Existence and Uniqueness of Path Wise Solutions for Stochastic

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. R¨ udiger** * Department of Statistics and Probability Michigan State University, East Lansing, MI 48824, USA ** - Mathematisches Institut, Universit¨at Koblenz-Landau, Campus Koblenz, Universit¨ atsstrasse 1, 56070 Koblenz, Germany; - SFB 611, Institut f¨ ur Angewandte Mathematik, Abteilung Stochastik, Universit¨ at 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 L´evy 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 c´ ad -l´ ag 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 c´ ad -l` ag 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 L´ evy 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 c` adl` ag.

An additive process is a L´evy 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 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 L´evy 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 L´evy 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 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 c´ ad -l` ag 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 c´ ad -l` ag. 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 c´ ad -l` ag. 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. c´ad -l`ag 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 - l´ag 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 L´evy 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 c´ ad -l` ag), then (Zt )0≤t≤T has no discontinuities of the second kind (resp. is c´ ad -l` ag). 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 c´ ad -l` ag), then (Zt )0≤t≤T has no discontinuities of the second kind (resp. is c´ ad -l` ag). (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 L´evy 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 c´ ad -l` ag 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 c´ ad -l` ag, 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 (Zˆt )t∈[0,T ] be two solutions with initial condition ˆ φ(ω), resp. φ(ω) satisfying a), b) and c) above. Then ˆ E[kZt (ω) − Zˆt (ω)k] ≤ E[kΦ(ω) − Φ(ω)k] Rt + 0 E[kA(s, Zs (ω), ω) − A(s, Zˆs (ω), ω)k]ds RtR +2 0 Λ E[kf (s, x, Zs (ω), ω) − fˆ(s, x, Zˆs (ω), ω)k]β(dx)ds Rt ˆ ≤ E[kΦ(ω) − Φ(ω)k] + 2L 0 E[kZs (ω) − Zˆs (ω)k]ds ,

(60)

where we used inequality (36) and Theorem 3.10. Let v(s) := E[kZs (ω) − Zˆs (ω)k] then 2Lt ˆ v(t) ≤ E[kΦ(ω) − Φ(ω)k]e

ˆ We get that if Φ(ω) = Φ(ω) then P (Zt (ω) = Zˆt (ω), t ∈ Q ∩ [0, T ]) = 1 where with Q we denote the rational numbers. By the c´ad -l`ag 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 c´ ad -l`ag, 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 c` ad -l` ag, then (Zt )0≤t≤T is c` ad -l` ag. 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 c`ad -l` ag, 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 c`ad -l`ag, 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 L´evy random measure q(dsdx)(ω) is translation invariant in time. I.e. if h > 0 and q˜(dsdx)(ω) denotes the unique σ -finite measure on B(IR+ × E \ {0}) which extends the pre -measure q˜(dsdx)(ω) on S(IR+ )×B(E\{0}), such that q˜((s, t], Λ) := q((s + h, t + h], Λ), for (s, t] × Λ ∈ S(IR+ )×B(E \ {0}), then q˜(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))˜ q (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 q˜(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 c`ad -l`ag. 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 L´evy 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. L´ evy 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 L´evy process. This implies in particular, by taking f (x) = x, that any compensated Poisson - L´evy measure is associated to a L´evy process. Let us apply the Ito formula [22] H(ξt (ω)) − H(ξτ (ω)) = RtR τ

+

RtR τ

01

0 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 01 (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