Modern Stochastics: Theory and Applications

A solution is given to generalized backward stochastic differential equations driven by a real-valued RCLL martingale on an arbitrary filtered probability space. The existence and uniqueness of a solution are proved via the Yosida approximation method when the generators are only stochastic monotone with respect to the y-variable and stochastic Lipschitz with respect to the z-variable, with different linear growth conditions.

Reflected generalized backward stochastic differential equations (BSDEs) with one discontinuous barrier are investigated when the noise is driven by a Brownian motion and an independent Poisson measure. The existence and uniqueness of the solution are derived when the generators are monotone and the barrier is right-continuous with left limits (rcll). The link is established between this solution and a viscosity solution for an obstacle problem of integral-partial differential equations with nonlinear Neumann boundary conditions.

This paper proves the existence and uniqueness of a solution to doubly reflected backward stochastic differential equations where the coefficient is stochastic Lipschitz, by means of the penalization method.