Existence and Uniqueness of Path Wise Solutions for
Stochastic Integral Equations Driven by non Gaussian
Noise on Separable Banach Spaces
V. Mandrekar, B. Rüdiger
no. 186
Diese Arbeit ist mit Unterstützung des von der Deutschen Forschungsgemeinschaft getragenen Sonderforschungsbereiches 611 an der Universität Bonn entstanden und als Manuskript vervielfältigt worden.
Bonn, Oktober 2004
Existence and uniqueness of path wise solutions
for stochastic integral equations driven by non
Gaussian noise on separable Banach spaces
V. Mandrekar*, B. Rüdiger**
* Department of Statistics and Probability Michigan State University, East
Lansing, MI 48824, USA
** - Mathematisches Institut, Universität Koblenz-Landau, Campus Koblenz,
Universitätsstrasse 1, 56070 Koblenz, Germany;
- SFB 611, Institut für Angewandte Mathematik, Abteilung Stochastik, Universität Bonn, Wegelerstr. 6, D -53115 Bonn, Germany
Abstract
The stochastic integrals of M- type 2 Banach valued random functions
w.r.t. compensated Poisson random measures introduced in [21] are discussed for general random functions. These are used to solve stochastic
integral equations driven by non Gaussian noise on such spaces. Existence and uniqueness of the path wise solutions are proven under local
Lipshitz conditions for the drift and noise coefficients on M- type 2 as
well as general separable Banach spaces. The continuous dependence of
the solution on the initial data as well as on the drift and noise coefficients
are shown. The Markov properties for the solutions are analyzed. Some
example where this theory can be used to solve applied problems, e.g.
related to finance, are provided.
AMS -classification (2000): 60H05, 60G51, 60G57, 46B09, 47G99
Keywords: Stochastic differential equations, stochastic integrals on separable
Banach spaces, M- type 2 Banach spaces, martingales measures, compensated
Poisson random measures, additive processes, random Banach valued functions
1
Introduction
In this article we shall study existence and uniqueness of the path wise solution
of the following Banach valued integral equation
Z t
Z tZ
Zt (ω) = φ(t, ω)+
A(s, Zs (ω), ω)ds+
f (s, x, Zs (ω), ω)(N (dsdx)(ω)−ν(dsdx))
0
0
Λ
(1)
on each time interval [0, T ], T > 0, where (Zt )t∈IR+ is a random process with
values in a separable Banach space F . N (dsdx)(ω) − ν(dsdx) is a compensated Poisson random measure (cPrm) associated to a Lévy process (Xt )t≥0
1
(Definition 2.7 in Section 2). The process (Xt )t≥0 has values in a separable Banach space E and is defined on a probability space (Ω, F, P ), so that
N (dsdx)(ω) (for each ω ∈ Ω fixed) and its compensator ν(dsdx) are σ -finite
measures on the σ -algebra B(IR+ × E \ {0}), generated by the product semiring
B(IR+ ) × B(E \ {0}) of the Borel σ -algebra B(IR+ ) and the trace σ -algebra
B(E \ {0}). We assume ν(dsdx)= dsβ(dx), where ds denotes the Lesbegue
measure on B(IR+ ) (see Section 2 for the precise definitions of the (random)
measures N (dsdx)(ω) , ν(dsdx) ). A(t, z, ω) , φ(t, ω) and f (t, x, z, ω) with
t ∈ IR+ , x ∈ E \ {0}, z ∈ F , ω ∈ Ω, are measurable in all their variables and
have values in F (see Section 3 for a precise definition). The stochastic inteRtR
grals 0 Λ g(s, x, ω)(N (dsdx)(ω) − ν(dsdx)), with T > 0, Λ ∈ B(E \ {0}) and
g(t, x, ω) with values in a Hilbert and Banach space, have been defined in [21].
Here we improve some of these results (see e.g. Theorem 3.13). From these
RtR
results it follows that the integral 0 Λ f (s, x, Zs (ω), ω)(N (dsdx)(ω) − ν(dsdx))
is well defined (see Section 4). We refer to Section 2, where the results of [21],
that we need in this article, are reported.
In Section 3 we shall prove that if φ(t, ω) does not depend on time, i.e. φ(t, ω) =
φ(ω), and satisfies the condition
E[kφ(ω)kp ] < ∞ ,
(2)
with p = 1, and A(t, z, ω) and f (t, x, z, ω) satisfy a Lipshitz - condition, there
is a unique cád -lág solution of (1) for (t ∈ (0, T ]) s.th.
Z T
E[kZs kp ] ds < ∞ ,
(3)
0
with p = 1. (We denote with E[·] the expectation w.r.t. the probability P ).
(See Section 3 for a precise statement.) If F is a separable Banach space of
M- type 2 the same results hold for p = 2. We remark that we do not require
that the integrand f (s, x, z, ω) or the solution Zs has a left continuous version
or is predictable in the sense of [12]. This is a consequence of the fact that we
have defined the stochastic integrals w.r.t. the compensated Poisson random
measures like in [21], i.e. in a stronger sense than in [12]. (See also Section 3
where such integrals are discussed and results of [21] are improved). If φ(t, ω)
depends also on time and satisfies the condition
Z T
E[kφ(s, ω)kp ] ds < ∞ ,
(4)
0
then uniqueness of a cád -làg solution (1) for t ∈ (0, T ] is also proven, but only
up to stochastic equivalence.
We recall here the definition of M -type 2 separable Banach space (see e.g. [18]).
2
Definition 1.1 A separable Banach space F , with norm k · k, is of M -type 2,
if there is a constant K2 , such that for any F - valued martingale (Mk )k∈1,..,n
the following inequality holds:
n
X
E[kMn k2 ] ≤ K2
E[kMk − Mk−1 k2 ] ,
(5)
k=1
with the convention that M−1 = 0.
Definition 1.2 A separable Banach space F is of type 2, if there is a constant
K2 , such that if {Xi }ni=1 is any finite set of centered independent F - valued
random variables, such that E[kXi k2 ] < ∞, then
n
n
X
X
E[k
Xi k2 ] ≤ K2
E[kXi k2 ]
(6)
i=1
i=1
We remark that any separable Banach space of M -type 2 is a separable Banach
space of type 2. Moreover, a separable Banach space is of type 2 as well as of
cotype 2 if and only if it is isomorphic to a separable Hilbert space [16], where
a Banach space of cotype 2 is defined by putting ≥ instead of ≤ in (6) (see [3],
or [17]).
2
Poisson and Lévy measures of additive processes on separable Banach spaces
We assume that a filtered probability space (Ω, F, (Ft )0≤t≤+∞ , P ), satisfying
the ”usual hypothesis”, is given:
i)
Ft contain all null sets of F, for all t such that 0 ≤ t < +∞
ii)
Ft = Ft+ , where Ft+ = ∩u>t Fu , for all t such that 0 ≤ t < +∞, i.e. the
filtration is right continuous
In this Section we introduce the compensated Poisson random measures associated to additive processes on (Ω, F, (Ft )0≤t≤+∞ , P ) with values in (E, B(E)),
where in the whole paper we assume that E is a separable Banach space with
norm k · k and B(E) is the corresponding Borel σ -algebra.
Definition 2.1 A process (Xt )t≥0 with state space (E, B(E)) is an Ft - additive
process on (Ω, F, P ) if
i)
(Xt )t≥0 is adapted (to (Ft )t≥0 )
ii)
X0 = 0 a.s.
iii)
(Xt )t≥0 has increments independent of the past, i.e. Xt − Xs is independent of Fs if 0 ≤ s < t
iv)
(Xt )t≥0 is stochastically continuous, i.e. ∀ > 0 lims→t P (kXs − Xt k >
) = 0.
3
v)
(Xt )t≥0 is càdlàg.
An additive process is a Lévy process if the following condition is satisfied
vi)
(Xt )t≥0 has stationary increments, that is Xt − Xs has the same distribution as Xt−s , 0 ≤ s < t.
Let (Xt )t≥0 be an additive process on (E, B(E)) (in the sense of Definition 2.1).
Set Xt− := lims↑t Xs and ∆Xs := Xs − Xs− .
The following results, i.e. Theorem 2.2, Theorem 2.3, Corollary 2.4, Theorem
2.5, Corollary 2.6 are known (see e.g. [10]). (The proofs of Theorem 2.3, Corollary 2.4, Theorem 2.5, Corollary 2.6 can be given following [2]).
Theorem 2.2 Let Λ ∈ B(E), 0 ∈ (Λ)c (where as usual Γ denotes the closure
of the set Γ and with Γc we denote the complementary of a set Γ), then
X
X
1t≥TnΛ
(7)
NtΛ :=
1Λ (∆Xs ) =
n≥1
0<s≤t
where
T1Λ := inf {s > 0 : ∆Xs ∈ Λ}
Λ
Tn+1
:= inf {s > TΛn : ∆Xs ∈ Λ},
NtΛ
n ∈ IN .
(8)
(9)
is an adapted counting process without explosions and
P (NtΛ = k) = exp(−νt (Λ))
(νt (Λ))k
k!
νt (Λ) := E[NtΛ ]
(10)
(11)
Theorem 2.3 Let B(E \ {0}) be the trace σ -algebra on E \ {0} of the Borel σ
-algebra B(E) on E, and let
F(E \ {0}) := {Λ ∈ B(E \ {0}) : 0 ∈ (Λ)c } ,
(12)
then F(E \ {0}) is a ring and for all ω ∈ Ω the set function
Nt· := Nt (ω, ·) : F(E \ {0}) → IR+
Λ→
NtΛ (ω)
is a σ -finite pre- measure (in the sense of e.g. [4]).
4
(13)
Corollary 2.4 For any ω ∈ Ω there is a unique σ -finite measure on B(E \{0})
Nt (ω, ·) : B(E \ {0}) → IR+
Λ→
(14)
NtΛ (ω)
which is the continuation of the σ -finite pre- measure on F(E \ {0}) given by
Theorem 2.3.
From Theorem 2.3, Corollary 2.4 it follows that Nt : Λ → NtΛ is a random
measure on (E \ {0}, B(E \ {0})).
Theorem 2.5 The set function νt (Λ) := E[NtΛ (ω)] ∈ IR, Λ ∈ F(E \ {0}),
ω ∈ Ω satisfies:
νt : F(E \ {0}) → IR+
Λ→
(15)
E[NtΛ (ω)]
and is a σ -finite pre -measure on ((E \ {0}), F(E \ {0}))
Corollary 2.6 There is a unique σ -finite measure on the σ -algebra B(E \{0})
νt : B(E \ {0}) → IR+
(16)
Λ → E[NtΛ (ω)]
which is the continuation to B(E \ {0}) of the σ -finite pre- measure νt on the
ring ((E \ {0}), F(E \ {0})), given by Theorem 2.5.
Let S(IR+ ) be the semi -ring of sets (t1 , t2 ], 0 ≤ t1 < t2 , and S(IR+ )×B(E \{0})
be the semi -ring of the product sets (t1 , t2 ] × Λ, Λ ∈ B(E \ {0}).
Let
N ((t1 , t2 ] × Λ)(ω) = Nt2 (Λ)(ω) − Nt1 (Λ)(ω) ∀Λ ∈ B(E \ {0}) ∀ω ∈ Ω (17)
For all ω ∈ Ω fixed, N (dtdx)(ω) is a σ -finite pre -measure on the product semi
-ring S(IR+ ) × B(E \ {0}).
Let us denote also by N (dtdx)(ω) the measure which is the unique extension of
the pre -measure to the σ -algebra B(IR+ × (E \ {0})) generated by S(IR+ ) ×
B(E \ {0}) (see e.g. [4] Satz 5.7, Chapt. I, §5, [15] Theorem 1, Chapt. V, §2 for
the existence of a unique minimal σ -algebra containing a product semi -ring).
Let
ν((t1 , t2 ] × Λ) = νt2 (Λ) − νt1 (Λ)
∀A ∈ B(E \ {0})
(18)
ν(dtdx) is a σ -finite pre -measure on S(IR+ ) × B(E \ {0}). Let us denote also by
ν(dtdx) the σ -finite measure, which is the unique extension of this pre -measure
on B(IR+ × (E \ {0})).
