Stability of stochastic differential equations with respect to time-changed Brownian motions

Abstract

In this paper, the stability behaviors of stochastic differential equations (SDEs) driven by time-changed Brownian motions are discussed. Based on the generalized Lyapunov method and stochastic analysis, necessary conditions are provided for solutions of time-changed SDEs to be stable in differential senses, such as stochastic stability, stochastically asymptotic stability and globally stochastically asymptotic stability. Also, a connection between the stability of the solution to the time-changed SDEs and that to their corresponding non-time-changed SDEs is revealed by applying the duality theorem. Finally, two examples are provided to illustrate the theoretical results.

arXiv comment22 pages

References in corpus (3)

• Stochastic Calculus for a Time-changed Semimartingale and the Associated Stochastic Differential Equations
• SDEs driven by a time-changed Lévy process and their associated time-fractional order pseudo-differential equations
• Stability analysis for a class of nonlinear time-changed systems

Cited by in corpus (4)

• Path stability of the solution of stochastic differential equation driven by time-changed Lévy noises
• Strong approximation of time-changed stochastic differential equations involving drifts with random and non-random integrators
• Stability of stochastic differential equation driven by time-changed Lévy noise
• Strong approximation of stochastic differential equations driven by a time-changed Brownian motion with time-space-dependent coefficients