BSE's, BSDE's and fixed point problems

TL;DR

This paper introduces a unified framework for backward stochastic equations (BSEs), extending classical BSDEs, and demonstrates how fixed point methods can be used to establish existence and uniqueness results for various complex equations.

Contribution

It develops a general approach translating BSEs into fixed point problems, enabling the use of fixed point theorems for a broad class of equations including non-Lipschitz and multidimensional cases.

Findings

Fixed point methods can solve a wide range of BSEs.

Existence and uniqueness results are established for equations with Lipschitz and non-Lipschitz coefficients.

The approach applies to path-dependent, anticipating, and McKean-Vlasov type equations.

Abstract

In this paper, we introduce a class of backward stochastic equations (BSEs) that extend classical BSDEs and include many interesting examples of generalized BSDEs as well as semimartingale backward equations. We show that a BSE can be translated into a fixed point problem in a space of random vectors. This makes it possible to employ general fixed point arguments to find solutions. For instance, Banach's contraction mapping theorem can be used to derive general existence and uniqueness results for equations with Lipschitz coefficients, whereas Schauder-type fixed point arguments can be applied to non-Lipschitz equations. The approach works equally well for multidimensional as for one-dimensional equations and leads to results in several interesting cases such as equations with path-dependent coefficients, anticipating equations, McKean-Vlasov type equations and equations with coefficients of superlinear growth.