC|x| ,
for a certain positive constant C.
t ∈]0, T ], x ∈ R,
(1.3.8)
1.3. Heat equation
13
For a complete proof of the theorem we refer to the book of Prof. Pascucci
[32].
In the case of the heat equation, for f satisfying the (1.3.6), the (1.3.5) is
a classical solution of the Cauchy problem and it verifies the (1.3.8), then,
among the functions satisfying the (1.3.8), it is the unique one.
Moreover, from a probabilistic point of view, we can write the unique solution
(1.3.5) as
u(t, x) = E[f (Bt + x)]
where Bt is a standard Brownian motion.
14
Chapter 1. Stochastic tools
Chapter 2
Anti-Wick Product and heat
equation
2.1
A new product for smooth random variables
We begin introducing a family of products.
In the previous chapter we have introduced a particular property of the Wick
product which expresses the Wick product between two random variables as
the sum of pointwise products:
X (−1)n Z
X Y =
Dtn1 ,...,tn X · Dtn1 ,...,tn Y dt1 ...dtn .
(2.1.1)
n!
n
[0,T ]
n≥0
Now we want to use this property in order to define a new class of products
n
between random variables, particularly we want to replace the coefficient (−1)
n!
with a generic coefficient an and then to investigate which conditions on an
allow this new object to be a product.
Definition 2.1.1. Let ϕ : R → R be a real analytic function such that
ϕ(0) = 1 and with expansion,
X
ϕ(x) =
ak xk , x ∈ R.
k≥0
For X, Y ∈ G define their ◦ϕ -product as
X Z
X ◦ϕ Y :=
an
Dtn1 ,...,tn X · Dtn1 ,...,tn Y dt1 ...dtn .
n≥0
[0,T ]n
16
Chapter 2. Anti-Wick Product and heat equation
To ease the notation we will write from now on
Z
Dtn X · Dtn Y dt
[0,T ]n
to denote the quantity
Z
[0,T ]n
Dtn1 ,...,tn X · Dtn1 ,...,tn Y dt1 ...dtn .
Remark 2.1.2. Observe that for ϕ(x) = 1 we get X ◦ϕ Y = X · Y while for
ϕ(x) = e−x we restore the (2.1.1), so we get X ◦ϕ Y = X Y .
Due to the properties of the Malliavin derivative, the ◦ϕ -product is clearly
commutative and distributive with respect to the sum. We have required the
condition ϕ(0) = 1 so that X ◦ϕ Y reduces to X · Y in the case where X or Y
is constant.
The next theorem provides a necessary condition on ϕ for the associativity of
the corresponding ◦ϕ -product.
Theorem 2.1.3. If the ◦ϕ -product is associative then ϕ(x) = eλx for some
λ ∈ R.
Proof. Assume that the ◦ϕ -product is associative; this means that for any
X, Y, Z ∈ G such that X ◦ϕ Y, Y ◦ϕ Z ∈ G, the equality
(X ◦ϕ Y ) ◦ϕ Z = X ◦ϕ (Y ◦ϕ Z)
(2.1.2)
holds true.
Fix f, g, h ∈ L2 ([0, T ]) and choose
X = E(f ), Y = E(g) and Z = E(h).
First of all we have:
X ◦ϕ Y
= E(f ) ◦ϕ E(g)
X Z
=
an
Dtn1 ,...,tn E(f ) · Dtn1 ,...,tn E(g)dt1 ...dtn
n≥0
=
=
X
[0,T ]n
Z
an E(f )E(g)
f (t1 )f (t2 ) · · · f (tn ) · g(t1 )g(t2 ) · · · g(tn )dt1 ...dtn
n≥0
[0,T ]n
X
an E(f )E(g)hf, gin
n≥0
= ϕ(hf, gi)E(f )E(g).
17
2.1. A new product for smooth random variables
So, with these choices the left-hand side of (2.1.2) becomes:
(E(f ) ◦ϕ E(g)) ◦ϕ E(h) = ϕ(hf, gi)(E(f )E(g)) ◦ϕ E(h)
= ϕ(hf, gi)(E(f + g)ehf,gi ) ◦ϕ E(h)
= ϕ(hf, gi)ϕ(hf + g, hi)ehf,gi E(f + g)E(h)
= ϕ(hf, gi)ϕ(hf + g, hi)E(f )E(g)E(h).
While proceeding as before the right-hand side reduces to:
E(f ) ◦ϕ (E(g) ◦ϕ E(h)) = ϕ(hf, g + hi)ϕ(hg, hi)E(f )E(g)E(h).
Therefore equation (2.1.2) now reads
ϕ(hf, gi)ϕ(hf + g, hi) = ϕ(hf, g + hi)ϕ(hg, hi),
or equivalently
ϕ(hf, gi)ϕ(hf, hi + hg, hi) = ϕ(hf, gi + hf, hi)ϕ(hg, hi).
Since f, g, h ∈ L2 ([0, T ]) were fixed but arbitrary we can choose f and g to be
orthogonal in L2 ([0, T ]) and obtain the equation (recall that by assumption
ϕ(0) = 1),
ϕ(hf, hi + hg, hi) = ϕ(hf, hi)ϕ(hg, hi).
Since the equation above is satisfied only by exponential functions, among the
class of continuous functions, the proof is complete.
2
We now focus our attention on one specific ◦ϕ -product which will be shown
to be related to the heat equation. We begin by proving a regularity result.
Lemma 2.1.4. Let X, Y ∈ G√2 . Then the series
X 1 Z
Dn X · Dtn Y dt,
n! [0,T ]n t
n≥0
converges in L1 (Ω). More precisely,
X 1 Z
n
n
D X · Dt Y dt ≤ kXkG√2 kY kG√2 .
n! [0,T ]n t
1
n≥0
(2.1.3)
18
Chapter 2. Anti-Wick Product and heat equation
Proof. Denote by k · k1 the norm in L1 (Ω). By means of the triangle and
Cauchy-Schwarz inequalities we get
Z
X 1 Z
X 1
n
n
Dt X · Dt Y dt ≤
Dtn X · Dtn Y dt
n! [0,T ]n
n! [0,T ]n
1
1
n≥0
n≥0
Z
X 1
≤
kDtn X · Dtn Y k1 dt
n! [0,T ]n
n≥0
X 1 Z
≤
kDtn Xk2 · kDtn Y k2 dt
n! [0,T ]n
n≥0
1
X 1 Z
2
≤
kDtn Xk22 dt
n! [0,T ]n
n≥0
1
Z
2
kDtn Y k22 dt
×
[0,T ]n
1
X 1 Z
2
kDtn Xk22 dt
≤
n! [0,T ]n
n≥0
X 1 Z
1
2
×
kDtn Y k22 dt .
n! [0,T ]n
n≥0
Let us now consider the quantity
X 1 Z
kDtn Xk22 dt.
n! [0,T ]n
n≥0
If
P
k≥0 Ik (hk )
is the Wiener-Itô chaos decomposition of X, then
kDtn Xk22 = E[|Dtn X|2 ]
X
= E[|
Dtn Ik |2 ]
k≥0
= E[|
X
k(k − 1) · · · (k − n + 1)Ik−n (hk (·, t1 , . . . , tn ))|2 ]
k≥n
= E[|
X
k≥n
=
X
k≥n
k!
Ik−n (hk (·, t1 , . . . , tn ))|2 ]
(k − n)!
k!2
|hk (·, t1 , . . . , tn )|2L2 ([0,T ]k−n )
(k − n)!
19
2.1. A new product for smooth random variables
where t1 , . . . , tn are frozen and the norm is taken on the remaining k − n arguments of hk . In the last passage we have applied the property of orthogonality
of multiple Wiener integrals. So we have
Z
Z
n
2
kDtn1 ,...,tn Xk22 dt1 . . . dtn
kDt Xk2 dt =
[0,T ]n
[0,T ]n
=
X
k≥n
=
X
k≥n
k!2
(k − n)!
Z
[0,T ]n
|hk (·, t1 , . . . , tn )|2L2 ([0,T ]k−n ) dt1 . . . dtn
k!2
|hk |2L2 ([0,T ]k ) .
(k − n)!
Hence
X 1 X k!2
X 1 Z
kDtn Xk22 dt =
|hk |2L2 ([0,T ]k )
n! [0,T ]n
n!
(k − n)!
n≥0
n≥0
=
X
=
X
k≥n
k!|hk |2L2 ([0,T ]k )
k
X
n=0
k≥0
k
k!2
k!
n!(k − n)!
|hk |2L2 ([0,T ]k )
k≥0
= E[|
X
2k Ik (hk )|2 ]
k≥0
√
= E[|Γ( 2)X|2 ]
= kXk2G√ .
2
The same reasoning can be carried for Y completing the proof.
2
In view of Remark 2.1.2 and Lemma 2.1.4 we make the following definition.
Definition 2.1.5. Let X, Y ∈ G√2 . The anti-Wick product of X and Y ,
denoted by X ◦ Y , is the element of L1 (Ω) defined as
X 1 Z
Dn X · Dtn Y dt.
X ◦ Y :=
n! [0,T ]n t
n≥0
Remark 2.1.6. The anti-Wick product corresponds to the ◦ϕ -product with
ϕ(x) = ex .
We can express the anti-Wick product as sum of Wick products:
20
Chapter 2. Anti-Wick Product and heat equation
Proposition 2.1.7. Let X, Y ∈ G√2 . Then
X 2n Z
Dn X Dtn Y dt.
X ◦Y =
n! [0,T ]n t
(2.1.4)
n≥0
Proof. We can combine the definition of the anti-Wick product and the
property that expresses pointwise products as sum of Wick products. Hence
we have:
X1Z
X ◦Y =
Dtj X · Dtj Y dt
j! [0,T ]j
j≥0
X 1 Z
X1Z
j+k
j+k
=
Ds,t
X Ds,t
Y ds dt
j! [0,T ]j
k! [0,T ]k
j≥0
k≥0
Z
Z
XX 1 1
j+k
=
Dj+k X Ds,t
Y dsdt
j! k! [0,T ]j [0,T ]k s,t
j≥0 k≥0
n X 1 Z
X
n
n
n
D X Dt Y dt
=
n! [0,T ]n t
k
n≥0
k=0
Z
X 2n
Dn X Dtn Y dt.
=
n! [0,T ]n t
n≥0
2
Let us introduce some examples.
Example 2.1.8. Let us consider two chaos of the first order X = I1 (h) and
Y = I1 (g) where h, g ∈ L2 ([0, T ]). Obviously X and Y are in G, and in
particular in G√2 , since the chaos expansion of X and Y is just the term of
first order.
The anti-Wick product of X and Y is:
Z T
X ◦ Y = I1 (h) · I1 (g) +
Dt1 I1 (h) · Dt1 I1 (g)dt1
0
Z T
= I1 (h) · I1 (g) +
h(t1 )g(t1 )dt1
0
Z T
= I2 (h ⊗ g) + 2
h(t1 )g(t1 )dt1
0
G√
and it is still in G, then in
2 , because its chaos expansion is a deterministic
term plus a term of second order.
21
2.1. A new product for smooth random variables
If h(s) = g(s) = 1[0,t[ (s), we obtain the anti-Wick product between two Brownian motions:
Z
T
Z
Bt ◦ Bt =
=
T
T
1[0,t[ (s)1[0,t[ (s)ds
1[0,t[ (s)dBs +
1[0,t[ (s)dBs
0
Bt2
Z
0
0
+t
On the other side, let us remember that Bt Bt = Bt2 − t.
Example 2.1.9. Let us consider the family of stochastic exponentials:
E(f ) = exp
nZ
0
T
1
f (s)dBs −
2
Z
T
o
f 2 (s)ds , f ∈ L2 ([0, T ]),
0
they are in G. The anti-Wick product between two stochastic exponentials is:
X 1 Z
Dn E(f ) · Dtn E(g)dt
n! [0,T ]n t
n≥0
X 1 Z
=
Dn
E(f ) · Dtn1 ,...,tn E(g)dt1 . . . dtn
n! [0,T ]n t1 ,...,tn
n≥0
X 1 Z
E(f ) · E(g)f (t1 ) · · · f (tn ) · g(t1 ) · · · g(tn )dt1 . . . dtn
=
n! [0,T ]n
n≥0
n
X 1 Z T
= E(f )E(g)
f (t) · g(t)dt
n!
0
E(f ) ◦ E(g) =
n≥0
RT
= E(f )E(g)e
0
f (s)g(s)ds
On the other side, we remember that
E(f ) E(g) = E(f + g)
= E(f )E(g)e−
RT
0
f (s)g(s)ds
Example 2.1.10. Let us find the anti-Wick product between two generic
multiple Wiener-Itô integrals In (hn ) and Im (gm ) where hn ∈ L2 ([0, T ]n ) and
gm ∈ L2 ([0, T ]m ) are two generic symmetric functions. For the same reason
as in the Example 2.1.8, these integrals are in G. From the Proposition 2.1.7
22
Chapter 2. Anti-Wick Product and heat equation
we have:
In (hn ) ◦ Im (gm ) =
n∧m
X
k=0
2k
k!
Z
n(n − 1) · · · (n − k + 1)In−k (hn (·, t1 , . . . , tk )) [0,T ]k
m(m − 1) · · · (m − k + 1)Im−k (gm (·, t1 , . . . , tk ))dt1 . . . dtk
Z
n∧m
X 2k
n!
m!
˜
In+m−2k (hn (·, t1 , . . . , tk )⊗
=
k! (n − k)! (m − k)! [0,T ]k
k=0
=
gm (·, t1 , . . . , tk ))dt1 . . . dtk
n∧m
X
m
k n
˜ k gm )
In+m−2k (hn ⊗
k!2
k
k
k=0
where k indicates the integrations over t1 , . . . , tk .
We observe that the anti-Wick product of two multiple Wiener-Itô integrals
is still in G because it is the sum of some other multiple Wiener-Itô integrals
that are in G.
The anti-Wick product is associative. We will
R T prove later this assertion.
Now, let us consider a Wiener integral X := 0 f (t)dBt . As we saw in the
previous examples the anti-Wick product of X by himself is still in G√2 , and,
since it is a multiple Wiener integral, if we Wick-multiply it by X again, it
still stays in G√2 .
Then we are able to define the n-th anti-Wick power of a Wiener integral X:
X ◦n := X
◦ · · · ◦ X}
| ◦ X {z
(2.1.5)
n
as an object in G√2 . Now we shall prove a connection between these anti-Wick
powers and a certain type of polynomials which have properties quite close to
Hermite polynomials.
RT
Proposition 2.1.11. Let X be a chaos of first order, X := 0 f (t)dBt , and
X̃ an independent Gaussian copy of X. Then
X ◦n = E[(X + X̃)n |X].
Proof. Let us prove by induction.
For n = 1:
X = E[X + X̃|X] = X + E[X̃] = X.
(2.1.6)
23
2.1. A new product for smooth random variables
Now suppose that (2.1.6) is true for n − 1 and let us prove it for n.
