Ito Formula for Stochastic Integrals w.r.t.Compensated Poisson Random Measures on

Separable Banach Spaces

B. Rüdiger, G. Ziglio

no. 187

Diese Arbeit ist mit Unterstützung des von der Deutschen Forschungs-

gemeinschaft getragenen Sonderforschungsbereiches 611 an der Univer-

sität Bonn entstanden und als Manuskript vervielfältigt worden.

Bonn, Oktober 2004

Ito formula for stochastic integrals w.r.t. com-pensated Poisson random measures on separableBanach spaces

B.Rudiger*, G. Ziglio*** Mathematisches Institut, Universitat Koblenz-Landau, Campus Koblenz, Uni-versitatsstrasse 1, 56070 Koblenz, Germany;* SFB 611, Institut fur Angewandte Mathematik, Abteilung Stochastik, Uni-versitat Bonn, Wegelerstr. 6, D -53115 Bonn, Germany.** Facolta di Scienze, Universita di Trento, Via Sommarive 14, I-38050 Povo(Trento) Italia

Abstract

We prove the Ito formula (3) for Banach valued stochastic processes withjumps, the martingale part given by a stochastic integral w.r.t. a com-pensated Poisson random measure. Such stochastic integrals have beendiscussed in [35]. The Ito formula is computed here for Banach valuedfunctions.

AMS -classification (2000): 60H05, 60G51, 60G57, 46B09, 47G99

Keywords: Stochastic integrals on separable Banach spaces, random martingalesmeasures, compensated Poisson random measures, additive processes, randomBanach valued functions, type 2 Banach spaces, M-type 2 Banach spaces, Itoformula, Levy -Ito decomposition theorem.

1 Introduction

In [35] the definition of stochastic integrals

Zt(ω) :=∫ t

0

∫A

f(s, x, ω) (N(dsdx)(ω)− ν(dsdx)) (1)

of Hilbert and Banach valued functions f(s, x, ω) , w.r.t. compensated Poissonrandom measures (cPrms) N(dsdx)(ω)−ν(dsdx), has been discussed. (See also[7],[8], [9], [10],[13], [19],[25], [24],[26], [31], [32], [33], [34], [37], [38] for the dis-cussion of stochastic integrals w.r.t. cPrms on infinite dimensional spaces).The state space of f is denoted with F , and is a separable Banach space.s, t ∈ IR+, x is defined on some separable Banach space E, A is a Borelset of E \ 0, and ω is defined on a probability space (Ω,F , P ). The cPrmq(dsdx)(ω) :=N(dsdx)(ω)−ν(dsdx) is associated to some E - valued additiveprocess (Xt)t∈IR+ on (Ω,F , P ) (see Section 2 for the concept of cPrm, andSection 3 for the discussion of the integrals (1)). The problem of giving an

1

appropriate meaning to (1) comes from the property of the cPrms to be only σ-finite (in general not finite) random measures. The cPrms are in general notfinite on the sets (0, T ]× Λ, if it does not hold that 0 /∈ Λ.For real valued functions this problem was already discussed e.g. in [14], wherethe integrals are defined by approximating the integral (1) in Lp(Ω,F , P ), p =1, 2, with integrals on the set A ∩ Bc(0, ε), where with Bc(0, ε) we denote thecomplementary of the ball of radius ε > 0, centered in 0 (we refer to Definition3.20 for a precise statement).The definition of the integrals (1) for the real valued case was discussed in adifferent way than [14] in [36] and [6]. In [36] and [6] the ”natural” definitionof ”Ito integrals” (consisting of an Lp(Ω,F , P ) -approximation of integrals ofappropriate ”simple functions”), is extended to the case where integration isperformed w.r.t. cPrms instead of the Brownian motion. (In [36] cPrm associ-ated to α -stable processes are considered, while [6] discusses general martingalemeasures on IRd).In [35] the definition of stochastic integrals (1) for real valued random functions,given in [14], and resp. [36], [6], has been generalized to the case of Hilbert andBanach valued random functions. We call them simple -p -, and resp. strong -p-integrals. It has been proven that the definition of ”strong -p- integral” (whichfor the real valued case coincides with the one used in [36], [6]) is stronger thanthe definition of ”simple -p- integral” (which in the real valued case coincideswith the one of [14]). (In a similar way like e.g. the definition of a ”Lebeguesintegral” is a stronger definition than the definition of ”Riemann integral”.) Infact, we need one more sufficient condition for the integrand f , if we requirethe existence of the ”simple -p-integral”. We need that f has a left -continuousversion. (In [14] it is required that for the existence of the simple -p -integral theintegrand has to be predictable, in the sense that it is in the smallest σ -algebragenerated by the left -continuous functions). In [35] it has been proven that ifhowever this condition holds, then the simple -2- and strong -2- integrals coincide(see also Section 3, for precise statements). The results in [35], beside being ageneralization to Hilbert and Banach spaces, unify therefore the definition of theintegral (1) proposed in [14] with the one of [36], [6], for the case of integrandswhich have a left continuous version.The definition of the integral (1) on infinite dimensional spaces, like Hilbert orBanach spaces, is necessary to discuss stochastic differential equations (SDEs)with non Gaussian noise on such spaces (see e.g. [4], [3], where this problem isdiscussed). In fact, from the Levy -Ito decomposition theorem [15], [7], [1] itfollows that ”additive non Gaussian noise” is naturally defined through integralsof the form (1). In [22], [23], existence and uniqueness of SDEs with non Gaus-sian additive noise has been proven on separable Banach spaces, under localLipshitz conditions for the drift- and noise - coefficients. (We refer to [22] forprecise statements). In [23] the continuous dependence of the solutions on thecoefficients of such SDEs has been shown and the Markov properties have been

2

discussed. Applications of such SDEs related to financial models, are given in[23], too.In this article we prove the Ito formula for the Banach valued processes

Yt(ω) := Zt(ω) +∫ t

0

g(s, ω)dh(s, ω) +∫ t

0

∫Λ

k(s, x, ω)N(dsdx)(ω) , (2)

Zt(ω) being the stochastic integral in (1). We assume that h(s, ω) : IR+×Ω → IRis a real valued, cadlag process, whose almost all paths are of bounded variation,g is a Banach valued random (cad -lag or cag -lad) function depending on time(see Section 4 for precise statements). We consider H : IR+ ×F → G, with G aseparable Banach space, H(s, y), s ∈ IR+ , y ∈ F twice Frechet differentiable.We shall prove (see Section 4, Section 5, Section 6 for precise statements) thatthe following Ito formula holds:

H(t, Yt(ω))−H(τ, Yτ (ω)) =

+∫ t

τ∂sH(s, Ys−(ω))ds +

∫ t

τ

∫AH(s, Ys−(ω) + f(s, x, ω))−H(s, Ys−(ω)) q(dsdx)(ω)

+∫ t

τ

∫AH(s, Ys−(ω) + f(s, x, ω))−H(s, Ys−(ω))− ∂yH(s, Ys−(ω))f(s, x, ω) ν(dsdx)

+∫ t

τ

∫ΛH(s, Ys−(ω) + k(s, x, ω))−H(s, Ys−(ω))N(dsdx)(ω) +

∫ t

τ∂yH(s, Ys−)g(s−, ω)dh(s, ω)

+∑

τ<s≤t H(s, Ys−(ω) + g(s−, ω)∆h(s, ω))−H(s, Ys−(ω))− ∂yH(s, Ys−(ω))g(s−, ω)∆h(s, ω)P − a.s. , (3)

where ∂sH(s, y) denotes the Frechet derivative of H w.r.t. s ∈ IR+ for y ∈ Ffixed, and ∂yH(s, y) denotes the Frechet derivative of H w.r.t. y ∈ F , fors ∈ IR+ fixed and

q(dsdx)(ω) := N(dsdx)(ω)− ν(dsdx)

Remark 1.1 The simplest example for a process (2), for which the Ito formula(3) can be applied, is when (Yt)t∈IR+ coincides with a pure jump additive process(At)t∈IR+ with values in a Banach space E . From the Levy -Ito decompositiontheorem [15], [7], [1] it follows that

At(ω) =∫ t

0

∫0<‖x‖≤1

x(N(dsdx)(ω)−ν(dsdx))+αt+∫ t

0

∫‖x‖>1

xN(dsdx)(ω) P−a.s.

(4)where αt ∈ E. The stochastic integral w.r.t. q(dsdx)(ω) is a strong -p andsimple -p -integral with p = 1, resp. p = 2, if∫ t

0

∫0<|x|≤1

‖x‖p ν(dsdx) < ∞ , (5)

3

and if, in case of p = 2, the Banach space is of type 2 [1]. (See Definition 3.19or [5], [20] for the definition of type 2 Banach spaces.) In the general case it isonly a simple -p -integral, as follows from [7] (see also final Remark in [1]). Aseries of other examples for which the Ito formula (3) can be applied on infinitedimensional Banach spaces are in [22], [23] [2] (see e.g. also [12], [25]).

Suppose F = G = IR, and H depends not on the time s ∈ IR. Suppose thatf is left continuous in time, so that the integral in (1) is a simple -2 -integral.Suppose also that the term

∫ t

0g(s, ω)dh(s, ω) in (2) is skipped. The Ito formula

in (41) coincides for this case with the Ito -formula found in [14]. In [25], [13]an Ito formula is also given for Hilbert -valued stochastic integrals of the form(1). The stochastic integrals (1) are there defined in a more abstract way, byassuming some conditions which control the second moment of the integralsof approximating functions . These conditions are satisfied by our strong -2 -integrals (used also in [36], [6] for the real valued case), so that it follows that theIto -formula in [25], [13] is satisfied by such integrals. The Ito formula in [25],[13] is however not given in terms of the cPrm q(dsdx)(ω) and compensatorν(dsdx), like done in this paper (3), where we also generalize to the case ofBanach valued functions. An Ito formula in terms of the cPrm q(dsdx)(ω) andcompensator ν(dsdx) is however necessary to establish the generator of somejump Markov process Yt(ω) of the form (2), like e.g. the problems discussed in[23].

2 Compensated Poisson random measures on sep-arable Banach spaces

We assume that a filtered probability space (Ω,F , (Ft)0≤t≤+∞, P ), satisfyingthe ”usual hypothesis”, is given:i) Ft contain all null sets of F , for all t such that 0 ≤ t < +∞ii) Ft = F+

t , where F+t = ∩u>tFu, for all t such that 0 ≤ t < +∞, i.e. the

filtration is right continuous.In this Section we recall the definition of compensated Poisson random mea-sures 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 Banachspace with norm ‖ · ‖ and B(E) is the corresponding Borel σ -algebra (see also[12], [1], [35]).

Definition 2.1 A process (Xt)t≥0 with state space (E,B(E)) is an Ft - additiveprocess on (Ω,F , P ) ifi) (Xt)t≥0 is adapted (to (Ft)t≥0)ii) X0 = 0 a.s.

4

iii) (Xt)t≥0 has increments independent of the past, i.e. Xt−Xs is indepen-dent of Fs if 0 ≤ s < tiv) (Xt)t≥0 is stochastically continuousv) (Xt)t≥0 is cadlag.An additive process is a Levy process if the following condition is satisfiedvi) (Xt)t≥0 has stationary increments, that is Xt −Xs has the same distri-bution as Xt−s, 0 ≤ s < t.

Remark 2.2 That any process satisfying i) -iv) has a cadlag version followsfrom its being, after compensation, a martingale (see e.g. [11], [16],[29]).

