Let $V^n(t,\omega)$ be a sequence of continuous, adapted and bounded variation processes such that with probability 1, $V^n$ converges to $B$ uniformly on compact intervals of $[0,\infty)$ ($B$ is standard 1 dimensional Brownian motion). For $X \in L^2$ and adapted, is it true that $\int_0^T X_t dV_n$ converges to $\int_0^TX_t dB_t$ in some sense?
2026-04-04 00:35:35.1775262935
Approximation of Stochastic integral with Stieltjes integrals
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The answer is negative. They converge to Stratonovich integral. References : Karatzas and Shreve book and http://www.sciencedirect.com/science/article/pii/0047259X8390043X