5
Definition 2.7 We call N (dtdx)(ω) the Poisson random measure associated to
the additive process (Xt )t≥0 and ν(dtdx) its compensator. We call N (dtdx)(ω)−
ν(dtdx) the compensated Poisson random measure associated to the additive
process (Xt )t≥0 . (We omit sometimes to write the dependence on ω ∈ Ω.)
Remark 2.8 Let N (dtdx)(ω) − ν(dtdx) be the compensated Poisson random
measure associated to an additive process (Xt )t≥0 with values in a Banach space
E defined on the measure space (E \ {0}, B(E \ {0}). (Xt )t≥0 is a Lévy process
iff ν(dtdx)= dtβ(dx), where dt denotes the Lesbegues measure on B(IR+ ), and
β(dx) is a σ -finite measure on (E \ {0}, B(E \ {0})), and is called Lévy measure
associated to (Xt )t≥0 .
3
Stochastic integrals w.r.t. compensated Poisson random measures
Let N (dtdx)(ω) − ν(dtdx) be the compensated Poisson random measure associated to an additive process (Xt )t≥0 defined on a (Ω, F, (Ft )t≥0 , P ) and with
values in a separable Banach space E (Definition 2.7 in Section 2).
Let F be a separable Banach space with norm k·kF . (When no misunderstanding
is possible we write k · k instead of k · kF .) Let Ft := B(IR+ × (E \ {0})) ⊗ Ft
be the product σ -algebra generated by the semi -ring B(IR+ × (E \ {0})) × Ft
of the product sets Λ × F , Λ ∈ B(IR+ × E \ {0}), F ∈ Ft . Let T > 0, and
M T (E/F ) :=
{f : IR+ × E \ {0} × Ω → F, such that f is FT /B(F ) measurable
f (t, x, ω) is Ft − adapted ∀x ∈ E \ {0}, t ∈ (0, T ]}
(19)
In this Section we shall introduce the stochastic integrals of random functions
f (t, x, ω) ∈ M T (E/F ) with respect to the compensated Poisson random measures q(dtdx)(ω) := N (dtdx)(ω) − ν(dtdx) associated to an additive process
(Xt )t≥0 discussed in [21]. (We omit sometimes to write the dependence on
ω ∈ Ω.)
There is a ”natural definition” of stochastic integral w.r.t. q(dtdx)(ω) on those
sets (0, T ] × Λ where the measures N (dtdx)(ω) (with ω fixed) and ν(dtdx) are
finite, i.e. 0 ∈
/ Λ. According to [21] (see also [2] for the case of deterministic
functions f (x) , x ∈ \{0} ) we give the following definition
6
Definition 3.1 Let t ∈ (0, T ], Λ ∈ F(E\{0}) (defined in (12)), f ∈ M T (E/F ).
Assume that f (·, ·, ω) is Bochner integrable on (0, T ] × Λ w.r.t. ν, for all ω ∈ Ω
fixed. The natural integral of f on (0, t] × Λ w.r.t. the compensated Poisson
random measure q(dtdx) := N (dtdx)(ω) − ν(dtdx) is
RtR
f (s, x, ω) (N (dsdx)(ω) − ν(dsdx)) :=
0 Λ
RtR
P
0<s≤t f (s, (∆Xs )(ω), ω)1Λ (∆Xs (ω)) − 0 Λ f (s, x, ω)ν(dsdx) ω ∈ Ω (20)
where the last term is understood as a Bochner integral, (for ω ∈ Ω fixed) of
f (s, x, ω) w.r.t. the measure ν.
It is more difficult to define the stochastic integral on those sets (0, T ] × Λ, Λ ∈
B(E \ {0}), s.th. ν((0, T ] × Λ) = ∞. For real valued functions this problem was
already discussed e.g. in [24] and [6], [12], [25] (for general martingale measures).
Different definitions of stochastic integrals were proposed. In [21] (and [2] for the
case of deterministic functions f (x) , x ∈ E \ {0} ) we introduced the definition
of ”strong -p - integral” (Definition 3.8). The strong -p -integral is the limit in
p
LF
p (Ω, F, P ) (the space of F -valued random variables Y , with E[kY k ] < ∞,
defined in Definition 3.7) of the ”natural integrals” (20) of the ”simple functions”
defined in (21). (We refer to Definition 3.8 for a precise statement.) If p = 2, this
concept generalizes the definition in [6] of stochastic integration of real valued
functions with respect to martingales measures on IRd , to the case of Banach
space valued functions, for the case where the martingale measures are given
by compensated Poisson random measures on general separable Banach spaces.
It generalizes also to the stochastic integral introduced in [24] for real valued
functions integrated w.r.t. compensated Poisson random measures associated to
α -stable Lévy processes on IR. In [21] it has also been proven that the concept
of strong -p -integral, with p = 1 or p = 2, is more general than the definition of
stochastic integrals w.r.t. point processes introduced (for the real valued case)
in [12], Chapt. II.3. In fact, for the existence of the strong -p -integral, with
p = 1 or p = 2, no predictability condition (in the sense of [12]) is needed for
the integrand. This condition is however needed for the stochastic integrals
introduced in [12], which in [21] are denoted with ”simple -p -integrals”. (This
concept has been generalized to the Hilbert or Banach valued case in [21], too.)
If the integrand is however left -continuous, then the strong -p-integral coincide
with the simple -p- integral (we refer to [21] for a precise statement).
We recall here the definition of ”strong -p -integral”, p ≥ 1, introduced in [21].
We first introduce the simple functions.
Definition 3.2 A function f belongs to the set Σ(E/F ) of simple functions ,
if f ∈ M T (E/F ), T > 0 and there exist n ∈ IN , m ∈ IN , such that
n−1
m
XX
f (t, x, ω) =
1Ak,l (x)1Fk,l (ω)1(tk ,tk+1 ] (t)ak,l
(21)
k=1 l=1
7
where Ak,l ∈ F(E \ {0}) (i.e. 0 ∈
/ Ak,l ), tk ∈ (0, T ], tk < tk+1 , Fk,l ∈ Ftk ,
ak,l ∈ F . For all k ∈ 1, ..., n − 1 fixed, Ak,l1 × Fk,l1 ∩ Ak,l2 × Fk,l2 = ∅ if l1 6= l2 .
Proposition 3.3 Let f ∈ Σ(E/F ) be of the form (21), then
Z TZ
n−1
m
XX
f (t, x, ω)q(dtdx)(ω) =
ak,l 1Fk,l (ω)q((tk , tk+1 ]∩(0, T ]×Ak,l ∩ A)(ω).
0
A
k=1 l=1
(22)
for all A ∈ B(E \ {0}), T > 0.
Remark 3.4 The random variables 1Fk,l in (22) are independent of q((tk , tk+1 ]∩
(0, T ] × Ak,l ∩ A) for all k ∈ 1...n − 1, l ∈ 1...m fixed.
Proof of Proposition 3.3: The proof is an easy consequence of the Definition
2.7 of the random measure q(dtdx)(ω).
We recall here the definition of strong -p -integral, p ≥ 1, (Definition 3.8 below)
given in [21] through approximation of the natural integrals of simple functions.
First we establish some properties of the functions f ∈ MνT,p (E/F ), where
Z TZ
MνT,p (E/F ) := {f ∈ M T (E/F ) :
E[kf (t, x, ω)kp ] ν(dtdx) < ∞} (23)
0
Theorem 3.5 [21] Let p ≥ 1. Suppose that the compensator ν(dtdx) of the
Poisson random measure N (dtdx) satisfies the following hypothesis A.
Hypothesis A: ν is a product measure ν = α ⊗ β on the σ -algebra generated by
the semi -ring S(IR+ ) × B(E \ {0}), of a σ -finite measure α on S(IR+ ), s.th.
α([0, T ]) < ∞ , ∀T > 0 , α is absolutely continuous w.r.t the Lesbegues measure
on IR+ , and a σ -finite measure β on B(E \ {0}).
Let T > 0, then for all f ∈ MνT,p (E/F ) and all Λ ∈ B(E \ {0}), there is a
sequence of simple functions {fn }n∈IN satisfying the following property :
Property P: fn ∈ Σ(E/F ) ∀n ∈ IN , fn converges ν ⊗ P -a.s. to f on (0, T ] ×
Λ × Ω, when n → ∞, and
Z TZ
lim
E[kfn (t, x) − f (t, x)kp ] dν = 0 ,
(24)
n→∞
0
Λ
i.e. kfn − f k converges to zero in Lp ((0, T ] × Λ × Ω, ν ⊗ P ), when n → ∞.
Definition 3.6 We say that a a sequence of functions fn are Lp -approximating
f if these satisfy property P, i.e. fn converge ν ⊗ P -a.s. to f on (0, T ] × Λ × Ω,
when n → ∞, and satisfy (24).
8
For the real valued case Theorem 3.5 has also been stated in in Chapt.2 Section
4 [24] or in [6], without proof.
Definition 3.7 Let p ≥ 1, LRF
p (Ω, F, P ) is the space of F -valued random variables, such that EkY kp = kY kp dP < ∞. We denote by k · kp the norm
given by kY kp = (EkY kp )1/p . Given (Yn )n∈IN , Y ∈ LF
p (Ω, F, P ), we write
limpn→∞ Yn = Y if limn→∞ kYn − Y kp = 0.
In [21] we introduce the following
Definition 3.8 Let p ≥ 1, t > 0. We say that f is strong -p -integrable on
(0, t] × Λ, Λ ∈ B(E \ {0}), if there is a sequence {fn }n∈IN ∈ Σ(E/F ), which
satisfies the property P in Theorem 3.5, and such that the limit of the natural
integrals of fn w.r.t. q(dtdx) exists in LF
p (Ω, F, P ) for n → ∞, i.e.
Z tZ
Z tZ
p
f (t, x, ω)q(dtdx)(ω) := lim
fn (t, x, ω)q(dtdx)(ω)
(25)
0
Λ
n→∞
0
Λ
exists. Moreover, the limit (25) does not depend on the sequence {fn }n∈IN ∈
Σ(E/F ), for which property P and (25) holds.
We call the limit in (25) the strong -p -integral of f w.r.t. q(dtdx) on (0, t] × Λ.
Remark 3.9 In [21] it has been proven that the strong -p -integrals are martingales. It could then be concluded that these have a cád -làg version (see e.g.
[9], [13],[20]). We shall propose in Proposition 3.15 another proof of this statement, for the case of the strong -p -integrals of functions f (t, x, ω), which are
in MνT,p (E/F ), with p = 1, or p = 2 in case where F is a Banach space of Mtype 2. The proof of 3.15 includes for these cases also the proof of the stronger
statement that under the above conditions the strong -p- integrals are cád -làg.
We now give sufficient conditions for the existence of the strong -p -integrals,
when p = 1, or p = 2. In the whole article we assume that hypothesis A in
Theorem 3.5 is satisfied.
Theorem 3.10 [21] Let f ∈ MνT,1 (E/F ), then f is strong -1 -integrable w.r.t.
q(dt, dx) on (0, t] × Λ, for any 0 < t ≤ T , Λ ∈ B(E \ {0}) . Moreover
Z tZ
Z tZ
E[k
f (s, x, ω)q(dsdx)(ω)k] ≤ 2
E[kf (s, x, ω)k]ν(dsdx)(ω) (26)
0
Λ
0
Λ
Remark 3.11 By definition of Bochner integral f ∈ MνT,1 (E/F ) iff f ∈ M T (E/F )
and f is Bochner integrable w.r.t. ν ⊗ P . Moreover, from Definition 3.8 and
Theorem 3.10 it also follows that f is strong -1 -integrable, iff f ∈ M T (E/F )
and f is Bochner integrable w.r.t. ν ⊗ P .
9
Theorem 3.12 [21] Suppose (F, B(F )):= (H, B(H)) is a separable Hilbert space.
