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A stochastic Fubini theorem: BSDE method
Journal of Inequalities and Applications volume 2017, Article number: 77 (2017)
Abstract
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.
1 Introduction and the main result
Given \(T>0\), let \((\Omega,\mathcal{F},\mathcal{F}_{t},P;t\geq0)\) be a complete filtration space and \(\mathbb{F}=\{\mathcal{F}_{t};{t \geq0}\}\) be a filtration satisfying the usual conditions which are generated by the following two mutually independent stochastic processes:
-
(i)
a d-dimensional Brownian motion \(\{B(t);t\geq0\}\);
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(ii)
a Poisson random measure N on \(\mathbb{R}^{+}\times E\), where \(E=\mathbb{R}^{l}-\{0\}\) with the Borel σ-field \(\mathcal{B}(E)\). λ is the intensity (Lévy measure) of N with the property that
$$\int_{E} \bigl(1\wedge \vert z\vert ^{2} \bigr) \lambda(dz)< \infty $$and μ is the compensator of N with \(\mu(dt,dz)=dt\lambda(dz)\). Then \(\{\widetilde{N}((0,t]\times A)=(N-\mu)((0,t]\times A), \mathcal{F}_{t};t\geq0\}\) is a compensated Poisson process which is a cà dlà g martingale for all \(A\in\mathcal{B}(E)\) satisfying \(\lambda(A)<\infty\).
Fubini theorems giving conditions for change of the order of integration in multiple integrals are useful in all forms of calculus. The first such result of stochastic Fubini theorems perhaps belongs to Doob [1]. 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 [2–5] and the references therein). In those works, suppose that M is a stochastic process and \((X,\Sigma,\mu)\) is a σ-finite measure space, and \(\phi: X\times[0,T]\times\Omega\rightarrow\mathbb{R}\) is a stochastic process satisfying certain measurability properties. Under some integrability conditions, the following stochastic Fubini theorem holds:
The usual technique to prove (1.1) is the approximation method, i.e., firstly, (1.1) is proved for a simple process \(\phi_{n}\) 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: \(X=[0,T]\). In this case ϕ is only a process in \([0,T]\), and does the Fubini theorem hold? The key point is that the Lebesgue integral should be \(\mathbb{F}\)-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 \(d=l=1\) throughout this paper; the general cases can be treated by a similar method. For any \(n\geq1\), denote by \(\vert x\vert \) the Euclidean norm of \(x\in\mathbb{R}^{n}\). Also, we define the following classes of processes which will be used in the sequel.
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For any \(t\in[0,T]\), \(L^{2}_{\mathcal{F}_{t}}(\Omega; \mathbb{R} ^{n})\) is the space of all \(\mathcal{F}_{t}\)-measurable and \(\mathbb{R}^{n}\)-valued random variables ξ satisfying \(\vert \xi \vert ^{2} _{L^{2}_{\mathcal{F}_{t}}(\Omega; \mathbb{R}^{n})}=\mathbb{E}\vert \xi \vert ^{2}< \infty\).
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\(L^{2}_{\mathbb{F}}(\Omega; D([0,T]; \mathbb{R}^{n}))\) is the space of all \(\mathbb{F}\)-adapted cà dlà g stochastic processes \(X(\cdot)\) satisfying \(\vert X(\cdot)\vert ^{2}_{L^{2}_{\mathbb{F}}( \Omega; D([0,T]; \mathbb{R}^{n}))}=\mathbb{E} (\sup_{t\in [0,T]}\vert X(t)\vert ^{2} )<\infty\).
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\(L^{2}_{\mathbb{F}}(\Omega; C([0,T]; \mathbb{R}^{n}))\) is the subspace of \(L^{2}_{\mathbb{F}}(\Omega; D([0,T]; \mathbb{R}^{n}))\) whose element has continuous paths a.s.
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\(L^{2}_{P,\mathbb{F}}(\Omega; L^{2}(0,T; \mathbb{R}^{n}))\) is the space of all \(\mathbb{F}\)-predictable and \(\mathbb{R}^{n}\)-valued stochastic processes \(K(\cdot,\cdot)\) satisfying \(\vert K(\cdot,\cdot)\vert ^{2} _{L^{2}_{P,\mathbb{F}}(\Omega; L^{2}(0,T; \mathbb{R}^{n}))}= \mathbb{E}\int_{0}^{T}\int_{E}\vert K(t,z)\vert ^{2}\lambda(dz)\,dt<\infty\).
