The Annals of Probability

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

Patrick Cheridito and Kihun Nam

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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


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