Let f ∈ MνT,2 (E/H), then f is strong 2 -integrable w.r.t. q(dtdx) on (0, t] × Λ,
for any 0 < t ≤ T , Λ ∈ B(E \ {0}). Moreover
Z tZ
Z tZ
E[kf (s, x, ω)k2 ]ν(dsdx)
(27)
f (s, x, ω)q(dsdx)(ω)k2 ] =
E[k
0
0
Λ
Λ
The following Theorem 3.13 was proven in [21] for the case of deterministic
functions on type 2 Banach spaces, and on M-type 2 spaces for functions which
do not depend on the random variable x. We extend it here to general random
functions on any M- type 2 Banach space.
Theorem 3.13 Suppose that F is a separable Banach space of M- type 2. Let
f ∈ MνT,2 (E/F ), then f is strong 2 -integrable w.r.t. q(dtdx) on (0, t] × Λ, for
any 0 < t ≤ T , Λ ∈ B(E \ {0}). Moreover
Z tZ
Z tZ
E[kf (s, x, ω)k2 ]ν(dsdx) . (28)
f (s, x, ω)q(dsdx)(ω)k2 ] ≤ K22
E[k
0
0
Λ
Λ
where K2 is the constant in the Definition 1.1 of M -type 2 Banach spaces.
Proof of Theorem 3.13:
First we prove that given f ∈ Σ(E/F ) the inequality (28) holds:
let f be of the form (21), it then follows from Proposition 3.3
RtR
E[k 0 Λ f (s, x, ω)q(dsdx)(ω)k2 ]
h P
i
n−1 Pn
= E k k=1 l=1 1Fk,l ak,l q((tk , tk+1 ] ∩ (0, t] × Ak,l ∩ Λ)k2
hP
i
n−1 Pn
2
≤ K2 E
k
1
a
q((t
,
t
]
∩
(0,
t]
×
A
∩
Λ)k
F
k,l
k
k+1
k,l
k,l
k=1
l=1
(29)
where
R t R in the inequality we used the M- type 2 condition and the fact that
f (s, x, ω)q(dsdx)(ω) is a martingale [21].
0 Λ
In order to bound the r.h.s. of (29) by putting the norm inside all the sum
operators, we decompose for each k ∈ 1, ..., n − 1 fixed the sets ∪m
l=1 Ak,l in
disjoint sets, so that we can then use once more the M -type 2 condition and
prove the inequality (28) as described below:
let 2{1,...,m} be the set of all subsets of the set {1, ..., m}, then
m
∪m
l=1 Ak,l = ∪{j1 ,...,jl }∈2{1,...,m} Aj1 ∩ ... ∩ Ajl \ ∪h=1,h6=j1 ,...jl Ah ,
(30)
and the sets Aj1 ∩...∩Ajl \∪m
h=1,h6=j1 ,...jl Ah , are two by two disjoint. To facilitate
the notations we define
10
qk,A (ω) := q((tk , tk+1 ] ∩ (0, T ] × A ∩ Λ)(ω) .
We then have
RtR
E[k 0 Λ f (s, x, ω)q(dsdx)(ω)k2 ]
hP
i
n−1 Pn
2
≤ K2 E
k=1 k
l=1 1Fk,l ak,l qk,Ak,l (ω)k
Pn−1 P
≤ K2 E[ k=1 k {j1 ,...,jl }∈2{1,...,m} ak,j1 1Fk,j1 (ω) + ... + ak,jl 1Fk,jl (ω) ×
qk,Ak,j1 ∩...∩Ak,jl \∪h=1,h6=j1 ,...,jl Ak,h (ω)k2 ]
Pn−1
P
= K2 E[ k=1 EFtk [k {j1 ,...,jl }∈2{1,...,m} ak,j1 1Fk,j1 (ω) + ... + ak,jl 1Fk,jl (ω) ×
qk,Ak,j1 ∩...∩Ak,jl \∪h=1,h6=j1 ,...,jl Ak,h (ω)k2 ]]
Pn−1
P
≤ K22 E[ k=1 EFtk [ {j1 ,...,jl }∈2{1,...,m} kak,j1 1Fk,j1 (ω) + ... + ak,jl 1Fk,jl (ω)k2 ×
(qk,Ak,j1 ∩...∩Ak,jl \∪h=1,h6=j1 ,...,jl Ak,h (ω))2 ]]
Pn−1
P
≤ K22 E[ k=1 EFtk [ {j1 ,...,jl }∈2{1,...,m} (kak,j1 k1Fk,j1 (ω) + ... + kak,jl k1Fk,jl (ω)k)2 ×
(qk,Ak,j1 ∩...∩Ak,jl \∪h=1,h6=j1 ,...,jl Ak,h (ω))2 ]]
Pn−1
P
= K22 E[ k=1 EFtk [ {j1 ,...,jl }∈2{1,...,m} kak,j1 k2 1Fk,j1 (ω) + ... + kak,jl k2 1Fk,jl (ω) ×
(qk,Ak,j1 ∩...∩Ak,jl \∪h=1,h6=j1 ,...,jl Ak,h (ω))2 ]]
Pn−1
Pm
= K22 E[ k=1 EFtk [ l=1 1Fk,l (ω)kak,l k2 |qk,Ak,l (ω)|2 ]]
Pn−1 Pm
= K22 k=1 l=1 P (Fk,l )kak,l k2 ν((tk , tk+1 ] ∩ (0, t] × Λ ∩ Ak,l )
RtR
= 0 Λ E[kf (s, xk2 ]ν(dsdx)
(31)
In the above calculations we used the M -type 2 condition applied to the independent random variables qk,Ak,j1 ∩...∩Ak,jl \∪h=1,h6=j1 ,...,jl Ak,h (ω). (In fact for this
passage the type 2 condition would be sufficient.) Moreover we used that
(kak,j1 k1Fk,j1 (ω) + ... + kak,jl k1Fk,jl (ω)k)2 (qk,Ak,j1 ∩...∩Ak,jl \∪h=1,h6=j1 ,...,jl Ak,h (ω))2 =
(kak,j1 k2 1Fk,j1 (ω) + ... + kak,jl k2 1Fk,jl (ω))(qk,Ak,j1 ∩...∩Ak,jl \∪h=1,h6=j1 ,...,jl Ak,h (ω))2(32)
which follows from
kak,jp k1Fk,jp (ω)kak,ji k1Fk,ji (ω)(qk,Ak,j1 ∩...∩Ak,jl \∪h=1,h6=j1 ,...,jl Ak,h (ω))2 = 0 ∀ji , jp ∈ j1 , ..., jl , i 6= p
and is a consequence of
Fk,i × Ak,i ∩ Fk,p × Ak,p = ∅
11
∀i, p ∈ 1, ..., m .
Now we prove that inequality (28) holds for any f ∈ MνT,2 (E/F ):
let {fn }n∈IN ∈ Σ(E/F ) be a sequence L2 -approximating f in (0, T ] × A × Ω
w.r.t. ν ⊗ P . Then
Z tZ
Z
2
2
E[k
(fn (s, ω)−fm (s, ω))q(dsdx)(ω)k ] ≤ K2
E[kfn (s, ω)−fm (s, ω)k2 ]ν(dsdx)
0
A
A
(33)
RtR
LF
2 (Ω, F, P )
so that 0 A fn (s, ω)q(dsdx)(ω) is a Cauchy sequence in
and the
limit (25) for p = 2 exists. Moreover the limit does not depend on the choice of
RtR
the sequence {fn }n∈IN . It follows that the strong - 2- integral 0 A f (s, x, ω)q(dsdx)(ω)
exists. Moreover
RtR
f (s, x, ω)q(dsdx)(ω)k2 ] = limn→∞ E[k 0 A fn (s, x, ω)q(dsdx)(ω)k2 ]
RtR
RtR
≤ limn→∞ K22 0 A E[kfn (s, x, ω)k2 ]ν(dsdx) = K22 0 A E[kf (s, x, ω)k2 ]ν(dsdx)
E[k
RtR
0
A
Remark 3.14 Let 0 ∈
/ Λ. Suppose that the hypothesis of Theorem 3.10, or of
Theorem 3.12, or of Theorem 3.13 are satisfied. Suppose that f (·, x, ω) is left
-continuous for all x ∈ E, and P -a.e. ω ∈ Ω. From Corollary 5.2 in [21] it
follows that the strong -p- integral (with p = 1 in case of Theorem 3.10, and
p = 2, in case of the Theorems 3.12 -3.13) coincides P -a.s. with the natural
integral. If the condition that f (·, x, ω) is left -continuous for all x ∈ E and P
-a.e. ω ∈ Ω is not satisfied, then this might be false (see the Proof of Theorem
5.1 in [21]).
Proposition
3.15 Let f satisfy the hypothesis of Theorem 3.10, or 3.13. Then
RtR
f
(s,
x,
ω)q(dsdx)(ω)
, t ∈ [0, T ] is an Ft -martingale with mean zero and
0 Λ
is cád -làg.
Proof of Proposition 3.15: Let p = 1 if the hypothesis of Theorem 3.10 are
satisfied, p = 2 if the hypothesis of Theorem
R t R3.13 are satisfied. From Proposition
3.3 it follows that the natural integral 0 Λ f (s, x, ω)q(dsdx)(ω) of a simple
function f (s, x, ω) is P -a.s. cád -làg in t and is an Ft -martingale. It follows
RtR
that k 0 Λ f (s, x, ω)q(dsdx)(ω)k is a submartingale. Using Doob’ s inequality
we get for p = 1, resp. p = 2,
P (sup0≤t≤T k
RtR
0
Λ
RT R
f (s, x, ω)q(dsdx)(ω)k > ) ≤ 1p E[k 0 Λ f (s, x, ω) q(dsdx)(ω)kp ]
RT R
≤ C 1p 0 Λ E[kf (s, x, ω)kp ] ,
(34)
12
where the constant C in the last inequality equals C = 2, in case of p = 1, and
C = K2 , in case of p = 2. The last inequality follows in fact from inequality
(26), resp. (28).
By linearity of the integral we get the above inequality for fn − fm , n, m ∈ IN ,
where kfn − f k converges to zero in Lp ([0, T ] × Λ × Ω, ν ⊗ P ). Hence we get
RtR
that 0 Λ f (s, x, ω) q(dsdx)(ω) is a martingale and is cad - lág a.e..
4
Existence and uniqueness of stochastic integral equations driven by non Gaussian noise
In this Section we use the same notations as in the previous Sections. We assume
in the whole paper, that the compensator ν(dtdx) of the Poisson random measure N (dtdx)(ω) is a product measure on B(IR+ ×E), i.e. ν(dtdx) = α(dt)β(dx),
such that hypothesis A in Theorem 3.5 is satisfied. Moreover, we assume starting from this Section that α(dt) = dt, i.e. N (dtdx)(ω) is a Poisson random
measure associated to an E -valued Lévy process.
We also denote by B([0, T ]×F ) the product σ -algebra generated by the product
semiring B([0, T ]) × B(F ) of the Borel σ -algebra B([0, T ]) and the Borel σ algebra B(F ) of F , by B([0, T ] × E \ {0} × F ) the product σ -algebra generated
by the product semiring B([0, T ] × E \ {0}) × B(F ). Given in general two σ
-algebras M and L, with measure m and resp. l, we denote by M ⊗ L the
product σ -algebra generated by the product semiring M × L, and by m ⊗ l the
corresponding product measure.
We make the following hypothesis:
A) f (t, x, z, ω) is a B([0, T ] × E \ {0} × F ) ⊗ FT /B(F ) -measurable function,
s.th. for all t ∈ [0, T ], x ∈ E and z ∈ F fixed, f (t, x, z, ·) is Ft -adapted,
B) A(t, z, ω) is a B([0, T ] × F )⊗FT /B(F ) -measurable function, s.th. for all
t ∈ [0, T ] and z ∈ F fixed, A(t, z, ·) is Ft -adapted,
C) φ(t, ω)∈ M T (E/F ) (and does not depend on the variable x ∈ E).
