Weak solutions of backward stochastic differential equations with continuous generator

TL;DR

This paper establishes the existence of weak solutions for a class of backward stochastic differential equations with continuous generators, using advanced probabilistic tools like Young measures and Skorokhod space topology.

Contribution

It introduces a novel approach to construct weak solutions for BSDEs with affine generators and sublinear growth, extending the solution framework to an extended probability space.

Findings

Existence of weak solutions for BSDEs with continuous generators.

Use of Young measures and Skorokhod topology in solution construction.

Solution includes a continuous martingale component orthogonal to Wiener processes.

Abstract

We prove the existence of a weak solution to a backward stochastic differential equation (BSDE) $$ Y_t=\xi+\int_t^T f(s,X_s,Y_s,Z_s)\,ds-\int_t^T Z_s\,d\wien_s$$ in a finite-dimensional space, where $f(t,x,y,z)$ is affine with respect to $z$, and satisfies a sublinear growth condition and a continuity condition This solution takes the form of a triplet $(Y,Z,L)$ of processes defined on an extended probability space and satisfying $$ Y_t=\xi+\int_t^T f(s,X_s,Y_s,Z_s)\,ds-\int_t^T Z_s\,d\wien_s-(L_T-L_t)$$ where $L$ is a continuous martingale which is orthogonal to any $\wien$. The solution is constructed on an extended probability space, using Young measures on the space of trajectories. One component of this space is the Skorokhod space D endowed with the topology S of Jakubowski.