A note on stochastic Fubini’s theorem and stochastic convolution

# A note on stochastic Fubini’s theorem and stochastic convolution

## Abstract

We provide a version of the stochastic Fubini’s theorem which does not depend on the particular stochastic integrator chosen as far as the stochastic integration is built as a continuous linear operator from an space of Banach space-valued processes (the stochastically integrable processes) to an space of Banach space-valued paths (the integrated processes). Then, for integrators on a Hilbert space , we consider stochastic convolutions with respect to a strongly continuous map , not necessarily a semigroup. We prove existence of predictable versions of stochastic convolutions and we characterize the measurability needed by operator-valued processes in order to be convoluted with . Finally, when is a -semigroup and the stochastic integral provides continuous paths, we show existence of a continuous version of the convolution, by adapting the factorization method to the present setting.

Key words: stochastic Fubini’s theorem, stochastic convolution.

AMS 2010 subject classification: 28C05, 28C20, 60B99, 60G99, 60H05.

## 1 Introduction

In this note we prove a stochastic Fubini’s theorem and apply it to obtain existence of predictable/continuous versions of stochastic convolutions. We do not choose any particular stochastic integrator. We look at the stochastic integration simply as a linear and continuous operator from an space of Banach space-valued processes, the stochastically integrable processes, to another space, containing functions whose values are the paths of the stochastic integrals. The paths do not need to be continuous. Within this setting, the continuity assumption on plays the role of Itō’s isometry or of the Burkholder-Davis-Gundy inequality in the standard construction of stochastic integrals with respect to square integrable continuous martingales.

For such an operator , we prove the stochastic Fubini’s theorem (Theorem 2.3). The result can be applied e.g. to stochastic integration in infinite dimensional spaces with respect to -integrable martingales ([10, Ch. 8]) or more general martingale-valued measures (for the finite dimensional case, see e.g. [2, Ch. 4]), generalizing standard results as [4, Theorem 4.33], [8, Theorem 2.8], [10, Theorem 8.14].

Secondly, we particularize the study to the case in which is defined on a space of -valued processes, where are separable Hilbert spaces and is the vector space of Hilbert-Schmidt operators from into . Denote by , in this particular case. For a strongly continuous map and for a process such that the composition belongs to the domain of , we consider the convolution process

 (I(1[0,t)(⋅)R(t−⋅)Φ))tt∈[0,T]. (1.1)

By using the stochastic Fubini’s theorem, we show that (1.1) admits a jointly measurable version (Theorem 3.5). The joint measurability of the stochastic convolution is of interest e.g. when its paths must be integrated, as it happens in the factorization formula ([4, Theorem 5.10]). We also provide a characterisation of the measurability needed by functions in order that has the necessary measurability required by the operator (Theorem 3.10). This measurability result turns out to be useful e.g. in order to understand what are the most general measurability conditions for coefficients of stochastic differential equations in Hilbert spaces for which mild solutions are considered.

Finally, in case takes values in a space of processes with continuous paths and is a -semigroups, by adapting the factorization method to the present setting, we show that (1.1) admits a continuous version (Theorem 3.13).

## 2 Stochastic Fubini’s theorem

Throughout this section, and are positive finite measure spaces, is a measurable space, and is a kernel from to , i.e.

 ν1:D1×D2→R+

is such that

1. is -measurable, for all ;

2. is a positive measure, for all .

We assume that

 C\coloneqq∫D2ν1(D1,x)ν2(dx)<∞.

Let . On , we define the meaure by

 ν(A)\coloneqq∫D2(∫D11A(x1,x2)ν1(dx1,x2))ν2(dx2),∀A∈D1⊗D2.

Notice that is finite.

Let be a given sub--algebra of . When we consider measurability or integrability with respect to (resp. , , , ), we always mean it with respect to the space (resp. , , ). According to that, if we write, for example , for some Banach space , we mean , and similarly for other spaces of integrable functions on , , , .

Let be a given Banach space. For , we denote by the space of measurable functions such that

1. there exists such that and is separable;

2. the following integrability condition holds:

 |f|p,q\coloneqq⎛⎝∫D2(∫D1|f(x,y)|pEdν1(dx,y))q/pν2(dy)⎞⎠1/q<∞.

It is not difficult to see that is a Banach space, with the usual identification if and only if -a.e.. Indeed, if is Cauchy in , then it is Cauchy also in . Passing to a subsequence if necessary, we may assume that -a.e., for some . Now Fatou’s lemma gives and in .

Finally, we use the short notation for the space

 L1((D×G,D⊗G,ν⊗μ),E).

