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November 2017 BSE’s, BSDE’s and fixed-point problems
Patrick Cheridito, Kihun Nam
Ann. Probab. 45(6A): 3795-3828 (November 2017). DOI: 10.1214/16-AOP1149


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.


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Patrick Cheridito. Kihun Nam. "BSE’s, BSDE’s and fixed-point problems." Ann. Probab. 45 (6A) 3795 - 3828, November 2017.


Received: 1 August 2015; Revised: 1 September 2016; Published: November 2017
First available in Project Euclid: 27 November 2017

zbMATH: 06838107
MathSciNet: MR3729615
Digital Object Identifier: 10.1214/16-AOP1149

Primary: 47H10 , 60H10

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

Rights: Copyright © 2017 Institute of Mathematical Statistics


Vol.45 • No. 6A • November 2017
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