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Stability and stochastic stabilization of numerical solutions of regime-switching jump diffusion systems. (English) Zbl 1286.65013

Necessary and sufficient conditions are derived for almost sure and for \(p\)th moment exponential stability of regime-switching jump diffusion systems of the form \[ dx(t)= f(x(t), r(t))\,dt+ g(x(t), r(t))\,dw(t)+ h(x(t-), r(t-))\,dN(t),\tag{1} \] where \(r(t)\) is a Markov chain, \(w(t)\) is a scalar Brownian motion, \(N(t)\) is a scalar Poisson process, and \(r\), \(w\), \(N\) are independent. Conditions yielding exponential stability of the Euler-Maruyama (EM) and the backward Euler-Maruyama (BEM) numerical approximation schemes for equation (1) are derived. It is shown that a linear growth condition on \(f\) is required to obtain almost sure exponential stability of the EM approximations, but is not required for the BEM approximations.
Circumstances are identified under which approximations that are obtained from these numerical methods retain stability properties of the solution of the original system.

MSC:

65C30 Numerical solutions to stochastic differential and integral equations
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
65L20 Stability and convergence of numerical methods for ordinary differential equations
34F05 Ordinary differential equations and systems with randomness
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