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Introduction
Backward stochastic differential equations (BSDEs in short) were first introduced by Pardoux and Peng [7]. They proved the celebrated existence and uniqueness result under Lipschitz assumption. This pioneer work was extensively used in many fields like stochastic interpretation of solutions of PDEs and financial mathematics.
Few years later, several authors investigated BSDEs with respect to fractional Brownian motion $\begin{array}{}
\big(B^H_t\big)_{t\geq 0}
\end{array}$ with Hurst parameter H. Since BH is not a semimartingale when H ≠ $\begin{array}{}
\frac{1}{2}
\end{array}$, we cannot use the beautiful classical theory of stochastic calculus to define the fractional stochastic integral. It is a significant and challenging problem to extend the results in the classical stochastic calculus to this fractional Brownian motion. Essentially, two different types of integrals with respect to a fractional Brownian motion have been defined and studied. The first one is the pathwise Riemann-Stieltjes integral (see Young [10]). This integral has a proprieties of Stratonovich integral, which leads to difficulties in applications. The second one, introduced in Decreusefond and Ustunel [3] is the divergence operator (or Skorohod integral), defined as the adjoint of the derivative operator in the framework of the Malliavin calculus. Since this stochastic integral satisfies the zero mean property and it can be expressed as the limit of Riemann sums defined using Wick products, it was later developed by many authors.
Recently, new classes of BSDEs driven by both standard and fractional Brownian motions were introduced by Fei et al [4]. They established the existence and uniqueness of solutions.
In this paper, our aim is to generalize the result established in [2] to the following equation called fractional BSDE under stochastic conditions on the generator:
where $\begin{array}{}
\left(B_{1,t}^{H_1}\right)_{t\geq 0}\text{
and }\left(B_{2,t}^{H_2}\right)_{t\geq 0}
\end{array}$ are two mutually independent fractional Brownian motions. The novelty in these types of stochastic equations lies in the fact of coupling two mutually independent fractional Brownian motions. In this work, the authors established some properties of solutions of a fractional BSDE with Lipschitz coefficients. By the help of the fixed point principle, we establish existence and uniqueness of solutions.
The paper is organized as follows: In Section 2, we introduce some preliminaries, before studying the solvability of our equation under Lipschitz conditions on the generator in Section 3. Using this result, we prove existence and uniqueness of the solution with a coefficient satisfying rather weaker conditions.
Fractional Stochastic calculus
Let us assume given two mutually independent fractional Brownian motions $\begin{array}{}
B^{H}\in\left\{B_1^{H_1}, B_2^{H_2}\right\}
\end{array}$ with Hurst parameter H ≥ $\begin{array}{}
\frac{1}{2}
\end{array}$ is given.
Let Ώ be a non-empty set, ℱ a σ−algebra of sets Ώ, P a probability measure defined on ℱ and {ℱt, t ∈ [0, T]} a σ−algebra generated by both fractional Brownian motions.
The triplet (Ώ, ℱ, P) defines a probability space and E the mathematical expectation with respect to the probability measure P.
The fractional Brownian motion BH is a zero mean Gaussian process with the covariance function
Note that, for any t ∈ [0, T], 〈ξ, η〉t is a Hilbert scalar product. Let 𝓗 be the completion of the set of continuous functions under this Hilbert norm ∥⋅∥t and (ξn)n be a sequence in 𝓗 such that 〈ξi,ξj〉T = δij.
Let $\begin{array}{}
\mathscr{P}_T^{H}
\end{array}$ be the set of all polynomials of fractional Brownian motion $\begin{array}{}
\big(B^H_t\big)_{t\geq 0}.
\end{array}$ Namely, $\begin{array}{}
\mathscr{P}_T^{H}
\end{array}$ contains all elements of the form
Let T > 0 be fixed throughout this paper. Let $\begin{array}{}
\left\{B_{1,t}^{H_1}\right\}_{t\in[0,T]}\text{ and }\left\{B_{2,t}^{H_2}\right\}_{t\in[0,T]}
\end{array}$ be two mutually independent fractional Brownian motions processes, with respectively H1 ≥ $\begin{array}{}
\frac{1}{2}
\end{array}$ and H2 ≥ $\begin{array}{}
\frac{1}{2}
\end{array}$, defined on a probability space (Ώ,ℱ,P). Let N denote the class of P-null sets of ℱ.
where for any process {ψt}t≥0, $\begin{array}{}
{\mathscr F}^\psi_{s,t}
\end{array}$ = σ{ψr − ψs, s ≤ r ≤ t} ∨ 𝒩.
For every ℱ-adapted random process α = (α(t))t≥0 with positive values, we define an increasing process (A(t))t≥0 by setting A(t) = $\begin{array}{}
\int_0^t\alpha^2(s)ds.
\end{array}$
For a fixed β > 0, we will use the following sets:
$\begin{array}{}
\mathscr{C}_{\mbox{pol}}^{1,2}\left([0, T]\times {\bf R}\right)
\end{array}$ is the space of all 𝒞1,2-functions over [0, T] × R, which together with their derivative is of polynomial growth.
