Trotter-Kato Approximations of Stochastic Differential Equations

The objective of this chapter is to study Trotter-Kato approximations of many classes of stochastic differential equations, including equations with delays, McKean-Vlasov equations, neutral stochastic partial differential equations, stochastic integrodifferential equations, uncertain stochastic systems and stochastic evolution equations with jumps in Hilbert spaces. Let us begin with the following semilinear stochastic evolution equation. This could well serve as a motivation to the rest of the book. Let \((X, \langle \cdot , \cdot \rangle _X)\) and \((Y, \langle \cdot , \cdot \rangle _Y)\) be real separable Hilbert spaces unless otherwise specified. Consider the semilinear stochastic evolution equation of the form \(dx(t)= [ Ax(t)+ f(x(t)) ]dt + g(x(t)) dw(t), \quad t>0, x(0) = x_{0},\) where A is the infinitesimal generator of a strongly continuous semigroup \( \{ S(t): t \ge 0 \} \) on X, \( f: X \rightarrow X\) and \( g: X \rightarrow L(Y,X) \) and w(t) is a Y-valued Q-Wiener process. Here, the initial condition \(x_{0}\) is \(\mathscr {F}_{0}\) -measurable with \(E||x_{0}||^p_X < \infty \) , \(p \ge 2\) .