We shall give in this Section sufficient conditions for the existence of a unique
solution of the integral equation (1), which is in the space LTp , with p = 1 or
p = 2, where
LTp := LTp ([0, T ] × Ω , (Ft )t∈[0,T ] ) :=
{(Zt (ω))t∈[0,T ] , ω ∈ Ω : ∀t ∈ [0, T ] , Zt (ω) is Ft − adapted
and as a function from [0, T ] × Ω to F , is B([0, T ]) ⊗ FT /B(F ) − measurable ,
(Zt (ω))t∈[0,T ] satisf ies (3)}
13
Definition 4.1 We say that two processes Zt1 (ω)∈ LTp and Zt2 (ω) ∈ LTp are
dt ⊗ P - equivalent if these coincide for all (t, ω) ∈ Γ, with Γ ∈ B([0, T ]) ⊗ FT ,
and dt ⊗ P (Γc ) = 0, where with dt we denote the Lesbegues measure on B(IR+ ).
We denote with LTp the set of dt ⊗ P -equivalence classes.
Remark 4.2 LTp , p ≥ 1, with norm
Z
kZt kF,T := (
T
E[kZs kp ]ds)1/p .
(35)
0
is a Banach space.
Theorem 4.3 Let 0 < T < ∞ and Λ ∈ B(E \ {0}). Let (4) be satisfied for
p = 1. Suppose that there is a constant L > 0, s.th.
R
kA(t, z, ω) − A(t, z 0 , ω)k + Λ kf (t, x, z, ω) − f (t, x, z 0 , ω)kβ(dx) ≤ Lkz − z 0 k
f or all t ∈ (0, T ] , z, z 0 ∈ F , and f or P − a.e. ω ∈ Ω
(36)
Assume also that there is a constant K > 0 such that
R
kA(t, z, ω)k + Λ kf (t, x, z, ω)kβ(dx) ≤ K(kzk + 1)
f or all t ∈ (0, T ] , z ∈ F and P − a.e. ω ∈ Ω.
(37)
then there is a unique process (Zt )0≤t≤T ∈ LT1 which satisfies (1).
Corollary 4.4 Suppose that all hypothesis of Theorem 4.3 are satisfied. Then
there is up to stochastic equivalence (see Definition 4.5 below) a unique process
(Zt )0≤t≤T ∈ LT1 which satisfies (1). Assume also that φ(t, ω) has P -a.s. no
discontinuities of the second kind (resp. is cád -làg), then (Zt )0≤t≤T has no
discontinuities of the second kind (resp. is cád -làg).
The following Definition is well known.
Definition 4.5 Two processes (Xt )t∈IR+ and (Yt )t∈IR+ are stochastic equivalent
if P (Xt = Yt ) = 1 ∀t ∈ IR+ .
Theorem 4.6 Suppose that F is a separable Banach space of M -type 2. Let
0 < T < ∞ and Λ ∈ B(E \ {0}). Let (4) be satisfied for p = 2. Suppose also
that there is a constant L > 0, s.th.
14
R
T kA(t, z, ω) − A(t, z 0 , ω)k2 + Λ kf (t, x, z, ω) − f (t, x, z 0 , ω)k2 β(dx) ≤ Lkz − z 0 k2
f or all t ∈ (0, T ] , z, z 0 ∈ F , and f or P − a.e. ω ∈ Ω
(38)
Assume also that there is a constant K > 0 such that
R
kA(t, z, ω)k2 + Λ kf (t, x, z, ω)k2 β(dx) ≤ K(kzk2 + 1)
f or all t ∈ (0, T ] , z ∈ F and P − a.e. ω ∈ Ω.
(39)
then there is a unique process (Zt )0≤t≤T ∈ LT2 which satisfies (1).
Corollary 4.7 Suppose that all hypothesis of Theorem 4.6 are satisfied. Then
there is up to stochastic equivalence a unique process (Zt )0≤t≤T ∈ LT2 which
satisfies (1). If φ(t, ω) has P -a.s. no discontinuities of the second kind (resp.
is cád -làg), then (Zt )0≤t≤T has no discontinuities of the second kind (resp. is
cád -làg).
(To prove Theorem 4.3 and 4.6 we follow the strategy proposed in [24] for the
case where Zt is real valued and N (dtdx)(ω)−ν(dtdx) is a compensated Poisson
random measure associated to an α -stable Lévy process.)
Proof of Theorem 4.3: Let us consider the mapping S
Z t
Z tZ
SZt (ω) := φ(t, ω) +
A(s, Zs (ω), ω)ds +
f (s, x, Zs (ω), ω)q(dsdx)(ω)
0
0
Λ
(40)
We first prove that, if (Zt (ω))t∈[0,T ] ∈
LT1 ,
then (SZt (ω))t∈[0,T ] ∈
LT1 .
First let us prove that the integrals in the r.h.s. of (40) are well defined.
That A(s, Zs (ω), ω) is Bochner integrable w.r.t. the Lesbegues measure ds on
[0, T ], for P -a.e. ω ∈ Ω, follows from
!
Z T
Z T
kA(s, Zs (ω), ω)kds ≤ K T +
kZs (ω)kds ,
(41)
0
0
which is a consequence of (37).
That f (s, x, Zs (ω), ω) is strong -1-integrable w.r.t. q(dsdx)(ω) on (0, T ] × Λ
follows from (37), Theorem 3.10, and the following inequality
!
Z TZ
Z T
E[kf (s, x, Zs )k]ν(dsdx) ≤ K T +
E[kZs (ω)k]ds ,
(42)
0
Λ
0
15
From (41) and (42) it also follows that (SZt (ω))t∈[0,T ] ∈ LT1 . In fact,
RT
RT
RT
Rt
dtE[kSZt k] ≤ 0 dtE[kΦ(t, ω)k] + 0 dtE[ 0 dskA(s, Zs (ω), ω)k]
RT
RtR
+ 0 dtE[k 0 Λ f (s, Zs (ω), x, ω)q(dsdx)(ω)k]
RT
RT Rt
RT RtR
≤ 0 dtE[kΦ(t, ω)k] + K(T 2 + 0 dt 0 dsE[kZs (ω)k]) + 2 0 dt 0 A E[kf (s, x, Zs )k]ν(dsdx)
RT
RT Rt
RT
≤ 0 dtE[Φ(t, ω)] + 3K(T 2 + 0 dt 0 dsE[kZs k]) ≤ 3K(T 2 + T 0 dsE[kZs k]) < ∞ , (43)
0
where we used again Theorem 3.10.
We shall prove that the operator Sn is a contraction operator on LT1 , for sufficiently large values of n ∈ IN .
Given two processes (Zt1 )0≤t≤T , (Zt2 )0≤t≤T ∈ LT1 we have
SZt1 (ω) − SZt2 (ω) =
(44)
RtR
1
2
1
2
[A(s, Zs (ω), ω) − A(s, Zs (ω), ω)] ds + 0 Λ [f (s, x, Zs (ω), ω) − f (s, x, Zs (ω), ω)] q(dsdx)(ω)
0
Rt
From (36) and the properties of Bochner integrals (see e.g. [28] , Chapt. V, §5
for such properties), resp. Theorem 3.10, it follows
Rt
(A(s, Zs1 (ω), ω) − A(s, Zs2 (ω), ω)) dsk
0
Rt
≤ 0 kA(s, Zs1 (ω), ω) − A(s, Zs2 (ω), ω)k ds
Rt
≤ L 0 kZs1 (ω) − Zs2 (ω)kds
k
≤
RtR
f (s, x, Zs1 ) − f (s, x, Zs2 ) q(dsdx)(ω)k]
0 Λ
RtR
2 0 Λ E[kf (s, x, Zs1 ) − f (s, x, Zs2 )k] ν(dsdx)
Rt
≤ 2L 0 E[kZs1 − Zs2 k]ds < ∞
E[k
(45)
(46)
From (44), (45), (46) it follows
E[ kSZt1 (ω) − SZt2 (ω)k ]
+2
≤
RtR
0
Λ
Rt
0
E[kA(s, Zs1 (ω), ω) − A(s, Zs2 (ω), ω)k] ds
E[kf (s, x, Zs1 (ω), ω) − f (s, x, Zs2 (ω), ω)k] ν(dsdx)
Rt
≤ 3L 0 E[kZs1 − Zs2 k]ds
It follows by induction that
16
(47)
RT
E[ kSn Zt1 (ω) − Sn Zt2 (ω)k ]dt
0
R
Rt
Rs
T
≤ 3n Ln 0 dt 0 ds1 0 1 ds2 ....E[kZs1n − Zs2n k]dsn
n RT
≤ (3L)n Tn! 0 E[kZs1 − Zs2 k]ds
(48)
(49)
(50)
From this we get that, for sufficiently large values of n ∈ IN , the operator Sn is
a contraction operator on LT1 and has therefore a unique fixed point. Suppose
that Sn0 is a contraction operator on LT1 with fixed point (Zt (ω))t≥0 . We get
RT
≤
RT
dtE[kZt − SZt k] = 0 dtE[kSkn0 Zt
0
RT
3kn0 Lkn0 T kn0
dtE[kZt − SZt k] → 0
kn0 !
0
− Skn0 +1 Zt k]
when
k → ∞,
(51)
(52)
so that (Zt (ω))t≥0 is a fixed point also for the operator S and solves equation
(1).
Proof of Corollary 4.4: The statement in Corollary 4.4 is a direct consequence
of Theorem 4.3 and Proposition
3.15. From the proof of Theorem 4.3 it follows
RtR
in fact that Mt (ω) := 0 Λ f (s, x, Zs (ω), ω)q(dsdx)(ω) is a strong -1- integral.
Proof of Theorem 4.6: Let (Zt (ω))t∈[0,T ] ∈ LT2 . Similar to the proof of Theorem 4.3 (by taking p = 2 instead of p = 1), it can be proven that A(s, Zs (ω), ω)
is Bochner integrable w.r.t. the Lesbegues measure ds on [0, T ], for P -a.e.
ω ∈ Ω. In fact,
!
Z T
Z T
2
2
kA(s, Zs (ω), ω)k ds ≤ K T +
kZs (ω)k ds .
(53)
0
0
That f (s, x, Zs (ω), ω) is strong -2-integrable w.r.t. q(dsdx)(ω) on (0, T ] × Λ follows from (39), Theorem 3.12, resp. Theorem 3.13, and the following inequality
!
Z TZ
Z T
2
2
E[kf (s, x, Zs (ω), ω)k ]ν(dsdx) ≤ K T +
E[kZs (ω)k ]ds . (54)
0
Λ
0
Let S denote the operator defined in (40). Similar to the proof of Theorem 4.3,
it can be proven (by taking LT2 instead of LT1 ), that (Zt (ω))t∈[0,T ] ∈ LT2 implies
(SZt (ω))t∈[0,T ] ∈ LT2 .
We shall prove that S is a contraction operator on LT2 .
Let Zt1 (ω) ∈ LT2 and Zt2 (ω) ∈ LT2 .
Let K2 = 1 if F is a separable Hilbert space and otherwise be the constant in
the definition of M -type 2 spaces. It follows from (38) and Theorem 3.12, resp.
Theorem 3.13, that
17
E[ kSZt1 (ω) − SZt2 (ω)k2 ]
≤ 2T
+2K2
RtR
0
Rt
0
1
E[kf
(s,
x,
Z
s)
Λ
E[kA(s, Zs1 (ω), ω) − A(s, Zs2 (ω), ω)k2 ] ds
Rt
− f (s, x, Zs2 )k2 ] ν(dsdx) ≤ (2 + 2K2 )L 0 E[kZs1 − Zs2 k2 ]ds
It follows by induction that
RT
E[ kSn Zt1 (ω) − Sn Zt2 (ω)k2 ]dt
RT Rt
Rs
≤ (2 + 2K2 )n Ln 0 dt 0 ds1 0 1 ds2 ....E[kZs1n − Zs2n k2 ]dsn
n RT
≤ (2 + 2K2 )n Ln Tn! 0 E[kZs1 − Zs2 k2 ]ds
0
(55)
(56)
(57)
The rest of the proof is similar to the proof of Theorem 4.3.