Let (Xt)t≥0 be an additive process on (E,B(E)) (in the sense of Definition 2.1).Set Xt− := lims↑tXs and ∆Xs := Xs −Xs− .We denote with N(dtdx)(ω) the Poisson random measure associated to theadditive process (Xt)t≥0, and with ν(dtdx) its compensator. (See e.g. [12] or[35].)

q(dtdx)(ω) := N(dtdx)(ω)− ν(dtdx) (6)

is the compensated Poisson random measure associated to (Xt)t≥0. (We some-times omit to write the dependence on ω). We refer to e.g. [12] or [35] for theproperties of q(dtdx)(ω). We recall however that N(dtdx)(ω), for each ω fixed,resp. ν(dtdx), are σ -finite measures on the σ -algebra B(IR+×E \ 0), gener-ated by the product semi -ring S(IR+)×B(E \ 0) of product sets (t1, t2] ×A,with 0 ≤ t1 < t2, and A ∈ B(E \ 0), where B(E \ 0) is the trace σ -algebraon E \ 0 of the Borel σ -algebra B(E) on E. If however Λ ∈ F(E \ 0), with

F(E \ 0) := Λ ∈ B(E \ 0) : 0 ∈ (Λ)c , (7)

then ∀ω ∈ ΩN((t1, t2]× Λ)(ω) =

∑t1<s≤t2

1Λ(∆Xs) < ∞ (8)

and

P (N((t1, t2]× Λ) = k) = exp(−ν((t1, t2]× Λ))(ν((t1, t2]× Λ))k

k!(9)

withν((t1, t2]× Λ) := E[N((t1, t2]× Λ)] < ∞ (10)

5

3 Stochastic integrals w.r.t. compensated Pois-son random measures on separable Banachspaces

Let F be a separable Banach space with norm ‖·‖F . (When no misunderstandingis possible we write ‖ · ‖ instead of ‖ · ‖F .) 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 A× F , A ∈ B(IR+ × E \ 0), F ∈ Ft. Let T > 0, and

MT (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 ] (11)

Here we recall the definition of stochastic integrals of random functions f(t, x, ω) ∈MT (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 introducedin [35] (and [1] for the case of deterministic functions f(x) not depending ontime t ∈ IR+ and on ω ∈ Ω).There is a ”natural definition” of stochastic integral w.r.t. N(dtdx)(ω), resp.q(dtdx)(ω), on those sets (0, T ] × Λ where the measures N(dtdx)(ω) (with ωfixed) and resp. ν(dtdx) are finite, i.e. 0 /∈ Λ. According to [35] we give thefollowing definition

Definition 3.1 Let t ∈ (0, T ], Λ ∈ F(E \0) (defined in (7)), f ∈ MT (E/F ).The natural integral of f on (0, t]×Λ w.r.t. Poisson random measure N(dtdx)(ω)is ∫ t

0

∫Λ

f(s, x, ω) N(dsdx)(ω) :=∑0<s≤t f(s, (∆Xs)(ω), ω)1Λ(∆Xs(ω)) ω ∈ Ω (12)

Definition 3.2 Let t ∈ (0, T ], Λ ∈ F(E \0) (defined in (7)), f ∈ MT (E/F ).Assume that f(·, ·, ω) is Bochner integrable on (0, T ]×Λ w.r.t. ν, for all ω ∈ Ωfixed. The natural integral of f on (0, t] × Λ w.r.t. the compensated Poissonrandom measure q(dtdx) := N(dtdx)(ω)− ν(dtdx) is∫ t

0

∫Λ

f(s, x, ω) (N(dsdx)(ω)− ν(dsdx)) :=∑0<s≤t f(s, (∆Xs)(ω), ω)1Λ(∆Xs(ω))−

∫ t

0

∫Λ

f(s, x, ω)ν(dsdx) ω ∈ Ω(13)

where the last term is understood as a Bochner integral, (for ω ∈ Ω fixed) off(s, x, ω) w.r.t. the measure ν.

6

It is more difficult to define the stochastic integral on those sets (0, T ] × A,A ∈ B(E \0), s.th. ν((0, T ]×A) = ∞. For real valued functions this problemwas already discussed e.g. in [36] and [6], [14], [39] (for general martingale mea-sures), where two different definitions of stochastic integrals were introduced. In[35] the definitions of stochastic integrals of Hilbert and Banach valued randomfunctions, which generalize (in a natural way) the definitions of such stochasticintegrals (of real valued random functions) have been given, for the case whereintegration is performed w.r.t. compensated Poisson random measures on sep-arable Banach spaces. Like in [35], we shall call them strong -p -, resp. simple-p -integrals. The strong -p -integral is the limit in LF

p (Ω,F , P ) (the space ofF -valued random variables Y , with E[‖Y ‖p] < ∞, defined in Definition 3.8) ofthe the ”natural integrals” (13) of the ”simple functions” defined in (14). (Werefer to Definition 3.9 for a precise statement.) The simple -p -integral is thelimit in LF

p (Ω,F , P ) of the ”natural integrals” over the sets (0, T ] × Λε, withT > 0 and Λε := A ∩ Bc(0, ε), when ε → 0, where with Bc(0, ε) we denote thecomplementary of the ball of radius ε > 0 centered in 0 (we refer to Definition3.20 for a precise statement). In [35] sufficient conditions for the strong -1 - andsimple -1- integral are found on any separable Banach space, and sufficient con-ditions of the strong -2- and simple -2 -integral are found on separable Banachspaces of M -type 2 and type 2 (see Theorems 3.12 -3.16 and Theorems 3.22-3.26). (See Definition 3.19 or [5], [20] for the definition of type 2 Banach spaces,Definition 3.17 or [27], [28] for the definition of M -type 2 Banach spaces). Herewe first introduce the notion of strong -p, as it turns out from [35] that thestrong -p integral is a stronger definition, than the simple -p -integral. In fact,if the sufficient conditions in Theorems 3.12 -3.16 for the existence of the strong-p -integrals for f(t, x, ω) are satisfied, and, moreover, f(·, x, ω) is left continu-ous on t ∈ IR+ for each x ∈ E \ 0 and ω ∈ Ω fixed, then f is also simple -pintegrable, and the simple -p and strong -p, -integrals coincide (see Theorems3.22 -3.26). We first introduce the simple functions and recall the definition of”strong -p -integral”, p ≥ 1.

Definition 3.3 A function f belongs to the set Σ(E/F ) of simple functions ,if f ∈ MT (E/F ), T > 0 and there exist n ∈ IN , m ∈ IN , such that

f(t, x, ω) =n−1∑k=1

m∑l=1

1Ak,l(x)1Fk,l

(ω)1(tk,tk+1](t)ak,l (14)

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.

Remark 3.4 Let f ∈ Σ(E/F ) be of the form (14), then the natural integraldefined in Definition 3.2 is given also in the following form:

7

∫ T

0

∫A

f(t, x, ω)q(dtdx)(ω) =n−1∑k=1

m∑l=1

ak,l1Fk,l(ω)q((tk, tk+1]∩(0, T ]×Ak,l ∩A)(ω).

(15)for all A ∈ B(E \ 0), T > 0. This is an easy consequence of the Definition ofthe random measure q(dtdx)(ω).

We recall here the definition of strong -p -integral (Definition 3.9 below) given in[35] through approximation of the natural integrals of simple functions. Beforewe establish some properties of the functions f ∈ MT,p

ν (E/F ), p ≥ 1, with

MT,pν (E/F ) := f ∈ MT (E/F ) :

∫ T

0

∫E[‖f(t, x, ω)‖p] ν(dtdx) < ∞ , (16)

where with E[f ] we denote the expectation with respect to the probabilitymeasure P .Let ν × P be the product measure on the semi -ring B(IR+ × (E \ 0))×F∞of the product sets A × F , A ∈ B(IR+ × (E \ 0)), F ∈ F∞. Let us alsodenote by ν ⊗ P the unique extension of ν × P on the product σ -algebraF∞ := B(IR+ × (E \ 0))⊗F∞ generated by B(IR+ × (E \ 0))×F∞.Starting from here we assume in the whole article that the following hypothesisA is satisfied:Hypothesis A: ν is a product measure ν = α ⊗ β on the σ -algebra generatedby the semi -ring S(IR+) × B(E \ 0), of a finite measure α on S(IR+) anda σ -finite measure β on B(E \ 0). α(dt) is absolutely continuous w.r.t. theLebesgues measure dt on B(IR+), i.e. α(dt) ≺ dt

Definition 3.5 Let f : IR+×(E \0)×Ω → F be given. A sequence fnn∈IN

of FT /B(F ) -measurable functions is Lp -approximating f on (0, T ]×A×Ωw.r.t. ν ⊗ P , if fn is ν ⊗ P -a.s. converging to f , when n →∞, and

limn→∞

∫ T

0

∫A

E[‖fn(t, x, ω)− f(t, x, ω)‖p] dν = 0 , (17)

i.e. ‖fn − f‖ converges to zero in Lp((0, T ]×A× Ω, ν ⊗ P ), when n →∞.

Theorem 3.6 [35] Let p ≥ 1, T > 0, then for all f ∈ MT,pν (E/F ) and all

A ∈ B(E \0), there is a sequence of simple functions fnn∈IN which satisfiesthe followingProperty P: fn ∈ Σ(E/F )∀n ∈ IN , and fn is Lp -approximating f on (0, T ]×A× Ω w.r.t. ν ⊗ P .

Remark 3.7 From the proofs of Theorem 3.6 in [35] it follows that if f ∈MT,p

ν (E/F ) ∩MT,qν (E/F ) with p, q ≥ 1, T > 0, then, for all ∈ B(E \ 0) there

is a sequence of simple functions fnn∈IN , fn ∈ Σ(E/F )∀n ∈ IN , s.th. fn isboth Lp - and Lq - approximating f on (0, T ]×A× Ω w.r.t. ν ⊗ P .

8

Definition 3.8 Let p ≥ 1, LFp (Ω,F , P ) is the space of F -valued random vari-

ables, such that E‖Y ‖p =∫‖Y ‖pdP < ∞. We denote by ‖ · ‖p the quasi -norm

([20]) given by ‖Y ‖p = (E‖Y ‖p)1/p. Given (Yn)n∈IN , Y ∈ LFp (Ω,F , P ), we

write limpn→∞ Yn = Y if limn→∞‖Yn − Y ‖p = 0

Definition 3.9 Let p ≥ 1, T > 0. We say that f is strong -p -integrable on(0, T ]×A w.r.t. q(dtdx)(ω), A ∈ B(E \ 0), if there is a sequence fnn∈IN ∈Σ(E/F ), s.th. fn is Lp -approximating f on (0, T ] × A × Ω w.r.t. ν ⊗ P ,and for any such sequence the limit of the natural integrals of fn w.r.t. q(dtdx)exists in ∈ LF

p (Ω,F , P ) for n →∞, i.e.∫ T

0

∫A

f(t, x, ω)q(dtdx)(ω) := limpn→∞

∫ T

0

∫A

fn(t, x, ω)q(dtdx)(ω) (18)

exists. Moreover, the limit (18) does not depend on the sequence fnn∈IN ∈Σ(E/F ), which is Lp -approximating f on (0, T ]×A× Ω w.r.t. ν ⊗ P .We call the limit in (18) the strong -p -integral of f w.r.t. q(dtdx) on (0, T ]×A.The strong -p -integral of f w.r.t. q(dtdx) on (τ, T ] × A, with 0 < τ < T , isdefined as∫ T

τ

∫A

f(t, x, ω)q(dtdx)(ω) :=∫ T

0

∫A

f(t, x, ω)q(dtdx)(ω)−∫ τ

0

∫A

f(t, x, ω)q(dtdx)(ω)

(19)

Remark 3.10 Let f ∈ MT,rν (E/F ) ∩MT,q

ν (E/F ) , r, q ≥ 1 . If f is both r-and q- strong -integrable on (0, T ]×A , r, q≥ 1 , then from Remark 3.7 it followsthat the strong -r -integral coincides with the strong -q -integral. In fact fromany sequence fn, for which the limit in (18) holds with p = r and p = q, it ispossible to extract a subsequence for which the convergence (18) holds also P -a.s..