It is easy to show that
X n−1 ◦ X = X n + (n − 1)kXk22 X n−2
(2.1.7)
In fact we have,
X n−1 ◦ X = X n +
Z
T
Dt X n−1 · Dt Xdt
0
Z T
n
n−2
f 2 (t)dt
= X + (n − 1)X
0
= X n + (n − 1)kXk22 X n−2
So
X ◦n = X ◦(n−1) ◦ X
n−1
X n − 1
for the inductive hypothesis =
E[X̃ k X n−1−k |X] ◦ X
k
k=0
n−1
X n − 1
=
E[X̃ k ]X n−1−k ◦ X
k
k=0
n−1
X n − 1
E[X̃ k ][X n−k + kXk22 (n − k − 1)X n−k−2 ]
for the (2.1.7) =
k
k=0
n−1
n−2
X n − 1
X n − 1
k
n−k
=
E[X̃ ]X
+
E[X̃ k ]kXk22 (n − k − 1)X n−k−2
k
k
k=0
k=0
n−1
n−2
X n n−k
X n − 1
k
n−k
=
E[X̃ ]X
+
E[X̃ k ]kXk22 (n − k − 1)X n−k−2
k
n
k
k=0
k=0
n
X n
=
E[X̃ k X n−k |X]
k
k=0
n n−2
X
X n − 1
n k
k
n−k
−
E[X̃ ]X
+
E[X̃ k ]kXk22 (n − k − 1)X n−k−2
k n
k
k=2
k=0
24
Chapter 2. Anti-Wick Product and heat equation
= E[(X̃ + X)n |X]
n−2
X
n − 1!
1
−
E[X̃ m+2 ]X n−m−2
m!n − m − 2! m + 1
+
m=0
n−2
X
k=0
n − 1!
E[X̃ k ]kXk22 X n−k−2
k!n − k − 2!
= E[(X̃ + X)n |X]
where in the last passage we used the property of Gaussian random variables
E[X n+2 ] = kXk22 (n + 1)E[X n ].
2
Now, let us take Wiener integrals with kXk2 = 1. By defining
pn (X) := X ◦n ,
we obtain a class of polynomials pn (x) very close to the Hermite polynomials.
In fact we have the following proposition:
Proposition 2.1.12.
pn (x) = e−
d x2
e2
dxn
x2
2
(2.1.8)
Proof. Let X be a chaos of the first order with kXk22 = 1, then, from the
(2.1.6), we have:
pn (x) = E[(x + X)n ]
= (−i)n E[(ix + iX)n )]
= (−i)n hn (ix)
= (−i)n (−1)n e
= e−
x2
2
(ix)2
2
1 dn − (ix)2
e 2
in dxn
d x2
e2.
dxn
2
We can also define the polynomials with parameter σ 2
x
pn,σ2 := σ n pn
σ
and we have that
X
◦n
= pn,kXk2
X
kXk2
25
2.1. A new product for smooth random variables
where X is a Wiener integral.
Let us take the Brownian motion Bt . Hence
Bt◦n = pn,t (Bt ) = E[(Bt + B̃t )n |Bt ].
Later we will see that we can obtain similar expressions not only for powers,
but for any real function satisfying some properties.
We can now easily derive some properties about these polynomials pn close to
the Hermite ones (see Appendix A).
Proposition 2.1.13. The polynomials pn satisfy these properties:
1. The generating function of the polynomials pn is:
∞ n
X
t
n=0
n!
t2
pn (x) = etx+ 2
2.
p0n (x) = npn−1 (x)
3. We have the recursive formula:
pn+1 (x) = xpn (x) + npn−1 (x)
4. The polynomials pn satisfy the differential equation:
2
d
d
+x
− n pn (x) = 0
dx2
dx
Proof. These properties come from Hermite polynomials’ properties, by
observing that
pn (x) = (−i)n hn (ix).
2
The main difference from Hermite polynomials is that they are not orthogonal.
We now prove a crucial result.
Theorem 2.1.14. Let X, Y ∈ G√2 . Then we have the representation,
1 √
√
X ◦ Y = Γ √ (Γ( 2)X · Γ( 2)Y ).
2
(2.1.9)
26
Chapter 2. Anti-Wick Product and heat equation
Proof. First of all we observe that due to inequality (2.1.3), if Xn converges
to X in G√2 , then Xn ◦ Y converges to X ◦ Y in L1 (Ω). Moreover the family
of stochastic exponentials
Z
nZ T
o
1 T 2
f (s)dBs −
E(f ) = exp
f (s)ds , f ∈ L2 ([0, T ]),
2 0
0
forms a total set in G√2 . Therefore by linearity and continuity it is sufficient to
prove the theorem for stochastic exponentials. But this is easily done; indeed
we have seen in the Example 2.1.9 that for any f, g ∈ L2 ([0, T ]),
RT
E(f ) ◦ E(g) = E(f )E(g)e
0
f (s)g(s)ds
,
and
1 √
1 √
√
√
Γ √ (Γ( 2)E(f ) · Γ( 2)E(g)) = Γ √ (E( 2f )E( 2g))
2
2
1 √
RT
= Γ √ E( 2(f + g))e2 0 f (s)g(s)ds
2
= E(f + g)e2
= E(f )E(g)e
RT
0
RT
0
f (s)g(s)ds
f (s)g(s)ds
.
2 The following proposition provides the associativity of the anti-Wick
product.
Corollary 2.1.15. The anti-Wick product is associative.
Proof. Let X, Y, Z ∈ G√2 be such that X ◦ Y, Y ◦ Z ∈ G√2 . We have to
prove that
(X ◦ Y ) ◦ Z = X ◦ (Y ◦ Z).
By the representation of Theorem 2.1.14, this follows immediately by a straightforward verification.
2
2.2
Application to the heat equation
In the representation (2.1.9) of the anti-Wick product, we can also define the
anti-Wick power for every X ∈ G√2 and n ≥ 1; in fact we easily have that
X ◦n := |X ◦ ·{z
· · ◦ X}
n−times
=
1 √
Γ √
(Γ( 2)X)n .
2
27
2.2. Application to the heat equation
√
Therefore if f : R → R is such that f (Γ( 2)X) ∈ L1 (Ω) we can define
1 √
f ◦ (X) := Γ √ f (Γ( 2)X),
2
as an element of L1 (Ω).
Example 2.2.1. Let f (x) = exp{x} and h ∈ L2 ([0, T ]). Then
o
o
1 n √ Z T
nZ T
h(s)dBs
h(s)dBs
= Γ √ exp Γ( 2)
exp◦
2
0
0
1 o
n√ Z T
= Γ √ exp
h(s)dBs
2
2
0
1 √
RT 2
= Γ √ E( 2h)e 0 h (s)ds
2
RT
= E(h)e 0
nZ
= exp
T
0
Observe the symmetry with the case
nZ T
o
nZ
exp
h(s)dBs = exp
0
T
0
h2 (s)ds
1
h(s)dBs +
2
1
h(s)dBs −
2
Z
Z
T
T
o
h(s)ds .
0
o
h(s)ds .
0
We now come to the main theorems of the present section which establish a
new probabilistic representation for the solution of the heat equation in terms
of anti-Wick products.
Theorem 2.2.2. Let f : R → R be a continuous function such that
γ
|f (y)| ≤ c1 ec2 |y| ,
c1 , c2 ∈ R, γ < 2,
and let u : [0, T ] × R → R be the unique solution of
n ∂ u(t, x) = 1 ∂ u(t, x), x ∈ R, t ∈]0, T ]
t
2 xx
u(0, x) = f (x), x ∈ R
among the class of functions satisfying the bound
2
|u(t, x)| ≤ CeC|x| ,
t ∈]0, T ], x ∈ R,
for a certain positive constant C. Then for t ∈ [0, T ],
u(t, Bt ) = f ◦ (Bt )
P − a.e.
(2.2.1)
28
Chapter 2. Anti-Wick Product and heat equation
Proof. We have seen by the Theorem 1.3.2 that among the functions satisfying the (2.2.1), there exists a solution to the heat equation, it is unique and
it has the well known representation
u(t, x) = Ẽ[f (B̃t + x)],
x ∈ R, t ∈ [0, T ].
(2.2.2)
Recall now that
1 √
f ◦ (Bt ) = Γ √ f (Γ( 2)Bt )
2
1 √
= Γ √ f ( 2Bt ).
2
Fix h ∈ L2 ([0, T ]); we will prove the theorem by showing that
E[u(t, Bt )E(h)] = E[f ◦ (Bt )E(h)],
for any h ∈ L2 ([0, T ]). Using the properties of second quantization operators
and the Girsanov theorem we get
h 1 √
i
E[f ◦ (Bt )E(h)] = E Γ √ f ( 2Bt ) E(h)
2
h √
h i
= E f ( 2Bt )E √
2
Z t
h √
i
= E f
2Bt +
h(s)ds .
0
√
The law of 2Bt is the same as the one of Bt + B̃t where B̃t is another Brownian motion independent of Bt and defined on an auxiliary probability space
(Ω̃, F̃, P̃). Therefore, denoting by Ẽ the expectation in this new probability
space, we obtain
Z t
Z t
h √
ii
i
h h E f
h(s)ds
2Bt +
h(s)ds
= E Ẽ f Bt + B̃t +
0
0
Z t
h i
= E u t, Bt +
h(s)ds
0
= E[u(t, Bt )E(h)],
where we used the (2.2.2). The proof is then complete.
2
Under the same assumptions on the initial condition we can also prove the
following.
2.2. Application to the heat equation
29
Theorem 2.2.3. Let u(t, x) be the unique solution of
n ∂ u(t, x) = 1 ∂ u(t, x), x ∈ R, t ∈]0, T ],
t
2 xx
u(0, x) = f (x), x ∈ R,
and v(t, x) be the unique solution of
n ∂ v(t, x) = 1 ∂ v(t, x), x ∈ R, t ∈]0, T ]
t
2 xx
v(0, x) = g(x), x ∈ R.
Moreover let w(t, x) be the unique solution of
n ∂ w(t, x) = 1 ∂ w(t, x), x ∈ R, t ∈]0, T ]
t
2 xx
w(0, x) = f (x)g(x), x ∈ R.
Then for each t ∈ [0, T ],
u(t, Bt ) ◦ v(t, Bt ) = w(t, Bt ).
Proof. By Theorem 2.1.14 we have that
1 √
√
u(t, Bt ) ◦ v(t, Bt ) = Γ √ (Γ( 2)u(t, Bt ) · Γ( 2)v(t, Bt ))
2
1 √
√
= Γ √ (f ( 2Bt )g( 2Bt ))
2
1 √
= Γ √ ((f · g)( 2Bt ))
2
= w(t, Bt ).
2
30
Chapter 2. Anti-Wick Product and heat equation
Chapter 3
Newton-Leibniz rule for
stochastic integrals and
approximations of SDE’s
3.1
α-products
In the previous chapter we have defined the ◦ϕ -product (see Definition 2.1.1)
and we have seen that if this product is associative an must be the coefficient
λn
n! of the Taylor expansion of the exponential function. In particular, by
choosing λ = 1, we have defined the anti-Wick product between two random
variables
X 1 Z
X ◦Y =
Dn X · Dtn dt
n! [0,T ]n t
n≥0
We have seen that this product is associative and we have proved several
properties about this product.
We want now to generalize the anti-Wick product by choosing a generic λ ∈
[−1, 1]. Before doing that, let us generalize the Lemma 2.1.4.
Lemma 3.1.1. Let u, v, w ∈ R be such that w 6= 0, u, v > 0 and
If X ∈ G√u−|w|+1
1 1
1
+ =
.
u v
|w|
and Y ∈ G√v−|w|+1 then the series
X wn Z
Dn X · Dtn Y dt
n! [0,T ]n t
n≥0
32Chapter 3. Newton-Leibniz rule for stochastic integrals and approximations of SDE’s
converges in L1 (Ω). More precisely,
X wn Z
Dtn X · Dtn Y dt ≤ kXkG√u−|w|+1 kY kG√v−|w|+1 .
n! [0,T ]n
1
(3.1.1)
n≥0
Proof. First of all observe that the condition
1 1
1
+ =
,
u v
|w|
implies that
u
v
−1
− 1 = 1.
|w|
|w|
Hence by following the same procedure as the Lemma 2.1.4 we get
Z
X wn Z
X |w|n Dtn X · Dtn Y dt ≤
Dtn X · Dtn Y dt
n! [0,T ]n
n!
1
1
n
[0,T ]
n≥0
n≥0
Z
X |w|n
kDtn X · Dtn Y k1 dt
≤
n! [0,T ]n
n≥0
X |w|n Z
≤
kDtn Xk2 · kDtn Y k2 dt
n! [0,T ]n
n≥0
n n
2
2
v
u
n
Z
1
X |w| |w| − 1
|w| − 1
2
kDtn Xk22 dt
≤
n!
n
[0,T ]
n≥0
Z
1
2
×
kDtn Y k22 dt
[0,T ]n
=
X (u − |w|) n2 (v − |w|) n2 Z
n!
n≥0
Z
×
[0,T ]n
≤
[0,T ]n
kDtn Y k22 dt
1
2
X (u − |w|)n Z
n≥0
n!
1
2
kDtn Xk22 dt
[0,T ]n
1
2
kDtn Xk22 dt
X (v − |w|)n Z
1
2
kDtn Y k22 dt .
×
n!
[0,T ]n
n≥0
33
3.1. α-products
Let us now consider the quantity
X (u − |w|)n Z
n≥0
If
P
k≥0 Ik (hk )
n!
[0,T ]n
kDtn Xk22 dt.
is the Wiener-Itô chaos decomposition of X, then
Z
[0,T ]n
kDtn Xk22 dt =
X
k≥n
k!2
|hk |2L2 ([0,T ]k ) ,
(k − n)!
and therefore
X (u − |w|)n Z
n≥0
n!
[0,T ]n
kDtn Xk22 dt =
X (u − |w|)n X
n!
n≥0
=
X
=
X
k≥n
k!|hk |2L2 ([0,T ]k )
k!2
|hk |2L2 ([0,T ]k )
(k − n)!
k
X
(u − |w|)n
n=0
k≥0
k!
n!(k − n)!
k!(u − |w| + 1)k |hk |2L2 ([0,T ]k )
k≥0
p
= E[|Γ( u − |w| + 1)X|2 ]
= kXk2G√
u−|w|+1
.
The same reasoning can be carried for Y completing the proof of (3.1.1).
2
Remark 3.1.2. The case for w = 0 is trivial since we get the pointwise
product.
Now we are able to define the α-product.
Definition 3.1.3. Let α ∈ [0, 2] and let X, Y ∈ G√|α−1|+1 . The α-product of
X and Y , denoted by X ◦α Y , is the element of L1 (Ω) defined as
X ◦α Y :=
X (α − 1)n Z
n≥0
n!
[0,T ]n
Dtn X · Dtn Y dt
(3.1.2)
As for the anti-Wick product, we can achieve a representation like (2.1.9)
for the α-product, too.
34Chapter 3. Newton-Leibniz rule for stochastic integrals and approximations of SDE’s
Lemma 3.1.4. Let X, Y ∈ G√|α−1|+1 . Then we have the representation,
X ◦α Y = Γ
1
√
α
√
√
(Γ( α)X · Γ( α)Y ).
