A stochastic Fubini theorem: BSDE method

In this paper, we prove a stochastic Fubini theorem by solving a special backward stochastic differential equation (BSDE, for short) which is different from the existing techniques. As an application, we obtain the well-posedness of a class of BSDEs with the Itô integral in drift term under a subtle Lipschitz condition.


Introduction and the main result
Given T > , let ( , F, F t , P; t ≥ ) be a complete filtration space and F = {F t ; t ≥ } be a filtration satisfying the usual conditions which are generated by the following two mutually independent stochastic processes:  Doob []. After that, there are two directions on this topic. One is to generalize the Itô integrals; the other is to study the theorem under the weaker integrability conditions (see [-] and the references therein). In those works, suppose that M is a stochastic process and (X, , μ) is a σ -finite measure space, and φ : X × [, T] × → R is a stochastic process satisfying certain measurability properties. Under some integrability conditions, the following stochastic Fubini theorem holds: The usual technique to prove (.) is the approximation method, i.e., firstly, (.) is proved for a simple process φ n which is used to approximate φ in appropriate processes space, then it is proved by taking the appropriate limit. In this work, we treat a special case: X = [, T]. In this case φ is only a process in [, T], and does the Fubini theorem hold?
The key point is that the Lebesgue integral should be F-adapted so that the Itô integral makes sense. In this paper, we want to prove this type of stochastic Fubini theorem by using the backward stochastic differential equation (BSDE, for short) method.
For simplicity, we consider only the case d = l =  throughout this paper; the general cases can be treated by a similar method. For any n ≥ , denote by |x| the Euclidean norm of x ∈ R n . Also, we define the following classes of processes which will be used in the sequel. • ) whose element has continuous paths a.s.
The main theorem of this paper is stated as follows.
Theorem . For any Y (·) ∈ L  F ( ; L  (, T; R n )), K(·, ·) ∈ L  P,F ( ; L  (, T; R n )) and any g(·), h(·) ∈ L  (, T), we have As an application, under suitable conditions, we obtain the well-posedness of the following two BSDEs: and y Ty(t) =  It seems that (.) belongs to the following backward stochastic Volterra integral equation: The paper is organized as follows. In Section , we present some fundamental results, well-posedness of BSDEs and prove Theorem . by virtue of BSDEs. In Section , we apply Theorem . to solve BSDEs (.) and (.) and get the well-posedness under subtle Lipschitz conditions.

Proof of the main result
The following BSDE with jump has been studied in some works, such as [, ]: where f satisfies f (·, , , ) ∈ L  F ( ; L  (, T; R n )), and The following lemma is about the well-posedness of (.). The proof can be found in [, ]. Hence, it is omitted. For simplicity, we denote admits a unique adapted solution (y(·), Y (·), K(·, ·)) ∈ H. Furthermore, there is a constant C > , depending only on L and T, such that In order to prove Theorem ., we first consider the following BSDE in [, T]: where δ is a positive constant. The well-posedness of (.) is presented in the following theorem.
where C is a constant depending on δ and T.
Proof We divide the proof into two steps.
Step . By Lemma ., we know that the following BSDE admits a unique solution (y(·), Y (·), K(·, ·)) satisfying where C depends only on T. Hence, by virtue of (.), it follows that Then, by (.) and (.), we deduce that where C depends on δ and T.
As a corollary, we give the proof of the stochastic Fubini theorem stated in Theorem ..
Proof of Theorem . Set where g + (·), h + (·) are the positive parts of g(·), h(·), respectively. By Hölder's inequality and the Itô isometry, we have By the proof of Theorem ., it is easy to get that

An application: well-posedness of two BSDEs
In this section, we consider only the BSDEs driven by one-dimensional Brownian motions.
The other cases such as BSDEs driven by high dimension Brownian motions and BSDEs with jumps can also be treated in a similar procedure. Let ( , F, F t , P; t ≥ ) be a complete filtration space and B(·) be a one-dimensional standard Brownian motion whose natural filtration is given by F = {F t } t≥ . As an application of Theorem ., we prove the wellposedness of the following two BSDEs: Here, generators f (·, ·) and g(·, ·, ·) satisfy the following assumptions: , a.s., and (H) g(·, , ) ∈ L  F ( ; L  (, T; R n )), where C is a constant depending on θ and T.
Proof We divide the proof into three steps.
Step . In this step, we check that for this Step . In this step, we show that (.) holds. By Jensen's inequality, Doob's maximal inequality and equation (.), we see that Applying Itô's formula to |y(·)|  , we get On the other hand, one has By |a + b|  ≥ (ε)|a|  +  -/ε|b|  , where ε > , and assumption (H), we can obtain Taking ε = T+θ T , by (.), we have Combining (.) and (.), one has where C is a constant depending on θ and T. That completes the proof.
We list two examples from which we can see that the Lipschitz constant /(T + θ ) cannot be improved. For any η ∈ L  F ((, θ ) × ; R), let It is easy to check that (y(·), Y η (·)) ∈ L  where θ and L are positive constants and L ≥ /(T + θ ). Then BSDE (.) admits a unique solution.
Proof Set h(t, x) = (Tt)f (t, x) + x. It is easy to see that By Theorem . in [], we conclude that (.) admits a unique solution.
Theorem . Under assumption (H), for any y T ∈ L  F T ( ; R n ), equation (.) admits a unique adapted solution (y(·), Y (·)) such that where C is a constant depending on L and T.
Proof We divide the proof into two steps.