A general theory of Finite State Backward Stochastic Difference Equations

Details A general theory of Finite State Backward Stochastic Difference Equations A general theory of Finite State Backward Stochastic Difference Equations

2008-10-27

arxiv.org/abs/0810.4957v2

Stochastic Processes and their Applications 120 (2010), pp. 442-466

Mathematics

Probability

25 pages, final preprint prior to refereeing

Scientific paper

10.1016/j.spa.2010.01.004

By analogy with the theory of Backward Stochastic Differential Equations, we define Backward Stochastic Difference Equations on spaces related to discrete time, finite state processes. This paper considers these processes as constructions in their own right, not as approximations to the continuous case. We establish the existence and uniqueness of solutions under weaker assumptions than are needed in the continuous time setting, and also establish a comparison theorem for these solutions. The conditions of this theorem are shown to approximate those required in the continuous time setting. We also explore the relationship between the driver $F$ and the set of solutions; in particular, we determine under what conditions the driver is uniquely determined by the solution. Applications to the theory of nonlinear expectations are explored, including a representation result.

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