Proposition 3.11 [35] Let p ≥ 1. Let f be strong -p -integrable on (0, T ]×A,A ∈ B(E \ 0). Then the strong -p -integral

∫ t

0

∫A

f(s, x)q(dsdx) , t ∈ [0, T ] ,is an Ft -martingale with mean zero.

Theorem 3.12 [35] Let f ∈ MT,1ν (E/F ), then f is strong -1 -integrable w.r.t.

q(dt, dx) on (0, t]×A, for any 0 < t ≤ T , A ∈ B(E \ 0) . Moreover

E[‖∫ t

0

∫A

f(s, x, ω)q(dsdx)(ω)‖] ≤ 2∫ t

0

∫A

E[‖f(s, x, ω)‖]ν(dsdx) (20)

9

Theorem 3.13 [35] Suppose (F,B(F )):= (H,B(H)) is a separable Hilbert space.Let f ∈ MT,2

ν (E/H), then f is strong -2 -integrable w.r.t. q(dtdx) on (0, t]×A,for any 0 < t ≤ T , A ∈ B(E \ 0). Moreover

E[‖∫ t

0

∫A

f(s, x, ω)q(dsdx)(ω)‖2] =∫ t

0

∫A

E[‖f(s, x, ω)‖2]ν(dsdx) (21)

Theorem 3.14 [23] Suppose that F is a separable Banach space of M -type 2.Let f ∈ MT,2

ν (E/F ), then f is strong 2 -integrable w.r.t. q(dtdx) on (0, t]×A,for any 0 < t ≤ T , A ∈ B(E \ 0). Moreover

E[‖∫ t

0

∫A

f(s, ω)q(dsdx)(ω)‖2] ≤ K22

∫ t

0

∫A

E[‖f(s, ω)‖2]ν(dsdx) . (22)

where K2 is the constant in the Definition 3.17 of M -type 2 Banach spaces.

Remark 3.15 Theorem 3.14 was first proven in [35], however only for func-tions f(t, x, ω) = f(t, ω), i.e. which do not depend on the variable x ∈ E.

Theorem 3.16 [35] Suppose that F is a separable Banach space of type 2.Let f ∈ MT,2

ν (E/F ), and f be a deterministic function, i.e. f(t, x, ω) = f(t, x),then f is strong 2 -integrable w.r.t. q(dtdx) on (0, t] × A, for any 0 < t ≤ T ,A ∈ B(E \ 0). Moreover inequality (22) holds, with K2 being the constant inthe Definition 3.19 of type 2 Banach spaces.

We recall here the definitions of M -type 2 and type 2 separable Banach spaces(see e.g. [27], [5]).

Definition 3.17 A separable Banach space F , with norm ‖ · ‖, 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:

E[‖Mn‖2] ≤ K2

n∑k=1

E[‖Mk −Mk−1‖2] (23)

with the convention that M−1 = 0.

We remark that a separable Hilbert space is in particular a separable Banachspace of M -type 2. In fact, any 2 -uniformly -smooth separable Banach spaceis of M -type 2 [28], [40]. We recall here the definition of 2 - uniformly -smoothseparable Banach space.

Definition 3.18 A separable Banach space F , with norm ‖ · ‖, is 2 -uniformly-smooth if there is a constant K2 > 0, s.th. for all x, y ∈ F

‖x + y‖2 + ‖x− y‖2 ≤ 2‖x‖2 + K2‖y‖2

10

Definition 3.19 A separable Banach space F is of type 2, if there is a constantK2, such that if Xin

i=1 is any finite set of centered independent F - valuedrandom variables, such that E[‖Xi‖2] < ∞, then

E[‖n∑

i=1

Xi‖2] ≤ K2

n∑i=1

E[‖Xi‖2] (24)

We remark that any separable Banach space of M -type 2 is a separable Banachspace of type 2. Moreover, a separable Banach space is of type 2 as well as ofcotype 2 if and only if it is isomorphic to a separable Hilbert space [18], wherea Banach space of cotype 2 is defined by putting ≥ instead of ≤ in (24) (see [5],or [20]).We recall the definition of ”simple -p -integrals” w.r.t. the compensated Poissonrandom measures q(dtdx), introduced in [35].

Definition 3.20 Let p ≥ 1, T > 0, A ∈ B(E \ 0). Let f ∈ MT (E/F ) and fbounded ν⊗P -a.s. on (0, T ]×A×Ω. f is simple -p integrable on (0, T ]×A×Ω,if for all sequences δnn∈IN , δn ∈ IR+, such that limn→∞ δn = 0, the limit

p

limn→∞

∑0<s≤T

1∆Xs(ω)∈Λδn∩A f(s,∆Xs(ω), ω)−∫ T

0

∫Λδn∩A

f(t, x, ω)ν(dtdx)

(25)with

Λδn:= x ∈ E \ 0 : δn < ‖x‖ (26)

exists and does not depend on the choice of the sequences δnn∈IN , satisfyingthe above properties.We call the limit (25) the simple -p integral of f on (0, T ] × A w.r.t. thecompensated Poisson random measure N(dtdx)− ν(dtdx), and denote it with∫ T

0

∫A

f(t, x, ω)(N(dtdx)(ω)− ν(dtdx)) (27)

Remark 3.21 If f is both r- and q- simple integrable on (0, T ] × A, p,q ≥ 1,then the limit in (25) with p = r coincides P -a.s. with the limit in (25) withp = q, as there exist in both cases a subsequence of a sequence δnn∈IN , suchthat limn→∞ δn = 0, for which the convergence (25) holds also P - a.s..

The following Theorems 3.22 - 3.27 and Corollary 3.28 have been proven in [35].

Theorem 3.22 Let A ∈ B(E \ 0), f ∈ MT,1ν (E/F ), and assume f is left

continuous in the time interval (0, T ] for every x ∈ A and P - a.e. ω ∈ Ω fixed.Then f is simple 1 -integrable on (0, T ]×A and the simple 1 -integral coincideswith the strong 1 -integral.

11

Theorem 3.23 Let F be a separable Hilbert space. Let A ∈ B(E \ 0), f ∈MT,2

ν E/F ), and let f be left continuous in the time interval (0, T ] for everyx ∈ A and P - a.e. ω ∈ Ω fixed. Then f is simple 2 -integrable on (0, T ] × A.The simple 2 -integral coincides with the strong 2 -integral.

Theorem 3.24 Let F be a separable Banach space of M -type 2. Let A ∈B(E \ 0), f ∈ MT,2

ν E/F ). Let f be left continuous in the time interval (0, T ]for P - a.e. ω ∈ Ω fixed. Then f is simple 2 -integrable on (0, T ] × A. Thesimple 2 -integral coincides with the strong 2 -integral.

Remark 3.25 Theorem 3.24 has been stated in [35] for functions f , such thatf(s, x, ω) = f(s, ω), i.e. such that f does not depend on the variable x ∈ E.The proof of the Theorem goes through however for general functions, like statedin 3.24, and is in this case a consequence of Theorem 3.14 (see Remark 3.15).

Theorem 3.26 Let F be a separable Banach space of type 2. Let A ∈ B(E \0), f ∈ MT,2

ν E/F ), and let f be left continuous in the time interval (0, T ] forevery x ∈ A fixed. Let f be a deterministic function, i.e. f(t, x, ω) = f(t, x).Then f is simple 2 -integrable on (0, T ] × A. The simple 2 -integral coincideswith the strong 2 -integral.

The statements in Theorems 3.22 -3.26 are proven in [35] by first proving thefollowing Theorem 3.27 and its Corollary 3.28, that we shall also use in the nextSection of this article.

Theorem 3.27 Let Λ ∈ F(E\0) (defined in (7)). Let f be uniformly boundedon (0, T ]×Λ×Ω, and left continuous in the time interval (0, T ] for every x ∈ Λand for P - a. e. ω ∈ Ω. Let p ≥ 1. ∀t ∈ (0, T ] the strong -p -integral of f ,coincides with the natural integral of f on (0, t]× Λ.

Corollary 3.28 Let Λ ∈ F(E \ 0), and let f be left continuous in the timeinterval (0, T ], for every x ∈ Λ and for P - a. e. ω ∈ Ω fixed. Suppose that oneof the following three hypothesis is satisfied:i) f ∈ MT,1

ν (E/F ),ii) f ∈ MT,2

ν (E/F ) and F is a separable Banach space of M -type 2.iii) f ∈ MT,2

ν (E/F ), f(s, x, ω) = f(s, x), i.e. f is a deterministic function,and F is a Banach space of type 2.Then the natural integral of f on (0, t]×Λ is ν⊗P - a.s. bounded ∀t ∈ (0, T ]and coincides with the strong 1 -integral of f , if condition i) is satisfied, withthe strong 2 -integral of f , if condition ii), or iii) is satisfied.

12

Remark 3.29 The condition ii) in Corollary 3.28 has been proven in [35] forthe case where f does not depend on the variable x ∈ E. In fact, ii) followsfrom Theorem 3.14, which in [35] was still proven only for this case (see Remark3.15. The same proof holds however for the general case, like stated in ii).

Similar to what is discussed for the Ito -integral w.r.t. the Brownian motion,the concept of strong -p -integral (1) can be generalized by approximating (1)with the natural integral of simple functions which converge to the integrandf in probability, instead of converging in Lp((0, T ] × A × Ω, ν ⊗ P ). This isdiscussed in [36] for the case of stochastic integrals (1) of real valued functionsw.r.t. cPrm associated to α -stable processes, and in [35] for Banach valuedfunctions and general cPrms. Such stochastic integrals are called in [35] ”strongintegrals of type p”, to distinguish these from the strong -p- integrals. We reporthere the definition of strong integrals of type p, and some related results [35]needed in Section 6, where we shall prove that the Ito formula (3) holds also forsuch integrals.

Definition 3.30 Let p ≥ 1, T > 0, A ∈ B(E\0). We say that f ∈ MT (E/F )is strong integrable on (0, T ]×A, with strong integral of type p, if for all N ∈ INfp

N (t, x, ω) := f(t, x, ω)1∫ t0

∫A‖f‖pdν≤N is strong p -integrable and the following

limit (28) exists

∫ T

0

∫A

f(t, x, ω)q(dtdx)(ω) :=

limN→∞∫ T

0

∫A

f(t, x, ω)1∫ t0

∫A‖f(s,y,ω)‖p ν(dsdy)≤Nq(dtdx)(ω)

in probability , (28)

where∫ T

0

∫A

f(t, x, ω)1∫ t0

∫A‖f‖p dν≤Nq(dtdx) is the strong -p integral of the func-

tion fpN (t, x, ω).

Remark 3.31 Let p ≥ 1. Let f, g be strong integrable on (0, T ] × A, A ∈B(E \ 0), T > 0, with strong integral of type p. For any a, b ∈ IR, a f + b g isstrong integrable on (0, T ]×A with strong integral of type p, and

a

∫ T

0

∫A

f(s, x, ω)q(dsdx)+ b

∫ T

0

∫A

g(s, x, ω)q(dsdx) =∫ T

0

∫A

(a f(s, x, ω)+b g(s, x, ω))q(dsdx)

(29)

Let

NT,pν (E/F ) := f ∈ MT (E/F ) :

∫ T

0

∫A

‖f(t, x, ω)‖p ν(dtdx) < ∞ P −a.s. .

(30)

13

Proposition 3.32 [35] Supposei’) f ∈ NT,1

ν (E/F ),then f is strong integrable on (0, t] × A with strong integral of type 1, for allt ∈ (0, T ], A ∈ B(E \ 0).

