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Introduction
Backward stochastic differential equations (BSDEs in short) were first introduced by Pardoux and Peng [4]. They proved an existence and uniqueness result under Lipschitz condition. Since then many efforts have been made in relaxing the Lipschitz assumption of the generator of the BSDEs (see among others Mao [3] and Wang and Huang [7]). Few years later, the same authors considered in [5] a new type of BSDEs, that is a class of backward doubly stochastic differential equations (BDSDEs in short) with two different directions of stochastic integrals. These equations are extensively used in the study of stochastic partial differential equations (SPDEs). Their link with SPDEs in the case of Lipchitzian drift was established in [5]. The key point of solvency of such equations is the martingale representation theorem. In this spirit, Bally and Matoussi [1] gave the probabilistic representation of the solution in Sobolev space of semilinear SPDEs in terms of BDSDEs.
On the other hand, Peng and Yang [6] introduced the following type of anticipated backward stochastic differential equations (ABSDEs in short)
where δ and ζ are given nonnegative deterministic functions. In these equations, the generator includes not only the values of solutions of the present but also the future. In [6], the authors obtained the existence and uniqueness of the solution of ABSDE under Lipschitz assumption, gave the comparison theorem for one dimensional ABSDEs and finally they solved a stochastic control problem by showing the duality between linear stochastic differential delay equations and ABSDEs.
This paper is devoted to the following anticipated BDSDE
where K is a positive constant, x., η: are given stochastic processes and δ,ζ : [0,T]➝R+ are continuous functions satisfying:
(A1) : t +δ(t) ≤ T +K, t +ζ (t) ≤ T +K:
(A2) : There exists M ≥ 0 such that for 0 ≤ t ≤ T and non negative integrable function h,
$$\int_{t}^{T}{h\left( r+\phi \left( r \right) \right)}dr\le M\int_{t}^{T+K}{h\left( r \right)dr,\ \ \ \phi \in \left\{ \delta ,\zeta \right\}}.$$
The paper is organized as follows. In section 2, we study first solvability of our equation in the case of Lipschtzian coefficients. Using this result, in section 3 we prove existence and uniqueness of solution with coefficients satisfying rather weaker conditions.
Preliminaries
Let Ω be a non-empty set, ℱ a σ‒algebra of sets of Ω and P a probability measure defined on ℱ. The triplet (Ω,ℱ, P) defines a probability space, which is assumed to be complete. For a fix real 0 < T ≤ ∞, we assume given two mutually independent processes:
where for any process ${{\left\{ {{\varphi }_{t}} \right\}}_{t\ge 0}},F_{s,t}^{\varphi }=\sigma \left\{ {{\varphi }_{r}}-{{\varphi }_{s}},s\le r\le t \right\}\vee N,F_{t}^{\varphi }=F_{0,t}^{\varphi }.$Here 𝒩 denotes the class of P‒null sets of F. Note that (Ft)0≤t≤T does not constitute a classical filtration.
For k ∈ N* we consider the following sets (where E denotes the mathematical expectation with respect to the probability measure P):
L2(ℊT,Rk) the space of ℊT -measurable random variable such that $\mathbf{E}\left[ {{\left| {{\xi }_{T}} \right|}^{2}} \right]<+\infty .$
$I_{\left[ 0,T \right]}^{2}\left( G,{{\text{R}}^{k}} \right)$the space of ℊt‒adapted càdlàg processes
$C_{G}^{2}\left( 0,T \right)=M_{\left[ 0,T \right]}^{2}\left( G,{{\mathbf{R}}^{k}} \right)\times M_{\left[ 0.T \right]}^{2}\left( G,{{\mathbf{R}}^{k\times d}} \right)$endowed with the norm
$$\left\| \left( Y,Z \right) \right\|_{C_{G}^{2}\left( 0,T \right)}^{2}=\left\| Y \right\|_{{{M}^{2}}\left( {{\mathbf{R}}^{k}} \right)}^{2}+\left\| Z \right\|_{{{M}^{2}}\left( {{\mathbf{R}}^{k\times d}} \right)}^{2}.$$
$B_{G}^{2}\left( 0,T \right)=I_{\left[ 0,T \right]}^{2}\left( G,{{\text{R}}^{k}} \right)\times M_{\left[ 0,T \right]}^{2}\left( G,{{\mathbf{R}}^{k\times d}} \right)$endowed with the norm
$$\left\| \left( Y,Z \right) \right\|_{B_{G}^{2}\left( 0,T \right)}^{2}=\left\| Y \right\|_{{{S}^{2}}\left( {{\mathbf{R}}^{k}} \right)}^{2}+{{\left\| Z \right\|}^{2}}_{{{M}^{2}}\left( {{\mathbf{R}}^{k\times d}} \right)}.$$
S be the set of all nondecreasing, continuous and concave function ρ(·) : R+➝ R+ satisfying ρ(0) = 0, $\rho \left( s \right)>0\ \text{for}\ s\ >\ 0\ \text{and }\int_{0+}{\frac{du}{\rho \left( u \right)}=+\infty .}$
Remark 2.1. For any ρ ∈ S, we can find a pair of positive constants a and b such that ρ(ν) ≤ a +bν for all ν ≥ 0.