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For general \(p,q\geq1\), \(L^{p}_{\mathbb{F}}(\Omega; L^{q}(0,T; \mathbb{R}^{n}))\) denotes the space of all \(\mathbb{F}\)-adapted processes \(Y(\cdot)\) satisfying \(\vert Y(\cdot)\vert ^{p}_{L^{p}_{ \mathbb{F}}(\Omega; L^{q}(0,T; \mathbb{R}^{n}))}=\mathbb{E} (\int _{0}^{T}\vert Y(t)\vert ^{q}\,dt)^{p/q}<\infty\).
The main theorem of this paper is stated as follows.
Theorem 1.1
For any \(Y(\cdot)\in L^{2}_{\mathbb{F}}(\Omega; L^{2}(0,T; \mathbb{R}^{n}))\), \(K(\cdot,\cdot)\in L^{2}_{P,\mathbb{F}}(\Omega; L^{2}(0,T; \mathbb{R}^{n}))\) and any \(g(\cdot),h(\cdot)\in L^{2}(0,T)\), we have
As an application, under suitable conditions, we obtain the well-posedness of the following two BSDEs:
and
The development of BSDEs theory has lasted for 40 years. Linear BSDEs were first introduced in 1973 by Bismut in [6] as the equations for the conjugate variable in the stochastic version of the Pontryagin maximum principle. Pardoux and Peng in [7] first studied the general nonlinear BSDEs of the following form in 1990:
Since 1990, there has appeared a large number of works published related to the theory and applications for BSDEs (see [8–15] for examples).
It seems that (1.4) belongs to the following backward stochastic Volterra integral equation:
which was studied in [15, 16]. Because of the special form, we can transform (1.3) and (1.4) into (1.5). Hence the second component Y of the solution just depends on one time variable.
The paper is organized as follows. In Section 2, we present some fundamental results, well-posedness of BSDEs and prove Theorem 1.1 by virtue of BSDEs. In Section 3, we apply Theorem 1.1 to solve BSDEs (3.1) and (3.2) and get the well-posedness under subtle Lipschitz conditions.
2 Proof of the main result
The following BSDE with jump has been studied in some works, such as [11, 17]:
where f satisfies \(f(\cdot,0,0,0)\in L^{2}_{\mathbb{F}}(\Omega; L ^{1}(0,T; \mathbb{R}^{n}))\), and
for any \(y, \widetilde{y}, Y, \widetilde{Y}\in\mathbb{R}^{n}\), \(K, \widetilde{K}\in L^{2}(E,\mathcal{B}(E),\lambda;\mathbb{R}^{n})\).
The following lemma is about the well-posedness of (2.1). The proof can be found in [11, 17]. Hence, it is omitted. For simplicity, we denote \(\mathcal{H}=L^{2}_{\mathbb{F}}(\Omega; D([0,T]; \mathbb{R}^{n}))\times L^{2}_{\mathbb{F}}(\Omega; L^{2}(0,T; \mathbb{R}^{n}))\times L^{2}_{P,\mathbb{F}}(\Omega; L^{2}(0,T; \mathbb{R}^{n}))\) with the canonical norm.
Lemma 2.1
For any \(y_{T}\in L^{2}_{\mathcal{F}_{T}}(\Omega;\mathbb{R}^{n})\), equation (2.1) admits a unique adapted solution \((y(\cdot),Y(\cdot),K(\cdot,\cdot))\in\mathcal{H}\). Furthermore, there is a constant \(C>0\), depending only on L and T, such that
In order to prove Theorem 1.1, we first consider the following BSDE in \([0,T]\):
\(g(\cdot)\) and \(h(\cdot)\in L^{1}(0,T)\) such that for any \(t\in[0,T]\),
where δ is a positive constant.
The well-posedness of (2.4) is presented in the following theorem.
Theorem 2.2
Under assumption (2.5), for any \(y_{T}\in L^{2} _{\mathcal{F}_{T}}(\Omega;\mathbb{R}^{n})\), equation (2.4) admits a unique adapted solution \((y(\cdot),Y(\cdot),K(\cdot,\cdot))\in\mathcal{H}\) such that
where C is a constant depending on δ and T.
Proof
We divide the proof into two steps.
Step 1. By Lemma 2.1, we know that the following BSDE
admits a unique solution \((y(\cdot),Y(\cdot),K(\cdot,\cdot))\) satisfying
where C depends only on T. Hence, by virtue of (2.5), it follows that
Then, by (2.8) and (2.9), we deduce that
where C depends on δ and T.