Proof of Corollary 4.7: Corollary 4.7 is a straight consequence of Theorem
4.6 and Remark 3.9.
Remark 4.8 Suppose that the conditions in Theorem 4.3, resp. 4.6 are satisfied. Using the inequalities (37), resp. (39), and Gronwall’s Lemma it follows
that there are constants CT,K and LT,K depending on T and K such that the
solution (Zt (ω))t∈[0,T ] of (59) satisfies
Z t
Z t
E[kZs (ω)kp ds ≤ LT,K
E[kΦ(s, ω)kp ] ds + CT,K
(58)
0
0
In the above theorems we have produced a solution which is Ft -adapted and
is in LTp , which has cád -làg version. Howevever in case Φ(t, ω)= Φ(ω) we can
produce a solution which is in D([0, T ], E) and the uniqueness in this case is in
the sense of P (Z1 (t, ω) = Z2 (t, ω), ∀t ∈ [0, T ])= 1. Let us consider the stochastic
differential equation
Z t
Z tZ
Zt (ω) = Φ(ω)+
A(s, Zs (ω), ω)ds+
f (s, x, Zs (ω), ω)(N (dsdx)(ω)−ν(dsdx))
0
0
Λ
(59)
Theorem 4.9 Fix p = 1 or p = 2. If p = 2 assume that F is a separable
Banach space of M -type 2. Let 0 < T < ∞. Suppose that
E[kφ(ω)kp ] < ∞
Suppose that there exists a constant L > 0 and a constant K > 0, s.th. (36)
and (37) (if p = 1), resp. (38) and (39) (if p = 2) holds. Then there exists a
unique process satisfying (59) s.th.
18
a)
b)
c)
Zt is cád -làg,
Zt is Ft -measurable,
RT
E[kZs kp ] < ∞
0
Proof of Theorem 4.9:
Proof of the Uniqueness:
Let p = 1. Let (Zt )t∈[0,T ] and (Ẑt )t∈[0,T ] be two solutions with initial condition
φ(ω), resp. φ̂(ω) satisfying a), b) and c) above. Then
E[kZt (ω) − Ẑt (ω)k] ≤ E[kΦ(ω) − Φ̂(ω)k]
Rt
+ 0 E[kA(s, Zs (ω), ω) − A(s, Ẑs (ω), ω)k]ds
RtR
+2 0 Λ E[kf (s, x, Zs (ω), ω) − fˆ(s, x, Ẑs (ω), ω)k]β(dx)ds
Rt
≤ E[kΦ(ω) − Φ̂(ω)k] + 2L 0 E[kZs (ω) − Ẑs (ω)k]ds ,
(60)
where we used inequality (36) and Theorem 3.10. Let
v(s) := E[kZs (ω) − Ẑs (ω)k]
then
v(t) ≤ E[kΦ(ω) − Φ̂(ω)k]e2Lt
We get that if Φ(ω) = Φ̂(ω) then
P (Zt (ω) = Ẑt (ω), t ∈ Q ∩ [0, T ]) = 1
where with Q we denote the rational numbers. By the cád -làg property we
get uniqueness. We get also continuity of the solution of (59) w.r.t. the initial
condition. The proof works in a similar way for p = 2 .
Proof of the Existence:
Let
Zt0 (ω) = Φ(ω) ∀t ∈ [0, T ]
(61)
Rt
= Φ(ω) + 0 A(s, Zsk (ω), ω) ds
RtR
+ 0 Λ f (s, Zsk (ω), x, ω)q(dsdx)(ω)
(62)
and
Ztk+1 (ω)
From (26) and (36) we get that for any k ∈ IN , t ∈ [0, T ],
Z t
E[kZtk+1 (ω) − Ztk (ω)k] ≤ 2L
E[kZsk (ω) − Zsk−1 k]ds
0
19
(63)
and
Rt
E[kZt1 (ω) − Zt0 (ω)k] ≤ 0 E[kA(s, Zs0 (ω), ω)k] ds
RtR
+E[k 0 Λ f (s, Zs0 (ω), x, ω)q(dsdx)(ω)k]
Rt
≤ 2K 0 (1 + E[kΦ(ω)k])ds ≤ 2Kt(1 + E[kΦ(ω)k])
(64)
Repeating in k, we get
E[kZtk+1 (ω) − Ztk (ω)k] ≤
(2t)k+1 k
L K(1 + E[kΦ(ω)k])
(k + 1)!
(65)
We have
RT
sup0≤t≤T kZtk+1 (ω) − Ztk (ω)k ≤ 0 kA(s, Zsk+1 (ω), ω) − A(s, Zsk (ω), ω)kds
RtR
+ sup0≤t≤T k 0 Λ (f (s, Zsk (ω), x, ω) − f (s, Zsk−1 (ω), x, ω)) q(dsdx)(ω)k(66)
It follows
P (sup0≤t≤T kZtk+1 (ω) − Ztk (ω)k > 2−k )
RT
≤ P ( 0 kA(s, Zsk (ω), ω) − A(s, Zsk−1 (ω), ω)kds > 2−k−1 )
RtR
+P (sup0≤t≤T k 0 Λ (f (s, Zsk (ω), x, ω) − f (s, Zsk−1 (ω), x, ω)) q(dsdx)(ω)k > 2−k−1 )
RT
≤ 2k+1 E[ 0 kA(s, Zsk (ω), ω) − A(s, Zsk−1 (ω), ω)kds]
RT R
+2k+1 E[k 0 Λ (f (s, Zsk (ω), x, ω) − f (s, Zsk−1 (ω), x, ω)) q(dsdx)(ω)k]
RT
≤ 2k+1 E[ 0 kA(s, Zsk (ω), ω) − A(s, Zsk−1 (ω), ω)kds]
RT R
+2k+1+1 0 Λ E[kf (s, Zsk (ω), x, ω) − f (s, Zsk−1 (ω), x, ω)k]dsβ(dx)
k+1
k
≤ 2k+1 (2t)
(k+1)! L K(1 + E[kΦ(ω)k])
(67)
where the last inequality is obtained in a similar way as (65). By Borel -Cantelli
Lemma we get that P -a.s. there exists k0 (ω) ∈ IN , s. th.
sup kZtk+1 (ω) − Ztk (ω)k ≤ 2−k ∀k ≥ k0 (ω)
(68)
0≤t≤T
Define
Ztn (ω) = Zt0 (ω) +
n−1
X
(Ztk+1 (ω) − Ztk (ω))
(69)
k=0
Then Ztn converges P - a.s. uniformly on [0, T ].
Let
Zt (ω) := lim Ztn (ω)
n→∞
20
(70)
then {Zt (ω)}t∈[0,T ] } is cád -làg, as each {Ztn (ω)}t∈[0,T ] } is, and the limit is
in sup norm. Zt is Ft -measurable as Zt1 , Zt2 ,..., Ztn ,...are Ft -measurable by
induction.
Note that for n > m
Pn−1
E[kZtn − Ztm k] ≤ E[k m (Ztk+1 − Ztk )k]
k+1
Pn−1
P∞
k
≤ m E[kZtk+1 − Ztk k] ≤ K(1 + E[kΦ(ω)k]) m (4t)
(k+1)! L
→ 0 as m → ∞
(71)
It follows that convergence holds also in L1 , i.e.
1
lim Ztn = Zt
n→∞
so that (Zt )t∈[0,T ] satisfies also c). The proof works in a similar way for p = 2.
Let us prove that (Zt )t∈[0,T ] satisfies (59).
Z t
Z tZ
Ztn+1 (ω) = Φ(ω)+
A(s, Zsn (ω), ω)ds+
f (s, x, Zsn (ω), ω)(N (dsdx)(ω)−ν(dsdx))
0
0
Λ
(72)
As (Ztn )t∈[0,T ] converges uniformly on [0, T ] P -a.e. to (Zt )t∈[0,T ] it follows from
Fatou’s LemmaZthat
Z
T
T
kZt − Ztn kdt] ≤ lim E[
E[
kZtm − Ztn kdt] = 0
m→∞
0
(73)
0
We get from (28) and the Lipshitz property
Z tZ
Z t
Z tZ
E[k
F (s, x, Zsn (ω), ω)q(dsdx)(ω)−
F (s, x, Zs (ω), ω)q(dsdx)(ω)k ≤ 2LE[
kZsn (ω)−Zs (ω)kds
0
Λ
0
0
(74)
so thatZ Z
Z tZ
t
1
n
lim
F (s, x, Zs (ω), ω)q(dsdx)(ω) =
F (s, x, Zs (ω), ω)q(dsdx)(ω)
n→∞
0
Λ
0
Λ
(75)
Similarly
1
Z
lim
n→∞
t
A(s, Zsn (ω), ω)ds
0
Z
=
t
A(s, Zs (ω), ω)ds
(76)
0
There exists a subsequence such that the convergence in (75) and (76) holds P
-a.s.. Thus (Zt )t∈[0,T ] solves P -a.e. (59). The proof works in a similar way for
p = 2.
Theorem 4.10 Fix p = 1 or p = 2. If p = 2 assume that F is a separable
Banach space of M -type 2. Let 0 < T < ∞. Suppose that for every constant
C > 0 there is a constant LC > 0, s.th.
21
Λ
R
T kA(t, z, ω) − A(t, z 0 , ω)kp + Λ kf (t, x, z, ω) − f (t, x, z 0 , ω)kp β(dx)
≤ LC kz − z 0 kp f or all t ∈ (0, T ] ,
f or all z, z 0 ∈ F s.th. kzk ≤ C, kz 0 k ≤ C ,
and P − a.e. ω ∈ Ω .
(77)
Assume also that there is a constant K such that
kA(t, z, ω)kp +
R
kf (t, x, z, ω)kp β(dx) ≤ K(kzkp + 1) f or all
f or all z ∈ F , and P − a.e. ω ∈ Ω .
Λ
t ∈ (0, T ] ,
(78)
Let
E[ sup kφ(t, ω)kp ] < ∞ .
(79)
t∈[0,T ]
Then there is, up to stochastic equivalence, a unique Ft -adapted process (Zt )0≤t≤T
∈ LTp , which satisfies (1). Assume also that (φ(t, ω))0≤t≤T has P -a.s. no discontinuities of the second kind, then (Zt )0≤t≤T has no discontinuities of the
second kind. If (φ(t, ω))0≤t≤T is càd -làg, then (Zt )0≤t≤T is càd -làg.
Suppose that φ(t, ω) = φ(ω), i.e. φ does not depend on time. Then there exists
a unique process which satisfies (1), s.th. a), b), and c) in Theorem 4.9 holds.
To prove this theorem we follow the strategy proposed in [27] to generalize the
proof of the existence of a unique solution of a Hilbert valued SDE driven by
a Gaussian noise with drift terms satisfying Lipshitz conditions, to the case of
local Lipshitz conditions.
Proof: We denote with Bn := B(0, n) a centered ball in F with radius n , and
with d(z, Bn ) the distance of z ∈ F from Bn . We also denote with
An (s, z, ω) := A(s,
z
, ω)
1 + d(z, Bn )
fn (s, x, z, ω) := f (s, x,
z
, ω)
1 + d(z, Bn )
(80)
(81)
An and fn satisfy conditions A), B), the Lipshitz condition (36) (resp. (38)),
and the growth condition (37) (resp. (39)), if p = 1 (resp. p = 2).
It follows from Corollary 4.4, resp. 4.7, that for each n ∈ IN there is, up to
stochastic equivalence, a unique Ft -adapted process (Ztn )0≤t≤T ∈ LTp , which
satisfies
Z t
Z tZ
Ztn (ω) = φ(t, ω)+
An (s, Zsn (ω), ω)ds+
fn (s, x, Zsn (ω), ω)(N (dsdx)(ω)−ν(dsdx))
0
0
Λ
(82)
22
Moreover, (Ztn (ω))0≤t≤T has no discontinuities of the second kind, resp. is càd
-làg, if this holds for (φ(t, ω))0≤t≤T . If φ(t, ω) = φ(ω) then for each n ∈ IN
there exists a unique process (Ztn (ω))0≤t≤T which satisfies (1), s.th. a), b), and
c) in Theorem 4.9 holds.