Proposition 3.33 [35] Suppose that one of the following two hypothesis is sat-isfied:ii’) f ∈ NT,2

ν (E/F ) and F is a separable Banach space of M -type 2,iii’) f ∈ NT,2

ν (E/F ), f(s, x, ω) = f(s, x), i.e. f is a deterministic function,and F is a Banach space of type 2.then f is strong integrable on (0, t] × A with strong integral of type 2, for allt ∈ (0, T ], A ∈ B(E \ 0).

Theorem 3.34 [35] Let T > 0, p ≥ 1, then for all f ∈ NT,pν (E/F ), for all

A ∈ B(E \ 0), there is a sequence of simple functions fnn∈IN satisfying thefollowing property :Property Q: fn ∈ Σ(E/F )∀n ∈ IN , fn converges ν ⊗ P -a.s. to f on (0, T ] ×A× Ω, when n →∞, and

limn→∞

∫ T

0

∫A

‖fn(t, x)− f(t, x)‖p dν = 0 P − a.s. (31)

Theorem 3.35 [35] Let T > 0, A ∈ B(E \ 0). Define p = 1 if the hypothesisi’) in Proposition 3.32 are satisfied, or p = 2, if the hypothesis ii’) or iii’) inProposition 3.33 are satisfied. Suppose that fnn∈IN ∈ NT,p

ν (E/F ) satisfiesproperty Q in Theorem 3.34, and there exists g ∈ NT,p

ν (E/F ), s.th.∫ T

0

∫A

‖fn‖pdν ≤∫ T

0

∫A

‖g‖pdν P − a.s. (32)

then∫ T

0

∫A

f(t, x)q(dtdx) = limn→∞

∫ T

0

∫A

fn(t, x)q(dtdx) in probability (33)

Remark 3.36 Taking fnn∈IN ∈ Σ(E/F ) instead of fnn∈IN ∈ NT,pν (E/F )

in Theorem 3.35, we can see that the strong integrals of type p have similarproperties compared with the strong -p- integrals.

Remark 3.37 Under the hypothesis ii’) Proposition 3.33 and Theorem 3.35have been proven in [35] for the case where f does not depend on the variablex ∈ E. In fact, under the condition ii’) the statements follow from Theorem3.14, which in [35] was still proven only for this case (see Remark 3.15). Thesame proofs hold however for the general case.

14

4 The Ito formula for Banach valued functions

Like in the previous Sections, we assume that E and F are separable Banachspaces, q(dtdx) = N(dtdx)(ω) − ν(dtdx) is the compensated Poisson randommeasure associated to the additive process (Xt)t≥0 with values on E .Let 0 < t ≤ T , A ∈ B(E \ 0) and

Zt(ω) :=∫ t

0

∫A

f(s, x, ω)q(dsdx)(ω) (34)

We assume that one of the following hypothesis is satisfied:i) f ∈ MT,1

ν (E/F ),ii) f ∈ MT,2

ν (E/F ) and F is a separable Banach space of M -type 2,iii) f ∈ MT,2

ν (E/F ), f(s, x, ω) = f(s, x), i.e. f is deterministic, and F is aBanach space of type 2.Zt(ω) is then the strong -p -integral of f w.r.t. q(dsdx)(ω), with p = 1 if i)holds, and p = 2 if ii) or iii) holds.In this and the coming Sections we analyze the Ito -formula for the F -valuedrandom process (Yt)t≥0, with

Yt(ω) := Zt(ω) +∫ t

0

g(s, ω)dh(s, ω) +∫ t

0

∫Λ

k(s, x, ω)N(dsdx)(ω) . (35)

We assume that h(s, ω) : IR+ × Ω → IR is a real valued, cadlag process, whosealmost all paths are of bounded variation, moreover h(s, ω)s∈[0,T ] is Fs -adaptedfor all s ∈ [0, T ], and jointly measurable in all its variables. The random processg(s, ω) : IR+ × Ω → F is Fs -adapted for all s ∈ [0, T ], and jointly measurablein all its variables, and is either cadlag or caglad (i.e. left continuous with rightlimit, for the definition see e.g. [30]). We assume that, for P -a.e. ω ∈ Ω ,g(s, ω) is Bochner integrable w.r.t. the signed Lesbegues measure induced bythe bounded variation process (h(s, ω))s≥0, i.e.∫ t

0

‖g(s, ω)‖ d|h(s, ω)| < ∞ P − a.e. (36)

where (|h(s, ω)|)s≥0 is the variation process associated to (h(s, ω))s≥0 (see e.g.[30] for the definition of variation process).Moreover, we assume that

Λ ∈ F(E \ 0) so that 0 /∈ Λ , (37)

k(s, x, ω) ∈ MT (E/F ) and k(s, x, ω) is finite P -a.s. for every s ∈ [0, T ], x ∈ Λ.Moreover k(s, x, ω) is cadlag or caglad.We also assume that ∀s ∈ [0, T ]1Λ(∆Xs(ω))∆k(s,∆Xs(ω), ω) = 0 1A(∆Xs(ω))∆f(s,∆Xs(ω), ω) = 0 P −a.s. ,

(38)

15

∀s ∈ [0, T ] ∆Xs(ω)∆h(s, ω) = 0 P − a.s. , (39)

where

∆h(s, ω) := limt↓s h(t, ω)− limt↑s h(t, ω) P − a.s.

∆k(s, x, ω) := limt↓s k(t, x, ω)− limt↑s k(t, x, ω) P − a.s.

∆f(s, x, ω) := limt↓s f(t, x, ω)− limt↑s f(t, x, ω) P − a.s.

Moreover∀s ∈ [0, T ] 1A(∆Xs(ω))1Λ(∆Xs(ω)) = 0 P − a.s. , (40)

Let H : F → G, G be a separable Banach space, 0 ≤ τ < t ≤ T . (In the nextSection we shall also consider H depending on time.) In this Section we shallprove that the following Ito formula holds (see Theorem 4.4 and Theorem 4.6for a precise statement):

H(Yt(ω))−H(Yτ (ω)) =∫ t

τ

∫AH(Ys−(ω) + f(s, x, ω))−H(Ys−(ω)) q(dsdx)(ω)

+∫ t

τ

∫AH(Ys−(ω) + f(s, x, ω))−H(Ys−(ω))−H′(Ys−(ω))f(s, x, ω) ν(dsdx)

+∫ t

τ

∫ΛH(Ys−(ω) + k(s, x, ω))−H(Ys−(ω))N(dsdx)(ω) +

∫ t

τH′(Ys−(ω))g(s−, ω)dh(s, ω)

+∑

τ<s≤t H(Ys−(ω) + g(s−, ω)∆h(s, ω))−H(Ys−(ω))−H′(Ys−(ω))g(s−, ω)∆h(s, ω)P − a.s. , (41)

where H(y) is twice Frechet differentiable ∀y ∈ F . H′(y) (resp. H′′(y)) denotesthe first (resp. second) Frechet derivative in y.Obviously H′ : F → L(F/G), and H′′ : F → L(F,L(F/G)), where, as usual,we denote with L(K/L) the Banach space of linear bounded operators from aBanach space K to a Banach space L, with the usual supremum norm. With‖·‖K we denote the norm of a Banach space K, so that in particular ‖F‖L(K/L)=supy∈K

‖F(y)‖L

‖y‖K, if F ∈ L(K/L). When no misunderstanding is possible, we shall

write simply ‖ · ‖, leaving the subscription which denotes the Banach space. Inthe above example we would e.g. write ‖F‖ instead of ‖F‖L(K/L), and ‖y‖,resp. ‖F(y)‖, instead of ‖y‖K , resp. ‖F(y)‖L.We shall use in this Section the following inequalities, which can be found e.g.in [17], Chapt. X.‖H(y)−H(y0)−H′(y0)(y− y0)‖ ≤ ‖y− y0‖ sup

0<θ≤1‖H′(y0 + θ(y− y0))−H′(y0)‖

(42)‖H(y)−H(y0)‖ ≤ ‖y − y0‖ sup

0<θ≤1‖H′(y0 + θ(y − y0))‖ (43)

16

‖H′(y)−H′(y0)‖ ≤ ‖y − y0‖ sup0<θ≤1

‖H′′(y0 + θ(y − y0))‖ (44)

We now prove the properties 1)-6) in the following Theorem 4.1 and Lemma4.3, which imply that all the integrals in equation (41) are well defined.

Theorem 4.1 Let the following hypothesis holdhypothesis B: (At least) one of the hypothesis i) -iii) is satisfied. G is a separableBanach space. If ii) or iii) is satisfied, we make the additional hypothesis thatG is a separable Banach space of M -type 2Let H : F → G, with H(y) twice Frechet differentiable for all y ∈ F . Letthe second Frechet derivative H′′ be uniformly bounded on each centered ball ofradius R, i.e. B(0, R) with R ≥ 0, and the Frechet derivative H′ be uniformlybounded.Then1) H(Ys−(ω) + f(s, x, ω)) − H(Ys−(ω)) − H′(Ys−(ω))f(s, x, ω) is P - a.s.Bochner integrable w.r.t ν on (0, T ]×A, ∀A ∈ B(E \ 0).2) H(Ys−(ω) + f(s, x, ω)) − H(Ys−(ω)) ∈ MT,1

ν (E/G), resp. ∈ MT,2ν (E/G),

if i), resp. ii) or iii), is satisfied.

Remark 4.2 Let C := supy∈F ‖H′(y)‖, then from (43) it follows that‖H(y)−H(y0)‖ ≤ C‖y − y0‖ (45)

From (42) and (44) it follows that for each R > 0, there exists a constant CR,depending on R, such that if y, y0 ∈ B(0, R),

‖H(y)−H(y0)−H′(y0)(y − y0)‖ ≤ CR‖y − y0‖2 (46)

Proof of Theorem 4.1:

Let us first prove property 1). Suppose that condition i) is satisfied, then prop-erty 1) follows from (45). In fact

∫ T

0

∫A‖H(Ys−(ω) + f(s, x, ω))−H(Ys−(ω))−H′(Ys−(ω))f(s, x, ω)‖ν(dsdx)

≤ 2∫ T

0

∫A

C‖f(s, x, ω)‖ν(dsdx) (47)

where the r.h.s. in (47) is P -a.s. finite, as f ∈ MT,1ν (E/F ).

Suppose that condition ii) or iii) is satisfied, then property 1) follows from (46).In fact, from Remark 4.2, it follows that there is a finite constant CR(ω) > 0depending on ω ∈ Ω, s.th.∫ T

0

∫A‖H(Ys−(ω) + f(s, x, ω))−H(Ys−(ω))−H′(Ys−(ω))f(s, x, ω)‖ν(dsdx)

≤∫ T

0

∫A

CR(ω)‖f(s, x, ω)‖2ν(dsdx) , (48)

17

where the r.h.s. in (41) is P -a.s. finite, as f ∈ MT,2ν (E/F ).

Property 2) follows from theorems 3.12 - 3.16, and

∫ T

0

∫A

E[‖H(Ys−(ω) + f(s, x, ω))−H(Ys−(ω))‖p]ν(dsdx) ≤∫ T

0

∫A

E[sup0<θ≤1 ‖H′(Ys−(ω) + θf(s, x, ω))‖p‖f(s, x, ω)‖p]ν(dsdx)

≤ Cp∫ T

0

∫A

E[‖f(s, x, ω)‖p]ν(dsdx). (49)

Lemma 4.3 Suppose that all the hypothesis in Theorem 4.1 are verified. Thefollowing properties hold

3)∫ T

0‖H′(Ys−)g(s−, ω)‖d|h(s, ω)| < ∞ P − a.s.

4)∑

0<s≤t ‖H(Ys−(ω) + g(s−, ω)∆h(s, ω))−H(Ys−(ω))‖ < ∞ P − a.s.