(3.1.3)
Proof. The proof comes from the same arguments as in the Theorem 2.1.14.
By following the same calculation, we just observe that
RT
E(f ) ◦α E(g) = E(f )E(g)e(α−1) 0 f (s)g(s)ds
1 √
√
= Γ √ (Γ( α)E(f ) · Γ( α)E(g)).
α
(3.1.4)
2
As a consequence of this representation the α-product is associative.
By means of a theorem due to Potthof and Timpel [31], we know that G is
an algebra under pointwise multiplication, then if X and Y are in G, their
multiplication is in G, too. Hence, by the representation (3.1.3), for X, Y ∈ G,
√
√
the product Γ( α)X · Γ( α)Y is in G, then the α-product is in G.
For X ∈ G it makes sense to define the n-th α-powers of X, that means the
n-th power of the α-product of X:
X ◦α n := X ◦α · · · ◦α X
|
{z
}
n−times
=
1 √
Γ √
(Γ( α)X)n
α
∈ G.
In the same way as in the previous chapter, we can relate the α-powers of a
Wiener integral X to a particular kind of polynomials. In particular we need
the following assertion:
RT
Proposition 3.1.5. Let X be a chaos of first order, X := 0 f (t)dBt , and X̃
a Gaussian copy independent of X. Then
√
X ◦α n = E[(X + α − 1X̃)n |X]
(3.1.5)
Proof. The proof is identical to the one proved for α = 2. We have just to
observe that, in order to achieve the result, we have to use the formula
X n−1 ◦α X = X n + (n − 1)(α − 1)kXk22 X n−2 .
(3.1.6)
2
35
3.2. A new representation for the Wick product
For α = 0 we have the useful representation for the Wick product and for the
Hermite polynomial
X n = hn,kXk22 (X) = E[(X + iX̃)n |X].
Let us define the α-polynomials of X with parameter σ 2 :
pαn,σ2 (x) := [(α − 1)σ 2 ]n e
−
x2
2(α−1)σ 2
x2
dn 2(α−1)σ
2
e
dxn
(3.1.7)
Example 3.1.6. Let us calculate the first 4 polynomials:
pα1,σ2 (x) = x ;
pα2,σ2 (x) = x2 + (α − 1)σ 2 ;
pα3,σ2 (x) = x3 + 3(α − 1)σ 2 x ;
pα4,σ2 (x) = x4 + 6(α − 1)σ 2 x2 + 3(α − 1)2 σ 4 .
For α = 0 we obtain the Hermite polynomials with parameter σ 2 , for α = 0
and σ 2 = 1 we obtain the usual Hermite polynomials.
We can finally prove the following statement:
RT
Proposition 3.1.7. Let X be a chaos of first order, X := 0 f (t)dBt . Then
X ◦α n = pαn,kXk2 (X)
2
Proof. The proposition is proved by following the same reasoning of the
analogue theorem for α = 2 in the previous chapter.
2
In particular
Bt◦α n = pαn,t (Bt )
3.2
A new representation for the Wick product
In this paragraph we prove an important property of the α-products.
As we mentioned before, if u1 + v1 = 1, X ∈ G√u and Y ∈ G√v , we can write
X Y =
X (−1)n Z
n≥0
n!
[0,T ]n
Dtn X · Dtn Y dt,
36Chapter 3. Newton-Leibniz rule for stochastic integrals and approximations of SDE’s
where the series converges in L1 (Ω).1 On the other hand Lemma 3.1.4 tells
that for any α > 0,
1 √
X (α − 1)n Z
√
X ◦α Y =
Dtn X · Dtn Y dt = Γ √ (Γ( α)X · Γ( α)Y ).
n!
α
[0,T ]n
n≥0
From a comparison between the previous two equalities one may expect that
1 √
√
X Y = lim Γ √ (Γ( α)X · Γ( α)Y ).
+
α
α→0
This is in fact the case as the following theorem shows.
Theorem 3.2.1. Let X ∈ G√u and Y ∈ G√v for some u, v > 1 satisfying
1 1
+ = 1.
u v
Then
X Y = lim X ◦α Y in L1 (Ω).
α→0+
Proof. Recall that, from the Lemma 3.1.1, for X ∈ G√u and Y ∈ G√v ,
kX Y k1 ≤ kXkG√u · kY kG√v .
From the (3.1.4), for any f, g ∈ L2 ([0, T ]) we have that
lim E(f ) ◦α E(g) =
α→0+
lim E(f )E(g)e(α−1)
RT
0
f (s)g(s)
α→0+
= E(f )E(g)e−
RT
0
f (s)g(s)
(3.2.1)
= E(f ) E(g)
In view of this continuity, since the family of random variables {E(f ), f ∈
L2 ([0, T ])} is total in Gλ for any λ ≥ 1, we can find two sequences {Xn }n≥0
and {Yn }n≥0 of linear combinations of stochastic exponentials such that
lim Xn Yn = X Y in L1 (Ω).
n→+∞
So, by using the (3.2.1), via a straightforward calculation one can see that
Xn Yn = lim Xn ◦α Yn ,
α→0+
1
If X, Y ∈ G the series converges in G
37
3.3. Stochastic integrals with generic evaluating point
which is the thesis of the theorem for linear combination of stochastic exponentials.
Therefore we can write
lim X ◦α Y
α→0+
lim lim Xn ◦α Yn
=
α→0+ n→∞
lim lim Xn ◦α Yn
=
n→∞ α→0+
lim Xn Yn
=
n→∞
= X Y
provided that we can interchange the two limits. But this can be done since
sup X ◦α Y − Xn ◦α Yn = sup X ◦α Y − X ◦α Yn + X ◦α Yn − Xn ◦α Yn 1
α∈[0,2]
1
α∈[0,2]
=
sup X ◦α (Y − Yn ) + (X − Xn ) ◦α Yn 1
α∈[0,2]
≤
sup {kXkG√|α−1|+1 kY − Yn kG√|α−1|+1 }
α∈[0,2]
+ sup {kX − Xn kG√|α−1|+1 kYn kG√|α−1|+1 }
α∈[0,2]
= kXkG√2 kY − Yn kG√2 + kX − Xn kG√2 kYn kG√2 .
since for µ > λ > 1 one has kXkGλ < kXkGµ This shows that the limit as n
goes to infinity is uniform with respect to α ∈ [0, 2]. This completes the proof.
2
3.3
Stochastic integrals with generic evaluating point
In this section we shall use the approach given in the book of Protter [33] in
order to define the α-integrals. Also see the book of Karatzas and Shreve [14].
Let X, Y be continuous semimartingales and α ∈ [0, 2]. Define the α-integral
RT
of Y with respect to X, denoted by 0 Yt dXtα , by
Z
0
T
Yt dXtα :=
Z
T
Yt dXt +
0
α
[Y, X]T ,
2
(3.3.1)
where the integral on the right is the usual Itô integral, and [·, ·] is the quadratic
covariation of X and Y .
38Chapter 3. Newton-Leibniz rule for stochastic integrals and approximations of SDE’s
Now, we want to know if these α-integrals have an interpretation as limits
of Riemann-like sums, as the Itô integral has one. This interpretation holds
with a supplementary hypothesis on the quadratic covariation. According to
[33] we get this result:
Theorem 3.3.1. Let X be a semimartingale and Y a continuous semimartingale. Let α ∈ [0, 2]. Further suppose that the quadratic covariation of X and
Y , namely [X, Y ], has absolute continuous paths. Let Π = {t0 , . . . , tn } a partition of [0, T ] with 0 = t0 < t1 < · · · < tn = T . Let f ∈ C 1 . Then
lim
n→∞
n
X
f Ytαi
Z
Xti+1 − Xti =
T
f (Yt )dXtα
(3.3.2)
0
i=0
where tαi := ti + α2 (ti+1 − ti ) and the convergence is in probability.
Remark 3.3.2. We see that the α-integral corresponds to a definition of a
stochastic integral in which we establish the evaluating point in the Riemann
sums being tαi between ti and ti+1 .
For α = 0 we get the Itô integral, for α = 1 we find the Stratonovich integral
and for α = 2 we get the backward integral.
The α-integral is then a semimartingale with the Itô integral as the martingale
term, plus an extra term of finite variation.
If f ∈ C 1 is a real function, as a consequence of the definition and the fact
that
Z T
[f (Y ), X]T =
f 0 (Yt )d[Y, X]t ,
0
we get
Z
T
f (Yt )dXtα
Z
=
0
0
T
α
f (Yt )dXt +
2
Z
T
f 0 (Yt )d[Y, X]t
In particular the covariation between an Itô integral
martingale X is
Z ·
Z T
[ Yt dXt , X· ]T =
Yt d[X, X]t
0
(3.3.3)
0
RT
0
Yt dXt and a semi(3.3.4)
0
Note that, in the Definition (3.3.1), if Y is a continuous martingale and X
is the Brownian motion B, then, due to the Kunita-Watanabe inequality2 ,
2
See the book of Protter [33]
39
3.4. Newton-Leibniz rule
[Y, B] is absolutely continuous, so the representation in Riemann sums in the
Theorem 3.3.1 is always valid. In particular, since [B, B]t = t, we get a formula
that will be useful later
Z T
Z ·
Yt dt
(3.3.5)
[ Yt dBt , B· ]T =
0
0
3.4
Newton-Leibniz rule
The aim of this section is to prove the fundamental theorem of calculus for
the α-integrals of the ”α-product versions” of a function f , that is
Z T
◦α
f ◦α (Bt )dBtα
(3.4.1)
F (Bt ) =
0
Rx
where F (x) = 0 f (y)dy is the primitive of f .
We must divide the problem in two cases, for 0 < α < 1 and for 1 < α ≤ 2.
This is due to the fact that, as we are going to see, the two cases are connected,
respectively, to the backward heat equation, and to the heat equation. Then,
in the first case, we work in a framework of an ill-posed problem, so the
problem for α < 1 becomes more complicated than the one for α > 1.
The cases α = 0 and α = 1 are known, in fact the first one is the Itô integral
and we know it satisfies the Newton-Leibniz rule with the Wick product [10].
The second case, for α = 1, deals with the Stratonovich integral and the
pointwise product.
3.4.1
Case 1 < α ≤ 2
In order to prove the Newton-Leibniz rule we need at first to prove a theorem
that links the heat equation and the α-product as the Theorem 2.2.2 does. As
√
in the case for α = 2, for X ∈ G√α , if f : R → R is such that f (Γ( α)X) ∈
L1 (Ω) we can define
1 √
f ◦α (X) := Γ √ f (Γ( α)X),
α
as an element of L1 (Ω). We can prove the following theorem.
Theorem 3.4.1. Let f : R → R be a continuous function such that
γ
|f (y)| ≤ c1 ec2 |y| ,
c1 , c2 ∈ R, γ < 2,
(3.4.2)
40Chapter 3. Newton-Leibniz rule for stochastic integrals and approximations of SDE’s
and let u : [0, T ] × R → R be the unique solution of
n ∂ u(t, x) = α−1 ∂ u(t, x), x ∈ R, t ∈]0, T ]
xx
t
2
u(0, x) = f (x), x ∈ R
among the class of functions satisfying the bound
2
|u(t, x)| ≤ CeC|x| ,
t ∈]0, T ], x ∈ R,
(3.4.3)
for a certain constant C. Then for t ∈ [0, T ],
u(t, Bt ) = f ◦α (Bt ).
Proof. We can prove the theorem in the same way as the Theorem 2.2.2,
by performing a change of variables. Let u : [0, T ] × R → R be the unique
solution of
n ∂ u(t, x) = α−1 ∂ u(t, x), x ∈ R, t ∈]0, T ]
t
xx
2
u(0, x) = f (x),
x∈R
By a change of variables the previous heat equation turns in
1
∂s u(s/(α − 1), x) = ∂xx u(s/(α − 1), x)
2
(3.4.4)
We can define the function v(s, x) := u(s/(α − 1), x). It is the unique solution
of
n ∂ v(t, x) = 1 ∂ v(t, x), x ∈ R, t ∈]0, T ]
t
2 xx
v(0, x) = f (x),
x∈R
For the Feynman-Kac formula
then
v(s, x) = E[f (Bs + x)] = u(s/(α − 1), x)
(3.4.5)
√
u(t, x) = E[f (Bt(α−1) + x)] = E[f ( α − 1Bt + x)]
(3.4.6)
Now, by following the proof of the Theorem 2.2.2, we obtain
E[f ◦α (Bt )E(h)] = E[u(t, Bt )E(h)],
(3.4.7)
for any h ∈ L([0, T ]), then
f ◦α (Bt ) = u(t, Bt )
(3.4.8)
2
41
3.4. Newton-Leibniz rule
Remark 3.4.2. This theorem is very important since it shows the deep connection between the heat equation and the α-products for α > 1. We can easily
see that for α < 1 the heat equation turns in the backward heat equation, so
the problem becomes ill-posed and more complicated. In fact we are not able
to prove this theorem in the case α < 1 since, by loosing the well-posedness
of the problem, we loose a Feynman-Kac formula too, important in order to
achieve the result, but we also loose an exact knowledge of the solution in the
same hypothesis.
In particular, for the backward heat equation, the initial condition f must be
at least analytic.
Let us prove now the Fundamental theorem of calculus.
Theorem 3.4.3. Let F ∈ C 1 (R) and f := F 0 be the first derivative of F . Let
f and F satisfy the (3.4.2). Then
F
◦α
Z
(BT ) =
T
f ◦α (Bt )dBtα
(3.4.9)
0
Proof. Let us consider the unique solution u : [0, T ] × R → R of
n ∂ u(t, x) = α−1 ∂ u(t, x),
t
xx
2
u(0, x) = F (x), x ∈ R
x ∈ R, t ∈]0, T ]
(3.4.10)
We have proved in the previous theorem that we can represent the solution in
this way
u(t, Bt ) = F ◦α (Bt )
Let us introduce the derivative of u w.r.t. the second argument x
v(t, x) := ∂x u(t, x)
We have that v(0, x) = f (x) and it trivially satisfies the heat equation
∂t v(t, x) =
α−1
∂xx v(t, x)
2
(3.4.11)
so we can apply the Theorem 3.4.1, the solution is unique, v(t, x), and we have
that v(t, Bt ) = f ◦α (Bt ).
Let us prove that
Z T
u(T, BT ) =
v(t, Bt )dBtα .
0
42Chapter 3. Newton-Leibniz rule for stochastic integrals and approximations of SDE’s
By the definition of the α-integral (3.3.1)
Z T
Z T
α
v(t, Bt )dBt + [v(·, B· ), B· ]T ,
v(t, Bt )dBtα =
2
0
0
(3.4.12)
For the Itô formula
Z t
Z t
Z
1 t
v 0 (s, Bs )dBs +
∂s v(s, Bs )ds
v(t, Bt ) =
v”(s, Bs )ds +
2 0
0
0
Z t
Z
α t
0
=
v (s, Bs )dBs +
v”(s, Bs )ds,
2 0
0
where the symbol 0 denotes the derivative w.r.t. to the second argument; in
the last passage we have used the (3.4.11).