Let Tn := sup{t : kZtn+1 (ω))k ≤ n}, then Tn is an Ft -stopping time and
R t∧T
n+1
n+1
Zt∧T
(ω) = φ(t ∧ Tn , ω) + 0 n An+1 (s, Zs∧T
(ω), ω)ds
n
n
R t∧Tn R
n+1
+ 0
f
(s, x, Zs∧Tn (ω), ω)(N (dsdx)(ω) − ν(dsdx))
Λ n+1
R t∧T
n+1
= φ(t ∧ Tn , ω) + 0 n A(s, Zs∧T
(ω), ω)ds
n
R t∧Tn R
n+1
+ 0
f (s, x, Zs∧Tn (ω), ω)(N (dsdx)(ω) − ν(dsdx))
Λ
R t∧T
n+1
= φ(t ∧ Tn , ω) + 0 n An (s, Zs∧T
(ω), ω)ds
n
R t∧Tn R
n+1
f (s, x, Zs∧T
+ 0
(ω), ω)(N (dsdx)(ω) − ν(dsdx)) .
Λ n
n
(83)
n+1
n
It follows that Zt∧T
= Zt∧T
P -a.s., ∀s ∈ [0, T ], which implies that Tn ≥
n
n
Tn−1 , P -a.s.. Moreover, we shall prove that
P (∀n , Tn < t) = 0
(84)
It then follows that there is a process (Zt (ω))t∈[0,T ] , which is Ft -adapted, is in
LTp , and s.th.
n+1
Zt∧Tn = Zt∧T
n
P − a.s.
(85)
lim ZTn+1
= Zt (ω) P − a.s. ∀t ∈ [0, T ]
n ∧t
(86)
R t∧T
Zt∧Tn (ω) = φ(t ∧ Tn , ω) + 0 n A(s, Zs∧Tn (ω), ω)ds
R t∧T R
+ 0 n Λ f (s, x, Zs∧Tn (ω), ω)(N (dsdx)(ω) − ν(dsdx)) .
(87)
n→∞
and
As a consequence (Zt )t∈[0,T ] is, up to stochastic equivalence, the unique Ft
-adapted process in LTp , which satisfies (1). Moreover, (Zt (ω))0≤t≤T has no discontinuities of the second kind, resp. is càd -làg, if this holds for (φ(t, ω))0≤t≤T .
Moreover if φ(t, ω) = φ(ω), then (Zt )t∈[0,T ] is the unique process which satisfies
(1), s.th. a), b), and c) in Theorem 4.9 holds.
We now prove (84):
23
P (∀n , Tn < t) ≤ P (Tn < t) ≤ P (sups≤t kZsn+1 k > n)
Rs
≤ P (sups≤t kφ(s, ω)k > n/3) + P (sups≤t k 0 An+1 (s0 , Zsn+1
(ω), ω)ds0 k > n/3)
0
RsR
+P (sups≤t k 0 Λ fn+1 (s0 , x, Zsn+1
(ω), ω)q(ds0 dx)(ω)k > n/3)
0
Rs
n+1
0
0 p
E[sups≤t kφ(s,ω)kp ]
p E[sups≤t k 0 An+1 (s ,Zs0 (ω),ω)ds k ]
+
3
np
np
R R
E[k 0s Λ fn+1 (s0 ,x,Zsn+1
(ω),ω)q(dsdx)(ω)kp ]
0
+3p sups≤t
np
R
kp ds0 ])
6p 2K(t+ 0t E[kZsn+1
E[sups≤t kφ(s,ω)kp ]
0
3p
+
→ 0 when n →
np
np
≤ 3p
≤
∞
(88)
where
inequality we used Doob’s inequality for the martingale
R s R in the0 fourth
f
(s , x, Zsn+1
(ω), ω)q(ds0 dx)(ω), in the last inequality we used the growth
0
0 Λ n+1
condition (37), resp. (39), and Theorems 3.10- 3.13.
5
Continuous dependence on initial data and
Markov property
In this Section we analyze the continuous dependence of the solutions of (1) from
the initial condition, as well as from the drift and noise coefficient (Theorem
5.1 below). We then analyze the Markov property (Theorem 5.2 below). We
assume again that (E, B(E)), and (F, B(F )) are separable Banach spaces and
use the same notations as in the previous Sections. We continue assuming in
the whole article that the compensator ν(dtdx) of the Poisson random measure
N (dtdx)(ω) is a product measure on B(IR+ × E), i.e. ν(dtdx) = α(dt)β(dx),
such that hypothesis A in Theorem 3.5 is satisfied, and that α(dt) = dt. We
also assume that f0 (t, x, z, ω) := f (t, x, z, ω), A0 (t, z, ω) := A(t, z, ω), φ0 (t, ω)
:= φ(t, ω) satisfy the conditions A), resp. B) resp. C) in Section 4. Moreover,
we assume that this holds also for fn (t, x, z, ω), resp. An (t, z, ω), resp. φn (t, ω),
where n ∈ IN . We prove the following result
Theorem 5.1 Fix p = 1 or p = 2. If p = 2 assume that F is a Banach space
of M -type 2. Let T > 0. Assume that there is a constant K > 0 such that for
all n ∈ IN 0 , t ∈ [0, T ] and z ∈ F
Z
kAn (t, z, ω)kp +
kfn (t, x, z, ω)kp β(dx) ≤ K(kzkp + 1) .
(89)
Λ
Assume that for any C > 0 there is a constant LC such that for all n ∈ IN 0 ,
t ∈ [0, T ] and z , z 0 ∈ F , with kzk < C, kz 0 k < C,
Z
T kAn (t, z, ω)−An (t, z 0 , ω)kp + kfn (t, x, z, ω)−fn (t, x, z 0 , ω)kp β(dx) ≤ LC kz−z 0 kp .
Λ
(90)
24
Moreover, assume that
sup E[ sup kφn (t, ω)kp ] < ∞ .
n∈IN 0
(91)
t∈[0,T ]
and that for all t ∈ [0, T ] and z ∈ F
kφn (t, ω) − φ(t, ω)k + kAn (t, z, ω) − A(t, z, ω)k
R
+ Λ kfn (t, x, z, ω) − f (t, x, z, ω)kβ(dx) → 0
in probability as n → ∞ .
(92)
Let us denote with Zn (t, ω) the solutions of (1) for the case where the initial
condition, resp. the drift, resp. the noise coefficient are φn (t, ω), An (t, z, ω),
fn (t, x, z, ω).
Then for each t ∈ [0, T ], Zn (t, ω) converge in probability to Z(t, ω) when n → ∞.
Proof of Theorem 5.1: We first remark that from the assumptions (89),(90),
(91), and Theorem 4.10 it follows that (Zn (t))t∈[0,T ] exists and is uniquely defined up to stochastic equivalence. Moreover from the assumptions (89), (91),
it follows that
E[ sup kZn (t)k] ≤ eCT E[ sup kφn (t)k]
t∈[0,T ]
(93)
t∈[0,T ]
Let us define
ψnN (t, ω)
= 1 if
= 0 if
kφn (s, ω)k + kφ(s, ω)k + kZn (s, ω)k + kZ(s, ω)k ≤ N ∀s ∈ [0, t]
kφn (s, ω)k + kφ(s, ω)k + kZn (s, ω)k + kZ(s, ω)k > N ∀s ∈ [0, t]
Let
(Zn (t, ω) − Z(t, ω))ψnN (t, ω) = (φn (t, ω) − φ(t, ω))ψnN (t, ω)
Rt
+ψnN (t, ω){ 0 (An (s, Zs (ω), ω) − A(s, Zs (ω), ω)ds
RtR
+ 0 Λ (fn (s, x, Zs (ω), ω) − f (s, x, Zs (ω), ω))q(dsdx)(ω)}
(94)
Since ψnN (t, ω) ≤ ψnN (s, ω) ∀s, t ∈ [0, T ], s ≤ t, it follows from the assumption
(89) that there is a constant C > 0, s.th.
E[kZn (t, ω) − Z(t, ω)kp ψnN (t, ω)] ≤
E[αnN (t, ω)]
Rt
+C 0 E[kZn (s, ω) − Z(s, ω)kψnN (s, ω)] ds
25
(95)
where
αnN (t) := kφn (t, ω) − φ(t, ω)kp ψnN (t, ω)
(96)
We remark that from (92) it follows that αnN (t, ω) → 0 in probability, as n →
∞ , uniformly in t ∈ [0, T ]. From the Lesbegues dominated convergence theorem
and
kφn (t, ω) − φ(t, ω)kp ψnN (t) ≤ 2p N p .
(97)
it follows that
E[αnN (t, ω)] → 0 as n → ∞
uniformly in t ∈ [0, T ]
(98)
By Gronwall’s Lemma (see e.g. [11]) it follows that
E[kZn (t, ω) − Z(t, ω)kp ψnN (t)] → 0 as n → ∞ .
(99)
Let > 0, then
P (kZn (t, ω)−Z(t, ω)k > ) ≤
1
E[ψnN (t, ω)kZn (t, ω)−Z(t, ω)k]+P (ψnN (t, ω) = 0)
(100)
P (ψnN (t, ω) = 0) ≤ P (sups∈[0,t] kφn (s, ω)k > N/4) + P (sups∈[0,t] kφ(s, ω)k > N/4)
+P (sups∈[0,t] kZn (s, ω)k > N/4) + P (sups∈[0,t] kZ(s, ω)k > N/4) ,
(101)
so that from (91), (93) it follows that
P (ψnN (t, ω) = 0) → 0 as N → 0
uniformly in n ∈ IN
(102)
and hence
≤
1
p
limN →∞ lim supn→∞ P (kZn (t, ω) − Z(t, ω)k > )
limN →∞ lim supn→∞ E[ψnN (t, ω)kZn (t, ω) − Z(t, ω)kp ]
+ limN →∞ lim supn→∞ P (ψnN (t, ω) = 0) ,
(103)
the left hand side equal zero, completing the proof.
Theorem 5.2 Let A(t, z, ω) = A(z) and φ(t, ω) =φ(ω), for all t ∈ [0, ∞) .
Fix p = 1 or p = 2. If p = 2 assume that F is a Banach space of M -type 2.
Let T > 0. Assume also that the hypothesis in Theorem 4.10 are satisfied and
let (Ztφ )t∈IR be the solution of (1). Let (Ztz )t∈IR , z ∈ F , be the solution of (1),
if φ(ω) = z ∀ω ∈ Ω.
Then
a)
(Ztφ )t≥0 is time homogenous
b)
(Ztz )t≥0 is Markov.
26
Proof of Theorem 5.2: Let us denote with (Zts,φ )t≥s the solution of
Z t
Z tZ
Zts,φ (ω) = φ(ω) +
A(Zus,φ (ω))du +
f (x, Zus,φ (ω))q(dudx)(ω)
s
s
(104)
Λ
Following the proof of Theorem 4.10 it can be checked that such solution exists
and is unique up to stochastic equivalence. Let us remark that the compensated
Lévy random measure q(dsdx)(ω) is translation invariant in time. I.e. if h > 0
and q̃(dsdx)(ω) denotes the unique σ -finite measure on B(IR+ × E \ {0}) which
extends the pre -measure q̃(dsdx)(ω) on S(IR+ )×B(E\{0}), such that q̃((s, t], Λ)
:= q((s + h, t + h], Λ), for (s, t] × Λ ∈ S(IR+ )×B(E \ {0}), then q̃(A) and q(A)
are equally distributed for all A ∈ B(IR+ × E \ {0}).