Step 2. By (2.7), we can easily obtain
Applying Itô’s formula to \((\int_{t}^{T}g(s)\,ds+1)\int _{0}^{t}Y(u)\,dB(u)\), one can get
Similarly, one has
Substituting (2.12) and (2.13) into (2.7) yields
Taking \(t=T\) in (2.14), one has
Subtracting (2.14) from (2.15), we just obtain (2.4). Hence \((y(\cdot),Y(\cdot),K( \cdot,\cdot))\) is a unique solution to (2.4). The desired estimate (2.6) follows from (2.10). That completes the proof. □
As a corollary, we give the proof of the stochastic Fubini theorem stated in Theorem 1.1.
Proof of Theorem 1.1
Set
where \(g^{+}(\cdot)\), \(h^{+}(\cdot)\) are the positive parts of \(g(\cdot)\), \(h(\cdot)\), respectively. By Hölder’s inequality and the Itô isometry, we have
By the proof of Theorem 2.2, it is easy to get that
with terminal condition \(y(T)=\xi\) admits a unique solution \((y(\cdot),Y(\cdot),K(\cdot,\cdot))\). And (2.18) is equivalent to
Combining (2.18) and (2.19), we can get
Similarly, we have
Addition of (2.20) and (2.21) gives (1.2), which completes the proof. □
3 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 \((\Omega,\mathcal{F},\mathcal{F}_{t},P;t\geq0)\) be a complete filtration space and \(B(\cdot)\) be a one-dimensional standard Brownian motion whose natural filtration is given by \(\mathbb{F}=\{ \mathcal{F}_{t}\}_{t\geq0}\). As an application of Theorem 1.1, we prove the well-posedness of the following two BSDEs:
and
Here, generators \(f(\cdot,\cdot)\) and \(g(\cdot,\cdot,\cdot)\) satisfy the following assumptions:
-
(H1)
\(f(\cdot,0)\in L^{2}_{\mathbb{F}}(\Omega; L^{2}(0,T; \mathbb{R}^{n}))\),
$$\bigl\vert f(t,Y)-f(t,\widetilde{Y}) \bigr\vert \leq\frac{1}{T+\theta} \vert Y-\widetilde{Y}\vert \quad \mbox{a.e. } t\in[0,T], \mbox{a.s.}, $$and
-
(H2)
\(g(\cdot,0,0)\in L^{2}_{\mathbb{F}}(\Omega; L^{2}(0,T; \mathbb{R}^{n}))\),
$$\bigl\vert g(t,y,Y)-g(t,\widetilde{y},\widetilde{Y}) \bigr\vert \leq L \vert y-\widetilde{y}\vert +\frac{1}{2(T+ \theta)}\vert Y-\widetilde{Y}\vert \quad \mbox{a.e. } t \in[0,T], \mbox{a.s.}, $$
respectively, where \(y, \widetilde{y}, Y, \widetilde{Y}\in \mathbb{R}^{n}\), and L and θ are positive constants.
Theorem 3.1
Under assumption (H1), for any \(y_{T}\in L^{2}_{\mathcal{F} _{T}}(\Omega;\mathbb{R}^{n})\), equation (3.1) admits a unique adapted solution \((y(\cdot),Y(\cdot))\in L^{2}_{\mathbb{F}}(\Omega; C([0,T]; \mathbb{R}^{n}))\times L^{2}_{\mathbb{F}}(\Omega; L^{2}(0,T; \mathbb{R}^{n}))\) such that
where C is a constant depending on θ and T.
Proof
We divide the proof into three steps.
Step 1. Suppose that \(f(\cdot,Y(\cdot))\in L^{2}_{ \mathbb{F}}(\Omega; L^{2}(0,T; \mathbb{R}^{n}))\). Then by Theorem 1.1 we rewrite equation (3.1) as follows:
We know that the following BSDE admits a unique adapted solution \((y(\cdot), Z(\cdot))\)
We define an operator \(S: L^{2}_{\mathbb{F}}(\Omega; L^{2}(0,T; \mathbb{R}^{n}))\rightarrow L^{2}_{\mathbb{F}}(\Omega; L^{2}(0,T; \mathbb{R}^{n}))\) by \(S(X(t))=Z(t)- (T-t)f(t,X(t))\) for a.e. t. For any \(X_{1}, X_{2}\in L^{2}_{\mathbb{F}}(\Omega; L^{2}(0,T; \mathbb{R} ^{n}))\) and a.e. t,
We see that S is contractive. Hence, by the Banach fixed point theorem, S admits a unique fixed point \(Y(\cdot)\in L^{2}_{ \mathbb{F}}(\Omega; L^{2}(0,T; \mathbb{R}^{n}))\) such that \(S(Y(\cdot))=Y(\cdot)\), i.e.,
By (3.5) and (3.6), we conclude that \((y(\cdot), Y(\cdot))\) is the unique solution to (3.1).