We denote A = Ω×[0,T]×Rk ×Rk×d, f(r,0) = h(r,0,0,0,0), for all x,y ∈ Rk |x| the Euclidean norm of x and denote by 〈x,y〉 the Euclidean inner product.
Definition 2.2. A pair of processes (Y,Z) is called a solution to ABDSDE (1.1), $if\left( Y,Z \right)\in \ B_{G}^{2}\left( 0,T+K \right)$and it satisfies eq (1.1).
First we investigate the case of lipschitz coefficients.
The case of Lipschitz coefficients
In this subsection, we will mainly study the existence and uniqueness of the solution to ABDSDE (1.1) with Lipschitz coefficients. For this purpose, we first make the following assumptions.
Assumptions
In the following, we assume that there exists ρ ∈ S such that f and g satisfy assumptions (H1).
(H1.1): There exists a constant c > 0 such that
$$\begin{align} & {{\left| f\left( t,y,z,\theta \left( r \right),\varphi \left( r \right) \right)-f\left( t,{y}',{z}',{\theta }'\left( r \right),{\varphi }'\left( r \right) \right) \right|}^{2}}\le c\left( {{\left| y-{y}' \right|}^{2}}+{{\left| z-{z}' \right|}^{2}} \right) \\ & +{{\mathbf{E}}^{{{F}_{t}}}}\left[ {{\left| \theta \left( r \right)-{\theta }'\left( r \right) \right|}^{2}}+{{\left| \varphi \left( r \right){{-}\varphi'}\left( r \right) \right|}^{2}} \right], \\ \end{align}$$
for all $\left( r,r' \right)\in \left[ t,T+K \right],\left( t,y,z,\theta \left( r \right),\varphi \left( r \right) \right),\left( t,{y}',{z}',{\theta }'\left( r \right),{\varphi }'\left( r \right) \right)\in A\times C_{G}^{2}\left( t,T+K \right).$
(H1.2): There exists a constant 0 < α1< 1 such that for any (t,y, z), (t,y′, z′) ∈ [0,T]×Rk ×Rk×d
Let us recall the following result which will be useful in the sequel (the proof is omitted since it is an adaptation of Theorem 3.1 in Xu [9]).
Proposition 3.2. Given ξT ∈ L2(GT,Rk), eq (3.7) has a unique solution${{\left( {{Y}_{t}},{{Z}_{t}} \right)}_{0\le t\le T}}\in C_{G}^{2}\left( 0,T \right).$
We are now in position to give our main results of this section.
Theorem 3.3
Assume that the assumptions (A1), (A2) and (H1) are true and let ξT ∈ L2(ℊT,Rk). Then for any$\left( \xi ,\eta \right)\in S_{\left[ T,T+K \right]}^{2}\left( G,{{\mathbf{R}}^{k}} \right)\times M_{\left[ T,T+K \right]}^{2}\left( G,{{\mathbf{R}}^{k\times d}} \right)$the ABDSDE (1.1) has a unique solution (Yt ,Zt)0≤t≤T∈$B_{G}^{2}\left( 0,T+K \right).$
Fix β ∈ R. The pair $\left( \overline{Y},\overline{Z} \right)$solves the ABDSDE
$$\left\{ \begin{array}& {{{\bar{Y}}}_{t}}=\int_{t}^{T}{\Delta f\left( r \right)dr+}\int_{t}^{T}{\Delta g\left( r \right)d{{B}_{r}}-\int_{t}^{T}{{{{\bar{Z}}}_{r}}d{{W}_{r}},\,\,\,\,\,}}\forall 0\le t\le T, \\ \forall t\,\in \left[ T,T+K \right],\,\,\,\,{{{\bar{Y}}}_{t}}=0,\,\,\,{{{\bar{Z}}}_{t}}=0,\\ \end{array} \right.$$
where for $\rho \in \left\{ Y,Z \right\},\bar{\rho }=\rho -\tilde{\rho },\Delta g\left( r \right)=g\left( r,{{y}_{r}},{{z}_{r}} \right)-g\left( r,{{{\tilde{y}}}_{r}},{{{\tilde{z}}}_{r}} \right)$and
$$\Delta f\left( r \right)=f\left( r,{{y}_{r}},{{z}_{r}},{{y}_{r+\delta \left( r \right)}},{{z}_{r+\zeta \left( r \right)}} \right)-f\left( r,{{{\tilde{y}}}_{r}},{{{\tilde{z}}}_{r}},{{{\tilde{y}}}_{r+\delta \left( r \right)}} \right).$$