By applying the (3.3.5) we have
Z T
v 0 (t, Bt )dt.
[v(·, B· ), B· ]T =
0
so the (3.4.12) becomes
Z T
Z
α
v(t, Bt )dBt =
0
0
T
α
v(t, Bt )dBt +
2
Z
T
v 0 (t, Bt )dt.
0
Then
α
2
Z
T
v 0 (t, Bt )dt =
0
Z
T
v(t, Bt )dBtα −
0
Z
T
v(t, Bt )dBt .
(3.4.13)
0
In the same way we can apply the Itô formula to u(T, BT )
Z
Z T
Z T
1 T
u(T, BT ) =
u0 (t, Bt )dBt +
u”(t, Bt )dt +
∂t u(t, Bt )dt
2 0
0
0
Z T
Z
α T
0
u (t, Bt )dBt +
u”(t, Bt )dt
=
2 0
0
Z T
Z
α T 0
=
v(t, Bt )dBt +
v (t, Bt )dt
2 0
0
Then, by substituting the (3.4.13) in the last equation, we get
Z T
u(T, BT ) =
v(t, Bt )dBtα
(3.4.14)
0
and the proof is complete.
2
43
3.4. Newton-Leibniz rule
3.4.2
Case 0 < α < 1
As we observed in the previous paragraph, for 0 < α < 1 the Theorem 3.4.1
is not valid anymore, since the Cauchy problem becomes ill-posed. Surely,
if we want to relate the α-product to the backward heat equation we must
consider at least analytic functions. We are going to see the connection between the backward heat equation and α-product versions of polynomials of
the Brownian motion. We have seen that
X ◦α n = pαn,kXk2 (X),
2
so X ◦α n is a polynomial in X. But, if X ∈ G, since G is an algebra under
pointwise multiplication
[31], X ◦α n is still in G.
PN
So, if Π(x) = n=0 an xn ; an ∈ R, x ∈ R is a polynomial, then we define its α
version
Π◦α : G → G
by
Π◦α (X) =
N
X
an X ◦α n
forX ∈ G
n=0
In particular we can define in this way, in G, the α versions of polynomials in
Bt , since the Brownian motion is in G.
By linearity, in order to prove that the α-integrals of the α version of a polynomial in Bt satisfy the fundamental theorem of calculus, it will be enough to
prove that it works for any n-th α-power of Bt , Bt◦α n . And that is the case as
the following theorem shows.
Theorem 3.4.4. Let Bt be a standard Brownian motion. Then,
Z
0
T
◦ (n+1)
Bt◦α n dBtα =
BTα
.
n+1
Proof. For the Proposition 3.1.7,
pn,α (t, Bt ) := Bt◦α n = pαn,t (Bt )
(3.4.15)
44Chapter 3. Newton-Leibniz rule for stochastic integrals and approximations of SDE’s
and we can apply the Itô formula to the polynomials pn,α . We obtain
BT◦n = pn,α (T, BT )
(3.4.16)
Z T
Z T
Z T
1
p0n,α (t, Bt )dBt +
∂t pn,α (t, Bt )dt
=
p”n,α (t, Bt )dt +
2 0
0
0
Z T
Z
Z
1 T
α−1 T
npn−1,α (t, Bt )dBt +
=
p”n,α (t, Bt )dt +
p”n,α (t, Bt )dt
2 0
2
0
0
Z
Z T
α T
n(n − 1)pn−2,α (t, Bt )dt
=
npn−1,α (t, Bt )dBt +
2 0
0
Z T
Z
α T
◦α (n−1)
◦ (n−2)
=
nBt
dBt +
n(n − 1)Bt α
dt.
2 0
0
In the last passages we have applied the formula for the polynomials pαn,t :
0
pαn,t (x) = npn−1,t (x)
and the property with respect to whom these polynomials satisfy the equation
∂t pn,α (t, x) =
α−1
∂xx pn,α (t, x)
2
for any 0 ≤ α ≤ 2.
We also have that
[B·◦α n , B· ]T =
Z
0
T
◦ (n−1)
nBt α
dt
From the previous calculus we also have
Z T
Z T
◦ (n+1)
BTα
◦ (n−1)
◦α n
−n
Bt α
dt.
Bt dBt =
n
+
1
0
0
Then, by definition, the α-integral of Bt◦α n is
Z T
Z T
α
Bt◦α n dBtα =
Bt◦α n dBt + [B·◦α n , B· ]T
2
0
0
(3.4.17)
(3.4.18)
(3.4.19)
By substituting (3.4.17) and (3.4.18) in (3.4.19) we achieve the desired result
Z T
Z T
Z T
◦ (n+1)
BTα
◦α (n−1)
◦ (n−1)
◦α n
α
Bt dBt =
−n
Bt
dt + n
Bt α
dt
n+1
0
0
0
◦ (n+1)
=
BTα
n+1
2
45
3.5. Wong-Zakai-type theorems
3.5
Wong-Zakai-type theorems
We shall give now a proof of a Wong-Zakai like theorem for the α-product.
Now we are going to show a series of lemmas and theorems that will be useful
for the proof. In the sequel we denote by Tf the so called translation operator
which is given by
Z Tf Y (ω) := Y ω + f
where Y ∈ G and f ∈ L2 (R). We refer to the book of Kuo [18] for the details
on this operator. The next result is known as Gjessing’s lemma [2]
Theorem 3.5.1. Let X ∈ G. Then
T−f X(ω) · exp
Z
T
T
Z
= X(ω) exp
f (t)dBt
(3.5.1)
f (t)dBt
0
0
In order to prove an analogue result for the α-product we need to show
this known result from Potthoff and Timpel [31].
Lemma 3.5.2. Let f ∈ L2 (R), X ∈ G. Then
X 1 Z
Tf X =
Dn
X · f ⊗n (t1 , . . . , tn )dt1 . . . dtn
n! [0,T ]n t1 ,...tn
(3.5.2)
n≥0
where the series converges in G.
Now we can prove the Gjessing’s fprmula for the α-products.
Theorem 3.5.3. Let X ∈ G. Then
T(α−1)f X(ω) · exp
◦β
Z
T
f (t)dBt
0
= X(ω) ◦α exp
◦β
Z
T
f (t)dBt
(3.5.3)
0
for every α, β ∈ [0, 2].
Proof. By the aid of the definition of the α-products the proof comes very
46Chapter 3. Newton-Leibniz rule for stochastic integrals and approximations of SDE’s
easy.
T
Z
◦β
X ◦α exp (
f (t)dBt ) =
0
X (α − 1)n Z
n≥0
=
n!
[0,T ]n
X (α − 1)n Z
n≥0
X 1
=
n!
n!
Z
[0,T ]n
[0,T ]n
n≥0
Dtn X
by using the (3.5.2) = T(α−1)f X · exp
Dtn X
·
Dtn exp◦β
T
Z
f (t)dBt dt
0
Dtn X · f ⊗n (t) exp◦β
Z
T
f (t)dBt dt
0
· ((α − 1)f )
⊗n
(t)dt · exp
◦β
Z
Z
T
f (t)dBt
0
2
Remark 3.5.4. For α = 0 we find a simple proof of the Gjessing’s lemma.
Moreover let us observe that this formula is valid for every β ∈ [0, 2], so the
type of translation the operator Tf applies on the variable X depends on the
kind of product we use.
The next proposition shows the differentiation of the α-products between
two Itô processes. Given two Itô processes
Z t
Z t
Z t
Z t
Xt =
as ds +
bs dBs and Yt =
cs ds +
ds dBs
(3.5.4)
0
0
0
0
by applying the Itô formula we have
Z t
Z t
Xt · Yt =
(bs Ys + ds Xs )dBs +
(as Ys + cs Xs + bs ds )ds
0
(3.5.5)
0
or in a differential formalism
d(Xt · Yt ) = Yt dXt + Xt dYt + d[X, Y ]t .
Proposition 3.5.5. Let X and Y be the Itô processes (3.5.4), with at , bt , ct , dt ∈
G√|α−1|+1 for every t ≥ 0 and α ∈]0, 2]. Then
Z
t
Z
(bs ◦α Ys +ds ◦α Xs )dBs +
X◦α Y =
0
t
(as ◦α Ys +cs ◦α Xs +α bs ◦α ds )ds (3.5.6)
0
or in differential formalism
d(Xt ◦α Yt ) = Yt ◦α dXt + Xt ◦α dYt + α (bt ◦α ds )ds
f (t)dBt
0
◦β
T
(3.5.7)
47
3.5. Wong-Zakai-type theorems
Proof. Let us observe that
Z T
Z
√
√
Γ( α)
at dBt = α
0
T
√
Γ( α)at dBt
0
It is very easy to prove this property by expliciting the chaos decomposition
of at .
For the representation (2.1.9) of the α-product, and by using the (3.5.5) we
have
1 √
√
Xt ◦α Yt = Γ √ (Γ( α)Xt · Γ( α)Yt )
α
Z t
1 h Z t √
√
√
= Γ √
Γ( α)as ds + α
Γ( α)bs dBs
α
0
0
Z t
Z t √
i
√
√
Γ( α)cs ds + α
Γ( α)ds dBs
·
0
0
1 h√ Z t √
√
√
√
= Γ √
α
Γ( α)bs Γ( α)Ys + Γ( α)ds Γ( α)Xs dBs
α
0
Z t
√
√
√
√
√
+ α
Γ( α)as Γ( α)Ys + Γ( α)cs Γ( α)Xs ds
0
Z t
i
√
√
+α
Γ( α)bs Γ( α)ds ds
0
Z t
Z t
=
(bs ◦α Ys + ds ◦α Xs )dBs +
(as ◦α Ys + cs ◦α Xs + α bs ◦α ds )ds
0
0
2
Let ρ ≥ 0 be a smooth function C ∞ -differentiable on R with compact support
on [0, T ] and such that
Z
ρ(t)dt = 1.
R
For k ∈ N define
φk (t) = kρ(kt)
and let
Wtk
Z
for t ∈ R
T
φk (s − t)dBs
:=
0
for t ∈ R. Then we can prove the existence and uniqueness of the solution of
the approximated equation in the following theorem:
48Chapter 3. Newton-Leibniz rule for stochastic integrals and approximations of SDE’s
Theorem 3.5.6. Let b : R → R be such that there exist C1 , C2 > 0 s.t.:
|b(x) − b(y)| ≤ C1 |x − y|,
∀x, y ∈ R
|b(x)| ≤ C2 (1 + |x|),
(3.5.8)
∀x ∈ R
Then for any K ≥ 1, the equation
n
Ẋt = b(Xtk ) + Xtk ◦α Wtk
X0k = x ∈ R
(3.5.9)
has a unique solution {Xtk }t∈[0,T ] such that for any p ≥ 1 and t ∈ [0, T ],
E[|Xtk |p ] < +∞
Proof. Let Btk =
Btk =
Z tZ
Rt
0
Wsk ds. In particular
T
Z
T
φ(s − u)dBu ds =
0
0
0
t
Z
Z
φ(s − u)ds dBu =
Z
Φ(t, u)dBu
0
0
where
T
t
φ(s − u)ds
Φ(t, u) :=
0
.
Denote for k ≥ 1 and t ∈ [0, T ],
Akt := exp◦α (−Btk )
and observe that
d k
A = −Akt ◦α Wtk .
dt t
In fact, by expliciting the α-version of the exponential we have
d
α−1
exp − Btk +
|Φ(t, ·)|2L2 ([0,T ])
dt
2
Z T
◦α
k
k
= exp (−Bt ) · − Wt + (α − 1)
Φ(t, u)φ(t − u)du
d
exp◦α (−Btk ) =
dt
0
and
−Akt ◦α Wtk = − exp◦α (−Btk )Wtk + (α − 1)
Z
T
Φ(t, u)φ(t − u)du
0
49
3.5. Wong-Zakai-type theorems
Therefore if Xtk is a solution of (3.5.9) then Ztk := Xtk ◦ Akt solves:
d k
d
Zt = X˙tk ◦α Akt + Xtk ◦α Akt
dt
dt
= (b(Xtk ) + Xtk ◦α Wtk ) ◦α Akt + Xtk ◦α (−Akt ◦α Wtk )
= b(Xtk ) ◦α Akt + Xtk ◦α Akt ◦α Wtk − Xtk ◦α Akt ◦α Wtk
= b(Xtk ) ◦α Akt
= b(Xtk ◦α Akt ◦α (Akt )◦α (−1) ) ◦α Akt
= b(Ztk ◦α (Akt )◦α (−1) ) ◦α Akt
where
(Akt )◦α (−1) = exp◦α (Btk ).
In the first passage we have applied the (3.5.7). By applying Gjessing’s formula
(3.5.3) for the antiWick-exponential we get
b(Ztk ◦α (Akt )◦α (−1) ) ◦α Akt = b(T−Φ (Ztk ◦α (Akt )◦α (−1) )) · Akt
= b(T−Φ (TΦ Ztk · (Akt )◦α (−1) )) · Akt
= b(Ztk · (Akt )−1 ) · Akt
since T−Φ (Akt )◦α (−1) = (Akt )−1 . Then we get
n
dZtk
k
k −1
k
dt = b(Zt (At ) )At
k
Z0 = x
(3.5.10)
Similarly we can show that if {Ztk }t∈[0,T ] is a solution of (3.5.10) then Xtk :=
Ztk ◦α (Akt )◦α (−1) is a solution of (3.5.9). This means that if equation (3.5.10)
has a unique solution, then also (3.5.9) does. But this is the case since b is
assumed to be Lipschitz-continuous and equation (3.5.10) can be solved as
an ordinary differential equation where ω ∈ Ω is a fixed parameter. Now let
{Ztk }t∈[0,T ] be the unique solution of (3.5.10).
50Chapter 3. Newton-Leibniz rule for stochastic integrals and approximations of SDE’s
Then:
|Ztk |
t
Z
= |x +
b(Zsk (Aks )−1 )Aks ds|
0
Z
t
≤ |x| +
|b(Zsk (Aks )−1 )Aks |ds
0
Z
t
C2 1 + |Zsk (Aks )−1 | Aks ds
0
Z t
Z t
k
C2 |Zsk |ds
C2 As ds +
= |x| +
0
0
Z t
Z T
k
C2 |Zsk |ds.
C2 As ds +
≤ |x| +
≤ |x| +
0
0
By the Gronwall inequality
|Ztk |
Z
≤ |x| +
T
C2 Aks ds
eC2 t
(3.5.11)
0
This shows that the LHS of (3.5.11) is in Lp (Ω), ∀p ≥ 1 and by the Gjessing
2
formula that also Xtk = Ztk ◦α (Akt )◦α (−1) is in Lp (Ω) for any p ≥ 1
In the next lemma we prove the existence and uniqueness of the solution of
the quasilinear SDE’s with α-integrals.
