BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS WITH STOCHASTIC MONOTONE COEFFICIENTS

We prove an existence and uniqueness result for backward stochastic differential equations whose coefficients satisfy a stochastic monotonicity condition. In this setting, we deal with both constant and random terminal times. In the random case, the terminal time is allowed to take infinite values. But in a Markovian framework, that is coupled with a forward SDE, our result provides a probabilistic interpretation of solutions to nonlinear PDEs.


Introduction
Backward stochastic differential equations (BSDEs), introduced by Pardoux and Peng [10], have been intensively studied in the last years.This class of equations is a powerful tool to give probabilistic formulas for solutions of semilinear partial differential equations (PDEs).We refer the reader to [8,9] for a good presentation of BSDEs and their connections to PDEs.These equations have found a broad area of applications, namely, in stochastic optimal control (see [7]), mathematical finance (see [6]).Many existence and uniqueness results have been proved in relaxing the uniform Lipschitz condition on the coefficient.Among others, we refer to those with monotonicity condition (see [1,3,4]).In this setting (in relaxing the Lipschitz condition), Bender and Kohlmann [2] recently considered the so-called stochastic Lipschitz condition introduced by El Karoui and Huang [5] and dealt with BSDEs with random terminal time.Indeed, the Lipschitz coefficient is allowed to be an Ᏺ t -adapted process.Doing so, one must reinforce the integrability conditions on the data as well as on the solutions.The interest in this type of extension of the classical existence and uniqueness result comes from the fact that, in many applications, the usual Lipschitz condition cannot be satisfied.For example, the pricing of a European claim is equivalent to solving the linear BSDE where ξ is the contingent claim, r(t) is the interest rate, θ(t) is the risk premium vector, and T is the terminal time.Both r(t) and θ(t) are not bounded in general.Therefore, the generator satisfies the so-called stochastic Lipschitz condition which means that the Lipschitz constant is a stochastic process.
In this paper, we continue this study by considering BSDEs with stochastic monotone coefficients.For example, our result treats generators of the following type: f (t, y,z) = µ(t)g(t, y) + h(t,0,z), (t, y,z) ∈ [0,T] × R × R d , where µ(t) is a nonnegative Ᏺ t -adapted process, g satisfies a monotonicity condition in y and h is stochastic Lipschitzian in z.It is not possible to apply the results of [2,4].Our aim is to prove the existence and uniqueness of solutions for both constant and random terminal times.When the terminal time is random, it is allowed to take values in [0,+∞].
The paper is organized as follows.In Section 2, we give some notations, state the assumptions, and define the BSDEs we are concerned with.Section 3 treats the nonrandom terminal time case, and Section 4 deals with the random one.

Notations, assumptions, and definitions
2.1.Notations.Let W = {W t ,Ᏺ t , t ≥ 0} be an n-dimensional Brownian motion defined on a probability space (Ω,Ᏺ,P).{Ᏺ t , t ≥ 0} stands for the natural filtration of W, augmented with the P-nul sets of Ᏺ.The inner product of R d is denoted by Let β > 0, τ be a positive real-valued random variable and a a nonnegative Ᏺ t -adapted process.We define the increasing process A(t) = t 0 a 2 (s)ds and consider the spaces: is a Banach space with the norm (Y ,Z) 2 β = aY 2 β + Z 2 β .We denote by ᏹ c (β,a,τ) the subspace of ᏹ(β,a,τ) defined as follows:

Assumptions and definitions. Let
) is progressively measurable, and let ξ be an R d -valued Ᏺ τ -measurable random variable.

Existence and uniqueness on fixed time interval
Throughout this section, τ is a fixed positive real number and C will denote a positive constant which may vary from line to line.
Then, for β sufficiently large, the following holds: where C(β) is a constant which depends on β.

Existence.
To reach our goal, we need first to establish the following technical result.
Part I. We set ξ = e (β/2)A(τ) |ξ| and assume that where ρ n : R d → R + is a sequence of smooth functions with compact support in the ball B(0,1) which approximate the Dirac measure at 0 and satisfy R d ρ n (u)du = 1.Clearly, h n (t,•) is a sequence of smooth functions with compact support satisfying the following: But, in view of (3.6) and (d), we have which justifies the choice of the integer q(n).The rest of this part is based on the following two lemmas.
Lemma 3.4.Under (H1), (H2), (H3), (H4), and (H5), for β sufficiently large, the following holds: where (3.18) We have Let C(δ,τ) denote a positive constant which may vary from line to line.We have where (3.36) The fact that I 1 tends to zero is obtained by a similar argument as in the proof of Lemma 3.5.
Proof.For a fixed (U,V ) ∈ ᏹ(β,a,τ), thanks to Proposition 3.3 and Corollary 3.2, the BSDE has a unique solution.So, we can define the mapping and (ii)(b) of Proposition 3.1 to obtain that Hence, if β is sufficiently large, Π is a contracting mapping and its unique fixed point solves our BSDE.

Random terminal time
In the sequel, we assume that (H1) to (H5) hold with τ being a random terminal time, which is allowed to take values in [0,+∞].
The existence result is based on the following sequence.
For each n We end this section by specifying what we call a solution of our BSDE.If τ is a random time, then a solution of the BSDE with data
Therefore, the second term on the right-hand side of (3.38) can be made arbitrarily small by choosing r large enough.Now, since y → h(•, y) is continuous, we deduce from (3.35) that for fixed s, X n s → 0 almost surely as n → ∞.So, it follows from (H2), (H3)(ii), Fubini's theorem, and Lebesgue dominated convergence theorem that the first term of (3.38) goes to zero as n → ∞.