It remains to prove that the above solution is in $B_{G}^{2}\left( 0,T+K \right).$Indeed, by Lemma 3.1, we have $Y\in S_{\left[ 0,T \right]}^{2}\left( G\text{,}{{\mathbf{R}}^{k}} \right).$Thus, we obtain ${{\left( {{Y}_{t}},{{Z}_{t}} \right)}_{0\le t\le T}}B_{G}^{2}\left( 0,T+K \right).$
(ii) Uniqueness. Let (Y,Z) and $\left( \tilde{Y},\tilde{Z} \right)$two solutions of eq.(1.1). Itô’s formula applied to eq.(3.9) yields, for 0 ≤ t ≤ T
$$\mathbf{E}\left[ {{\left| {{{\bar{Y}}}_{t}} \right|}^{2}} \right]+\mathbf{E}\int_{t}^{T}{{{\left| {{{\bar{Z}}}_{r}} \right|}^{2}}dr\le 2\mathbf{E}}\int_{t}^{T}{\left\langle {{{\bar{Y}}}_{r}},\Delta \left( r \right) \right\rangle dr+\mathbf{E}\int_{t}^{T}{{{\left| \Delta g\left( r \right) \right|}^{2}}}}dr.$$
Hence if we choose ξ = ξ0 satisfying $\bar{\alpha }=\frac{1}{{{\varepsilon }_{0}}}\left( c+M \right)+{{\alpha }_{1}}<1$and denote $\bar{c}=\frac{c\left( {{\varepsilon }_{0}}+1 \right)+M}{{{\varepsilon }_{0}}}+{{\varepsilon }_{0}},$then using the above inequalities, from (3.11), we obtain
Then we can use Gronwall’s inequality to deduce $\bar{Y}=0\,\text and \,\bar{Z}=\text{0}\text{.}$This completes the proof.
The case of non-Lipschitz coefficients
In this subsection, we will mainly study the existence and uniqueness of the solution to ABDSDE (1.1) with non-Lipschitz coefficients. For this purpose, we first make the following assumptions.
Assumptions
In the following, we assume that there exists ρ ∈ S such that f and g satisfy assumptions (H2).
(H2.1): There exists a constant c > 0 such that
$$\begin{align}& {{\left| f\left( t,y,z,\theta \left( r \right),\varphi \left( r \right) \right)-f\left( t,{y}',{z}',{\theta }'\left( r \right),{\varphi }'\left( r \right) \right) \right|}^{2}}\le c\left( \rho \left( {{\left| y-{y}' \right|}^{2}} \right)+{{\left| z-{z}' \right|}^{2}} \right) \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+{{E}^{{{F}_{t}}}}\left[ \rho \left( {{\left| \theta \left( r \right)-{\theta }'\left( r \right) \right|}^{2}} \right)+{{\left| \varphi \left( r \right)-{\varphi }'\left( r \right) \right|}^{2}} \right], \\ \end{align}$$
$$\text{for}\,\text{all}\left( r,{r}' \right)\in \left[ t,T+K \right],\left( t,y,z,\theta \left( r \right),\varphi \left( r \right) \right),\,\left( t,{y}',{z}',{\theta }'\left( r \right),{\varphi }'\left( r \right) \right)\in A\times C_{G}^{2}\left( t,T+K \right).$$
(H2.2): There exists a constant 0 < α1< 1 such that for any $\left( t,y,z \right),\,\left( t,{y}',{z}' \right)\in \left[ 0,T \right]\times {{\mathbf{R}}^{k}}\times {{\mathbf{R}}^{k\times d}}$
Thanks to Theorem 3.3, this sequence is well defined since the generators $f\left( r,Y_{r}^{n-1},\cdot ,Y_{r+\delta \left( r \right)}^{n-1},\cdot \right)\,\text{and}\,\text{g}\left( r,Y_{r}^{n-1},\cdot \right)$are Г-Lipschitz. Let us state the following previous result
Lemma 4.1
Assume that the assumptions (A1), (A2) and (H2) are true and let ξT ∈ L2(ℊT,Rk). Then for any$\left( \xi ,\eta \right)\in S_{\left[ T,T+K \right]}^{2}\left( G,{{\mathbf{R}}^{k}} \right)\times M_{\left[ T,T+K \right]}^{2}\left( G,{{\mathbf{R}}^{k\times d}} \right)$there exists a positive constant C′ such that