5)∑

0<s≤t ‖H′(Ys−(ω)g(s−, ω)∆h(s, ω)‖ < ∞ P − a.s.

6)∫ T

0

∫Λ‖H(Ys−(ω) + k(s, x, ω))−H(Ys−(ω))‖N(dsdx)(ω) < ∞ P − a.s.

Proof of Lemma: Property 3) follows from the assumption that H′ is uni-formly bounded on F , which implies∫ T

0

‖H′(Ys−)g(s−, ω)‖d|h(s, ω)| ≤ C

∫ T

0

‖g(s−, ω)‖d|h(s, ω)| < ∞ (50)

Property 4) follows from inequality (45), as this implies∑0<s≤t ‖H(Ys−(ω) + g(s−, ω)∆h(s, ω))−H(Ys−(ω))‖ ≤ C

∑0<s≤t ‖g(s−, ω)∆h(s, ω))‖

≤ C∫ T

0‖g(s−, ω)‖d|h(s, ω)| (51)

Property 5) is proven by similar arguments, and property 6) follows from theassumption that H′ is uniformly bounded on F , the assumption that k(s, x, ω)is finite P -a.s. for every s ∈ [0, T ], x ∈ Λ and the definition of natural integralw.r.t. N(dtdx)(ω).

Theorem 4.4 Suppose that the hypothesis in Theorem 4.1 are all satisfied. Letf(·, x, ω) be left continuous on (0, T ], for all x ∈ A and P -a.e. ω ∈ Ω fixed.Then on (0, T ]×A H(Ys−(ω)+f(s, x, ω))−H(Ys−(ω)) is strong -p and simple-p- integrable with p = 1, resp. p = 2, if i), resp. ii) or iii), is satisfied.Moreover (41) holds P -a.s..

18

Remark 4.5 The integral defined in (34) is, according to Theorems 3.12 - 3.14and Theorems 3.22 -3.26, a strong -p and simple -p -integral, with p = 1, ifcondition i) is satisfied, resp. p = 2, if condition ii), or iii) is satisfied. From theinequalities (20), (21), (22) it follows that Z·(ω) is P -a.e. uniformly boundedon [0, T ].

Proof of Theorem 4.4: That on (0, T ] × A, H(Ys−(ω) + f(s, x, ω)) −H(Ys−(ω)) is strong -p- and simple -p- integrable with p = 1 (resp. p = 2)if i) (resp. ii), or iii)) is satisfied, follows from Theorem 3.22- 3.26 and Theorem4.1.We now prove that (41) holds P -a.s.. First we assume the additional hypothesisthat 0 /∈ A, i.e. that A ∈ F(E \ 0).From hypothesis i), or ii), or iii), the above additional hypothesis A ∈ F(E\0),and Corollary 3.28, it follows that

Zt(ω) =∑

0<s≤t

f(s, (∆Xs)(ω), ω)1A(∆Xs(ω))−∫ t

0

∫A

f(s, x, ω)ν(dsdx) . (52)

From 2) in Theorem 4.1 and Corollary 3.28 it follows that the natural integral(Definition 3.2) of H(Ys−(ω)+f(s, x, ω))−H(Ys−(ω)) exists, coincides with thestrong -p -integral, with p = 1, resp. p = 2, and has the following form∫ t

τ

∫AH(Ys−(ω) + f(s, x, ω))−H(Ys−(ω)) q(dsdx)(ω)

=∑

τ<s≤tH(Ys−(ω) + f(s,∆Xs(ω), ω))−H(Ys−(ω))1A(∆Xs(ω))

−∫ t

τ

∫AH(Ys−(ω) + f(s, x, ω))−H(Ys−(ω)) ν(dsdx) P − a.e. (53)

Moreover, it is easily checked, thatH′(Ys−(ω))f(s, x, ω) andH(Ys−(ω)+f(s, x, ω))−H(Ys−(ω)) are Bochner integrable for P -q.e. ω ∈ Ω.Let

Γnω(A) := k ∈ 0, ..., 2n − 1 : ∃s ∈ (τn

k , τnk+1] : ∆Xs(ω) ∈ A (54)

with τnk := τ + k(t−τ)

2n , (55)

Γnω(Λ) := k ∈ 0, ..., 2n − 1 : ∃s ∈ (τn

k , τnk+1] : ∆Xs(ω) ∈ Λ (56)

γnω := k ∈ 0, ..., 2n − 1 : ∃s ∈ (τn

k , τnk+1] : ∆h(s, ω) 6= 0 (57)

then, for all ω ∈ Ω,

H(Yt(ω))−H(Yτ (ω))

=∑2n−1

k=0 H(Yτnk+1

(ω))−H(Yτnk(ω))

=∑

k∈Γnω(A)∪Γn

ω(Λ)∪γnωH(Yτn

k+1(ω))−H(Yτn

k(ω))

+∑

k/∈Γnω(A)∪Γn

ω(Λ)∪γnωH(Yτn

k+1(ω))−H(Yτn

k(ω)) (58)

19

As H is continuous on F , it follows

limn→∞∑

k∈Γnω(A)∪Γn

ω(Λ)∪γnωH(Yτn

k+1(ω))−H(Yτn

k(ω))

=∑

τ<s≤tH(Ys−(ω) + f(s,∆Xs(ω), ω))−H(Ys−(ω))1A(∆Xs(ω))+

∑τ<s≤tH(Ys−(ω) + k(s,∆Xs(ω), ω))−H(Ys−(ω))1Λ(∆Xs(ω))+

∑τ<s≤tH(Ys−(ω) + g(s−, ω)∆h(s−, ω))−H(Ys−(ω))

=∫ t

τ

∫AH(Ys−(ω) + f(s, x, ω))−H(Ys−(ω)) q(dsdx)(ω)

+∫ t

τ

∫ΛH(Ys−(ω) + k(s, x, ω))−H(Ys−(ω))N(dsdx)(ω)

+∑

τ<s≤tH(Ys−(ω) + g(s−, ω)∆h(s−, ω))−H(Ys−(ω))

+∫ t

τ

∫AH(Ys−(ω) + f(s, x, ω))−H(Ys−(ω)) ν(dsdx) (59)

P − a.s. .

(41) follows once shown that for some subsequence of nn∈IN , that for simplicitywe still denote with n, the following convergence holds for n →∞

limn→∞∑

k/∈Γnω(A)∪Γn

ω(Λ)∪γnωH(Yτn

k+1(ω))−H(Yτn

k(ω))

= −∫ t

τ

∫AH′(Ys−(ω))f(s, x, ω)ν(dsdx)

+∫ t

τH′(Ys−)g(s−, ω)dh(s, ω)

−∑

τ<s≤tH′(Ys−(ω))g(s−, ω)∆h(s, ω) P − a.s. (60)

Proof of (60):We make a Taylor expansion of the function H(y):∑

k/∈Γnω(A)∪Γn

ω(Λ)∪γnωH(Yτn

k+1(ω))−H(Yτn

k(ω))

=∑

k/∈Γnω(A)∪Γn

ω(Λ)∪γnωH′(Yτn

k(ω))(Yτn

k+1(ω)− Yτn

k(ω))

+∑

k/∈Γnω(A)∪Γn

ω(Λ)∪γnω

ernk (ω)

=∑

k/∈Γnω(A)∪Γn

ω(Λ)∪γnωH′(Yτn

k(ω))(Zτn

k+1(ω)− Zτn

k(ω))

+∑2n−1

k=1 H′(Yτnk(ω))(

∫ τnk+1

τnk

g(s, ω)dh(s, ω))

−∑

k∈Γnω(A)∪Γn

ω(Λ)∪γnωH′(Yτn

k(ω))(

∫ τnk+1

τnk

g(s, ω)dh(s, ω))

+∑

k/∈Γnω(A)∪Γn

ω(Λ)∪γnω

ernk (ω) (61)

We first prove that

limn→∞

∑k/∈Γn

ω(A)∪Γnω(Λ)∪γn

ω

ernk (ω) = 0 P − a.e. (62)

From Remark 4.5 it follows that for P -a.e. ω ∈ Ω there is a finite constantR(ω) such that Yt(ω) ≤ R(ω), ∀t ∈ (τ, T ] . From (46) it follows that for P -a.e.ω ∈ Ω there is a finite constant C(ω) > 0, s.th.

20

‖ernk (ω)‖ ≤ C(ω)‖Yτn

k+1(ω)− Yτn

k(ω)‖2 (63)

and ∑k/∈Γn

ω(A)∪Γnω(Λ)∪γn

ωC(ω)‖Yτn

k+1(ω)− Yτn

k(ω)‖2

≤ 2C(ω)∑

k/∈Γnω(A)∪Γn

ω(Λ)∪γnω‖Zτn

k+1(ω)− Zτn

k(ω)‖2

+2C(ω)∑

k/∈Γnω(A)∪Γn

ω(Λ)∪γnω‖

∫ τnk+1

τnk

g(s, ω)dh(s, ω)‖2

≤ 2C(ω) supk/∈Γnω(A)∪Γn

ω(Λ)∪γnω‖

∫ τnk+1

τnk

∫A

f(s, x, ω)ν(dsdx)‖

×∑2n−1

k=0 ‖∫ τn

k+1τn

k

∫A

f(s, x, ω)ν(dsdx)‖

+2C(ω) supk/∈Γnω(A)∪Γn

ω(Λ)∪γnω‖

∫ τnk+1

τnk

g(s, ω)dh(s, ω)‖

×∑

k/∈Γnω(A)∪Γn

ω(Λ)∪γnω‖

∫ τnk+1

τnk

g(s, ω)dh(s, ω)‖ (64)

From i), or ii), or iii), and the assumption that A ∈ F(E \ 0), it follows thatf(s, x, ω) is Bochner integrable P -a.e. w.r.t. ν, it follows

lim supn→∞ supk/∈Γnω(A)∪Γn

ω(Λ)∪γnω‖

∫ τnk+1

τnk

∫A

f(s, x, ω)ν(dsdx)‖

≤ lim supn→∞ supk/∈Γnω(A)∪Γn

ω(Λ)∪γnω

∫ τnk+1

τnk

∫A‖f(s, x, ω)‖ν(dsdx) = 0

P − a.s. (65)

2n−1∑k=0

‖∫ τn

k+1

τnk

∫A

f(s, x, ω)ν(dsdx)‖ ≤∫ t

τ

∫A

‖f(s, x, ω)‖ν(dsdx) (66)

Moreover we have that

supk/∈Γnω(A)∪Γn

ω(Λ)∪γnω‖

∫ τnk+1

τnk

g(s, ω)dh(s, ω)‖

×∑

k/∈Γnω(A)∪Γn

ω(Λ)∪γnω‖

∫ τnk+1

τnk

g(s, ω)dh(s, ω)‖

≤ supk/∈Γnω(A)∪Γn

ω(Λ)∪γnω‖

∫ τnk+1

τnk

g(s, ω)dh(s, ω)‖∫ t

τ‖g(s, ω)‖dh(s, ω)|

P − a.s. (67)

and

lim supn→∞ supk/∈Γnω(A)∪Γn

ω(Λ)∪γnω‖

∫ τnk+1

τnk

∫A

g(s, ω)dh(s, ω)‖

≤ lim supn→∞ supk/∈Γnω(A)∪Γn

ω(Λ)∪γnω

∫ τnk+1

τnk

∫A‖g(s, ω)‖d|h(s, ω)| = 0

P − a.s. (68)

21

It follows thatlim

n→∞

∑k/∈Γn

ω(A)∪Γnω(Λ)∪γn

ω

‖Yτnk+1

(ω)− Yτnk(ω)‖2 = 0 P − a.s. (69)

and hence from (63) it follows that (62) holds.We have

limn→∞∑

k/∈Γnω(A)∪Γn

ω(Λ)∪γnωH′(Yτn

k(ω))(Zτn

k+1(ω))− Zτn

k(ω))

