Günter Last
Institut für Stochastik
Universität Karlsruhe (TH)
On the chaos expansion of Poisson functionals
Günter Last (Karlsruhe)
joint work with Mathew Penrose (Bath)
talk presented at the
53rd Annual Meeting of the Australian Mathematical Society
University of South Australia, Adelaide
28.09.–01.10.2009
Günter Last
On the chaos expansion of Poisson functionals
1. Poincaré’s inequality
Theorem: Chernoff (81) Let X be a standard normal random vector
and let g ∈ L2 (PX ) be differentiable. Then
Var[g(X)] ≤ Ek∇g(X)k2 .
Equality holds iff g is a linear function.
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Günter Last
On the chaos expansion of Poisson functionals
Theorem: Wu (00) Let η be a Poisson process on some measurable
space (Y, Y) with intensity measure λ. Then for any f ∈ L2 (Pη ),
Z
Var[f (η)] ≤ E (f (η + δy ) − f (η))2 λ(dy).
Remark: Equality holds iff f (η) is a linear function of η. This will
be made more precise later.
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Günter Last
On the chaos expansion of Poisson functionals
Corollary: Chen (85) Let X be an infinitely purely non-Gaussian
infinitely divisible random vector with Lévy measure λ and let g ∈
L2 (PX ). Then
Z
Var[g(X)] ≤ E (g(X + y) − g(X))2 λ(dy).
Equality holds iff ∆2y1 ,y2 g(X) = 0 P-a.s. and for λ2 -a.e. (y1 , y2 ).
Here ∆y g(x) := g(x + y) − g(x) and ∆2y1 ,y2 := ∆y2 ◦ ∆y1 .
Proof: Let η be a Poisson process with intensity measure λ. Let
Z
Z
h(η) ≡ c +
x(η − λ)(dx) +
xη(dx),
|x|≤1
|x|>1
d
where c ∈ Rd is chosen such that h(η) = X. Apply Poincaré’s
inequality with
f (η) = g(h(η)).
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Günter Last
On the chaos expansion of Poisson functionals
2. Clark-Ocone type representation
Setting: η is a Poisson process on the product R+ × X of R+ and a
Borel space X.
Assumption: The intensity measure λ of η satisfies
λ({t} × X) = 0,
t ≥ 0.
Consequently we have a.s. that
η({t} × X) ∈ {0, 1},
t ≥ 0.
Notation: Let η̂ := η − λ be the compensated Poisson process. Integrals with respect to η̂ are to be understood in a stochastic sense.
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Günter Last
On the chaos expansion of Poisson functionals
Example: Let (Xt )t≥0 be a pure jump Lévy process in Rd . Then
Z
Z t
Z
Z t
Xt =
xη̂(ds, dx) +
xη(ds, dx), t ≥ 0,
|x|≤1
0
|x|>1
0
for a Poisson process η on R+ × (Rd \ {0}). The intensity measure
λ of η is the product of Lebesgue measure and the Lévy measure ν
of (Xt ) on Rd \ {0} satisfying
Z
(|x|2 ∧ 1)ν(dx) < ∞.
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Günter Last
On the chaos expansion of Poisson functionals
Notation: For t ∈ [0, ∞] let ηt (resp. ηt− ) be the restriction of η to
[0, t] × X (resp. [0, t) × X).
Theorem: Picard (96), Wu (00), L. and Penrose (09) Let f ∈ L2 (Pη ).
Then
Z
E E[f (η + δ(s,x) ) − f (η)|ηs− ]2 λ(d(s, x)) < ∞
and we have for any t ∈ [0, ∞] that P-a.s.
Z
E[f (η)|ηt ] = Ef (η) +
E[f (η + δ(s,x) ) − f (η)|ηs− ]η̂(d(s, x)).
[0,t]×X
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Günter Last
On the chaos expansion of Poisson functionals
Theorem: Wu (00), L. and Penrose (09) For any f, g ∈ L2 (Pη ),
Cov[f (η), g(η)]
Z
= E E[f (η + δ(s,x) ) − f (η)|ηs ]E[g(η + δ(s,x) ) − g(η)|ηs ]λ(d(s, x)).
