On the stability of some stochastic integro partial differential equations

Abstract

Stochastic integro partial differential equations of the form; du(x, t )= n i=1 ∂ 2 u(x, t) ∂x 2 dt + F (u(x, t) ,x , t)dt + t 0 K(t − θ)u(x, θ)dθdt +[ f (t)u(x, t )+ g(x, t)]dW (t), are considered, where {W (t ): t ≥ 0} is a standard one-dimensional Wiener process and the kernel K decreases to zero non-exponentially. The behavior of solutions and their convergence to zero are studied. It is proved under suitable conditions that lim t→∞ u(x, t) K(t) = ∞ , almost surely. The considered stochastic integro partial differential equations arise if we consider the Black-Scholes market consists of a bank account or a bond and a stock. These stochastic models can also applied to population dynamics in biology. Mathematics Subject Classifications: 34D20, 60H10, 65H20, 45N05