= − limn→∞∑

k/∈Γnω(A)∪Γn

ω(Λ)∪γnωH′(Yτn

k(ω))

∫ τnk+1

τnk

∫A

f(s, x, ω)ν(dsdx)

= −∫ t

τ

∫AH′(Ys−(ω))f(s, x, ω)ν(dsdx) P − a.e. , (70)

where the last equality in (70) follows from the following estimates and thehypothesis A that ν(dsdx) = α(ds)β(dx), with α(ds) ≺ ds:

lim supn→∞∑

k/∈Γnω(A)∪Γn

ω(Λ)∪γnω‖

∫ τnk+1

τnk

∫AH′(Yτn

k(ω))f(s, x, ω)ν(dsdx)

−∫ τn

k+1τn

k

∫AH′(Ys−(ω))f(s, x, ω)ν(dsdx)‖

≤ C(ω)∑

k/∈Γnω(A)∪Γn

ω(Λ)∪γnω

∫ τnk+1

τnk

∫A‖Yτn

k(ω)− Ys−(ω)‖‖f(s, x, ω)‖ν(dsdx)

≤ C(ω)supk/∈Γnω(A)∪Γn

ω(Λ)∪γnω

∫ τnk+1

τnk

∫A‖f(s, x, ω)‖ν(dsdx)

+ supk/∈Γnω(A)∪Γn

ω(Λ)∪γnω

∫ τnk+1

τnk

‖g(s, ω)‖d|h(s, ω)|

×∑2n−1

k=0

∫ τnk+1

τnk

∫A‖f(s, x, ω)‖ν(dsdx) = 0 P − a.s. (71)

where the first inequality and convergence to zero in (71) holds by the samearguments as in the proof of (62).(60) is proven, once we have shown that

∑2n−1

k=1 H′(Yτnk(ω))(

∫ τnk+1

τnk

g(s, ω)dh(s, ω))

−∑

k∈Γnω(A)∪Γn

ω(Λ)∪γnωH′(Yτn

k(ω))(

∫ τnk+1

τnk

g(s, ω)dh(s, ω))

=∫ t

τH′(Ys−)g(s−, ω)dh(s, ω)

−∑

τ<s≤tH′(Ys−(ω))g(s−, ω)∆h(s, ω) P − a.s. (72)

From the continuity of H′ it follows that

limn→∞∑

k∈Γnω(A)∪Γn

ω(Λ)∪γnωH′(Yτn

k(ω))(

∫ τnk+1

τnk

g(s, ω)dh(s, ω))

=∑

τ<s≤tH′(Ys−(ω)g(s−, ω)∆h(s, ω) P − a.s. , (73)

22

while from Theorem 21 Chapt. II, Paragraph 5 of [30] it follows

limn→∞∑2n−1

k=1 H′(Yτnk(ω))(

∫ τnk+1

τnk

g(s, ω)dh(s, ω))

=∫ t

τH′(Ys−)g(s−, ω)dh(s, ω) , (74)

where convergence holds in probability (see [30] for a stronger statement). Itfollows that for some subsequence of nn∈IN , that for simplicity we still denotewith n, convergence in (74) and hence in (60) holds P -a.s.. It follows that(41) holds for every A ∈ F(E \ 0), i.e. such that 0 /∈ A.In order to prove that (41) holds for any A ∈ B(E \ 0), we first consider (41)with a set A ∩ Bc(0, ε) instead of A, and then take the limit ε → 0. In fact,from Theorem 3.22 - 3.26, it follows that for some sequence εnn∈IN , such thatεn → 0 when n →∞, the following limit holds.

limn→∞∫ t

τ

∫A∩Bc(0,εn)

(H(Ys−(ω) + f(s, x, ω))−H(Ys−(ω))q(dsdx)(ω)

→∫ t

τ

∫A

(H(Ys−(ω) + f(s, x, ω))−H(Ys−(ω))q(dsdx)(ω)when n →∞ P − a.e.

and from the definition of Bochner integral it follows that

limε→0

∫ t

τ

∫A∩Bc(0,ε)

(H(Ys−(ω) + f(s, x, ω))−H(Ys−(ω)−H′(Ys−(ω))f(s, x, ω))ν(dsdx)

=∫ t

τ

∫A

(H(Ys−(ω) + f(s, x, ω))−H(Ys−(ω)−H′(Ys−(ω))f(s, x, ω))ν(dsdx)P − a.e.

so that (41) holds P -a.e.

In the next theorem we leave out the hypothesis that f(·, x, ω) is left continuous.

Theorem 4.6 Suppose that the same hypothesis as in Theorem 4.1 hold, thenH(Ys−(ω)+ f(s, x, ω))−H(Ys−(ω)) is strong -1, resp. strong -2, - integrable on(0, T ]×A if condition i), resp. ii) or iii), is satisfied. (41) holds P -a.s..

Remark 4.7 Zt(ω), defined in (34), is in Theorem 4.6, according to Theorems3.12 -3.16, a strong -p -integral, with p = 1, if condition i) is satisfied, resp.p = 2, if condition ii) or iii), is satisfied. Moreover E[‖Zs‖] is uniformly boundedon [0, T ].

Proof of Theorem 4.6:

23

ThatH(Ys−(ω)+f(s, x, ω))−H(Ys−(ω)) is strong -1, resp. strong -2, - integrableon (0, T ] × A if condition i), resp. ii) or iii), is satisfied, follows from Theorem3.12 -3.16 and Theorem 4.1.Let us prove that (41) holds P -a.s.. Let fnn∈IN ∈ Σ(E/F ). Suppose thatfnn∈IN is Lp -approximating f on (0, T ] × A × Ω w.r.t. ν ⊗ P with p = 1(resp. p = 2) if condition i) (resp. ii) or iii)) is satisfied. Let

Znt (ω) :=

∫ t

0

∫A

fn(s, x, ω)q(dsdx)(ω) (75)

Y nt (ω) :=

∫ t

0

∫A

fn(s, x, ω)q(dsdx)(ω)+∫ t

0

g(s, ω)dh(s, ω)+∫ t

0

∫Λ

k(s, x, ω)N(dsdx)(ω) .

(76)then from Theorem 4.4 it follows that Y n

t (ω) satisfies (41) P- a.s. for all 0 ≤τ ≤ t ≤ T .We know from Theorem 3.12 - Theorem 3.16 that limp

n→∞ Znt (ω) = Zt(ω). It

follows that there is a subsequence, that for simplicity we denote again withZn

t (ω)n∈IN , such thatlim

n→∞Zn

t (ω) = Zt(ω) P − a.s. for all 0 ≤ τ ≤ t ≤ T , (77)

so thatlim

n→∞Y n

t (ω) = Yt(ω) P − a.s. for all 0 ≤ τ ≤ t ≤ T , (78)

and hencelim

n→∞H(Y n

t (ω))−H(Y nτ (ω)) = H(Yt(ω))−H(Yτ (ω)) P−a.s. for all 0 ≤ τ ≤ t ≤ T ,

(79)

limn→∞∫ t

τ

∫ΛH(Y n

s−(ω) + k(s, x, ω))−H(Y ns−(ω))N(dsdx)(ω)

=∫ t

τ

∫ΛH(Ys−(ω) + k(s, x, ω))−H(Ys−(ω))N(dsdx)(ω)

P − a.s. for all 0 ≤ τ ≤ t ≤ T , (80)

limn→∞∑

τ<s≤t H(Y ns−(ω) + g(s−, ω)∆h(s, ω))−H(Y n

s−(ω))−H′(Y ns−(ω)g(s−, ω)∆h(s, ω)

= H(Ys−(ω) + g(s−, ω)∆h(s, ω))−H(Ys−(ω))−H′(Ys−(ω)g(s−, ω)∆h(s, ω)P − a.s. for all 0 ≤ τ ≤ t ≤ T . (81)

We shall prove that there is a subsequence such that the following properties(82), (83) hold.

limn→∞

∫ t

τ

H′(Y ns−)g(s−, ω)dh(s, ω) =

∫ t

τ

H′(Ys−)g(s−, ω)dh(s, ω) P − a.s. ,

(82)

24

limn→∞∫ t

τ

∫AH(Y n

s−(ω) + fn(s, x, ω))−H(Y ns−(ω))−H′(Y n

s−(ω))fn(s, x, ω) ν(dsdx)

=∫ t

τ

∫AH(Ys−(ω) + f(s, x, ω))−H(Ys−(ω))−H′(Ys−(ω))f(s, x, ω) ν(dsdx)

P − a.s. . (83)

Moreover, we shall prove that for some subsequence

limpn→∞

∫ t

τ

∫AH(Y n

s−(ω) + fn(s, x, ω))−H(Y ns−(ω)) q(dsdx)(ω)

=∫ t

τ

∫AH(Ys−(ω) + f(s, x, ω))−H(Ys−(ω)) q(dsdx)(ω) . (84)

It then follows that there is a subsequence, so that the limit in (84) holds alsoP -a.s..From (79)- (84) it then follows that (41) holds P -a.s..Proof of (82):Let us first prove the followingstatement: Y n

s (ω) converges for some subsequence, that for simplicity we denotestill with Y n

s (ω), ds⊗P - a.s. to Ys(ω) on [0, T ]×Ω (ds denoting the Lesbeguesmeasure on IR+).This is a consequence of Doob’s inequality applied to the sub -martingales‖Zn

t (ω) − Zt(ω)‖. (That ‖Znt (ω) − Zt(ω)‖ is a sub -martingale follows from

Proposition 3.11 and Lemma 8.11, Chapt. II, Part I, [21].) In fact,

supt∈[0,T ]

‖Y nt (ω)− Yt(ω)‖ = sup

t∈[0,T ]

‖Znt (ω)− Zt(ω)‖

and if i) holds, then

P (supt∈[0,T ] ‖Znt (ω)− Zt(ω)‖ ≥ ε) ≤ 1

ε supt∈[0,T ] E[‖Znt (ω)− Zt(ω)‖]

≤ 1ε

∫ T

0

∫A

E[‖fn(s, x, ω)− f(s, x, ω)‖] , (85)

while if ii) or iii) holds, then

E[sups∈[0,T ] ‖Zns (ω)− Zs(ω)‖2] ≤ sups∈[0,T ] E[‖Zn

s (ω)− Zs(ω)‖2] ≤∫ T

0

∫A

E[‖fn(s, x)− f(s, x)‖2ν(dsdx) → 0 when n →∞ , (86)

The statement is then a consequence of Lesbegue’s dominated convergence the-orem. (82) follows then from inequality (44), which implies the existence of aconstant R(ω) depending on ω, such that

‖∫ t

τH′(Y n

s−)g(s−, ω)dh(s, ω)−∫ t

τH′(Ys−)g(s−, ω)dh(s, ω)‖

≤ R(ω)∫ t

τ‖Y n

s−(ω)− Ys−(ω)‖ sups∈[0,T ] |g(s, ω)|d|h(s, ω)| (87)

25

Proof of (83):Suppose condition i) is satisfied. Let C := supz∈F ‖H′(z)‖. Then using (42),we obtain:∫ t

τ

∫A‖H(Y n

s−(ω) + fn(s, x, ω))−H(Y ns−(ω))−H′(Y n

s−(ω))fn(s, x, ω)−H(Ys−(ω) + f(s, x, ω))−H(Ys−(ω))−H′(Ys−(ω))f(s, x, ω)‖ ν(dsdx)

≤ 4C∫ t

τ

∫A‖fn(s, x, ω)− f(s, x, ω)‖ (88)

Since

limn→∞

E[∫ t

0

∫A

‖fn(s, x, ω)− f(s, x, ω)‖] = 0

by hypothesis, it follows that there is some subsequence, that we denote forsimplicity again with fn, such that

