Convergence of stochastic integrals with respect to Hilbert-valued semimartingales

摘要

For sequences of stochastic integrals ∫_0~· K_(s-)~n dX_s~n, functional limit theorems are presented. And stability of strong solutions of stochastic differential equations of type X~n = H~n + ∫_0~· f(X_(s-)~n)dY_s~n, any n ≥ 1 is discussed under jointly weak convergence of driving processes {(H~n,Y~n}_(n ≥ 1). Where Y~n is an H-valued semimartingale, H~n is a G-valued cadlag adapted process, K~n is an L(H, G)-valued cadlag adapted process and f : G → L(H, G) satisfies a Lipschitz condition.