We will prove the stochastic Fubini’s theorem first for simple functions and then for the general case through approximation. We need the following preparatory lemma.

###### Lemma 2.1.

Let and . If , assume that

 C(p,q)\coloneqq(∫D2(ν1(D1,x))q(p−1)p(q−1)ν2(dx))q−1q<∞. (2.1)

If and , assume that

 C(p,1)\coloneqqsupx∈D2(ν1(D1,x))p−1p<∞. (2.2)

Define . Then there exist measurable functions

 ~f:(D×G,D⊗G)→E (2.3) ~fn:(D×G,D⊗G)→E, n∈N (2.4)

such that

 ~f(⋅,y)∈Lp,qD(E), ∀y∈G, (2.5) G→Lp,qD(E), y↦~f(⋅,y) is measurable (2.6) ~f(⋅,y)=f(y) in Lp,qD(E) μ-a.e.\ y∈G, (2.7)
 ~fn(⋅,y)∈Lp,qD(E), ∀y∈G, ∀n∈N, (2.8) G→Lp,qD(E), y↦~fn(⋅,y) is a simple function, ∀n∈N, (2.9) limn→∞∫G⎛⎝∫D2(∫D1|~fn((x1,x2),y)−~f((x1,x2),y)|pEν1(dx1,x2))q/pν2(dx2)⎞⎠1/qμ(dy)=0 (2.10)
 ~f(x,⋅)∈L1(G,E), ∀x∈D, (2.11) D→L1(G,E), x↦~f(x,⋅), % belongs to\ Lp,q(D,L1(G,E)) (2.12)
 ~fn(x,⋅)∈L1(G,E), ∀x∈D, ∀n∈N, (2.13) D→L1(G,E), x↦~fn(x,⋅), %belongstoLp,q(D,L1(G,E)) (2.14) limn→∞∫D2(∫D1∣∣~fn((x1,x2),⋅)−~f((x1,x2),⋅)∣∣pL1(G,E)ν1(dx1,x2))q/pν2(dx2)=0. (2.15)
###### Proof.

Since is Bochner integrable, without loss of generality we can assume that is separable. Then there exists a sequence of -valued simple functions such that

 limn→∞fn(y)=f(y) in\ Lp,qD(E), ∀y∈G, (2.16) limn→∞|fn−f|L1(G,Lp,qD(E))=0. (2.17)

Each can be written in the form

 fn(y)=M(n)∑i=11Ani(y)φn,i∀y∈G, (2.18)

where , , and is a fixed representant of its equivalence class in . For , define

 ~fn:(D×G,D⊗G)→E, (x,y)↦fn(y)(x).

By using (2.18), we have the measurability of (2.4), and (2.8), (2.9), (2.13), (2.14), are immediately verified.

We claim that the sequence is Cauchy in . Indeed, since , we have , for every . Moreover, by Hölder’s inequality,

 ∫D×G|~fn−~fm|Ed(ν⊗μ)==∫G(∫D2(∫D1|~fn((x1,x2),y)−~fm((x1,x2),y)|Eν1(dx1,x2))ν2(dx2))μ(dy)≤C(p,q)∫G⎛⎝∫D2(∫D1|~fn((x1,x2),y)−~fm((x1,x2),y)|pEν1(dx1,x2))q/pν2(dx2)⎞⎠1/qμ(dy)=C(p,q)|fn−fm|L1(G,Lp,qD(E)),

and the last member tends to as and tend to , by (2.17). Then there exists such that, after replacing by a subsequence if necessary,

 limn→∞~fn(x,y)=~f(x,y) ∀(x,y)∈(D×G)∖N (2.19) limn→∞~fn=~f in L1(D×G,E), (2.20)

where is a -null set. We redefine on by for . After such a redefinition, the partial results of the theorem till now proved still hold true.

By (2.19), since we can assume that each has separable range, we see that the range of is separable. By measurability of sections of real-valued measurable functions and by Pettis’s measurability theorem (use the fact that the range of is separable and then use Hahn-Banach theorem to extend continuous linear functionals on the space generated by the range of to the whole space ), we have that

 (D,D)→E, x↦~f(x,y′)   and   (G,G)→E, y↦~f(x′,y)

are measurable, for all and all . Since

 ∫Gliminfm→∞∣∣~f(⋅,y)−~fm(⋅,y)∣∣p,qμ(dy)≤limm→∞∫G⎛⎝∫D2(∫D1∣∣~f((x1,x2),y)−~fm((x1,x2),y)∣∣pEν1(dx1,x2))q/pν2(dx2)⎞⎠1/qμ(dy)≤limm→∞liminfn→∞|fn−fm|L1(G,Lp,qD(E))=0, (2.21)

we have

 liminfm→∞∣∣~f(⋅,y)−~fm(⋅,y)∣∣p,q=0μ-a.e.\ y∈G. (2.22)

By recalling that for all , (2.22) shows that the map

 D→E, x↦~f(x,y)

belongs to for all , where is a -null set. We redefine on by for . Again, we notice that the partial results of the theorem till now proved still hold true after the redefinition on . In addition,

 ∀y∈G, the map\ D→E, x↦~f(x,y), belongs to\ Lp,qD(E).

