## The Annals of Probability

- Ann. Probab.
- Volume 45, Number 6A (2017), 3795-3828.

### BSE’s, BSDE’s and fixed-point problems

Patrick Cheridito and Kihun Nam

#### 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 establish the existence of a solution. 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.

#### Article information

**Source**

Ann. Probab., Volume 45, Number 6A (2017), 3795-3828.

**Dates**

Received: August 2015

Revised: September 2016

First available in Project Euclid: 27 November 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1511773664

**Digital Object Identifier**

doi:10.1214/16-AOP1149

**Mathematical Reviews number (MathSciNet)**

MR3729615

**Zentralblatt MATH identifier**

06838107

**Subjects**

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 47H10: Fixed-point theorems [See also 37C25, 54H25, 55M20, 58C30]

**Keywords**

Backward stochastic equation backward stochastic differential equation path-dependent coefficients anticipating equations McKean–Vlasov-type equations coefficients of superlinear growth

#### Citation

Cheridito, Patrick; Nam, Kihun. BSE’s, BSDE’s and fixed-point problems. Ann. Probab. 45 (2017), no. 6A, 3795--3828. doi:10.1214/16-AOP1149. https://projecteuclid.org/euclid.aop/1511773664