where C′ is a positive constant(which may vary from line to line). In particular, putting ${{q}_{n}}\left( t \right)={{\sup }_{n\in \mathbb{N}}}\mathbf{E}\left[ {{\left| Y_{t}^{n} \right|}^{2}} \right],$we have
$${{q}_{n}}\left( t \right)\le {C}'\left( E\left[ X \right]+1 \right)+{C}'\int_{t}^{T}{{{q}_{n}}\left( r \right)dr},\,\,\,\,\,0\le t\le T.$$
Assume that the assumptions (A1), (A2) and (H2) are true and let ξT ∈ L2(GT,Rk). Then for any$\left( \xi ,\eta \right)\in S_{\left[ T,T+K \right]}^{2}\left( G,{{\mathbf{R}}^{k}} \right)\times M_{\left[ T,T+K \right]}^{2}\left( G,{{\mathbf{R}}^{k\times d}} \right)$the ABDSDE (1.1) has a unique solution$\left( Y,Z \right)\in B_{G}^{2}\left( 0,T+K \right).$
Proof. (i) Existence. We consider the sequence defined in eq.(4.1). For a process ρ ∈ {Y,Z}, and n ∈ N,m ∈ N, $\begin{align}& \rho _{t}^{-n,m}=\rho _{t}^{n}-\rho _{t}^{m},\Delta {{g}^{\left( n,m \right)}}\left( r \right)=f\left( r,Y_{r}^{n-1},Z_{r}^{n} \right)-g\left( r,Y_{r}^{m-1},Z_{r}^{m} \right)\text{and} \\ & \Delta {{f}^{\left( n,m \right)}}\left( r \right)=f\left( r,Y_{r}^{n-1},Z_{r}^{n},Y_{r+\delta \left( r \right)}^{n-1},Z_{r+\zeta \left( r \right)}^{n} \right)-f\left( r,Y_{r}^{m-1},Z_{r}^{m},Y_{r+\delta \left( r \right)}^{m-1},Z_{r+\zeta \left( r \right)}^{m} \right). \\ \end{align}$
Note that the pair (Yn,m,Zn,m) solves the following equation
Applying Fatou’s lemma and the fact that ρ ∈ S, we deduce that
$$q\left( t \right)\le {C}'\int_{t}^{T+K}{\rho \left( q\left( r \right) \right)dr},\,\,\,0\le t\le T+K$$
where $q\left( t \right)={{\lim }_{n,m\to \infty }}\sup \mathbf{E}{{\left| \bar{Y}_{t}^{n,m} \right|}^{2}},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\le t\le T+K.$Therefore, we can use Bihari’s inequality to get q(t) = 0, i.${{\lim }_{n,m\to \infty }}\sup \mathbf{E}{{\left| \bar{Y}_{t}^{n,m} \right|}^{2}}=0$for all 0 ≤ t ≤ T + K.
This shows that $\left( Y,Z \right)\in B_{G}^{2}\left( 0,T+K \right)$solves ABDSDE (1.1). The proof of existence is complete.
(ii) Uniqueness. Let $\left( {{Y}^{i}},{{Z}^{i}} \right)\in B_{G}^{2}\left( 0,T+K \right),\,\,\,i=1,2$be two solutions of ABDSDE (1.1).
Define ${{\bar{Y}}_{t}}=Y_{t}^{1}-Y_{t}^{2},\,\,\,\,\,\,{{\bar{Z}}_{t}}=Z_{t}^{1}-Z_{t}^{2},\Delta g\left( r \right)=g\left( r,Y_{r}^{1},Z_{r}^{1} \right)-g\left( r,Y_{r}^{2},Z_{r}^{2} \right)$and $\Delta f\left( r \right)=f\left( r,Y_{r}^{1},Z_{r}^{1},Y_{r+\delta \left( r \right)}^{1},Z_{r+\zeta \left( r \right)}^{1} \right)-f\left( r,Y_{r}^{2},Z_{r}^{2},Y_{r+\delta \left( r \right)}^{2},Z_{r+\zeta \left( r \right)}^{2} \right).$
We obtain the following equation
$$\left\{ \begin{array}& {{{\bar{Y}}}_{t}}=\int_{t}^{T}{\Delta f\left( r \right)dr+\int_{t}^{T}{\Delta g\left( r \right)d{{B}_{r}}-\int_{t}^{T}{{{{\bar{Z}}}_{r}}d{{W}_{r}},\,\,\,\,\,\,\,0\le t\le T,}}} \\ {{{\bar{Y}}}_{t}}=0,\,\,\,\,{{{\bar{Z}}}_{t}}=0,\,\,\,\,\,\,\,\,\,\,T\le t\le T+K. \\ \end{array}\right.$$
By the same computations as in Lemma 4.1 , we obtain