Existence and Uniqueness of Solutions for BSDEs with Locally Lipschitz Coefficient

Abstract

We deal with multidimensional backward stochastic differential equations (BSDE) with locally Lipschitz coefficient in both variables y, z and an only square integrable terminal data. Let L N be the Lipschitz constant of the coefficient on the ball B(0, N ) of R d × R dr . We prove that if L N = O( √ log N ), then the corresponding BSDE has a unique solution. Moreover, the stability of the solution is established under the same assumptions. In the case where the terminal data is bounded, we establish the existence and uniqueness of the solution also when the coefficient has an arbitrary growth (in y) and without restriction on the behaviour of the Lipschitz constant L N .

Prevalence of backward stochastic differential equations with unique solution

“…Since a set of the second category in a Baire space contains "almost all" the points of the space, it may be thought of as the topological analogue of the measure-theoretical concept of a set whose complement is of measure zero. Our results state that, in some sense, almost all BSDEs with bounded continuous coefficient have solutions which satisfy the above properties (1), (2), and (3). We do not impose any boundedness condition on the terminal data ξ which is assumed to be square-integrable only.…”

“…Up to our knowledge, except for the papers [2,3,15,20], no results are known about when the coefficient is nonuniformly Lipschitz in the two variables (y,z). Moreover, in [15,20], the assumptions imposed on the coefficient are global.…”

Multidimensional BSDEs with super-linear growth coefficient: Application to degenerate systems of semilinear PDEs

“…The existence of solutions is deduced a suitable approximation of (ξ, f ) and an appropriate localization procedure which is close to those given in [1][2][3]. However, in contrast to [3], we do not use the L 2 -weak compactness of the approximating sequence (Y n , Z n ).…”

“…In other hand, the comparison methods (used for one-dimensional BSDEs) do not work in multidimensional case. The present Note is a natural development of [1][2][3] where some results on existence and uniqueness, as well as the stability of strong solutions are established for multidimensional BSDEs with local assumptions on the coefficient f in the two variables y and z. To begin, let ξ be a p-integrable (with 1 < p < 2) and consider the following example of BSDE,…”

Lp solutions of multidimensional BSDEs with weak monotonicity and general growth generators

“…is (F t )-progressively measurable for each (y, z), called the generator of BSDE (1). This BSDE is usually denoted by the BSDE (ξ, T, g).…”