This provides (2.5). Moreover, since can be chosen such that (2.22) holds for all and since , is measurable, for all , also (2.6) is proved. From the last inequality of (2.21), (2.10) follows. From (2.16) and (2.22), (2.7) follows as well.

By Hölder’s inequality, we have . Then, after redefining on a set , where is a -null set, by for , we have

 ∀x∈D, the map\ G→E, y↦~f(x,y), belongs to L1(G,E).

This provides (2.11). By applying Minkowski’s inequality for integrals twice (see [7, p. 194, 6.19]), we have

 limn→∞⎛⎝∫D2(∫D1(∫G|~f((x1,x2),y)−~fn((x1,x2),y)|Eμ(dy))pν1(dx1,x2))q/pν2(dx2)⎞⎠1/q≤limn→∞⎛⎜⎝∫D2⎛⎝∫G(∫D1|~f((x1,x2),y)−~fn((x1,x2),y)|pEν1(dx1,x2))1/pμ(dy)⎞⎠qν2(dx2)⎞⎟⎠1/q≤limn→∞∫G⎛⎝∫D2(∫D1|~f((x1,x2),y)−~fn((x1,x2),y)|pEν1(dx1,x2))q/pν2(dx2)⎞⎠1/qμ(dy).

Since the latter member tends to because of the second inequality in (2.21), the estimate above provides (2.12) and (2.15), after redefining on a set , where is a suitably chosen -null set, by for .

Let and let be a short notation for the Borel -algebra on . We recall that, if is a topological space, then denotes the Borel -algebra of (1). Let be a complete filtered probability space. We endow the product space with the -algebra of predictables sets associated to the filtration and the measurable space with the product measure , where denotes the Lebesgue’s measure. We need to introduce some further notation.

• is a Banach space;

• is a closed subspace (with respect to the norm ) such that

 T×[0,T]→F, (x,t)↦x(t); (2.23)

is Borel measurable, when is endowed with the product -algebra (and not just with the Borel -algebra of the product topology!).

• is a given sub--algebra of such that, for all with , .

• is the vector space of measurable functions

 X:(ΩT,P′)→F

such that, for -a.e. , the path

 X(ω):[0,T]→F, t↦Xt(ω)

belongs to , and the -a.e. defined map

 (Ω,FT)→T, ω↦X(ω) (2.24)

is measurable, when is endowed with the Borel -algebra induced by the norm .

• For , denotes the space of (equivalence classes of) such that (2.24) has separable range and

Then is a Banach space.

###### Remark 2.2.

The space can be e.g. , because in such a case (2.23) is continuous, hence measurable. This permits also to consider as the space of left-limited right-continuous functions, because, if is real valued and continuous with support and if , then converges pointwise to everywhere on as , after extending by continuity beyond . We finally observe that (2.23) is measurable whenever is separable: this comes from a straightforward application of [1, Lemma 4.51].

We now provide the main result of this section.

###### Theorem 2.3 (Stochastic Fubini’s theorem).

Let , . Let

 L:Lp,qD(E)→LrP′(T)

be a linear and continuous operator. Then there exist measurable functions

 X1:(D×G,D⊗G)→E X2:(ΩT×G,(FT⊗BT)⊗G)→F

such that

 X1(x,⋅)∈L1(G,E), ∀x∈D, and X2((ω,t),⋅)∈L1(G,F), ∀(ω,t)∈ΩT D→L1(G,E), x↦X1(x,⋅), ∈Lp,qD(L1(G,E)) (ΩT,PT)→L1(G,F), (ω,t)↦X2((ω,t),⋅), is measurable X1(⋅,y)∈Lp,qD(E), ∀y∈G G→Lp,qD(E), y↦X1(⋅,y), ∈L1(G,Lp,qD(E)) X1(⋅,y)=g(y) in Lp,qD(E) for μ-a.% e.\ y∈G X2(⋅,y)∈LrP′(T), ∀y∈G X2(⋅,y)=Lg(y) in LrP′(T), μ-a.e.\ y∈G (2.25a)

and such that

 for P-a.e.\ ω∈Ω,\ (LY)(ω,t)=∫GX2((ω,t),y)μ(dy), ∀t∈[0,T], (2.26)

where

 Y(x)\coloneqq∫GX1(x,y)μ(dy), ∀x∈D.
###### Proof.

