Stability of Numerical Methods for Jump Diffusions and Markovian Switching Jump Diffusions

摘要

This work focuses on stability analysis of numerical solutions to jumpdiffusions and jump diffusions with Markovian switching. Due to the use ofPoisson processes, using asymptotic expansions as in the usual approach oftreating diffusion processes does not work. Different from the existingtreatments of Euler-Maurayama methods for solutions of stochastic differentialequations, we use techniques from stochastic approximation. We analyze thealmost sure exponential stability and exponential $p$-stability. The benchmarktest model in numerical solutions, namely, one-dimensional linear scalar jumpdiffusion is examined first and easily verifiable conditions are presented.Then Markovian regime-switching jump diffusions are dealt with. Moreover,analysis on stability of numerical methods for linearizable andmulti-dimensional jump diffusions is carried out.