Lemma 3.5.7. Let b : R → R be such that there exist C1 , C2 > 0 s.t.:
|b(x) − b(y)| ≤ C1 |x − y|,
|b(x)| ≤ C2 (1 + |x|),
∀x, y ∈ R
∀x ∈ R
Denote by {Xt }t∈[0,T ] the unique strong solution of:
n dX = b(X )dt + X dB α ,
t
t
t
t
X0 = x ∈ R
t ∈]0, T ]
(3.5.12)
Then
Zt := Xt ◦α At
with At := exp◦α (−Bt ),
solves
n
dZt
dt = b(Zt
Z0 = x
· A−1
t )At ,
t ∈]0, T ]
t ∈ [0, T ]
51
3.5. Wong-Zakai-type theorems
Proof. Observe that, by virtue of the definition of the α-integral (3.3.1) and
(3.3.5), equation (3.5.12) can be rewritten as:
α dXt = b(Xt ) + Xt dt + Xt dBt
2
where the stochastic integration is in the Ito’s sense. Moreover
dAt = −At dBt +
α
At dt.
2
Hence, from (3.5.7)
d(Xt ◦α At ) = At ◦α dXt + Xt ◦α dAt + α dXt ◦α dAt
α = At ◦α b(Xt ) + Xt dt + Xt dBt
2
α
At dt − At dBt − αAt ◦α Xt dt
+Xt ◦α
2
= b(Xt ) ◦α At dt
◦ (−1)
= b((Xt ◦α At ) ◦α At α
) ◦α At dt
By defining Zt := Xt ◦α At and by using Gjessing’s formula in the same way
as in the proof of the previous theorem, we get
dZt = b(Zt · A−1
t )At dt
2
Finally we can prove the “α-version” of the Wong-Zakai theorem.
Theorem 3.5.8. Let {Xtk }t∈[0,T ] , k ≥ 1 be the unique solution of (3.5.9) and
let {Xt }t∈[0,T ] be the unique solution of (3.5.12) where as before the function
b : R → R is assumed to satisfy conditions (3.5.8). Then for any p ≥ 1, we
have
lim Xtk = Xt
in Lp (Ω)
k→∞
Proof. Using the notation of the previous Theorem and Lemma, let {Ztk }
and {Zt } be the solution of
Z˙tk = b(Ztk (Akt )−1 )Akt ,
Z0k = x
and
Żt = b(Zt A−1
t )At ,
Z0 = x,
52Chapter 3. Newton-Leibniz rule for stochastic integrals and approximations of SDE’s
respectively. We will first prove that
lim Ztk = Zt
k→∞
in Lp (Ω),
∀p ≥ 1.
Indeed,
|Ztk
Z
− Zt | = |
t
b(Zsk (Aks )−1 )Aks ds
t
Z
−
b(Zs A−1
s )As ds|
0
0
Z t
k
−1
k
−1
= | (b(Zsk (Aks )−1 )Aks − b(Zs A−1
s )As + b(Zs As )As − b(Zs As )As ds|
0
Z t
Z t
k
k
k −1
−1
k
|b(Zs A−1
|b(Zs (As ) ) − b(Zs As )|As ds +
≤
s )||As − As |ds
0
0
Z t
Z t
k
k
k −1
−1 k
C2 (1 + |Zs A−1
C1 |Zs (As ) − Zs As |As ds +
≤
s |)|As − As |ds
0
0
Z t
k
|Zsk (Aks )−1 − Zs (Aks )−1 |Aks + |Zs (Aks )−1 − Zs A−1
≤ C1
s |As ds
0
Z t
k
(1 + |Zs |A−1
+C2
s )|As − As |ds
0
t
Z
|Zsk
= C1
Z
− Zs |ds + C1
0
Z
t
k
|Zs ||(Aks )−1 − A−1
s |As ds
0
t
k
+C2
(1 + |Zs |A−1
s )|As − As |ds
0
Z T
Z t
k
|Zs ||(Aks )−1 − A−1
|Zsk − Zs |ds + C1
≤ C1
s |As ds
0
0
Z T
k
+C2
(1 + |Zs |A−1
s )|As − As |ds
0
Z t
= Λ k + C1
|Zsk − Zs |ds
0
where
Z
Λk := C1
T
|Zs ||(Aks )−1
−
k
A−1
s |As ds
Z
+ C2
0
T
k
(1 + |Zs |A−1
s )|As − As |ds.
0
By Gronwall inequality we deduce that
|Ztk − Zt | ≤ Λk eC1 t ,
t ∈ [0, T ]
53
3.5. Wong-Zakai-type theorems
and
E[|Ztk − Zt |p ] ≤ epC1 t E[Λpk ]
for p ≥ 1
Since
|(Aks )−1 − A−1
s | −→k→∞ 0
|Aks
− As | −→k→∞ 0
in Lp (Ω)
∀s ∈ [0, T ]
Lp (Ω)
∀s ∈ [0, T ]
in
k
Zs , A−1
s , As
∈
Lp (Ω),
∀p ≥ 1
by applying Jensen and Hölder inequalities we can see that
E[Λpk ] −→k→∞ 0
and hence that
||Ztk − Zt ||p −→k→∞ 0,
∀p ≥ 1, ∀t ∈ [0, T ]
To prove that
||Xtk − Xt ||p −→k→∞ 0
observe that
Xtk − Xt = Ztk ◦α Akt − Zt ◦α At
= Ztk ◦α Akt − Zt ◦α Akt + Zt ◦α Akt − Zt ◦α At
= (Ztk − Zt ) ◦α Akt + Zt ◦α (Akt − At )
And that by Gjessing formula,
||Xtk − Xt ||p ≤ Ck,t ||Ztk − Zt ||q + Dk,t ||Zt ||q
∀q ≥ p
where
sup Ck,t < +∞ and Dk,t −→k→∞ 0.
k
2
54Chapter 3. Newton-Leibniz rule for stochastic integrals and approximations of SDE’s
Chapter 4
Sharp inequalities for
Gaussian Wick products
In order to achieve the results we will work on a finite dimensional probability
space. Passing from the finite dimensional case to the infinite dimensional
case will be done by using the Fatou’s lemma.
4.1
A connection between the Gaussian Wick product and Lebesgue convolution product
Let d be a fixed positive
integer. Let dN x denote the normalized Lebesgue
√
measure on Rd , (1/ 2π)d dx, and µ the standard Gaussian probability measure on Rd , i.e., dµ = e−hx,xi/2 dN x. If Xi : Ω → R, i = {1, 2, . . . , d} are
independent standard normal random variables, and F is the sigma–algebra
generated by them, then any random variable Y : Ω → C, that is measurable
with respect F, can be written as Y = g(X1 , X2 , . . . , Xd ), where g : Rd → C
is a Borel measurable function. From now on we will write g(x) with a lower
case x instead of the upper case X, and do the computations of integrals in
terms of the probability distribution µ of X, where X := (X1 , X2 , . . . , Xd ).
Observe that in terms of distributions, for any p ≥ 1, the Lp norm of a function
f (x) with respect to the Gaussian measure µ is the same as the Lp norm of
f (x)e−hx,xi/(2p) with respect to the normalized Lebesgue measure dN x. This
simple fact will be used throughout this paper. Everything will be done using
the normalized Lebesgue measure. Throughout this paper, for any p ∈ [1, ∞],
we will denote by k · kp and k| · |kp the Lp norms with respect to the Gaussian
56
Chapter 4. Sharp inequalities for Gaussian Wick products
measure µ and normalized Lebesgue measure dN x, respectively.
We are now presenting a connection between the Gaussian Wick product and convolution product ? with respect to the normalized Lebesgue measure.
Lemma 4.1.1. Let u and v be positive numbers, such that: (1/u) + (1/v) = 1.
√
√
Then for any ϕ and ψ in L1 (Rd , µ), we have that Γ(1/ u)ϕΓ(1/ v)ψ belongs
to L1 (Rd , µ) and:
hx,xi
hx,xi
1
1
− 2v
− 2u
ϕ √ x e
? ψ √ x e
v
u
1
1
1
1
√ x e− 2uv hx,xi ,
= Γ √
ϕΓ √
ψ
(4.1.1)
u
v
uv
where the convolution product in the left–hand side is computed with
respect to the normalized Lebesgue measure dN x.
We prove first an easier version of this lemma.
P
Lemma 4.1.2. Let E = { ni=1 ci ehξi ,xi−(1/2)hξi ,ξi i | n ∈ N, ci ∈ C, ξi ∈ Cd , ∀i ∈
{1, 2, . . . , n}}. Let u and v be positive numbers, such that (1/u) + (1/v) = 1.
For any ϕ and ψ in E, we have:
√
x
√
x
hx,xi
hx,xi
Γ( u)ϕ √
e− 2v
? Γ( v)ψ √
e− 2u
v
u
hx,xi
x
= (ϕ ψ) √
e− 2uv .
(4.1.2)
uv
√
√
In particular, replacing ϕ and ψ by Γ(1/ u)ϕ and Γ(1/ v)ψ, respectively, we
obtain that (4.1.1) holds for any two functions that are linear combinations of
exponential functions.
Proof. Since both sides of (4.1.2) are bilinear with respect to ϕ and ψ,
it is enough to check the relation for ϕ(x) = ehξ,xi−(1/2)hξ,ξi and ψ(x) =
ehη,xi−(1/2)hη,ηi , where ξ and η are arbitrarily fixed vectors in Cd . Indeed,
for these functions we have:
√
√
h·,·i
h·,·i
·
·
− 2v
− 2u
Γ( u)ϕ √
e
? Γ( v)ψ √
e
(x)
v
u
√
√
h·,·i
h·,·i
u
v
√ hξ,·i− u hξ,ξi−
√ hη,·i− v hη,ηi−
2
2v
2
2u
v
u
= e
? e
(x)
Z
√
√
hx−y,x−yi
hy,yi
u
v
√ hξ,x−yi− u hξ,ξi−
√ hη,yi− v hη,ηi−
2
2v
2
2u d y
=
e v
·e u
N
Rd
D √
E
Z
√
√
v
u
√ hξ,xi− u hξ,ξi− v hη,ηi− 1 hx,xi
− 1 hy,yi+ − √u
ξ+ √u
η+ v1 x,y
v
2
2
2v
= e v
e 2
dN y.
Rd
4.1. A connection between the Gaussian Wick product and Lebesgue convolution product57
We now perform the classic trick of completing the square, in the exponential,
√ √
√ √
√ √
by subtracting and adding (1/2)h−( u/ v)ξ+( v/ u)η+(1/v)x, −( u/ v)ξ+
√ √
( v/ u)η + (1/v)xi, a factor that does not depend on the variable of integration y, and can be taken out of the integral. Thus, we obtain:
√
√
h·,·i
h·,·i
·
·
− 2v
− 2u
Γ( u)ϕ √
e
? Γ( v)ψ √
e
(x)
v
u
D √
E
√
√
√
√
u
v
u
v
u
1
√ ξ+ √ η+ 1 x,− √ ξ+ √ η+ 1 x
√ hξ,xi− u hξ,ξi− v hη,ηi− 1 hx,xi
−
2
v
v
v
u
v
u
2
2
2v
v
D
E
√
√
√
√
u
v
u
v
1
1
1
√
√
√
√
− 2 y− v ξ+ u η+ v x,y− v ξ+ u η+ v x
·e
= e Z
×
e
dN y
Rd
√
1
1− v1 )hξ,ξi−hξ,ηi− v2 (1− u
−u
)hη,ηi+ √uv (1− v1 )hξ,xi+ √1uv hη,xi− 2v1 (1− v1 )hx,xi
2(
= eZ
e
− 21 hz,zi
×
dN z
Rd
√
1 1
· 1 hξ,ξi−hξ,ηi− v2 · v1 hη,ηi+ √u
· 1 hξ,xi+ √1uv ·hη,xi− 2v
· u hx,xi
−u
2 u
v u
= e
·1
1
− 21 hξ+η,ξ+ηi+hξ+η, √1uv xi− 2uv
·hx,xi
= e
= (ϕ ψ)
x
√
uv
e−
hx,xi
2uv
,
since (ϕ ψ)(x) = ehξ+η,xi−(1/2)hξ+η,ξ+ηi .
2
A complete proof of Lemma 4.1.1 is the following:
Proof. Since E is dense in L1 (Rd , µ), there exist two sequences {fn }n≥1 and
{gn }n≥1 of elements of E, such that: k fn − ϕ k1 → 0 and k gn − ψ k1 → 0, as
n → ∞. For each n ≥ 1, we have:
hx,xi
hx,xi
x
x
− 2v
− 2u
fn √
e
? gn √
e
v
u
hx,xi
1
1
x
√
= Γ √
fn Γ √
gn
e− 2uv .
u
v
uv
hx,xi
hx,xi
x
x
− 2v
− 2u
e
? ψ √
e
, in
The left–hand side converges to ϕ √
v
u
L1 (Rd , µ), as n → ∞, by Young inequality for the normalized Lebesgue measure. The right–hand side converges to
hx,xi
1
1
x
√
Γ √
ϕΓ √
ψ
e− 2uv ,
u
v
uv
by Lanconelli–Stan inequality, from [22], about the L1 norms for Wick products.
2
58
4.2
Chapter 4. Sharp inequalities for Gaussian Wick products
Hölder-Young-Lieb inequality
Theorem 4.2.1. (Finite dimensional Hölder inequality for Gaussian
Wick products.) Let d ∈ N and p ∈ [1, ∞] be fixed. Let µ denote the
standard Gaussian probability measure on Rd . Let u and v be positive numbers,
√
such that (1/u) + (1/v) = 1. Then for any ϕ and ψ in Lp (Rd , µ), Γ(1/ u)ϕ √
Γ(1/ v)ψ ∈ Lp (Rd , µ) and the following inequality holds:
Γ √1 ϕ Γ √1 ψ ≤ kϕkp · kψkp .
(4.2.1)
u
v
p
Proof. Let p0 be the conjugate of p, i.e., (1/p) + (1/p0 ) = 1. Let’s multiply
0
both sides of formula (4.1.1) by ehx,xi/(2p uv) . We get:
hx,xi 1
x
1
−
1− p10
√
ϕΓ √
ψ
e 2uv
Γ √
u
v
uv
hx,xi
hx,xi
hx,xi
x
x
− 2v
− 2u
0 uv
2p
= e
ϕ √
e
? ψ √
e
v
u
hx−y,x−yi Z hx−y,x−yi
hy,yi
y
x−y
−
−
− 2pu
0v
2pv
2p
√
√
e
e
ψ
e
·
=
ϕ
v
u
Rd
hy,yi
− 2p0 u
hx,xi
· e 2p0 uv dN y.
√
Let f (x) := ϕ((1/ v)x) exp(−[1/(2pv)]hx, xi)
√
and g(x) := ψ((1/ u)x)e(−[1/(2pu)]hx, xi). With these notations we get:
hx,xi 1
1
1
x
−
·
√
Γ √
ϕΓ √
ψ
e 2uv p
u
v
uv
Z
− 1 y− 1 x,y− v1 xi
dN y.
=
f (x − y)g(y)e 2p0 h v
e
Rd
Putting the modulus in both sides, then introducing it in the integral in the
right (triangle inequality), and then applying the Hölder inequality to the pair
(p, p0 ), we get:
hx,xi
x
− 2puv Γ √1 ϕ Γ √1 ψ
√
e
u
v
uv
Z
1 Z 10
p 0
p
p
1
1
1
− 2p
p
0 hy− v x,y− v xi
≤
|f (x − y)g(y)| dN y
e
dN y
Rd
Z
≤
Rd
Rd
|f (x − y)|p |g(y)|p dN y
1
p
· 1.