ℒ2(β, ℱt, R) = {ξ : Ώ → R ∣ ξ is ℱt − measurable, E [eβA(T)∣ξ∣2] < +∞},
$\begin{array}{}
\widetilde{\cal V}_{[0,T]}^{\beta} \text{ and }\widetilde{\cal V}_{[0,T]}^{a,\beta}
\end{array}$ are the completion of 𝒱[0,T] under the following norm
$\begin{array}{}
{\mathscr B}^2{([0,T],{\bf R})}=\widetilde{\cal V}_{[0,T]}^{\alpha,\beta}\times\widetilde{\cal V}_{[0,T]}^{\beta}\times\widetilde{\cal V}_{[0,T]}^{\beta}
\end{array}$ is a Banach space with the norm
Let us define for a process δ ∈ {Y, Z1, Z2, U, V1, V2}, δ = δ − δ′ where $\begin{array}{}
\delta'\in\widetilde{\mathscr V}_{[0,T]}^{\beta}
\end{array}$ and the function
It is known that, by Proposition 3.3, $\begin{array}{}
\mathbb{D}^{H_1}_s\overline Y_s=\frac{\hat{\sigma}_1(s)}{\sigma_1(s)}\overline Z_{1,s}\text{ and }\mathbb{D}^{H_2}_s\overline Y_s=\frac{\hat{\sigma}_2(s)}{\sigma_2(s)}\overline Z_{2,s}.
\end{array}$
Thus, the mapping (U, V1, V2) ⟼ Γ(U, V1, V2) = (Y, Z1, Z2) determined by the fractional BSDE (3.2) is a strict contraction on ℬ2([0, T],R). Using the fixed point principle, we deduce the solution to the fractional BSDE (3.2) that exists uniquely. This completes the proof.□
The case of weak stochastic Lipschitz coefficient
Aassumptions
In the following, we assume that f satisfies assumptions (H2):
There exist three non-negative processes {μ(t)}0≤t≤T, {ν(t)}0≤t≤T and {ϑ(t)}0≤t≤T such that:
for any t ∈ [0, T], μ(t), ν(t), ϑ(t) are ℱt-measurable,
for any t ∈ [0, T], $\begin{array}{}
\displaystyle
x,y,y',z_1,z'_1,z_2,z'_2
\end{array}$ ∈ R, we have
Let us define for a process δ ∈ {Y, Z1, Z2}, n, m ≥ 1, δn,m = δn+m − δn and the function Δ f(n,m) (s) = $\begin{array}{}
\displaystyle
f(s, \eta_s, Y_s^{n+m-1}, Z_{1,s}^{n+m}, Z_{2,s}^{n+m}) - f(s, \eta_s, Y_s^{n-1}, Z_{1,s}^n, Z_{2,s}^n)
\end{array}$.
Then, it is obvious that $\begin{array}{}
\displaystyle
(\overline Y^{n,m}, \overline Z_1^{n,m}, \overline Z_2^{n,m})
\end{array}$ solves the fractional BSDE
which means that {φn(t), t ∈ [T1, T]}n≥1 is an equicontinuous family of function. Therefore, by the Arzelá-Ascoli theorem, we can define by φ(t) the limit function of (φn(t))n≥1.
Exploiting the argument developed in [[1], Theorem 3.9] we prove that the sequence $\begin{array}{}
\displaystyle
\left(Y^n,Z_1^n,Z_2^n\right)
\end{array}$ is a Cauchy sequence in ℬ2([T1, T], R). Letting n → + ∞ in eq.(3.11), we obtain
In other words, we have shown the existence of the solution (Y, Z1, Z2) to fractional BSDE (3.2) on [T1, T]. Finally, by iteration, one can deduce the existence on [T − λ(T − T1), T], for each λ, and therefore the existence on the whole [0, T].
Let $\begin{array}{}
\displaystyle
\left(Y_t^i, Z_{1,t}^i, Z_{2,t}^i\right)_{0\leq t\leq T}
\end{array}$, i = 1, 2, be two solutions of fractional BSDE (3.2).
Using the same method as in the proof of Lemma (3.5), we have
From the comparison theorem of ODE, we know that $\begin{array}{}
\displaystyle
{\bf E}\left[e^{\beta t}|Y_t^1 - Y_t^2|^2\right]\leq r(t)
\end{array}$, where r(t) is the maximum of solution of (3.10) on [0, T]. As a consequence, we have $\begin{array}{}
\displaystyle
Y_t^1=Y_t^2
\end{array}$ for t ∈[0, T]. From (3.19), we deduce $\begin{array}{}
\displaystyle
\left(Z_{1,t}^1, Z_{2,t}^1\right)= \left(Z_{1,t}^2, Z_{2,t}^2\right)
\end{array}$ for t ∈[0, T]. This completes the proof. □