Convergence of stochastic integrals with respect to Hilbert-valued semimartingales

For sequences of stochastic integrals ∫ 0 ⋅ 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 + ∫ 0 ⋅ f ( X s - n ) d Y s n , ∀ 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 càdlàg adapted process, K n is an ℒ ( H , G ) -valued càdlàg adapted process and f : G ↦ ℒ ( H , G ) satisfies a Lipschitz condition.