4.2. Hölder-Young-Lieb inequality
59
Let us first raise the last inequality to the power p, then integrate it with
respect to x, and apply Fubini (actually Tonelli) theorem. We obtain:
p
Z hx,xi 1
1
x
−
Γ √
√
ϕΓ √
ψ
e 2puv dN x
u
v
uv
Rd
Z Z
p
p
≤
|f (x − y)| |g(y)| dN y dN x
Rd
Rd
Z
Z
p
p
|f (x − y)| dN x dN y
=
|g(y)|
Rd
Rd
Z
|g(y)|p k|f |kpp dN y
=
d
R
Z
p
|g(y)|p dN y
= k|f |kp
Rd
= k|f |kpp · k|g|kpp .
Raising both sides of this inequality to the power 1/p, we get:
Z p
1/p
− hx,xi
x
Γ √1 ϕ Γ √1 ψ
√
e 2uv dN x
u
v
uv Rd
≤ k|f |kp · k|g|kp
1/p Z 1/p
Z x p − hx,xi
x p − hx,xi
·
.
=
ψ √u e 2u dN x
ϕ √v e 2v dN x
Rd
Rd
√
Making now the changes of variable: x1 := (1/ uv)x in the integral
p
Z − hx,xi
x
Γ √1 ϕ Γ √1 ψ
e 2uv dN x,
√
u
v
uv Rd
√
√
x0 := (1/ v)x and x00 := (1/ u)x in the integrals:
p
Z hx,xi
ϕ √x e− 2v dN x
v
Rd
and
Z
p
hx,xi
ψ √x e− 2u dN x,
u d
R
respectively, and dividing both sides by (uv)d/(2p) ,
since dµ = exp(−hx, xi/2)dN x, we get:
Γ √1 ϕ Γ √1 ψ ≤ kϕkp · kψkp .
u
v
p
60
Chapter 4. Sharp inequalities for Gaussian Wick products
2
Theorem 4.2.2. (General Hölder inequality for Gaussian Wick products.) Let H be a separable Gaussian Hilbert space, p ∈ [1, ∞], and u and
v positive numbers, such that (1/u) + (1/v) = 1. Let F(H) be the sigma–
algebra generated by the random variables h from H. Then for any ϕ and
√
√
ψ in Lp (Ω, F(H), P ), Γ(1/ u)ϕ Γ(1/ v)ψ ∈ Lp (Ω, F(H), P ) and the
following inequality holds:
Γ √1 ϕ Γ √1 ψ ≤ kϕkp · kψkp .
(4.2.2)
u
v
p
Proof. Let {en }n≥1 be an orthonormal basis of H. Then {en }n≥1 is a set of
independent, normally distributed random variables with mean 0 and variance
1. For every d ≥ 1, let Fd denote the sigma–algebra generated by the random
variables e1 , e2 , . . . , ed . It is well–known that every function f from Lp (Ω,
F(H), P ) can be approximated in the p norm by a sequence of functions fn
from Lp (Fn ) := Lp (Ω, Fn , P ), n ≥ 1. This is due to the fact that the sigma–
algebra F(H) is generated by the cylinder sets, and every cylinder set is in
one of the sigma–algebras Fd , for some d ≥ 1. There exist two sequences
{ϕn }n≥1 and {ψn }n≥1 contained in ∪n≥1 Lp (Fn ), such that kϕn − ϕkp → 0 and
kψn − ψkp → 0, as n → ∞. These facts imply two things: first kϕn kp → kϕkp
and kψn kp → kψkp , as n → ∞, and second ϕn → ϕ and ψn → ψ, in L1 , as n →
√
√
√
√
∞. We know from [22] that Γ(1/ u)ϕn Γ(1/ v)ψn → Γ(1/ u)ϕΓ(1/ v)ψ,
in L1 , as n → ∞. Since L1 convergence implies almost sure convergence for
a subsequence, working eventually with a subsequence, we may assume that
√
√
√
√
Γ(1/ u)ϕn Γ(1/ v)ψn → Γ(1/ u)ϕ Γ(1/ v)ψ, almost surely, as n → ∞.
Using now Fatou’s lemma and the finite dimensional inequality proven in the
previous theorem, we have:
p
p Γ √1 ϕ Γ √1 ψ = E Γ √1 ϕ Γ √1 ψ u
v
u
v
p
p 1
1
= E lim inf Γ √
ϕn Γ √
ψn n→∞
u
v
p 1
1
ϕn Γ √
ψn ≤ lim inf E Γ √
n→∞
u
v
p
p
≤ lim inf {E [|ϕn | ] · E [|ψn | ]} .
n→∞
= E [|ϕ|p ] · E [|ψ|p ]
= kϕkpp · kψkpp .
61
4.2. Hölder-Young-Lieb inequality
2
We are now presenting an interesting connection between the Young inequality with the best constant, and Nelson hypercontractivity. The Young
inequality with the best constant is the following:
Theorem 4.2.3. Let p, q, r ≥ 1, such that (1/p) + (1/q) = (1/r) +
1. There
p Rd , dx and g ∈
exists a constant
C
>
0,
such
that,
for
any
f
∈
L
p,q,r;d
Lq Rd , dx , we have:
Z
r
|(f ? g)(x)| dx
Rd
1/r
Z
≤ Cp,q,r;d
1/p Z
|f (x)| dx
·
p
Rd
The sharp constant Cp,q,r;d equals (Cp Cq /Cr
any k ≥ 1, where k 0 is the conjugate of k.
q
|g(x)| dx
Rd
)d ,
where
Ck2
1/q
=
.
(4.2.3)
for
0
k 1/k /k 01/k ,
See [1], [4], or [25] (Theorem 4.2) for a proof. Let us make the observation
that if we replace the Lebesgue measure dx, on Rd , by cdx, where c is any
positive constant, then, by convoluting with respect to cdx, the best constant
Cp,q,r:d from inequality (4.2.3) does not change. This is due to the fact that
the left–hand side of (4.2.3) is multiplied by c · c1/r = c1+(1/r) , while the
right–hand side must be multiplied by c1/p · c1/q = c(1/p)+(1/q) , and fortunately
1 + (1/r) = (1/p) + (1/q). Thus, the inequality (4.2.3) remains valid, with the
same sharp constant Cp,q,r;d , even for the normalized Lebesgue measure dN x.
Theorem 4.2.4. Let d be a fixed natural number. Let p and r be positive
numbers such that 1 <√p ≤ r. Then for any√ϕ in Lp (Rd , µ) and any ψ in
√
√
L∞ (Rd , µ), Γ( p − 1/ r − 1)ϕ Γ( r − p/ r − 1)ψ belongs to Lr (Rd , µ)
and the following inequality holds:
√
√
Γ √p − 1 ϕ Γ √r − p ψ ≤ kϕkp · kψk∞ . (4.2.4)
r−1
r−1
r
In particular, if we choose ψ = 1 (the constant random variable equal to 1),
then we get the classic Nelson hypercontractivity inequality:
√
Γ √p − 1 ϕ ≤ kϕkp .
(4.2.5)
r−1
r
Proof. Let u := (r − 1)/(p − 1) and v := (r − 1)/(r − p). Then we have
(1/u) + (1/v) = 1. Let p0 and r0 be the conjugates of p and r, respectively .
62
Chapter 4. Sharp inequalities for Gaussian Wick products
Let’s go back to the identity (4.1.1) and multiply both sides of that relation
by exp(−hx, xi/(2uvr0 )). We get:
hx,xi
1
1
1
x
√
Γ √
ϕΓ √
ψ
e− 2uv (1− r0 )
u
v
uv
hx,xi
hx,xi
hx,xi
x
x
ϕ √
e− 2v ? ψ √
e− 2u
= e 2r0 uv
v
u
hx−y,x−yi
Z hx−y,x−yi
x−y
−
−
2pv
2p0 v
e
ϕ √
=
e
×
v
Rd
hx,xi
hy,yi
y
− 2u
√
ψ
e
· e 2r0 uv dN y.
u
√
Let f (x) := ϕ(x/ v) exp(−hx, xi/(2pv)). We have:
≤
≤
=
=
hx,xi
x
− 2uv (1− r10 ) Γ √1 ϕ Γ √1 ψ
√
e
u
v
uv
Z
hx−y,x−yi hy,yi
hx,xi
−
ψ √y e− 2u · e 2r0 uv dN y
2p0 v
|f (x − y)|e
u
d
ZR
hx−y,x−yi
hx,xi
hy,yi
−
2p0 v
|f (x − y)|e
kψk∞ e− 2u · e 2r0 uv dN y
Rd
Z
hx−y,x−yi
hx,xi
hy,yi
−
2p0 v
· e− 2u · e 2r0 uv dN y
kψk∞
|f (x − y)| · e
d
h
i
ZR
hx−y,x−yi
hx,xi
hy,yi
−
+ 2u − 2r0 uv
0v
2p
kψk∞
|f (x − y)| · e
dN y.
Rd
Let us observe that the expression:
E(x, y) =
hx − y, x − yi hy, yi hx, xi
+
− 0
2p0 v
2u
2r uv
(4.2.6)
4.2. Hölder-Young-Lieb inequality
is a perfect square. Indeed, the coefficient of hx, xi in E(x, y) is:
a =
=
=
=
=
=
=
=
1
1
− 0
0
2p v 2r uv
1
1
1 1
− ·
2v p0 r0 u
1 p−1 r−1 p−1
−
·
2v
p
r
r−1
p−1 1 1
−
2v
p r
p−1 r−p
·
2v
pr
p−1 r−1 r−p 1
·
·
·
p
r
r − 1 2v
1 1 1 1
· · ·
p0 r0 v 2v
1
.
2p0 r0 v 2
The coefficient of hy, yi in E(x, y) is:
1
1
+ 0
2u 2p v
1 1
1 1
+ 0·
2 u p v
1 p−1 p−1 r−p
+
·
2 r−1
p
r−1
c =
=
=
=
=
=
=
=
p−1
r−p
1+
2(r − 1)
p
r
p−1
·
2(r − 1) p
1
r
p−1
·
·
2 r−1
p
1 0 1
·r · 0
2
p
0
r
.
2p0
63
64
Chapter 4. Sharp inequalities for Gaussian Wick products
The coefficient of hx, yi is E(x, y) is:
b = −
1
.
p0 v
Thus we have:
E(x, y) = ahx, xi + bhx, yi + chy, yi
1
r0
1
hx,
xi
−
hx,
yi
+
hy, yi
=
2p0 r0 v 2
p0 v
2p0
1
hx − r0 vy, x − r0 vyi.
=
2p0 r0 v 2
It follows now from (4.2.6) that:
hx,xi
x
− 2uv · r1 Γ √1 ϕ Γ √1 ψ
√
e
u
v
uv
Z
− 1 hx−r0 vy,x−r0 vyi
|f (x − y)| · e 2p0 r0 v2
≤ kψk∞
dN y.
Rd
Let us make the change of variable t := x − y in the last integral. We obtain:
hx,xi x
− 2uvr Γ √1 ϕ Γ √1 ψ
√
e
u
v
uv
Z
− 1 hx−r0 v(x−t),x−r0 v(x−t)i
≤ kψk∞
dN t
|f (t)| · e 2p0 r0 v2
d
R
Z
− 1 h(1−r0 v)x+r0 vt,(1−r0 v)x+r0 vti
dN t
= kψk∞
|f (t)| · e 2p0 r0 v2
Rd
E
D 0
Z
r 0 v−1
− 0 10 2 ·(−r0 v)2 r rv−1
0 v x−t, r 0 v x−t
2p
r
v
= kψk∞
|f (t)| · e
dN t
d
ZR
0
− r · 1 x−t, s10 x−ti
= kψk∞
|f (t)| · e 2p0 h s0
dN t,
Rd
where s := r0 v and s0 is the conjugate of s. Let us observe that the last integral
is a convolution product. Indeed, if we define g(x) := exp(−[r0 /(2p0 )] · hx, xi),
then:
hx,xi x
− 2uvr Γ √1 ϕ Γ √1 ψ
√
e
u
v
uv
Z
0
− r · 1 x−t, s10 x−ti
≤ kψk∞
|f (t)| · e 2p0 h s0
dN t
Rd
1
= kψk∞ · [f ? g] 0 x ,
s
4.2. Hölder-Young-Lieb inequality
65
for all x ∈ Rd . Replacing x by s0 x, in the last inequality, we obtain:
0 s02 hx,xi sx
− 2uvr Γ √1 ϕ Γ √1 ψ
√
e
≤ kψk∞ · [f ? g] (x),
u
v
uv
for all x ∈ Rd . Raising this inequality to the power r and integrating with
respect to the normalized Lebesgue measure dN x, we get:
r
0 1/r
Z s02 hx,xi 1
s
x
1
−
Γ √
√
ϕΓ √
ψ
e 2uvr dN x
u
v
uv
Rd
Z
1/r
r
≤ kψk∞ ·
|(f ? g)(x)| dN x
.
Rd
√
Making, the change of variable x0 := (s0 / uv)x in the integral from the left,
we obtain:
√ d/r uv
Γ √1 ϕ Γ √1 ψ · e− 2r1 h·,·i ≤ kψk∞ · k|f ? g|kr .
0
s
u
v
r
The left hand–side is a Gaussian Lr –norm, and so, we get:
√ d/r uv
Γ √1 ϕ Γ √1 ψ ≤ kψk∞ · k|f ? g|kr .
s0
u
v
r
Since r ≥ p, (1/r) + 1 − (1/p) ≤ 1. Thus there exists q ≥ 1, such that
(1/r) + 1 = (1/p) + (1/q). We apply now the Young inequality with the sharp
constant, in the right side of (4.2.7), and obtain:
0 d/r
s
1
1
Γ √
√
≤
ϕΓ √
ψ
· kψk∞ · k|f ? g|kr
u
v
uv
r
0 d/r
s
√
≤
· kψk∞ · (Cp Cq /Cr )d k|f |kp · k|g|kq ,
uv
√
0
where Cp2 = p1/p /p01/p . Since f (x) = ϕ(x/ v) exp(−hx, xi/(2pv)), it is easy
to see that:
√
k|f |kp = ( v)d/p kϕkp .
(4.2.7)
Because g(x) := exp(−[r0 /(2p0 )] · hx, xi), it is not hard to see that:
s !d/q
p0
k|g|kq =
.
qr0
66
Chapter 4. Sharp inequalities for Gaussian Wick products
Thus, inequality (4.2.7) becomes:
0 d/r
√
s
Γ √1 ϕ Γ √1 ψ ≤
√
(Cp Cq /Cr )d ( v)d/p
u
v
uv
r
s
p0
qr0
!d/q
×
kϕkp kψk∞ .
Therefore, to prove (4.2.4), we just need to show that:
s0
√
uv
d/r
√
· (Cp Cq /Cr )d · ( v)d/p ·
s
p0
qr0
!d/q
= 1,
(4.2.8)
which (by raising both sides to the power 2/d) is equivalent to:
Cp2 Cq2 s02/r p01/q v 1/p−1/r
Cr2 u1/r q 1/q r01/q
= 1.