By Lemma 2.1 , there exist measurable functions

 ~f:(D×G,D⊗G)→E, ~fn:(D×G,D⊗G)→E,n∈N

satisfying (2.11)–(2.15). For , has the form

 ~fn(x,y)=M(n)∑i=11Ani(y)φn,i(x)∀x∈D, ∀y∈G,

where is a fixed representant of its class in . For all , the function defined by

 ~f(μ)n:D→E x↦∫G~fn(x,y)μ(dy)=M(n)∑i=1φn,i(x)μ(Ani)

belongs to . Then, if we define

 ~f(μ):D→E, x↦∫G~f(x,y)μ(dy),

due to (2.15), we have

 limn→∞~f(μ)n=~f(μ) in Lp,qD(E). (2.27)

By linearity of , we have

 L~f(μ)n=M(n)∑i=1μ(Ani)Lφn,i in LrP′(T). (2.28)

By continuity of , (2.27) and (2.28) give

 limn→∞M(n)∑i=1μ(Ani)Lφn,i=L~f(μ) in LrP′(T). (2.29)

For , we now consider the measurable function

 ~f(L)n:(ΩT×G,P′⊗G)→F, ((ω,t),y)↦M(n)∑i=11Ani(y)(Lφn,i)(ω,t)

where here is a fixed representant of its class in . For all , is a representant of the class of in . Moreover,

 ∫G~f(L)n((ω,t),y)μ(dy)=M(n)∑i=1μ(Ani)(Lφn,i)(ω,t)∀(ω,t)∈ΩT, ∀n∈N.

By (2.28), we obtain

 for P-a.e.\ ω∈Ω,\ ∫G~f(L)n((ω,t),y)μ(dy)=(L~f(μ)n)(ω,t) ∀t∈[0,T]. (2.30)

We now show that we can pass to the limit in (2.30). By (2.10),

 limn→∞∫G∣∣~fn(⋅,y)−~f(⋅,y)∣∣Lp,qD(E)μ(dy)=0,

hence, by continuity of ,

 limn→∞∫G∣∣L(~fn(⋅,y))−L(~f(⋅,y))∣∣LrP′(T)μ(dy)=0. (2.31)

Since is a closed subspace of

 Lr(Ω,T)\coloneqqLr((Ω,FT,P),(T,|⋅|∞)),

the map

 (G,G)→Lr(Ω,T), y↦L(~f(⋅,y)) (2.32)

is measurable and integrable (the range of (2.32) is separable). By applying Lemma 2.1 again, now to (2.32), we have that there exists a measurable function

 g:(Ω×G,FT⊗G)→T (2.33)

such that, for some with ,

 g(⋅,y)=L(~f(⋅,y)) in% Lr(Ω,T), ∀y∈A. (2.34)

Define

Notice that, since is -measurable for all (by definition of ) and since contains the sets when and , we have, by (2.34), that is -measurable for all . Moreover, since the evaluation map (2.23) is assumed to be measurable, by measurability of (2.33) and by definition of we have that

 X2:(ΩT×G,(FT⊗BT)⊗G)→F

is measurable. By (2.31), we can write

 limn→∞∫Ω(∫Gsupt∈[0,T]∣∣~f(L)n((ω,t),y)−X2((ω,t),y)∣∣Fμ(dy))P(dω)≤limn→∞∫G(∫Ωsupt∈[0,T]∣∣~f(L)n((ω,t),y)−X2((ω,t),y)∣∣rFP(dω))1/rμ(dy)=limn→∞∫G∣∣L(~fn(⋅,y))−L(~f(⋅,y))∣∣LrP′(T)μ(dy)=0, (2.35)

where the measurability of , jointly in , is due to the measurability of (2.33), to the definition of , and to the definition of . By (2.35), by considering a subsequence if necessary, it follows that

 limn→∞supt∈[0,T]∣∣∣∫G~f(L)n((ω,t),y)μ(dy)−∫GX2((ω,t),y)μ(dy)∣∣∣F=0P-a.e.\ ω∈Ω. (2.36)

By (2.28), (2.29), (2.30), and (2.36), we conclude that, for -a.e. ,

 (L~f(μ))(ω,t)=∫GX2((ω,t),y)μ(dy), ∀t∈[0,T