Comparisons for backward stochastic differential equations on Markov chains and related no-arbitrage conditions

Comparisons for backward stochastic differential equations on Markov chains and related no-arbitrage conditions

Samuel N. Cohen and Robert J. Elliott

Source: Ann. Appl. Probab. Volume 20, Number 1 (2010), 267-311.

Abstract

Most previous contributions to BSDEs, and the related theories of nonlinear expectation and dynamic risk measures, have been in the framework of continuous time diffusions or jump diffusions. Using solutions of BSDEs on spaces related to finite state, continuous time Markov chains, we develop a theory of nonlinear expectations in the spirit of [Dynamically consistent nonlinear evaluations and expectations (2005) Shandong Univ.]. We prove basic properties of these expectations and show their applications to dynamic risk measures on such spaces. In particular, we prove comparison theorems for scalar and vector valued solutions to BSDEs, and discuss arbitrage and risk measures in the scalar case.

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Primary Subjects: 60H10

Secondary Subjects: 91B70

Keywords: Backward stochastic differential equation; Markov chains; nonlinear expectation; dynamic risk measures; comparison theorem

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aoap/1262962324
Digital Object Identifier: doi:10.1214/09-AAP619
Zentralblatt MATH identifier: 05678705
Mathematical Reviews number (MathSciNet): MR2582649

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