Since 1/p − 1/r = 1 − 1/q and 1 − 1/q = 1/q 0 , we have to prove that:
Cp2 Cq2 s02/r p01/q v 1/q
0
= 1.
Cr2 u1/r q 1/q r01/q
(4.2.9)
Let:
C :=
Cp2 Cq2 s02/r p01/q v 1/q
Cr2 u1/r q 1/q r01/q
0
.
(4.2.10)
To prove (4.2.9) we will write u, v, and s0 in terms of p, q, r and their conjugates. We have:
u =
=
r−1
p−1
r 1−
·
p 1−
1
r
1
p
1
r0
1
p0
=
r
·
p
=
rp0
.
pr0
(4.2.11)
67
4.2. Hölder-Young-Lieb inequality
We also have:
r−1
r−p
r 1 − 1r
pr p1 − 1r
v =
=
1 1 − 1r
·
.
p p1 − 1r
=
Let’s remember that (1/p) − (1/r) = 1/q 0 . Thus, we obtain:
v =
1 1 − 1r
·
p p1 − 1r
1
r0
1
q0
=
1
·
p
=
q0
.
pr0
(4.2.12)
Finally, we have:
s0 =
=
s
s−1
r0 v
.
r0 v − 1
Replacing v by q 0 /(pr0 ) we get:
s0 =
=
=
r0 v
r0 v − 1
q0
r0 pr
0
0
q
r0 pr
0 − 1
q0
p
q0
p
−1
.
Dividing both the numerator and denominator of the last fraction by q 0 we
68
Chapter 4. Sharp inequalities for Gaussian Wick products
get:
1
p
s0 =
=
=
1
p
−
1
p
− 1 − 1q
1
p
1
p
1
r
=
+
1
q0
1
p
1
p
1
q
−1
r
.
p
=
(4.2.13)
Let us substitute Cp2 , Cq2 , Cr2 , u, v, and s0 , in the formula (4.2.10). We have:
Cp2 Cq2 s02/r p01/q v 1/q
C =
0
Cr2 u1/r q 1/q r01/q
0 1/q0
2/r
p1/p
q 1/q
q
r
01/q ·
·
p
0 · 01/q 0 ·
01/p
p
pr0
p
q
.
0 1/r
rp
r1/r
1/q
01/q
·q ·r
0 ·
pr0
r01/r
=
0
Observe that the factors q 1/q , q 01/q , and r2/r cancel. Collecting the powers of
p, p0 , and r0 , we obtain:
1
C = pp
= p
− r1 − q10
1
− r1 −
p
= p
1
+ 1q
p
·p
0 1q − p10 − r1
1− 1q
·p
−( r1 +1)
·r
0 r10 + r1 − q10 − 1q
0 1q − 1− p1 − r1
·p
0
1
+ 1q
p
·r
0( r10 + r1 )− q10 + 1q
−( r1 +1)
· r01−1
= p0 · p00 · r00
= 1.
2
Passing from the finite dimensional case to the infinite dimensional case,
can be done in the same way as before, using Fatou’s lemma.
We now present a more general Hölder inequality. The proof of this inequality uses the following theorem of Lieb (see [24] or [25] (page 100)).
69
4.2. Hölder-Young-Lieb inequality
Theorem 4.2.5. Fix k > 1, integers n1 , . . . , nk and numbers p1 , . . . , pk ≥ 1.
Let M ≥ 1 and let Bi (for i = 1, . . . , k) be a linear mapping from RM to Rni .
Let Z : RM → R+ be some fixed Gaussian function,
Z(x) = exp [−hx, Jxi]
with J a real, positive–semidefinite M × M matrix (possible zero).
For functions fi in Lpi (Rni ) consider the integral IZ and the number CZ :
Z
IZ (f1 , . . . , fk ) =
Z(x)
RM
CZ
k
Y
fi (Bi x)dx
(4.2.14)
i=1
:= sup{IZ (f1 , . . . , fk ) | k|fi |kpi = 1 f or i = 1, . . . , k}. (4.2.15)
Then CZ is determined by restricting the f ’s to be Gaussian functions, i.e.,
CZ
= sup{IZ (f1 , . . . , fk ) | k|fi |kpi = 1 and fi (x) = ci exp[−hx, Ji xi]
with ci > 0, and Ji a real, symmetric,
positive − def inite ni × ni matrix}.
Corollary 4.2.6. Let p, q, r ≥ 1. Let B1 and B2 be linear maps from R2 to
R2 , and J a real, positive–semidefinite 2 × 2 matrix (possible zero). For f in
Lp (R2 ) and g in Lq (R2 ), we consider the product:
Z
(f ?B1 ,B2 ,J g) (x) =
f (B1 (x, y))g(B2 (x, y))e−h(x,y),J(x,y)i dN y.
R
We define:
C := sup{k|f ?B1 ,B2 ,J g|kr | k|f |kp = k|g|kq = 1}.
(4.2.16)
Then C is determined by restricting f and g to be Gaussian functions.
Proof. Let r0 be the conjugate of r. For any k ≥ 1, we denote by Gk the set
of Gaussian functions of Lk –norm equal to 1. Using the duality between Lr
70
Chapter 4. Sharp inequalities for Gaussian Wick products
0
and Lr , Lieb’s theorem, and Hölder inequality, we have:
C = sup{k|f ?B1 ,B2 ,J g|kr | k|f |kp = k|g|kq = 1}
Z
= sup (f ?B1 ,B2 ,J g)(x)h(x)dN x | k|f |kp = k|g|kq = k|h|kr0 = 1
ZR Z
−h(x,y),J(x,y)i
f (B1 (x, y))g(B2 (x, y))h(x)e
dN xdN y |
= sup R
R
k|f |kp = k|g|kq = k|h|kr0 = 1}
n Z Z
−h(x,y),J(x,y)i
f (B1 (x, y))g(B2 (x, y))h(x)e
dN xdN y ≤ sup R R
o
| f ∈ Gp , g ∈ Gq , h ∈ Gr0
≤ sup {k|f ?B1 ,B2 ,J g|kr · k|h|kr0 | f ∈ Gp , g ∈ Gq , h ∈ Gr0 }
= sup {k|f ?B1 ,B2 ,J g|kr · 1 | f ∈ Gp , g ∈ Gq } .
2
Theorem 4.2.7. (Full Hölder inequality for Gaussian Hilbert spaces.)
Let H be a separable Gaussian Hilbert space. Let u, v, p, q, and r be numbers
greater than 1, such that:
1 1
+
u v
= 1
and
1
1
+
u(p − 1) v(q − 1)
=
1
.
r−1
(4.2.17)
√
Then for all ϕ in Lp (Ω, F(H), P ) and ψ in Lq (Ω, F(H), P ), Γ(1/ u)ϕ √
Γ(1/ v)ψ belongs to Lr (Ω, F(H), P ) and the following inequality holds:
√
√
Γ(1/ u)ϕ Γ(1/ v)ψ ≤ kϕkp · kψkq .
r
(4.2.18)
Proof. Let p0 , q 0 , and r0 be the conjugates of p, q, and r, respectively. Since:
1
p−1
p
−1
p−1
= p0 − 1
=
4.2. Hölder-Young-Lieb inequality
71
and similarly 1/(q − 1) = q 0 − 1, and 1/(r − 1) = r0 − 1, condition (4.2.17) is
equivalent to:
r0 − 1 =
=
=
=
=
1
r−1
1
1
+
u(p − 1) v(q − 1)
1 0
1 0
p −1 +
q −1
u
v 1 0 1 0
1 1
·p + ·q −
+
u
v
u v
1 0 1 0
· p + · q − 1.
u
v
That means, we have:
p0 q 0
+
= r0 .
(4.2.19)
u
v
Following the same steps as before, it is enough to check the inequality in
the finite dimensional case. Multiplying both sides of the convolution identity
(4.1.1) by exp(1/(2uvr0 )hx, xi), we obtain:
hx,xi
1
1
1
x
√
Γ √
ϕΓ √
ψ
e− 2uv (1− r0 )
u
v
uv
hx,xi
hx,xi
hx,xi
x
x
= e 2r0 uv
ϕ √
e− 2v ? ψ √
e− 2u
v
u
hx−y,x−yi hy,yi
Z hx−y,x−yi
hy,yi
hx,xi
y
x−y
−
−
−
− 2qu
0v
2pv
2p
e
ψ √
e 2q0 u · e 2r0 uv dN y.
=
ϕ √
e
e
v
u
Rd
√
√
Let f (x) := ϕ(x/ v) exp(−hx, xi/(2pv)) and g(x) := ψ(x/ u) exp(−hx, xi/(2qu)).
We have:
hx,xi
x
− 2uv · r1 Γ √1 ϕ Γ √1 ψ
√
e
u
v
uv
Z
hx−y,x−yi
hy,yi
hx,xi
−
−
2p0 v
≤
|f (x − y)|e
|g(y)|e 2q0 u · e 2r0 uv dN y
d
ZR
hx−y,x−yi
hy,yi
hx,xi
−
− 2q0 u + 2r0 uv
2p0 v
=
|f (x − y)| · |g(y)| · e
dN y.
(4.2.20)
Rd
As before, we are now showing that the expression:
E(x, y) =
hx − y, x − yi hy, yi hx, xi
+
− 0
2p0 v
2q 0 u
2r uv
72
Chapter 4. Sharp inequalities for Gaussian Wick products
is a perfect square. Indeed, the coefficient of hx, xi in E(x, y) is:
1
1
a =
− 0
0
2p v 2r uv
1
p0
0
=
r −
2p0 r0 v
u
0
1
q
from (4.2.19) =
·
0
0
2p r v v
0 2
q
1
·
.
=
0
0
0
2p q r
v
The coefficient of hy, yi in E(x, y) is:
c =
=
from (4.2.19) =
=
1
1
+ 0
0
2q u 2p v
0
1
p
q0
+
2p0 q 0 u
v
1
· r0
2p0 q 0
1
· r02 .
2p0 q 0 r0
The coefficient of hx, yi is E(x, y) is:
1
b = − 0
pv
0
1
q
0
= − 0 0 0·
·r .
pqr
v
Thus we have:
E(x, y) = ahx, xi + bhx, yi + chy, yi
" #
0
q0 2
q
1
hx, xi − 2
· r0 hx, yi + r02 hy, yi
=
2p0 q 0 r0
v
v
0
q
q0
1
0
0
x − r y, x − r y .
=
2p0 q 0 r0 v
v
It follows now from (4.2.20) that:
hx,xi x
− 2uvr Γ √1 ϕ Γ √1 ψ
√
e
u
v
uv
D 0
E
Z
0
q
x−r0 y, qv x−r0 y
− 1
≤
|f (x − y)| · |g(y)| · e 2p0 q0 r0 v
dN y.
Rd
(4.2.21)
73
4.2. Hölder-Young-Lieb inequality
Claim 1: For all d ≥ 1, f ∈ Lp (Rd , dN x), and g ∈ Lq (Rd , dN y), we have:
Z Z
r
1/r
|f (x − y)| · |g(y)| · Jd (x, y)dN y dN x
≤ C d k|f |kp · k|g|kq ,
Rd
Rd
(4.2.22)
where:
− 2p01q0 r0
D
Jd (x, y) = e
E
0
q0
x−r0 y, qv x−r0 y
v
and
1
C2 = v r
− p1
1
ur
− 1q
.
To prove this claim, we reduce the problem to the one–dimensional case, via
Minkowski’s inequality, copying the argument from [25] (page 201). Namely,
let us assume that (4.2.22) holds for two dimensions d1 = m and d2 = n. We
can prove that (4.2.22) holds for d = m + n, using Minkowski’s inequality
in the form in which the discrete summation is replaced by the continuous
integration, in the following way. Let x = (xm , xn ) be a generic vector in
Rm+n , where xm and xn are generic vectors in Rm and Rn , respectively. Let
us observe that, for all x = (xm , xn ) and y = (ym , yn ) in Rm+n , we have:
Jm+n (x, y) = Jm (xm , ym ) · Jn (xn , yn ).
We have:
Z
r
Z
m+n
m+n
ZR Z
RZ
=
Rm
Rn
Z
Rm
Rn
Z
Rm
Z
Rm
Rn
Rn
or
o1/r
× Jn (xn , yn )dN yn )r dN xn ]1/r dN ym dN xm
r
1/r
Jm (xm , ym )C n k|f (xm − ym , ·)|kp · k|g(ym , ·)|kq dN ym dN xm
≤
= C
|f (x − y)| · |g(y)| · Jm+n (x, y)dN y dN x
Z
|f (xm − ym , xn − yn )| · |g(ym , yn )|
× Jm+n ((xm , xn ), (ym , yn )) dN yn dN ym ]r dN xn dN xm }1/r
Z
Z Z
Jm (xm , ym )
|f (xm − ym , xn − yn )| · |g(ym , yn )|
≤
n
Rm
1/r
Rm
Z
Jm (xm , ym )k|f (xm − ym , ·)|kp · k|g(ym , ·)|kq dN ym
Rm
m
1/r
r
Z
Rm
≤ C n · C k|f |kp · k|g|kq
= C m+n k|f |kp · k|g|kq .
dN xm
74
Chapter 4. Sharp inequalities for Gaussian Wick products
This shows that in order to prove (4.2.22), it is enough to prove it for the
dimension d = 1 only. To achieve this, since the function (x, y) 7→ [(q 0 /v)x −
r0 y]2 is non–negative, according to Lieb theorem, it is enough to check it
for exponential functions of the form f (x) = c1 exp[−(s/2)x2 ] and g(x) =
c2 exp[−(t/2)x2 ], where s > 0, t > 0, and c1 and c2 are positive constants
chosen such that k|f |kp = k|g|kq = 1. Let us first compute the values of c1
and c2 . We have:
Z
1/p
ps 2
e− 2 x dN x
k|f |kp = c1
R
0
(let x :=
√
1/p
Z
02
1
− x2
0
e
= c1 √
dN x
ps R
1
= c1 √ 1/p .
( ps)
psx)
Thus, in order to have k|f |kp = 1, we must have:
√ 1
c1 = ( ps) p .
(4.2.23)
Similarly, in order to have k|gk|q = 1, we must have:
√ 1
c2 = ( qt) q .
(4.2.24)
Hence, we have:
Z
0
2
− qv ·−r0 y
f
(·
−
y)g(y)e
d
y
N R
r
#r
)1/r
(Z "Z
2
0
q
1
0
√ 1/p √ 1/q
− 2s (x−y)2 − 2t y 2 − 2p0 q0 r0 v x−r y
= ( ps)
e
dN y dN x
.
qt
e
e
R
R
√
√
√
Let α := q 0 /(v p0 q 0 r0 ), β := r0 / p0 q 0 r0 , and γ := p0 /(u p0 q 0 r0 ). Let us observe
first that α + γ = β, since (p0 /u) + (q 0 /v) = r0 . We have:
Z
0
2
q
0y
−
·−r
dN y f (· − y)g(y)e v
R
r
Z Z
r
1/r
s
√ 1/p √ 1/q
− 2 (x−y)2 − 2t y 2 − 12 (αx−βy)2
e
e
dN y dN x
e
= ( ps) ( qt)
R
√
= ( ps)
1/p
√
( qt)1/q
(Z "
R
−
e
R
(s+α2 ) x2 Z
2
R
−
e
(s+t+β2 ) y2 +(s+αβ)xy
2
)1/r
#r
dN y
dN x
.
