We study backward stochastic difference equations (BS$\Delta$Es) driven by a d-dimensional stochastic process on a lattice, whose increments take only $d+1$ possible values that generate the lattice. Interpreting the driving process as a d-dimensional asset price process, we provide applications to an optimal investment problem and to a market equilibrium analysis, where utility functionals are defined via BS$\Delta$Es.

This article considers a dynamic version of risk measures for stochastic asset processes and gives a mathematical benchmark for required capital in a solvency regulation framework. Some dynamic risk measures, based on the expected discounted penalty function launched by Gerber and Shiu, are proposed to measure solvency risk from the company’s going-concern point of view. This study proposes a novel mathematical justification of a risk measure for stochastic processes as a map on a functional path space of future loss processes.

A useful result when dealing with backward stochastic differential equations is the comparison theorem of Peng (1992). When the equations are not based on Brownian motion, the comparison theorem no longer holds in general. In this paper we present a condition for a comparison theorem to hold for backward stochastic differential equations based on arbitrary martingales. This theorem applies to both vector and scalar situations. Applications to the theory of nonlinear expectations are also explored.