Convergence of stochastic integrals with respect to Hilbert-valued semimartingales

Abstract

For sequences of stochastic integrals ∈t -0^• K^n-{s-}dX^n-s, functional limit theorems are presented. And stability of strong solutions of stochastic differential equations of type
kip3pt X^n=H^n+∈t -0^• f(X^n-{s-})dY^n-s, \quad ∀ n≥q 1
is discussed under jointly weak convergence of driving processes { (H^n, Y^n)}-{n≥q 1}. Where Y^n is an \bm{H}-valued semimartingale, H^n is a \bm{G}-valued càdlàg adapted process, K^n is an {\mathscr L}(\bm{H}, \bm{G})-valued càdlàg adapted process and f:\bm{G}\mapsto {\mathscr L}(\bm{H}, \bm{G}) satisfies a Lipschitz condition.