4.2. Hölder-Young-Lieb inequality
75
p
In the last integral we make the change of variable y 0 =
s + t + β 2 · y.
Completing the square, we obtain:
Z
0
2
− qv ·−r0 y
dN y f (· − y)g(y)e
R
r
#r
(Z "
)1/r
Z
2
1 2 √ s+αβ
s+α
(
)
xy
−
y
+
1
√ 1/p √ 1/q
2
s+t+β 2
dN y dN x
= ( ps) ( qt)
e− 2 x p
e 2
2
s
+
t
+
β
R
R
 
1/r
r
s
2
Z
Z


s+αβ
1
s+α2 )
(s+αβ)2
1/p
1/q
(
2
√
− 2 y−
x
(ps) (qt)
s+t+β 2
e− 2 x2 · e 2(s+t+β2 ) x
 dN x
e
=
.
d
y
N

s + t + β2  R
R
Therefore,
=
=
=
=
=
=
Z
0
2
− qv ·−r0 y
dN y f (· − y)g(y)e
R
r
s
(Z "
#
)1/r
2
(s+α ) x2 + (s+αβ)2 x2 r
(ps)1/p (qt)1/q
−
2
2(s+t+β 2 )
e
dN x
s + t + β2
R
s
(Z "
#
)1/r
(s+α2 )(s+t+β2 )+(s+αβ)2 x2 r
(ps)1/p (qt)1/q
−
2
2(s+t+β )
e
dN x
s + t + β2
R
s
(Z " s(β−α)2 +tα2 +st #r
)1/r
[
] x2
(ps)1/p (qt)1/q
−
2
2(s+t+β )
e
dN x
s + t + β2
R
s
(Z
)1/r
r (sγ 2 +tα2 +st)
2
(ps)1/p (qt)1/q
−
x
2(s+t+β 2 )
e
dN x
s + t + β2
R
s
s
1/p
1/q
(ps) (qt)
(s + t + β 2 )1/r
·
2
s+t+β
r1/r (sγ 2 + tα2 + st)1/r
s
s
p1/p q 1/q
s1/p t1/q
·
.
r1/r
(s + t + β 2 )1/r0 (γ 2 s + α2 t + st)1/r
To finish our proof we need to show that:
(
)
1
p1/p q 1/q
s1/p t1/q
−1 1−1
sup
·
= v r p ur q .
0
1/r
1/r
1/r
2
2
2
r
(s + t + β ) (γ s + α t + st)
s>0,t>0
(4.2.25)
76
Chapter 4. Sharp inequalities for Gaussian Wick products
Before we compute this supremum, we would like to outline the intuition
behind what we are going to do next. Let us observe that the numerator
s1/p t1/q , being a product, is somehow like a geometric mean, while the factors
from the denominator (s + t + β 2 ) and (γ 2 s + α2 t + st), being sums, are
like arithmetic means. We know from the inequality between the geometric
and arithmetic means of positive numbers, that the geometric mean is always
dominated by the arithmetic mean, and this classic inequality is based on the
concavity of the logarithmic function.
Let S := (pv)s and T := (qu)t. We have:
p1/p q 1/q
s1/p t1/q
·
r1/r
(s + t + β 2 )1/r0 (γ 2 s + α2 t + st)1/r
=
=
p1/p q 1/q
·
r1/r
1
pv
1
v 1/p u1/q r1/r
·S+
·
1
pv
1
qu
·T +
·S+
1
qu
1
S 1/p (qu)11/q T 1/q
(pv)1/p
1/r0 2
γ
α2
β2 · 1
pv · S + qu
·T +
S 1/p T 1/q
1/r0 2
γ
· T + β2 · 1
pv · S +
α2
qu
1
pquv
·T +
· ST
1
pquv
1/r
· ST
1/r .
(4.2.26)
Let us observe that:
1
1
+
+ β 2 = 1.
pv qu
(4.2.27)
Indeed, we have:
1
1
+
+ β2 =
pv qu
1
1
r0
+
+ 0 0
pv qu p q
0
=
=
=
=
=
0
p
+q
1
1
+
+ u 0 0v
pv qu
pq
1
1
1
1
+
+ 0 + 0
pv qu q u p v
1 1
1
1 1
1
+
+
+
v p p0
u q q0
1
1
·1+ ·1
v
u
1.
77
4.2. Hölder-Young-Lieb inequality
Let us also observe that:
γ 2 α2
1
+
+
pv qu pquv
=
1
.
uvr
(4.2.28)
Indeed, we have:
=
=
=
=
=
=
=
=
=
=
γ 2 α2
1
+
+
pv qu pquv
1
1
p0
q0
1
·
·
+
+
2
0
0
2
0
0
u q r pv v p r qu pquv
0
p
1
q0
1
0
+
+
·r
uvr0 puq 0 qvp0 pq
0
0
1
p
q0
1
p
q0
+
+
·
+
uvr0 puq 0 qvp0 pq
u
v
0
0
0
0
p
q
p
1
q
+
+
+
0
0
0
uvr puq
qvp
pqu pqu
0 1
p
1
1
q0 1
1
+
+
+
uvr0 pu q 0 q
qv p0 p
0
p
q0
1
·
1
+
·1
uvr0 pu
qv
1
p/(p − 1) q/(q − 1)
+
uvr0
pu
qv
1
1
1
+
uvr0 u(p − 1) v(q − 1)
1
1
·
uvr/(r − 1) r − 1
1
.
uvr
We go back to the denominator of formula (4.2.26) and apply the Jensen
inequality for the strictly concave downward function L(x) = ln(x). From
(4.2.27) we conclude that:
ln
1
1
·S+
· T + β2 · 1
pv
qu
1
1
ln(S) +
ln(T ) + β 2 ln(1)
pv
qu
= ln S 1/(pv) T 1/(qu) .
≥
78
Chapter 4. Sharp inequalities for Gaussian Wick products
Exponentiating both sides of this inequality and then rasing them to the power
1/r0 , we obtain:
1
1
·S+
· T + β2 · 1
pv
qv
1/r0
0
0
≥ S 1/(pvr ) T 1/(qur ) .
(4.2.29)
Formula (4.2.28) shows that in order to obtain a convex combination in the
sum:
γ2
α2
1
·S+
·T +
· ST
pv
qu
pquv
we need first to multiply it by K := uvr. After doing this, applying again the
strict concavity of the function ln, we obtain:
α2
1
γ2
α2
γ2
·S+K
·T +K
· ST
≥ K ln(S) + K
ln(T ) +
ln K
pv
qu
pquv
pv
qu
1
K
ln(ST )
pquv
Kγ 2
K
Kα2
K
+ pquv
+ pquv
pv
qu
= ln S
T
.
This inequality is equivalent to:
α2
1
γ2
·S+
·T +
· ST
pv
qu
pquv
1/r
≥
=
Kγ 2
Kα2
K
K
1
+ pquvr
+ pquvr
pvr
qur
S
T
1/r
K
uγ 2
vα2
1
1
1
+ pq
+ pq
p
q
T
.
S
(uvr)1/r
(4.2.30)
Going now back to the formula (4.2.26) and using the inequalities (4.2.29) and
(4.2.30), we obtain:
p1/p q 1/q
s1/p t1/q
·
r1/r
(s + t + β 2 )1/r0 (γ 2 s + α2 t + st)1/r
=
≤
1
v 1/p u1/q r1/r
1
v 1/p u1/q r1/r
·
·
1
pv
·S+
1
qv
S 1/p T 1/q
1/r0 2
γ
· T + β2 · 1
pv · S +
α2
qu
·T +
1
pquv
· ST
S 1/p T 1/q
.
1
(uγ 2 )/p+1/(pq) T (vα2 )/q+1/(pq)
S 1/(pvr0 ) T 1/(qur0 ) (uvr)
1/r S
1/r
79
4.2. Hölder-Young-Lieb inequality
The exponent of S in the denominator of the last fraction is:
1
p0
1
+
+
0
0
0
pvr
upq r
pq
=
=
=
=
=
=
=
1
pr0
1
pr0
1
pr0
1
pr0
1
pr0
1
pr0
1
.
p
1
v
1
v
1
v
1
v
1
v
+
+
+
+
+
p0
1 0
+ ·r
uq 0 q
p0
1 p0 q 0
+
+
uq 0 q u
v
0
q0 1
p 1
1
+ ·
+
u q0 q
v q
0
0
1
p
q
1− 0
·1+
u
v
q
0
0
p
q
1
+ −
u
v
v
· r0
Similarly, the exponent of T in the denominator of the same fraction is 1/q.
Hence, for all s, t > 0, we have:
s1/p t1/q
p1/p q 1/q
·
r1/r
(s + t + β 2 )1/r0 (γ 2 s + α2 t + st)1/r
≤ v (1/r)−(1/p) u(1/r)−(1/q) ·
S 1/p T 1/q
S 1/p T 1/q
= v (1/r)−(1/p) u(1/r)−(1/q) .
(4.2.31)
The equality in (4.2.31) holds if and only if S = T = 1, due to the strict
concavity of the function y = ln(x). This is equivalent to s = 1/(pv) and
t = 1/(qu), since S = (pv)s and T = (qu)t. Thus, we have:
(
sup
s>0,t>0
p1/p q 1/q
s1/p t1/q
·
r1/r
(s + t + β 2 )1/r0 (γ 2 s + α2 t + st)1/r
v (1/r)−(1/p) u(1/r)−(1/q) .
)
=
80
Chapter 4. Sharp inequalities for Gaussian Wick products
Going back to the inequality (4.2.21), we conclude that:
r
1/r
hx,xi
x
− 2uv · r1 Γ √1 ϕ Γ √1 ψ
√
e
dN x
u
v
uv
Rd
r
1/r
Z Z
D 0
E
0
− 2p01q0 r0 qv x−r0 y, qv x−r0 y
dN y dN x
≤
|f (x − y)| · |g(y)| · e
Rd
Rd
q
Z
d
− pd
d
− dq
k|f |kp k|g|kq
Z 1/p Z 1/q
d
d
d
d
x p − hx,xi
x q − hx,xi
− 2p
− 2q
2r
2r
2v
2u
√
√
e
e
= v
u
dN x
dN x
.
ϕ
ψ
v u Rd
Rd
≤
vr
ur
This inequality is equivalent to:
r
1/r
− hx,xi
1
1
x
2uv
√
e
dN x
(uv)
Γ √u ϕ Γ √v ψ
uv Rd
Z Z 1/p
1/q
d
d
x p − hx,xi
x q − hx,xi
− 2p
− 2q
≤ v
u
.
ϕ √v e 2v dN x
ψ √u e 2u dN x
Rd
Rd
d
− 2r
Z
√
√
Making now the changes of variable x0 := x/ uv in the left, and x1 := x/ v
√
and x2 := x/ u in the right, and moving back to the Gaussian norms, we
obtain:
Γ √1 ϕ Γ √1 ψ ≤ kϕkp · kψkq .
u
v
r
To prove the inequality in the infinite dimensional case we proceed in the
following way. Let H be a separable Gaussian Hilbert space. Let {en }n≥1 be
an orthonormal basis of centered Gaussian random variables from H. For all
d ≥ 1, let
Hd := Ce1 ⊕ Ce2 ⊕ · · · ⊕ Ced .
Let Fd := F(Hd ), i.e., the smallest sigma–algebra with respect to which e1 ,
e2 , . . . , ed are measurable. If ϕ ∈ Lp (Ω, F(H), P ) and ψ ∈ Lq (Ω, F(H), P ),
and if we denote the conditional expectations of ϕ and ψ, with respect to Fd ,
by ϕd and ψd , respectively, i.e., ϕd := E[ϕ | Fd ] and ψd := E[ψ | Fd ], then it
is not hard to see that:
√
√
E[Γ(1/ u)ϕ | Fd ] = Γ(1/ u)ϕd ,
(4.2.32)
4.2. Hölder-Young-Lieb inequality
√
√
E[Γ(1/ v)ψ | Fd ] = Γ(1/ v)ψd ,
81
(4.2.33)
and
√
√
√
√
E[Γ(1/ u)ϕ Γ(1/ v)ψ | Fd ] = Γ(1/ u)ϕd Γ(1/ v)ψd .
Since {Fd }d≥1 is an increasing family of sigma–algebras and the sigma–algebra
generated by them is F(H), using the Martingale Convergence Theorem we
conclude that:
√
√
√
√
E[Γ(1/ u)ϕ Γ(1/ v)ψ | Fd ] → Γ(1/ u)ϕ Γ(1/ v)ψ,
E[ϕ | Fd ] → ϕ,
and
E[ψ | Fd ] → ψ,
as d → ∞, both almost surely and in L1 (Ω, F(H), P ). Using now the fact
that the result is true in the finite dimensional case and Fatou’s Lemma as
before, we conclude that:
√
√
Γ(1/ u)ϕ Γ(1/ v)ψ ≤ kϕkp · kψkq .
r
2
82
Chapter 4. Sharp inequalities for Gaussian Wick products
Appendix A
Hermite polynomials
The Hermite polynomials hn (x) are defined by
hn (x) = (−1)n e
x2
2
dn − x2
e 2.
dxn
Thus the first Hermite polynomials are
h0 (x) = 1,
h1 (x) = x,
h2 (x) = x2 − 1, h3 (x) = x3 − 3x
and so on. They have the orthogonal property
Z
+∞
−∞
x2
n n!, if n = m
e− 2
.
hn (x)hm (x) √ dx =
0 if n 6= m
2
Thus {hn (x), n ≥ 1} forms an orthogonal basis for L2 (R, µ(dx)) if µ(dx) =
2
x
√1 e− 2
2π
dx. They satisfy the following properties:
1. The generating function of the polynomials hn is:
∞ n
X
t
n=0
n!
22
hn (x) = etx+−f ract
2.
h0n (x) = nhn−1 (x)
3. We have the recursive formula:
hn+1 (x) = xhn (x) − nhn−1 (x)
84
Appendix A. Hermite polynomials
4. The polynomials hn satisfy the differential equation:
2
d
d
−x
+ n hn (x) = 0.
dx2
dx
Moreover they can be represented
Z +∞
y2
1
hn (x) =
(x + iy)n √ e− 2 dy
2π
−∞
or from a probabilistic point of view
hn (x) = E[(x + iX)n ]
where X is a random variable with normal standard distribution.
At last, we can define the Hermite polynomials of order n and parameter σ 2
hn,σ2 (x) = σ n hn
x
σ
x2
= (σ 2 )n e 2σ2
dn − x22
e 2σ .
dxn
hn,t satisfy the backward heat equation
∂t hn,t (x) + ∂xx hn,t (x) = 0
for any n ≥ 1, t > 0, x ∈ R
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