Stability for multidimensional jump-diffusion processes
Consider an \(n\)-dimensional jump-diffusion process \(\{X^x_t\}\) satisfying \[ X^x_t=x+\int _0^t\mu (X^x_{s-}) ds + \int _0^t\sigma (X^x_{s-}) dB_s + \int _0^t \int c(X^x_{s-},u)\widetilde \nu (ds,du), \] where \(\mu (x)\) and \(c(x,u)\) are \(R^n\)-valued and \(\sigma (x)\) is \(n\times m\)-matrix valued for \(x,u\in R^n\), \(\{B_t\}\) denotes a standard \(m\)-dimensional Brownian motion and \(\widetilde \nu (ds,dy)\) a compensated Poisson random measure. Sufficient conditions for existence and uniqueness of an invariant measure, for stability in distribution of solutions, for stochastic stability, stochastic asymptotic stability and a.s. exponential stability of an equilibrium solution are given.
scientific article; zbMATH DE number 1554298
scientific article