LpL^p Solutions of Backward Stochastic Differential Equations with Jumps

Given $p \in (1, 2)$, we study $L^p$-solutions of a multi-dimensional backward stochastic differential equation with jumps (BSDEJ) whose generator may not be Lipschitz continuous in $(y,z)-$variables. We show that such a BSDEJ with a p-integrable terminal data admits a unique $L^p$ solution by approximating the monotonic generator by a sequence of Lipschitz generators via convolution with mollifiers and using a stability result.Comment: Keywords: Backward stochastic differential equations with jumps, $L^p$ solutions, monotonic generators, convolution with mollifier

On Zero-Sum Stochastic Differential Games

We generalize the results of Fleming and Souganidis (1989) on zero sum stochastic differential games to the case when the controls are unbounded. We do this by proving a dynamic programming principle using a covering argument instead of relying on a discrete approximation (which is used along with a comparison principle by Fleming and Souganidis). Also, in contrast with Fleming and Souganidis, we define our pay-off through a doubly reflected backward stochastic differential equation. The value function (in the degenerate case of a single controller) is closely related to the second order doubly reflected BSDEs.Comment: Key Words: Zero-sum stochastic differential games, Elliott-Kalton strategies, dynamic programming principle, stability under pasting, doubly reflected backward stochastic differential equations, viscosity solutions, obstacle problem for fully non-linear PDEs, shifted processes, shifted SDEs, second-order doubly reflected backward stochastic differential equation

Quadratic Reflected BSDEs with Unbounded Obstacles

In this paper, we analyze a real-valued reflected backward stochastic differential equation (RBSDE) with an unbounded obstacle and an unbounded terminal condition when its generator $f$ has quadratic growth in the $z$-variable. In particular, we obtain existence, comparison, and stability results, and consider the optimal stopping for quadratic $g$-evaluations. As an application of our results we analyze the obstacle problem for semi-linear parabolic PDEs in which the non-linearity appears as the square of the gradient. Finally, we prove a comparison theorem for these obstacle problems when the generator is convex or concave in the $z$-variable.Comment: Key Words: Quadratic reflected backward stochastic differential equations, convex/concave generator, $\th$-difference method, Legenre-Fenchel duality, optimal stopping problems for quadratic $g$-evaluations, stability, obstacle problems for semi-linear parabolic PDEs, viscosity solution

On Quadratic g-Evaluations/Expectations and Related Analysis

In this paper we extend the notion of g-evaluation, in particular g-expectation, to the case where the generator g is allowed to have a quadratic growth. We show that some important properties of the g-expectations, including a representation theorem between the generator and the corresponding g-expectation, and consequently the reverse comparison theorem of quadratic BSDEs as well as the Jensen inequality, remain true in the quadratic case. Our main results also include a Doob-Meyer type decomposition, the optional sampling theorem, and the up-crossing inequality. The results of this paper are important in the further development of the general quadratic nonlinear expectations.Comment: 27 page