A new approach to BSDE

Key words: Backward stochastic differential equation, semimartingale, comparison\ud theorem, ordinary functional differential equation, stochastic differential equation, local\ud condition, homogeneous property, K-Lipschitz conditio

Density estimates for solutions to one dimensional Backward SDE's

In this paper, we derive sufficient conditions for each component of the solution to a general backward stochastic differential equation to have a density for which upper and lower Gaussian estimates can be obtained

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

On backward stochastic differential equations and strict local martingales

We study a backward stochastic differential equation whose terminal condition is an integrable function of a local martingale and generator has bounded growth in $z$. When the local martingale is a strict local martingale, the BSDE admits at least two different solutions. Other than a solution whose first component is of class D, there exists another solution whose first component is not of class D and strictly dominates the class D solution. Both solutions are $\mathbb{L}^p$ integrable for any $0<p<1$. These two different BSDE solutions generate different viscosity solutions to the associated quasi-linear partial differential equation. On the contrary, when a Lyapunov function exists, the local martingale is a martingale and the quasi-linear equation admits a unique viscosity solution of at most linear growth.Comment: Keywords: Backward stochastic differential equation, strict local martingale, viscosity solution, comparison theore

Differentiability of backward stochastic differential equations in Hilbert spaces with monotone generators

The aim of the present paper is to study the regularity properties of the solution of a backward stochastic differential equation with a monotone generator in infinite dimension. We show some applications to the nonlinear Kolmogorov equation and to stochastic optimal control

Forward-backward systems for expected utility maximization

In this paper we deal with the utility maximization problem with a general utility function. We derive a new approach in which we reduce the utility maximization problem with general utility to the study of a fully-coupled Forward-Backward Stochastic Differential Equation (FBSDE)