A stopped Doob inequality for stochastic convolution integrals and stochastic evolution equations

Abstract

A stopped Doob inequality is proved for stochastic convolution integrals in Hilbert space, where M is a square integrable Hilbert space valued cadlag martingale, ⊘ an operator valued predictable function and U(t, s) a contraction-type evolution operator. This allows to obtain the mild solution for evolution equations (with memory) where A(t) is the quasigenerator of U(t,s), V(t) a bounded variation process, and Z(t) a semimartingale, under the same weak assumptions on B and D as for stochastic differential equations with bounded coefficients, i .e ., (A( t) = 0) . Moreover, a weak convergence criterion for is derived.

Keywords:

• Stochastic Convolution Integral
• Doob Inequality
• Stochastic Evolution Equation
• Mild Solution
• Weak Convergence