∫ t

τ

∫A

‖fn(s, x, ω)− f(s, x, ω)‖ ν(dsdx) → 0 P − a.s. when n →∞ (89)

Suppose condition ii), or iii) is satisfied (i.e. p = 2). By hypothesis fnn∈IN

converges ν ⊗ P - a.s. to f on [0, T ] × A × Ω. Moreover Y ns (ω) converges for

some subsequence ν⊗P - a.s. to Ys(ω). This is a consequence of the hypothesisA that ν(dsdx) = α(ds)β(dx) with α(ds) ≺ ds and Doob’s inequality (86). Itfollows that there is a subsequence s.th. the integrand in (83) converges ν⊗P -a.s. to zero when n →∞. Moreover using (46) and Remark 4.5 it follows thatthere is a constant R(ω) depending on ω ∈ Ω, s.th. for some subsequence∫ t

τ

∫A‖H(Y n

s−(ω) + fn(s, x, ω))−H(Y ns−(ω))−H′(Y n

s−(ω))fn(s, x, ω)−H(Ys−(ω) + f(s, x, ω))−H(Ys−(ω))−H′(Ys−(ω))f(s, x, ω)‖ ν(dsdx)

≤ 4R(ω)∫ t

τ

∫A(‖fn(s, x, ω)‖2 + ‖f(s, x, ω)‖2)ν(dsdx)

≤ 16R(ω)∫ t

τ

∫A‖fn(s, x, ω)− f(s, x, ω)‖2ν(dsdx) + 16R(ω)

∫ t

τ

∫A‖f(s, x, ω)‖2ν(dsdx)

→ 16R(ω)∫ t

τ

∫A‖f(s, x, ω)‖2ν(dsdx) P − a.s. when n →∞ (90)

where convergence in (90) holds for some subsequence, by a similar argumentas in (89). (83) is then a consequence of the Lebegues dominated convergencetheorem.Proof of (84):

Let p = 1 if condition i) is satisfied, or p = 2 if condition ii) or iii) is satisfied.From the Theorems 3.12- 3.16 and inequality (45) it follows

26

E[‖∫ t

τ

∫AH(Y n

s−(ω) + fn(s, x, ω))−H(Y ns−(ω)) q(dsdx)(ω)−∫ t

τ

∫AH(Ys−(ω) + f(s, x, ω))−H(Ys−(ω)) q(dsdx)(ω)‖p] ≤

2pKp

∫ t

τ

∫A‖H(Y n

s−(ω) + fn(s, x, ω))−H(Y ns−(ω))

−H(Ys−(ω) + f(s, x, ω))−H(Ys−(ω))‖p ν(dsdx)

≤ 4pCpKp

∫ t

τ

∫A

E[‖fn(s, x, ω)‖p + ‖f(s, x, ω)‖pν(dsdx)

≤ 8pCpKp

∫ t

τ

∫A

E[‖fn(s, x)− f(s, x)‖p]ν(dsdx) +

8pCpK2p

∫ t

τ

∫A

E[‖f(s, x)‖p]ν(dsdx)

→ 8pCpKp

∫ t

τ

∫A

E[‖f(s, x)‖p]ν(dsdx) when n →∞ (91)

where for p = 1, Kp = 1, while in case p = 2, Kp is the constant in the Definition3.17 of M -type 2, if condition ii) holds, resp. Kp is the twice the constant inthe Definition 3.19 of type 2 Banach space, if condition iii) holds. That (84)holds for some subsequence follows from the Lebegues dominated convergencetheorem and the fact that the integrand in the l.h.s. of (91) converges for somesubsequence ν ⊗ P - a.s. to zero, on [0, T ]×A× Ω.

5 The Ito formula for time dependent Banachvalued functions

We shall consider in this article also the case where H depends (continuously)on the time variable, too, i.e.

H : IR+ × F → G (92)(s, y) → G

We shall prove the Ito formula (3).

Theorem 5.1 Suppose that hypothesis B (given in Theorem 4.1) is satisfied.Suppose that the Frechet derivatives ∂sH(s, y) and ∂yH(s, y) exists and are uni-formly bounded on [τ, t]×F , and all the second Frechet derivatives ∂s∂sH(s, y),∂s,∂yH(s, y), ∂y∂sH(s, y) and ∂y∂yH(s, y) exists and are uniformly bounded on[τ, t]×B(0, R), for all R ≥ 0. Then1) H(s, Ys−(ω) + f(s, x, ω))−H(s, Ys−(ω))− ∂yH(s, Ys−(ω))f(s, x, ω) is P -a.s. Bochner integrable w.r.t ν on (0, T ]×A, ∀A ∈ B(E \ 0).2) H(s, Ys−(ω)+f(s, x, ω))−H(s, Ys−(ω)) ∈ MT,1

ν (E/G), resp. ∈ MT,2ν (E/G),

if i), resp. ii) or iii), is satisfied.

3)∫ T

0‖∂yH(s, Ys−)g(s−, ω)‖d|h(s, ω)| < ∞ P -a.s.

27

4)∑

0<s≤t ‖H(s, Ys−(ω) + g(s−, ω)∆h(s, ω))−H(s, Ys−(ω))‖ < ∞ P -a.s.5)

∑0<s≤t ‖∂yH(s, Ys−(ω))g(s−, ω)∆h(s, ω)‖ < ∞ P -a.s.

6)∫ T

0

∫Λ‖H(s, Ys−(ω) + k(s, x, ω)) −H(s, Ys−(ω))‖N(dsdx)(ω) < ∞ P

-a.s.7) ∂sH(s, Ys−(ω)) is P -a.s. Bochner integrable w.r.t. the Lebegues measureds in [τ, t]

Moreover (3) holds P -a.s..

Remark 5.2 The stochastic integral w.r.t. q(dsdx)(ω) in (3) is a strong -p-integral, with p = 1 if i) holds, resp. p = 2 if ii) or iii) holds. If f(·, x, ω) is leftcontinuous on (0, T ], for all x ∈ A and P -a.e. ω ∈ Ω fixed, then it is a simple-p- integral, too.

Proof of Theorem 5.1: The properties 1) -6) are proven like in Theorem 4.1.Property 7) is an obvious consequence of the fact that ∂yH(s, y) is uniformlybounded.To prove that (3) holds P -a.s. we proceed in a similar way to the proof ofthe Theorems 4.4 and 4.6. We first make the assumption that f(·, x, ω) is leftcontinuous on (0, T ], for all x ∈ A and P -a.e. ω ∈ Ω fixed, and that 0 /∈ A, i.e.A ∈ F(E \ 0). Similar to the proof of Theorem 4.4 we obtain than that forall ω ∈ Ω,

H(t, Yt(ω))−H(τ, Yτ (ω))

=∑2n−1

k=0 H(τnk+1, Yτn

k+1(ω))−H(τn

k , Yτnk(ω))

=∑

k∈Γnω(A)∪Γn

ω(Λ)∪γnωH(τn

k+1, Yτnk+1

(ω))−H(τnk , Yτn

k(ω))

+∑

k/∈Γnω(A)∪Γn

ω(Λ)∪γnωH(τn

k+1, Yτnk+1

(ω))−H(τnk , Yτn

k(ω)) (93)

where Γnω(A),Γn

ω(Λ) and γnω are defined like in (54) -(57).

Similar to (59) it can be checked that

limn→∞∑

k∈Γnω(A)∪Γn

ω(Λ)∪γnωH(τn

k+1, Yτnk+1

(ω))−H(τnk , Yτn

k(ω))

=∫ t

τ

∫AH(s, Ys−(ω) + f(s, x, ω))−H(s, Ys−(ω)) q(dsdx)(ω)

+∫ t

τ

∫ΛH(s, Ys−(ω) + k(s, x, ω))−H(s, Ys−(ω))N(dsdx)(ω)

+∑

τ<s≤tH(s, Ys−(ω) + g(s−, ω)∆h(s−, ω))−H(s, Ys−(ω))

+∫ t

τ

∫AH(s, Ys−(ω) + f(s, x, ω))−H(s, Ys−(ω)) ν(dsdx) (94)

P − a.s. .

(3) follows once shown that for some subsequence of nn∈IN , that for simplicitywe still denote with n, the following convergence holds for n →∞

28

limn→∞∑

k/∈Γnω(A)∪Γn

ω(Λ)∪γnωH(τn

k+1, Yτnk+1

(ω))−H(τnk , Yτn

k(ω))

=∫ t

τ∂sH(s, Ys(ω))ds−

∫ t

τ

∫A

∂yH(s, Ys−(ω))f(s, x, ω)ν(dsdx)

+∫ t

τ∂yH(s, Ys−)g(s−, ω)dh(s, ω)

−∑

τ<s≤t∂yH(s, Ys−(ω))g(s−, ω)∆h(s, ω) P − a.s. (95)

Proof of (95):We make a Taylor expansion of the function H(s, y):

∑k/∈Γn

ω(A)∪Γnω(Λ)∪γn

ω(H(τn

k+1, Yτnk+1

(ω))−H(τnk , Yτn

k(ω)))

=∑

k/∈Γnω(A)∪Γn

ω(Λ)∪γnω

∂yH(τnk , Yτn

k(ω))(Yτn

k+1(ω)− Yτn

k(ω))

+∑

k/∈Γnω(A)∪Γn

ω(Λ)∪γnω

∂sH(τnk , Yτn

k)(τn

k+1 − τnk )

+∑

k/∈Γnω(A)∪Γn

ω(Λ)∪γnω

ernk (ω) (96)

We remark that

limn→∞

∑k/∈Γn

ω(A)∪Γnω(Λ)∪γn

ω

∂sH(τnk , Yτn

k)(τn

k+1−τnk ) =

∫ t

0

∂sH(s, Ys−(ω))ds P −a.s.

(97)while ∑

k/∈Γnω(A)∪Γn

ω(Λ)∪γnω

∂yH(τnk , Yτn

k(ω))(Yτn

k+1(ω)− Yτn

k(ω))

=∑

k/∈Γnω(A)∪Γn

ω(Λ)∪γnω

∂yH(τnk , Yτn

k(ω))(Zτn

k+1(ω)− Zτn

k(ω))

+∑2n−1

k=1 ∂yH(τnk , Yτn

k(ω))(

∫ τnk+1

τnk

g(s, ω)dh(s, ω))

−∑

k∈Γnω(A)∪Γn

ω(Λ)∪γnω

∂yH(τnk , Yτn

k(ω))(

∫ τnk+1

τnk

g(s, ω)dh(s, ω)) (98)

That

limn→∞∑

k/∈Γnω(A)∪Γn

ω(Λ)∪γnω

∂yH(τnk , Yτn

k(ω))(Zτn

k+1(ω))− Zτn

k(ω))

= −∫ t

τ

∫A

∂yH(s, Ys−(ω))f(s, x, ω)ν(dsdx) P − a.e. , (99)

is proven in a similar way as in (70). Moreover in a similar way as in the proofof (72) it can be shown that

∑2n−1

k=1 ∂yH(τnk , Yτn

k(ω))(

∫ τnk+1

τnk

g(s, ω)dh(s, ω))

−∑

k∈Γnω(A)∪Γn

ω(Λ)∪γnω

∂yH(τnk , Yτn

k(ω))(

∫ τnk+1

τnk

g(s, ω)dh(s, ω))

=∫ t

τ∂yH(s, Ys−)g(s−, ω)dh(s, ω)

−∑

τ<s≤t∂yH(s, Ys−(ω))g(s−, ω)∆h(s, ω) P − a.s. (100)

29

(95) is proven, once shown that

limn→∞

∑k/∈Γn

ω

ernk (ω) = 0 P − a.e. (101)

The proof of (101) is similar to the proof of (62), however, instead of gettinginequality (63), we get