Step 2. In this step, we check that for this \(Y(\cdot)\), \(f(\cdot,Y(\cdot))\in L^{2}_{\mathbb{F}}(\Omega; L^{2}(0,T; \mathbb{R}^{n}))\)
Step 3. In this step, we show that (3.3) holds.
By Jensen’s inequality, Doob’s maximal inequality and equation (3.5), we see that
Applying Itô’s formula to \(\vert y(\cdot)\vert ^{2}\), we get
On the other hand, one has
By \(\vert a+b\vert ^{2}\geq(1-\varepsilon)\vert a\vert ^{2}+1-1/\varepsilon \vert b\vert ^{2}\), where \(\varepsilon>1\), and assumption (H1), we can obtain
Taking \(\varepsilon=\frac{T+\theta}{T}\), by (3.10), we have
(3.11), together with (3.9), yields
Combining (3.8) and (3.12), 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 \(1/(T+\theta)\) cannot be improved.
Example 3.2
For any \(\theta>0\), suppose that \(y_{T}=\int_{\theta}^{T}(s- \theta)/(T-\theta)\,dB(s)\) and \(f(s,x)=-\chi_{[0,\theta]}(s)x/(T-s)- \chi_{(\theta,T]}(s)x/(T-\theta)\) with Lipschitz constant \(1/(T-\theta)\). Then equation (3.1) can be written as
For any \(\eta\in L^{2}_{\mathbb{F}}((0,\theta)\times\Omega; \mathbb{R})\), let
It is easy to check that \((y(\cdot),Y_{\eta}(\cdot))\in L^{2}_{ \mathbb{F}}(\Omega;C([0,T];\mathbb{R}))\times L^{2}_{\mathbb{F}}( \Omega;L^{2}(0,T;\mathbb{R}))\) is an adapted solution of (3.1). Since η is arbitrary, the solution of (3.1) is not unique.
Example 3.3
Set \(y_{T}=B(T)\) and \(f(t,x)=-x/T\) whose Lipschitz constant is \(1/T\). If equation (3.1) has an adapted solution \((y,Y)\in L^{2}_{\mathbb{F}}(\Omega;C([0,T];\mathbb{R})) \times L^{2}_{\mathbb{F}}(\Omega;L^{2}(0,T;\mathbb{R}))\), then by Theorem 1.1, (3.1) becomes
Since \((B(\cdot),1)\) is the unique adapted solution of \(y_{T}-y(t)= \int_{t}^{T}Z(t)\,dB(t)\), we get that
It is clear that \(\vert Y(\cdot)\vert _{ L^{2}_{\mathbb{F}}(\Omega;L^{2}(0,T; \mathbb{R}))}=+\infty\). Hence (3.1) with \(f(t,x)=-x/T, y_{T}=B(T)\) does not have a square integrable adapted solution.
From the previous two examples, we know that in equation (3.1), Lipschitz constant \(1/(T+\theta)\) cannot be improved. But in the one-dimensional case, we can obtain a better result.
Theorem 3.4
Suppose that in equation (3.1), \(y_{T}\in L^{2}_{\mathcal{F}_{T}}(\Omega;\mathbb{R})\), \(f(\cdot,0) \in L^{2}_{\mathbb{F}}(\Omega; L^{2}(0,T; \mathbb{R}))\), and
where θ and L are positive constants and \(L\geq1/(T+\theta)\). Then BSDE (3.1) admits a unique solution.
Proof
Set \(h(t,x)=(T-t)f(t,x)+x\). It is easy to see that
By Theorem 4.1 in [7], we conclude that (3.1) admits a unique solution. □
For \(\beta>0\) and \(X\in L^{2}_{\mathbb{F}}(\Omega; L^{2}(0,T; \mathbb{R}^{n}))\), \(\Vert X\Vert _{\beta}^{2}\) denotes \(\mathbb{E}\int _{0}^{T}e^{\beta t}\vert X(t)\vert ^{2}\,dt\). \(H_{\beta}\) denotes the space \(L^{2}_{\mathbb{F}}(\Omega; L^{2}(0,T; \mathbb{R}^{n}))\) endowed with the norm \(\Vert \cdot \Vert _{\beta}\).
