For sequences of stochastic integrals , functional limit theorems are presented. And stability of strong solutions of stochastic differential equations of type is discussed under jointly weak convergence of driving processes . Where is an -valued semimartingale, is a -valued càdlàg adapted process, is an -valued càdlàg adapted process and satisfies a Lipschitz condition.
Yingchao XIE. "Convergence of stochastic integrals with respect to Hilbert-valued semimartingales." J. Math. Soc. Japan 57 (3) 735 - 751, July, 2005. https://doi.org/10.2969/jmsj/1158241933