Exponential stability of stochastic differential equations driven by discontinuous semimartingales

Abstract

Both almost sure exponential stability and exponential stability in mean square for stochastic differential equations driven by general semimartingales, which may not be continuous, are investigated under various hypotheses. New methods are introduced to incorporate the discontinuity of semimartingales. The Doleans Dade stochastic exponential, the convergence of nonnegative special semimartingales established by Liptser & Shiryayey [5] as well as the lto formula will play a great role in this paper