It follows that
R s+h
R s+h R
s,φ
Zs+h
(ω) = φ(ω) + s A(Zus,φ (ω))du + s
f (x, Zus,φ (ω))q(dudx)(ω)
Λ
Rh
RhR
u,φ
s,φ
= φ(ω) + 0 A(Zs+u
(ω))du + 0 Λ f (x, Zs+u
(ω))q̃(dudx)(ω)
(105)
From Theorem 4.10 it follows
Z h
Z
0,φ
0,φ
Zh (ω) = φ(ω) +
A(Zu (ω))du +
0
0
h
Z
f (x, Zu0,φ (ω))q(dudx)(ω) (106)
Λ
As the solutions of (105) and (106) are unique up to stochastic equivalent and
q(dsdx) and q̃(dsdx) are equal distributed, it follows that (Zh0,φ (ω))h≥0 and
s,φ
(Zs+h
(ω))h≥0 are stochastic equivalent. This proves property a).
We remark that from Theorem 4.10 it follows that (Zt0,φ (ω))t≥0 is càd -làg.
Let T ≥ 0. We denote by Qφ the distribution induced by (Zt0,φ (ω))t∈[0,T ] on
the Skorohod space D([0, T ], F ), and by Eφ the corresponding expectation. We
also remark that the σ -algebra Ftφ := σ{Zs0,φ , s ≤ t} ⊆ Ft , where (Ft )t≥0 ,
denotes the natural filtration of the compensated Lévy measure q(dsdx)(ω),
and σ{Zs0,φ , s ≤ t} is the σ -algebra generated by (Zs0,φ )s≤t .
Let us consider now the solution (Zr (ω))r∈[t,T ] of
Z r
Z uZ
Zr (ω) = Zt (ω) +
A(Zu (ω))du +
f (x, Zu (ω))q(dudx)(ω)
(107)
t
t
A
From Theorem 4.10 it follows that (Zr (ω))r∈[t,T ] is stochastic equivalent to
(Zrt,Zt (ω))r∈[t,T ] . Let us define H(z, t, r, ω):= Zrt,z (ω), r ∈ [t, T ]. We remark
that H(z, t, r, ω) is independent of Ft .
Consider γ bounded, real valued measurable function on F . Then we can write
Ez [γ(Zt+h )|Ft ](ω) = E[γ(H(Zt , t, t + h, ω))|Ft ](ω)
27
(108)
and
EZt (ω) [γ(Zh (ω))] = E[γ(H(z, 0, h, ω))]z=Zt (ω) ,
(109)
where E[·]z=Zt (ω) := E[·|Zt (ω) = z]. We shall prove that
E[γ(H(Zt , t, t + h, ω))|Ft ](ω) = E[γ(H(z, t, t + h, ω))]z=Zt (ω)
(110)
It then follows by property a) and (109) that
E[γ(H(Zt (ω), t, t + h, ω))|Ft ](ω) = E[γ(H(z, t, t + h, ω))]z=Zt (ω)
= E[γ(H(z, 0, h, ω))]z=Zt (ω) = EZt (ω) [γ(Zh (ω))]
(111)
and hence, using (108), it follows
Ez [γ(Zt+h (ω)|Ft ] = EZt (ω) [γ(Zh (ω))]
(112)
and, since FtZt ⊆ Ft this gives
Ez [γ(Zt+h (ω)|FtZt ] = EZt (ω) [γ(Zh (ω))]
(113)
Proof of (110):
Put g(z, ω) = γ(H(z, t, t + h, ω)) . Clearly g(z, ·) is measurable in ω , and
z → g(z, ω) is continuous by the continuity with respect to the initial condition.
Thus g(z, ω) is separately measurable, since F is separable. It follows that
g(z, ω) is jointly measurable. Clearly g is bounded.
We can approximate g
Pm
pointwise boundedly by functions of the form k=r φk (z)ψk (ω).
Pm
E[g(Zt (ω), ω)|Ft ] = limm→∞ k=1 φk (Zt (ω))E[ψk (ω)|Ft ]
Pm
= limm→∞ k=1 E[φk (z)ψk (ω)|Ft ]z=Zt (ω) = E[g(z, ω)]z=Zt (ω)
(114)
where in the first inequality we used that φk (Zt (ω)) is Ft -measurable.
6
APPENDIX: Stochastic integrals w.r.t. Lévy
processes
We assume again that (E, B(E)), and (F, B(F )) are separable Banach spaces
and use the same notations as in the previous Sections. Let p = 1, or p = 2 and
(F, B(F )) be a separable Banach space of type 2. We assume that
Z
kf (x)kp β(dx) < ∞
(115)
E\{0}
28
then
ξt :=
RtR
0
=
RtR
f (x)q(dsdx) + 0 kf (x)k>1 f (x)N (dsdx)(ω)
RtR
+ 0 kf (x)k>1 f (x)β(dx)ds
kf (x)k≤1
RtR
f (x)q(dsdx)
0 E\0
(116)
is a Markov process (Theorem 5.2). In this Section we show that any such
solution (ξt )t≥0 is a Lévy process. This implies in particular, by taking f (x) = x,
that any compensated Poisson - Lévy measure is associated to a Lévy process.
Let us apply the Ito formula [22]
H(ξt (ω)) − H(ξτ (ω)) =
RtR
τ
+
RtR
τ
0<kf (x)k≤1
{H(ξs− (ω) + f (x)) − H(ξs− (ω))} q(dsdx)(ω)
{H(ξs− (ω) + f (x)) − H(ξs− (ω)) − H0 (ξs− (ω))f (x)} dsβ(dx)
0<kf (x)k≤1
RtR
+ τ {kf (x)k>1} {H(ξs− (ω) + f (x)) − H(ξs− (ω))} N (dsdx)(ω)
P − a.s. ,
(117)
?
to H(ξt (ω)) = eix (ξt ) , for x? ∈ E ? , E ? denoting the dual space of E. (In [22]
the Ito formula is given in a much more general statement) We get
E[eix
RtR
0
E\{0}
?
(ξ(s))
?
(ξ(t))
E[eix
Define φt (x? ) = E[eix
?
eix
?
(ξ(t))
(f (x))
− 1] =
− 1 − ix? (f (x))1kf (x)k≤1 dsβ(dx) (118)
], then we get
d
φt (x? ) = cφt (x? )
dt
φ0 (ξ ? ) = 1
(119)
(120)
with
Z
c :=
eix
?
− 1 − ix? (f (x))1kf (x)k≤1 β(dx)
(f (x))
(121)
E\{0}
Solving the above equation, we get
?
t
φt (x ) = e
R
E\{0}
? (f (x))
eix
−1−ix? (f (x))1kf (x)k≤1 β(dx)
(122)
Thus (ξt )t≥0 , solution
of (116) is a Lévy process with Lévy measure βf (dx), such
R
that βf (A) = A β(f −1 (dx)) for any A ∈ B(E \ {0}). Conversely, from the Lévy
-Ito decomposition Theorem [2] it follows that, given any Lévy process (ξt )t≥0 ,
29
such that the associated Lévy measure β(dx) satisfies (115) for f (x)= x , then
(ξt )t≥0 satisfies (116), with f (x) = x. (If however we interpret the stochastic
integral w.r.t the compensated Poisson measure q(dsdx) to be a simple -p integral, in the sense of [21], for any p ≥ 1, then the Lévy -Ito decomposition
(116), with f (x) = x holds for any Levy process having values on any separable
Banach space, the condition (115) not being necessary [8]).
7
APPENDIX: Examples of applied problems
coming from finance
We now prove that the equation (124) below, which comes as a model for volatility in Finance [5], can be studied on general Banach spaces. Also for the case
where the state space is the real line, some of the results can be obtained in a
more direct way than [23] and [14] (see Remark 7.4 below).
Let p = 1, or p = 2 and (F, B(F )) be a separable Banach space of type 2. We
assume that
Z
kxkp β(dx) < ∞
(123)
0<kxk≤1
We analyze
dηt (ω) = −aηt (ω)dt + dξt (ω)
(124)
where
Z tZ
Z tZ
ξt (ω) :=
xq(dsdx)(ω) +
0
0<kxk≤1
xN (dsdx)(ω) ,
(125)
kxk>1
0
a > 0, ν(dsdx) = dsβ(dx) and with the initial condition η0 being independent
of the filtration (Ft )t≥0 of (ξt )t∈[0,T ] , and where we define
Z
dξt (ω) =
Z
xq(dsdx)(ω) +
xN (dsdx)(ω)
(126)
kxk>1
0<kxk≤1
From the results in the previous Sections (Theorem 4.10, Theorem 5.2) we know
that for every T > 0 there is a unique path wise solution (ξt )t∈[0,T ] of (124) with
initial condition η0 . Moreover if η0 = x, x ∈ F , then (ξt )t∈[0,T ] is Markov. It
can be shown, using Ito formula ([22]), that the solution is
Z t
ηt (ω) = e−at (η0 (ω) +
eas dξs (ω))
(127)
0
In fact, applying the Ito formula [22]
30
H(t, Yt (ω)) − H(τ, Yτ (ω)) =
Rt
RtR
+ τ ∂s H(s, Ys− (ω))ds + τ 0<kxk≤1 {H(s, Ys− (ω) + f (s, x, ω)) − H(s, Ys− (ω))} q(dsdx)(ω)
RtR
+ τ 0<kxk≤1 {H(s, Ys− (ω) + f (s, x, ω)) − H(s, Ys− (ω)) − ∂y H(s, Ys− (ω))f (s, x, ω)} dsβ(dx)
RtR
+ τ kxk>1 {H(s, Ys− (ω) + f (s, x, ω)) − H(s, Ys− (ω))} N (dsdx)(ω)
P − a.s.
f or
any
0 < τ < t ≤ T,
(128)
to
H(s, z) := e−as z
Z
Ys (ω) := η0 (ω) +
(129)
t
eas dξs (ω))
(130)
0
f (s, x, ω) := eas x
(131)
we obtain
H(t, Yt (ω)) − H(τ, Yτ (ω)) =
Rt
−a τ e−as Ys (ω)ds
+
RtR
+
τ
0<kxk≤1
{e−as [Ys− (ω) + eas x] − e−as Ys− (ω) − x}dsβ(dx)
RtR
+
τ
0<kxk≤1
{e−as [Ys− (ω) + eas x] − e−as Ys− (ω)}q(dsdx)(ω)
RtR
{e−as [Ys− (ω) + eas x] − e−as Ys− (ω)}N (dsdx)(ω)
Rt
= −a τ e−as Ys (ω)ds
RtR
RtR
+ τ 0<kxk≤1 xq(dsdx)(ω) + τ kxk>1 xN (dsdx)(ω)
τ
kxk>1
(132)
and hence (127).
−at
dηt (ω) = −ae
t
Z
Z t
−at
as
e dξs (ω) dt + e d
e dξs (ω)
as
η0 (ω) +
0
where
Z
d
t
(133)
0
e dξs (ω) = eas dξs (ω)
as
(134)
0
Let us denote for x ∈ E \ {0}, A ∈ B(E \ {0})
Pt (x, A) = P (ηt ∈ A/η0 = x)
the transition probability of ηt . Then
Pt (x, A) = δe−at x ? Qt (A) ∀A ∈ E \ {0}
31
(135)
(136)
where with ? we denote the convolution, δ is the Dirac measure, and Qt (·)
Rt
is the distribution of 0 e−(a(t−s) dξs . We remark that this coincides with the
Rt
distribution of 0 e−as dξs , as ξs has stationary independent increments. If ηt
has invariant measure, i.e. Zthere exists a measure µ on E s.th.
Pt (x, A)µ(dx) = µ(A)
(137)
δe−at x ? Qt (A)µ(dx) = µ(A)
(138)
E
then
Z
E
Theorem 7.1 Let η0 be independent of the filtration (Ft )t≥0 of (ξt )t≥0 , and
Z t
eas dξs .
(139)
ηt = e−at η0 +
0
Let (Pt )t≥0 denote the Markov semigroup associated to (ηt )t≥0 . Suppose that µ
is a corresponding invariant measure, and let L(X) denote the law of a random
Rt
variable X, then if 0 e−as dξs converges in law, then
Z t
lim L(
e−as dξs ) = µ
(140)
t→∞
0
Proof of Theorem 7.1:
−at
µ = Pt ? µ = µ(e
Z t
·) ? L(
e−a(t−s) dξs )
(141)
0
We note that
Z t
Z t
L(
e−as dξ(s)) = L(
e−a(t−s) dξs )
0
(142)
0
Since a > 0, e−at → 0 when t → ∞, gives µ(e−at ·) → δ0 when t → ∞, so that
there is a measure ν (see e.g. [17]), s.th.