Idea of the Proof: The martingales
Mt := E[f (η)|ηt ],
Nt := E[g(η)|ηt ]
satisfy
Mt Nt = M0 N0 +
Z
t
Ns− dMs +
0
Z
t
Ms− dNs +
0
30.09.2009, 53rd Annual Meeting of the Australian Mathematical Society
X
∆Ms ∆Ns .
s≤t
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Günter Last
On the chaos expansion of Poisson functionals
Corollary: For any f ∈ L2 (Pη ),
Z
Var[f (η)] = E E[f (η + δ(s,x) ) − f (η)|ηs ]2 λ(d(s, x)).
In particular,
Var[f (η)] ≤ E
Z
(f (η + δ(s,x) ) − f (η))2 λ(d(s, x)).
Proof of the general Poincaré inequality: Randomization!
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On the chaos expansion of Poisson functionals
3. Fock space representation
Setting: η is a Poisson process on some measurable space (Y, Y) with
intensity measure λ. This process can be interpreted as a random
element in the space N of all integer-valued σ-finite measures on Y,
equipped with the usual (product) σ-field.
Definition: For n ∈ N let Hn be the space of symmetric functions
in L2 (λn ), and let H0 := R. The Fock space H associated with η
(or λ) is the direct sum of the spaces Hn , n ≥ 0, equipped with the
scalar product
hf, giH
∞
X
1
hfn , gn in ,
:=
n!
n=0
f = (fn ), g = (gn ) ∈ H,
where h·, ·in is the scalar product in L2 (λn ). This is a Hilbert space.
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Günter Last
On the chaos expansion of Poisson functionals
Definition: For a measurable functions f : N → R and y ∈ Y we
define a function Dy f : N → R by
Dy f (µ) := f (µ + δy ) − f (µ).
For y1 , . . . , yn ∈ Y we define Dyn1 ,...,yn f : N → R inductively by
Dyn1 ,...,yn f := Dy11 Dyn−1
f,
2 ,...,yn
where D 1 := D and D 0 f = f .
Lemma: For any f ∈ L2 (Pη )
Tn f (y1 , . . . , yn ) := EDyn1 ,...,yn f (η),
defines a function Tn f ∈ Hn .
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Günter Last
On the chaos expansion of Poisson functionals
Theorem: L. and Penrose (09) Let f, g ∈ L2 (Pη ). Then
∞
X
1
Ef (η)g(η) = (Ef (η))(Eg(η)) +
hTn f, Tn gin .
n!
n=1
That is,
Ef (η)g(η) = hT f, T giH ,
where T f := (Tn f ) and T g := (Tn g).
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Günter Last
On the chaos expansion of Poisson functionals
Idea of the Proof:
1. Check the result for functions of the form
h Z
i
µ 7→ exp − v(y)µ(dy) ,
where v : Y → R+ vanishes outside a set of finite λ-measure.
2. The set G of all linear combinations of functions of the above
type is dense in L2 (Pη ).
3. Use Hilbert space and completeness arguments to extend the
result from G to L2 (Pη ).
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Günter Last
On the chaos expansion of Poisson functionals
4. Back to the Poincaré inequality
It follows from the Fock space representation that any f ∈ L2 (Pη )
satisfies
∞
X
1
hTn f, Tn f in
Var f (η) =
n!
n=1
Z
= (EDy f (η))2 λ(dy)
ZZ
∞
X
1
2 n−1
(EDyn−1
+
D
f
(η))
λ
(d(y2 , . . . , yn ))λ(dy)
y
2 ,...,yn
n!
n=2
Z
≤ (EDy f (η))2 λ(dy)
ZZ
∞
X
1
(EDyn1 ,...,yn Dy f (η))2 λn (d(y1 , . . . , yn ))λ(dy).
+
n!
n=1
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Günter Last
On the chaos expansion of Poisson functionals
Assuming w.l.o.g. that
Z
E (f (η + δy ) − f (η))2 λ(dy) < ∞
and applying the Fock representation to Dy f for λ-a.e. y ∈ Y gives
Z
Var f (η) ≤ E(Dy f (η))2 λ(dy).
