Mild Solution of Stochastic Equations with Lèvy Jumps: Existence, Uniqueness, Regularity and Stability (Journal of Partial Differential Equations, 2013)
The existence and uniqueness of mild solution to stochastic equations with jumps are established, a stochastic Fubini theorem and a type of Burkholder-Davis-Gundy inequality are proved, and the two formulas are used to study the regularity property of the mild solution of a general stochastic evolution equation perturbed by Levy process. Then the authors prove the moment exponential stability, almost sure exponential stability and comparison principles of the mild solution. As applications, the stability and comparison principles of stochastic heat equation with Levy jump are given.
Keywords: Stochastic evolution equation; Levy processes; mild solution; stability