Asymptotic Stability of Stochastic Differential Equations Driven by Lévy Noise

Abstract

Using key tools such as Itô's formula for general semimartingales, Kunita's moment estimates for Lévy-type stochastic integrals, and the exponential martingale inequality, we find conditions under which the solutions to the stochastic differential equations (SDEs) driven by Lévy noise are stable in probability, almost surely and moment exponentially stable.

Approximate solutions for a class of doubly perturbed stochastic differential equations

“…If α = β = 0, then DPSDEwJs and DPSDEwMS will become SDEs with jumps and SDEs with Markovian switching which were investigated by [23][24][25][26][27][28][29][30][31][32][33][34]. Similarly, we can also give the Carathéodory approximate solution and show that the Carathéodory approximate solution converges to the solution of SDEs with jumps and SDEs with Markovian switching under our assumptions.…”

Stochastic Stabilization of Dynamical Systems Using Lévy Noise

“…We will need the following technical result from Applebaum, Siakalli [2] to ensure that the solution of (4) can never reach the origin provided that x 0 = 0.…”

“…If (5) does not hold, we may apply the stopping time argument that is used at the end of the proof of Theorem 3.7 in [2]. 2…”

“…Applebaum [1] and references therein). The work presented here is a sequel to that in Applebaum and Siakalli [2] where we studied asymptotic stability for a class of stochastic differential equations (SDEs) with jumps.…”

Almost Sure and Moment Exponential Stability of Regime-Switching Jump Diffusions

“…For processes with Lévy type jumps, additional assumptions are needed to handle the jumps to obtain the "nonzero" property. For instance, Applebaum and Siakalli (2009) and Wee (1999) contain different sufficient conditions. The differences are essentially on the assumptions concerning the jumps.…”