  This work focuses on stability analysis of numerical solutions to jump
diffusions and jump diffusions with Markovian switching. Due to the use of
Poisson processes, using asymptotic expansions as in the usual approach of
treating diffusion processes does not work. Different from the existing
treatments of Euler-Maurayama methods for solutions of stochastic differential
equations, we use techniques from stochastic approximation. We analyze the
almost sure exponential stability and exponential $p$-stability. The benchmark
test model in numerical solutions, namely, one-dimensional linear scalar jump
diffusion 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 and
multi-dimensional jump diffusions is carried out.
