Weak compactness of weak solutions sets of forward-backward stochastic differential inclusions

Abstract

The main result of the article deals with weak compactness of weak solutions sets of forward-backward stochastic differential inclusions. The main result is preceded by existence theorem for considered forward-backward stochastic differential inclusions. More precisely, we shall prove that by given set-valued mappings there exists a system consisting of a complete probability space $(\Omega ,\mathcal{F},P)$, an d-dimensional continuous stochastic process X, an $\mathcal{l}$-dimensional càdlàg process Y, and an m-dimensional Brownian motion B defined on the space $(\Omega ,\mathcal{F},P)$ satisfying considered forward-backward stochastic differential inclusion and that each ${\mathbb{F}}^X$-martingale is an ${\mathbb{F}}^{XY}$-martingale.

Keywords:

• Brownian motion
• backward stochastic differential equations
• forward-backward stochastic differential inclusions
• continuous and càdlàg stochastic processes
• weak compactness of sequences of stochastic processes

2001 Mathematics Subject Classification:

• 60H05
• 28B20
• 47H04

Acknowledgments

I am deeply indebted to my collegues Mariusz Michta and Jerzy Motyl for valueble discussions improving the text.

Disclosure statement

No potential conflict of interest was reported by the author