It is well-known from e.g. Kurtz-Protter 1996 that if $(H_n,X_n)$ is a sequence of semimartingales converging a.s. to $(H,X)$ in the Skorokhod topology, then (under some measurability conditions) we have

$$ \int_0^t H_n(s)d X_n(s) \to \int_0^t H(s)d X(s) $$

in probability.

I need a related result. Let $H$ be a measurable cadlag process. Let $(L^n)$ be a sequence of Levy processes and let $(\widetilde N^n)$ be a sequence of corresponding compensated Poisson random measures, that is

$$ \widetilde N^n(t,A):=\sum_{s\le t} \Delta L^n_s \mathrm{I}( \Delta L^n_s\in A)-E\sum_{s\le t} \Delta L^n_s \mathrm{I}( \Delta L^n_s\in A) . $$

Consider a stochastic integral with respect to the compensated Poisson random measure:

$$ I^n(t):=\int_0^t\int_{\mathbb{R}} f(X(s-),x)\widetilde N^n(ds,dx), $$ where $f$ is some bounded smooth function.

Assume that $(H,L^n)$ converges "nicely" (e.g., almost surely in the Skorokhod topology) to $(H,L)$, where $L$ is a Levy process. Can we say that the corresponding stochastic integrals $I_n$ also converge (almost surely or in probability) to the stochastic integral $$ I(t):=\int_0^t\int_{\mathbb{R}} f(H(s-),x)\widetilde N(ds,dx)? $$


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