Z t
L(
e−as dξs ) → ν when t → ∞
(143)
0
and ν = µ.
Let us discuss when Theorem 7.1 can be used to find the invariant measure
of the solution (127) of (124), (125). We first prove that Pt (x, ·) is infinitely
divisible
32
Lemma 7.2
?
R
E\{0}
exp
hR R
t
0
eix
(y)
Pt (x, dy) = exp [e−at ix? (x)]
i
− 1 − iea(t−s) x? (y))dsβ(dy)
i
a(t−s) ?
x (y)
exp 0 kyk>1 (eie
− 1)dsβ(dy)
h
i
Rt
= exp e−at ix? (x) + 0 ψ(e−as x? )ds
(eie
hR R
t
a(t−s)
x? (y)
0<kyk≤1
(144)
where exp(tψ(x? )) is the Fourier transform of ξt .
Proof of Lemma: The proof is obtained by applying Ito formula to (127),
similar to what is done in (122).
Remark 7.3 Lemma 7.2 implies that Pt (x, ·) is infinitely divisible and the corresponding Lévy measure βt (·) is such that
Z
t
Z
βt (A) =
1A (e−as y)ds ,
β(dy)
E\{0}
A ∈ B(IR+ × E \ {0}) ,
while the corresponding
shift Zis
Z
t
γ := e−at x +
β(dy)
e−as y(10<kyk≤1 e−as y − 10<kyk≤1 (y)ds .
E\{0}
(145)
0
(146)
0
(See e.g. [23] for the definitions and properties related to infinitely divisible
laws on IRd , Chapt. 3, Pragraph 17 in particular for such computations, [17]
on Banach spaces).
Remark 7.4 Lemma 7.2 was proven in Lemma 17.1 in Chapt.3, Paragr. 17,
[23], for the real valued case, however using an approximation by simple functions. The existence of the solution of (124) has been proven in Paragr. 17 of
[23], only for the case where (ξt )t∈[0,T ] is of bounded variation, (i.e. has big
jumps), while it has been proven on the real line in [14] in an ad hoc way, by
defining the stochastic integral of e−as w.r.t. dξs by assuming that an integration
by part formula holds, which follows now as a consequence of the Ito formula
[22].
Following [23] Theorem 17.5,
Z Chapt. 3, Pragraph 17, it can be shown that if
log kykβ(dy) < ∞
(147)
kyk>2
then the limit distribution µ in (140) exists and its Fourier transform µ̂ is
Z ∞
µ̂ =
ψ(e−as x? )ds
(148)
0
33
We note that µ is infinitely divisible as the limit of infinitely divisible Qt . Similar
to [23] one can show that the corresponding
measure β̃ and shift γ is
Z
Z Lévy
∞
1
1A (e−s y)ds ,
(149)
β(dy)
β̃(A) := lim
t→∞ a E
0
Z
1
y
γ :=
β(dy) .
(150)
a kyk>1 kyk
8
APPENDIX: Examples coming from finance
and insurance
We shall now consider another example [7] where the use of stochastic differential
equations with respect to non -Gaussian additive noise play a role in finance and
insurance. Let (ξt )t∈IR+ be a real valued Lévy process with Lévy measure β,
like in (125). Using Ito formula [22] we get a SDE for eξt as follows.
RtR
(eξt − 1) =
+
0
0<kxk≤1
RtR
0
eξs− {ex − 1 − x}dsβ(dx)
0<kxk≤1
RtR
+
kxk>1
0
eξs− {ex − 1}q(dsdx)
eξs− {ex − 1}N (dsdx)
(151)
In fact (151) is obtained
(128) to H(s, ξs ) =H(ξs ) = eξs . We note
R by applying
y
x
that with a(s, y) = e 0<kxk≤1 {e − 1 − x}β(dx), f (s, y, x)= ey (ex − 1) this
equation is of the form studied in Skorohod ([24], page 45). Because a(s, y),
f (s, y, x) satisfy Lipshitz conditions given there, we get eξt is a unique solution
of (151) and is Markov. Let (ηt )t∈IR+ be another pure jump Lévy process. Then
using Ito formula [22]
Z t
ζt = eξt ζ0 +
e−ξs dηs
(152)
0
is a solution of the equation
dζt = deξt + dηt
(153)
with ζ0 independent of {ξt , ηt , t ≥ 0}. Defining like in the previous Section dηt
through the Lévy -Ito decomposition, we get
deξt
= eξt−
R
0<kxk≤1
(ex − 1)q(dtdx)
R
+eξt− kxk>1 (ex − 1)N (dtdx)
R
+eξt− 0<kxk≤1 (ex − 1 − x)β(dx)
34
(154)
Rt
(deξt ) ζ0 + 0 eξs− dηs
Rt
= eξt− ζ0 + 0 eξs− dηs ×
R
[ 0<kxk≤1 (ex − 1)q(dtdx)
R
R
+ kxk>1 (ex − 1)N (dtdx) + 0<kxk≤1 (ex − 1 − x)β(dx)]
R
= ζt− [ 0<kxk≤1 (ex − 1)q(dtdx)
R
R
+ kxk>1 (ex − 1)N (dtdx) + 0<kxk≤1 (ex − 1 − x)β(dx)]
(155)
giving
R
(dζt ) = ζt− 0<kxk≤1 (ex − 1)q(dtdx)
R
+ζt− kxk>1 (ex − 1)N (dtdx)
R
+ζt− 0<kxk≤1 (ex − 1 − x)β(dx)
+dηt
(156)
The SDE (156) has a unique Markov solution with initial condition ζ0 = 0 .
The corresponding transition functions are constant in x.
Hence (ζt )t≥0 is a Markov process and because of continuous dependence on
initial condition [24], we get that the transition semigroup is Feller. Further
using stationary independent increment property of ξt and ηt we get
Z t
ξt −ξs
L
e
dηs = L(Qt ) ,
(157)
0
where
Z
t
Qt :=
eξs dηs .
(158)
0
Let us denote with Pt (x, A):= E[ζt ∈ A|ζ0 = x], A ∈ B(IR) and with PXt the
distribution of a process (Xt )t∈IR+ at time t, then
Pt (x, A) = Peξt x+R t eξs dηs
(159)
0
If we assume that Peξt x → δ0 , when t → ∞, then we get that if the invariant
measure µ exists for ζt then µ = PR0∞ eξs dηs . In particular if ηs = s for all
R∞
s ∈ IR+ , we get that µ= L 0 eξs ds which is called perpetuity as the invariant
measure of the solution of
dζt = deξt + dt
ζo = x .
To compute the infinitesimal generator A of
35
(160)
ζt = eξt
Z
ξ0 +
t
e−ξs ds
.
(161)
0
is useful, e.g. to compute the invariant measure analytically. To compute A we
apply the Ito formula [22] to ζt . Let F a real valued function, F ∈ C 2 (IR).
F (ζt ) − F (ζ0 )
=
RtR
+
+
E[F (ζt ) − F (ζ0 )]
RtR
0
=
0
0<kxk≤1
RtR
0
kxk>1
[F (ζs− + ζs− (ex − 1)) − F (ζs− )]q(dsdx)
[F (ζs− + ζs− (ex − 1)) − F (ζs− )]N (dsdx)
[F (ζs− + ζs− (ex − 1)) − F (ζs− ) − F 0 (ζs− )ζs− (ex − 1))]dsβ(dx)
Rt
R
+ 0 F 0 (ζs− )[ζs− 0<kxk≤1 (ex − 1 − x)β(dx) + 1]ds
(162)
0<kxk≤1
RtR
0
[F (ζs− + ζs− (ex − 1)) − F (ζs− ) − F 0 (ζs− )ζs− (ex − 1)]β(dx)ds
h
i
Rt
R
+ 0 F 0 (ζs− ) ζs− 0<kxk≤1 (ex − 1 − x)β(dx) + 1 ds
(163)
0<kxk≤1
It follows that the infinitesimal generator A acts on the functions F ∈ C 2 (IR)
as
R
AF (y) = 0<kxk≤1 [F (y + y(ex − 1)) − F (y) − F 0 (y)(ex − 1)y]β(dx)
R
+F 0 (y)[y 0<kxk≤1 (ex − 1 − x)β(dx) + 1]
(164)
AF (y) =
R
[F (ex y) − F (y) − F 0 (y)(ex − 1)y]β(dx)
R
+yF (y) 0<kxk≤1 (ex − 1 − x)β(dx) + F 0 (y)
0<kxk≤1
0
(165)
Suppose that
Z
kxkβ(dx) < ∞
(166)
[F (ex y) − F (y)]β(dx)
0<kxk≤1
R
−yF 0 (y) 0<kxk≤1 xβ(dx) + F 0 (y)
(167)
0<kxk≤1
then
AF (y) =
R
Acknowledgements We thank Sergio Albeverio for many useful discussions
related to this work. The second author thanks Marc Veraar for useful discussions related to this work in connection to [21] and Carlo Marinelli for useful
comments. The support and hospitality of the ”Sonderforschungsbereich” SFB
611 in Bonn, as well as of the University Koblenz -Landau, is gratefully acknowledged.
36
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39
Bestellungen nimmt entgegen:
Institut für Angewandte Mathematik
der Universität Bonn
Sonderforschungsbereich 611
Wegelerstr. 6
D - 53115 Bonn
Telefon:
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E-mail:
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Spline-Wavelets
173. Djah, Sidi Hamidou; Gottschalk, Hanno; Ouerdiane, Habib: Feynman Graphs for nonGaussian Measures
174. Djah, Sidi Hamidou; Gottschalk, Hanno; Ouerdiane, Habib: Feynman Graph Representation
of the Perturbation Series for General Functional Measures
175. not published
176. Albeverio, Sergio; Shelkovich, Vladimir M.: Delta-Shock Waves in Multidimensional NonConservative System of Zero-Pressure Gas Dynamics
177. Albeverio, Sergio; Kuzhel, Sergej: η-Hermitian Operators and Previously Unnoticed
Symmetries in the Theory of Singular Perturbations
178. Albeverio, Sergio; Alimov, Shavkat: On Some Integral Equations in Hilbert Space with an
Application to the Theory of Elasticity; eingereicht bei: Oper. Th. and Int. Eqts.
179. Albeverio, Sergio; Galperin, Gregory; Nizhnik, Irena L.; Nizhnik, Leonid P.: Generalized
Billiards Inside an Infinite Strip with Periodic Laws of Reflection Along the Strip’s
Boundaries; eingereicht bei: Regular and Chaotic Dynamics
180. Albeverio, Sergio; Torbin, Grygoriy: Fractal Properties of Singular Probability Distributions
with Independent Q*-Digits; eingereicht bei: Bull. Sci. Math.
181. Melikyan, Arik; Botkin, Nikolai; Turova, Varvara: Propagation of Disturbances in Inhomogeneous Anisotropic Media
182. Albeverio, Sergio; Bodnarchuk, Maksim; Koshmanenko, Volodymyr: Dynamics of Discrete
Conflict Interactions between Non-Annihilating Opponents
2
183. Albeverio, Sergio; Daletskii, Alexei: L -Betti Numbers of Infinite Configuration Spaces
2
184. Albeverio, Sergio; Daletskii, Alexei: Recent Developments on Harmonic Forms and L -Betti
Numbers of Infinite Configuration Spaces with Poisson Measures
185. Hildebrandt, Stefan; von der Mosel, Heiko: On Lichtenstein’s Theorem About Globally Conformal Mappings
186. Mandrekar, Vidyadhar; Rüdiger, Barbara: Existence and Uniqueness for Stochastic Integral
Equations Driven by non Gaussian Noise on Separable Banach Spaces