Theorem 3.5
Under assumption (H2), for any \(y_{T}\in L^{2}_{\mathcal{F} _{T}}(\Omega;\mathbb{R}^{n})\), equation (3.2) admits a unique adapted solution \((y(\cdot),Y(\cdot))\) such that
where C is a constant depending on L and T.
Proof
We divide the proof into two steps.
Step 1. For any \(\bar{y}(\cdot),\bar{Y}(\cdot)\in L^{2} _{\mathbb{F}}(\Omega; L^{2}(0,T; \mathbb{R}^{n}))\), by virtue of the martingale representation theorem, the following BSDE admits a unique solution \((y(\cdot), Y(\cdot))\)
Step 2. We use the Banach fixed point theorem for the mapping Φ from \(H_{\beta}\times H_{\beta}\) to itself, which maps \((\bar{y}(\cdot),\bar{Y}(\cdot))\) onto \((y(\cdot),Y(\cdot))\), where β is determined later.
For any \((\bar{y}_{i}(\cdot),\bar{Y}_{i}(\cdot))\), suppose that the corresponding solution to (3.16) is \((y_{i}(\cdot),Y_{i}(\cdot))\), \(i=1,2\).
In order to use Theorem 1.1, we first check that \(g(\cdot,\bar{y_{i}}(\cdot),\bar{Y_{i}}(\cdot))\in L^{2}_{ \mathbb{F}}(\Omega; L^{2}(0,T; \mathbb{R}^{n}))\). Indeed, applying the same method as that in Step 2 of Theorem 3.1, one has
Hence, by Theorem 1.1, for \(i=1, 2\), (3.16) turns to
Set \(\Delta y=y_{1}-y_{2}\), \(\Delta Y=Y_{1}-Y_{2}\), \(\Delta\bar{y}= \bar{y}_{1}-\bar{y}_{2}\), \(\Delta\bar{Y}=\bar{Y}_{1}-\bar{Y}_{2}\) and \(\Delta g(\cdot)=g(\cdot,\bar{y}_{1}(\cdot),\bar{Y}_{1}(\cdot))-g( \cdot,\bar{y}_{2}(\cdot),\bar{Y}_{2}(\cdot))\). Applying Itô’s formula to \(e^{\beta\cdot}(y_{1}(\cdot)-y_{2}(\cdot))^{2}\), we have
Setting \(y_{1}(T)=y_{2}(T)\) and taking expectation, we get
Let \(t=0\),
which is equivalent to
Take \(\varepsilon_{1}\), \(\varepsilon_{2}\) and β satisfying
We know these \(\varepsilon_{1}\), \(\varepsilon_{2}\) and β exist. For example, one can take \(\varepsilon_{1}=\frac{2(1+\alpha)}{ \alpha}\), \(\varepsilon_{2}=2\), and \(\beta=\frac{2\varepsilon_{1}L ^{2}T^{2}(T+\theta)^{2}}{(T+\theta)^{2}+T^{2}}\), where \(\alpha =\frac{2T\theta+\theta^{2}}{4T^{2}}\). For this β, we see that Φ is a contraction and that there exists a fixed point, which is the unique solution of BSDE (3.2).
By the same method used in Step 3 of Theorem 3.1, we can show that (3.15) holds. That completes the proof. □
4 Conclusions
In this paper, we consider a stochastic Fubini theorem with two time variables. In that Fubini theorem, one time variable is related to the Lebesgue integral, and the other is related to the Itô integral, in which the adapted property is crucial. We prove the result by solving a BSDE which is different from the existing method. As an application, we apply the theorem to study the well-posedness of two special BSDEs under subtle Lipschitz conditions. Besides this, we provide two examples to state that aforementioned Lipschitz conditions cannot be improved.
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Acknowledgements
This work is supported in part by the National Natural Science Foundation of China (11526167), the Fundamental Research Funds for the Central Universities (SWU113038, XDJK2014C076), the Natural Science Foundation of CQCSTC (2015jcyjA00017).
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Wang, Y. A stochastic Fubini theorem: BSDE method. J Inequal Appl 2017, 77 (2017). https://doi.org/10.1186/s13660-017-1358-3
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DOI: https://doi.org/10.1186/s13660-017-1358-3