A general theory of Finite State Backward Stochastic Difference Equations

Abstract

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.

25 pages, final preprint prior to refereeing

References in corpus (2)

• Comparisons for backward stochastic differential equations on Markov chains and related no-arbitrage conditions
• Dynamically Consistent Nonlinear Evaluations and Expectations

Cited by in corpus (14)

• Comparisons for backward stochastic differential equations on Markov chains and related no-arbitrage conditions
• Existence, uniqueness and comparisons for BSDEs in general spaces
• A primal-dual algorithm for BSDEs
• BSΔEs and BSDEs with non-Lipschitz drivers: Comparison, convergence and robustness
• Sublinear Expectations and Martingales in discrete time
• Time consistency and moving horizons for risk measures
• Solvability of one kind of forward-backward stochastic difference equations
• Reflections on BSDEs
• Solvability of finite state forward-backward stochastic difference equations
• Maximum principle for stochastic optimal control problem of forward-backward stochastic difference systems
• Maximum principle for stochastic optimal control problem of finite state forward-backward stochastic difference systems
• Reflected Backward Stochastic Difference Equations with Finite State and their applications
• Controllability Metrics on Networks with Linear Decision Process-type Interactions and Multiplicative Noise
• Backward stochastic difference equations on lattices with application to market equilibrium analysis