Proposition: We have equality in the Poincaré inequality iff
Dy f (η) = EDy f (η)
P-a.s., λ-a.e. y.
This is equivalent to
Dy1 ,y2 f (η) = 0
P-a.s., λ2 -a.e. (y1 , y2 ).
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Günter Last
On the chaos expansion of Poisson functionals
5. Chaos expansion
Definition: Let n ∈ N and g ∈ Hn . Then
Z
In (g) ≡ g dη̂ n
denotes the multiple Wiener-Itô integral of g w.r.t. the compensated
Poisson process η̂ = η − λ. For c ∈ R let I0 (c) := c. These integrals
have the properties
Z
EIn (g)In (h) = n! ghdλn , n ∈ N0 ,
and
EIm (g)In (h) = 0,
m 6= n.
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Günter Last
On the chaos expansion of Poisson functionals
Remarks: (i) Let g1 , . . . , gn ∈ L1 (λ) ∩ L2 (λ) have disjoint supports
and let f be the symmetrization of g1 ⊗ . . . ⊗ gn , where
g1 ⊗ . . . ⊗ gn (y1 , . . . , yn ) := g1 (y1 ) · . . . · gn (yn ),
y1 , . . . , yn ∈ Y.
Then
In (f ) =
n
Y
(η(gi ) − λ(gi )).
i=1
(ii) Let g ∈ L1 (λ) ∩ L2 (λ) and f := g ⊗n . Then
n X
n (k) ⊗k
η (h )(−λ(h))n−k ,
In (f ) =
k
k=0
where η (k) is the kth factorial moment measure associated with η.
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Günter Last
On the chaos expansion of Poisson functionals
Theorem: Wiener (38), Itô (56) Assume that λ is diffuse and let
f ∈ L2 (Pη ). Then there are functions fn ∈ Hn , n ∈ N0 , such that
f (η) =
∞
X
In (fn ),
n=0
where the series converges in L2 (P).
Theorem: Y. Ito (88), L. and Penrose (09) For any f ∈ L2 (Pη ),
∞
X
1
In (Tn f ).
f (η) =
n!
n=0
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Günter Last
On the chaos expansion of Poisson functionals
6. Variance inequalities
Theorem: Houdré and Perez-Abreu (95), L. and Penrose (09) Let
f ∈ L2 (Pη ) and k ∈ N be such that
EkD n f (η)k2n < ∞,
n = 1, . . . , 2k.
Then
2k
X
(−1)n+1
EkD n f (η)k2n ≤ Var[f (η)]
n!
n=1
≤
2k−1
X
n=1
(−1)n+1
EkD n f (η)k2n .
n!
The equality cases can be characterized as before.
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On the chaos expansion of Poisson functionals
6. References
• Chen, L. (1985). Poincaré-type inequalities via stochastic
integrals. Z. Wahrscheinlichkeitstheorie verw. Gebiete 69,
251-277.
• Houdré, C. and Perez-Abreu, V. (1995). Covariance identities
and inequalities for functionals on Wiener space and Poisson
space. Ann. Probab. 23, 400-419.
• Itô, K. (1956). Spectral type of the shift transformation of
differential processes with stationary increments. Trans. Amer.
Math. Soc. 81, 253-263.
• Ito, Y. (1988). Generalized Poisson functionals. Probab. Th.
Rel. Fields 77, 1-28.
• Last, G. and Penrose, M.D. (2009). Fock space representation,
chaos expansion and covariance inequalities for general Poisson
processes. submitted for publication.
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Günter Last
On the chaos expansion of Poisson functionals
• Last, G. and Penrose, M.D. (2009). Martingale representation
for Poisson processes with applications to minimal variance
hedging. in preparation.
• Wiener, N. (1938). The homogeneous chaos. Am J. Math. 60,
897-936.
• Wu, L. (2000). A new modified logarithmic Sobolev inequality
for Poisson point processes and several applications. Probab.
Theory Related Fields 118, 427-438.
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