‖ernk (ω)‖ ≤ C(ω)(‖Yτn

k+1(ω)− Yτn

k(ω)‖2 + |τn

k+1 − τnk |2 , (102)

which is obtained in a similar way as (62), however by the following estimates

‖H(s, y)−H(s0, y0)− ∂sH(s0, y0)(s− s0)− ∂yH(s0, y0)(y − y0)‖ ≤‖H(s, y)−H(s, y0)− ∂yH(s, y0)(y − y0)‖+‖H(s, y0)−H(s0, y0)− ∂sH(s0, y0)(s− s0)‖+

‖(∂yH(s, y0)− ∂yH(s0, y0))(y − y0)‖ ≤sup0<θ≤1 ‖∂sH(s0 + θ(s− s0), y0)− ∂sH(s0, y0)‖|(s− s0)|+sup0<θ≤1 ‖∂yH(s, y0 + θ(y − y0))− ∂yH(s, y0)‖‖(y − y0)‖+

‖∂yH(s, y0)(y − y0)− ∂yH(s0, y0)(y − y0)‖ ≤sup0<θ≤1 ‖∂s∂sH(s0 + θ(s− s0), y0)‖|s− s0|2 +

sup0<θ≤1 ‖∂y∂yH(s, y0 + θ(y − y0), y0)‖‖y − y0‖2 +sups∈[τ,t] ‖∂s∂yH(s, y0)‖|s− s0|‖y − y0| (103)

(101) follows then from (69),(102) and2n−1∑k=0

|τnk+1 − τn

k |2 ≤(t− τ)2

2n(104)

The rest of the proof of Theorem 5.1 is done by leaving out the hypothesisA ∈ F(E \0) and the hypothesis that f is left continuous in the time variable.This is done similar to the proof of the Theorems 4.1 and Theorem 4.4.

6 The Ito formula for the strong integrals oftype p

We now prove the Ito formula for the strong integrals of type p on separableBanach spaces. Instead of assuming that i) or ii) or iii) is satisfied, we assumethat either i’) in Proposition 3.32, or ii’) or iii’) in Proposition 3.33 is satisfied:Zt(ω) in (1) is then the strong integral of type p, with p = 1 , if i’) holds, andp = 2, if ii’) or iii’) holds. We analyze the Ito -formula for the F -valued randomprocess (Yt)t≥0 in (35). We assume again that the assumptions (36) - (40) hold,and prove that the Ito formula (3) holds. We prove in fact the following twotheorems:

30

Theorem 6.1 We make the followinghypothesis C: (At least) one of the hypothesis i’) -iii’) is satisfied. G is a sepa-rable Banach space. If ii’) or iii’) is satisfied, we make the additional hypothesisthat G is a separable Banach space of M -type 2Let

H : IR+ × F → G (105)(s, y) → G

Suppose that the Frechet derivatives ∂sH(s, y) and ∂yH(s, y) exists and are uni-formly bounded on [τ, t]×F , and all the second Frechet derivatives ∂s∂sH(s, y),∂s,∂yH(s, y), ∂y∂sH(s, y) and ∂y∂yH(s, y) exists and are uniformly bounded on[τ, t]×B(0, R), for all R ≥ 0.Then the properties 1),and 3)-7) in Theorem 5.1 hold. Moreover2’) H(s, Ys−(ω)+f(s, x, ω))−H(s, Ys−(ω)) ∈ NT,1

ν (E/G), resp. ∈ NT,2ν (E/G),

if i’), resp. ii’) or iii’), is satisfied.

Proof of Theorem 6.1: Suppose i’), resp. ii’) or iii’) is satisfied. 1) and 3)-7) are proven like in Theorem 5.1. 2’) is proven in a similar way as in Theorem5.1, by using (43).

Theorem 6.2 Suppose that the same hypothesis as in Theorem 6.1 hold, thenH(s, Ys−(ω) + f(s, x, ω))−H(s, Ys−(ω)) is strong integrable on (0, T ]×A, withstrong integral of type 1, resp. strong integral of type 2, if condition i’), resp.ii’) or iii’), is satisfied. (3) holds P -a.s..

Proof of Theorem 6.2: The statement about the strong integrability followsfrom Proposition 3.32, Proposition 3.33 and Theorem 6.1. Let p = 1, if i’) holds,and p = 2, if ii’) or iii’) holds. Let fn(s, x, ω) := f(s, x, ω)1∫ s

0

∫A‖f‖pdν≤n. From

Theorem 4.6 it follows that the Ito formula (3) holds for Y nt (ω) defined like in

(76), where for this case Znt denotes the strong -p- integral of fn, with p = 1,

resp. p = 2. As ∫ T

0

∫A

‖fn‖p dν ≤∫ T

0

∫A

‖f‖p dν P − a.s.

and fn satisfies property Q in Theorem 3.34, it follows from Theorem 3.35 that(77)- (78) hold for some subsequence. Moreover, it followslim

n→∞H(t, Y n

t (ω))−H(τ, Y nτ (ω)) = H(t, Yt(ω))−H(τ, Yτ (ω)) P−a.s. for all 0 ≤ τ ≤ t ≤ T ,

(106)

limn→∞∫ t

τ

∫ΛH(s, Y n

s−(ω) + k(s, x, ω))−H(s, Y ns−(ω))N(dsdx)(ω)

=∫ t

τ

∫ΛH(s, Ys−(ω) + k(s, x, ω))−H(s, Ys−(ω))N(dsdx)(ω)

P − a.s. for all 0 ≤ τ ≤ t ≤ T , (107)

31

limn→∞∑

τ<s≤t H(s, Y ns−(ω) + g(s−, ω)∆h(s, ω))−H(s, Y n

s−(ω))− ∂yH(s, Y ns−(ω)g(s−, ω)∆h(s, ω)

=∑

τ<s≤t H(s, Ys−(ω) + g(s−, ω)∆h(s, ω))−H(s, Ys−(ω))− ∂yH(s, Ys−(ω)g(s−, ω)∆h(s, ω)P − a.s. for all 0 ≤ τ ≤ t ≤ T . (108)

In a similar way, as in the proof of Theorem 4.6, where it has been shownthat (82)-(83) hold for some subsequence, it can be proven that the properties(109)-(110) below hold. We leave the proof as an exercise.

limn→∞

∫ t

τ

∂yH(s, Y ns−(ω))g(s−, ω)dh(s, ω) =

∫ t

τ

∂yH(s, Ys−(ω))g(s−, ω)dh(s, ω) P−a.s. ,

(109)

limn→∞∫ t

τ

∫AH(s, Y n

s−(ω) + fn(s, x, ω))−H(s, Y ns−(ω))− ∂yH(s, Y n

s−(ω))fn(s, x, ω) ν(dsdx)

=∫ t

τ

∫AH(s, Ys−(ω) + f(s, x, ω))−H(s, Ys−(ω))− ∂yH(s, Ys−(ω))f(s, x, ω) ν(dsdx)

P − a.s. . (110)

Moreover the following property holds

limn→∞

∫ t

τ

∂sH(s, Y ns−(ω))ds =

∫ t

τ

∂sH(s, Ys−(ω))ds P − a.s. . (111)

The whole statement in Theorem 6.2 is then proven, once shown that

limn→∞∫ t

τ

∫AH(s, Y n

s−(ω) + fn(s, x, ω))−H(s, Y ns−(ω)) q(dsdx)(ω)

=∫ t

τ

∫AH(s, Ys−(ω) + f(s, x, ω))−H(s, Ys−(ω)) q(dsdx)(ω)

P − a.s. . (112)

Proof of (112):As

∫ T

0

∫A

‖H(s, Y ns−(ω)+fn(s, x, ω))−H(s, Y n

s−(ω))‖p ν(dsdx) ≤ C

∫ T

0

∫A

‖f(s, x, ω)‖pν(dsdx)

(113)with C := supy∈F ‖∂yH(s, y)‖, and as H(s, Y n

s−(ω) + fn(s, x, ω))−H(s, Y ns−(ω))

satisfies property Q in Theorem 3.34, it follows that (112) holds.

Acknowledgements We thank Luciano Tubaro for many useful discussionsrelated to this work and Sergio Albeverio for useful comments. The supportand hospitality of the ”Sonderforschungsbereich” SFB 611 in Bonn, as well asthe Mathematics Department of the University of Trento, is gratefully acknowl-edged.

32

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36

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167. Schätzle, Reiner: Lower Semicontinuity of the Willmore Functional for Currents

168. Ebmeyer, Carsten; Urbano, José Miguel: The Smoothing Property for a Class of DoublyNonlinear Parabolic Equations; erscheint in: Trans. Am. Math. Soc.

169. Albeverio, Sergio; Lakaev, Saidakhmat N.; Djumanova, Ramiza Kh.: On the Essential andDiscrete Spectrum of a Model Operator

170. Albeverio, Sergio; Lakaev, Saidakhmat N.; Abdullaev, Janikul I.: On the Spectral Propertiesof Two-Particle Discrete Schrödinger Operators

171. Albeverio, Sergio; Lakaev, Saidakhmat N.; Makarov, K.A.; Muminov, Z.I.: Low-EnergyEffects for the Two-Particle Operators on a Lattice

172. Castaño, Daniel; Kunoth, Angela: Robust Progression of Scattered Data with AdaptiveSpline-Wavelets

173. Djah, Sidi Hamidou; Gottschalk, Hanno; Ouerdiane, Habib: Feynman Graphs for non-Gaussian Measures

174. Djah, Sidi Hamidou; Gottschalk, Hanno; Ouerdiane, Habib: Feynman Graph Representationof the Perturbation Series for General Functional Measures

175. not published

176. Albeverio, Sergio; Shelkovich, Vladimir M.: Delta-Shock Waves in Multidimensional Non-Conservative System of Zero-Pressure Gas Dynamics

177. Albeverio, Sergio; Kuzhel, Sergej: η-Hermitian Operators and Previously UnnoticedSymmetries in the Theory of Singular Perturbations

178. Albeverio, Sergio; Alimov, Shavkat: On Some Integral Equations in Hilbert Space with anApplication to the Theory of Elasticity; eingereicht bei: Oper. Th. and Int. Eqts.

179. Albeverio, Sergio; Galperin, Gregory; Nizhnik, Irena L.; Nizhnik, Leonid P.: GeneralizedBilliards Inside an Infinite Strip with Periodic Laws of Reflection Along the Strip’sBoundaries; eingereicht bei: Regular and Chaotic Dynamics

180. Albeverio, Sergio; Torbin, Grygoriy: Fractal Properties of Singular Probability Distributionswith Independent Q*-Digits; eingereicht bei: Bull. Sci. Math.

181. Melikyan, Arik; Botkin, Nikolai; Turova, Varvara: Propagation of Disturbances in Inhomo-geneous Anisotropic Media

182. Albeverio, Sergio; Bodnarchuk, Maksim; Koshmanenko, Volodymyr: Dynamics of DiscreteConflict Interactions between Non-Annihilating Opponents

183. Albeverio, Sergio; Daletskii, Alexei: L2-Betti Numbers of Infinite Configuration Spaces

184. Albeverio, Sergio; Daletskii, Alexei: Recent Developments on Harmonic Forms and L2-Betti

Numbers of Infinite Configuration Spaces with Poisson Measures

185. Hildebrandt, Stefan; von der Mosel, Heiko: On Lichtenstein’s Theorem About Globally Con-formal Mappings

186. Mandrekar, Vidyadhar; Rüdiger, Barbara: Existence and Uniqueness for Stochastic IntegralEquations Driven by non Gaussian Noise on Separable Banach Spaces

187. Rüdiger, Barbara; Ziglio, Giacomo: Ito Formula for Stochastic Integrals w.r.t. CompensatedPoisson Random Measures on Separable Banach Spaces