Convergence of stochastic integrals with respect to Hilbert-valued semimartingales

Abstract

For sequences of stochastic integrals ${\int }_0^{\cdot }{K}_{s-}^{n}d{X}_s^n$, functional limit theorems are presented. And stability of strong solutions of stochastic differential equations of type $$X^n=H^n+{\int }_0^{\cdot }f\left(X_{s-}^n\right)d{Y}_s^n,\forall n\ge 1$$ is discussed under jointly weak convergence of driving processes $\left\{\right(H^n,Y^n){\}}_{n\ge 1}$. Where $Y^n$ is an $\mathbf{H}$-valued semimartingale, $H^n$ is a $\mathbf{G}$-valued càdlàg adapted process, $K^n$ is an $ℒ(\mathbf{H},\mathbf{G})$-valued càdlàg adapted process and $f:\mathbf{G}\mapsto ℒ(\mathbf{H},\mathbf{G})$ satisfies a Lipschitz condition.