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Suppose we are given two filtered probability spaces $(\Omega, \mathcal F, (\mathcal F_t)_t, \mathbb P)$ and $(\Omega', \mathcal F', (\mathcal F'_t)_t, \mathbb P')$. On $\Omega$ (respectively $\Omega'$) we are a given a standard Wiener process $W$ (respectively $W'$) adapted to the filtration. Suppose also to have a stochastic process $(X_t)_t$ on $\Omega$ such that it makes sense to define its Ito integral $\int_0^T X(t)dW(t)$. Suppose similarly to have a stochastic process $(X'_t)_t$ on $\Omega'$ such that it makes sense to define its Ito integral $\int_0^T X'(t)dW'(t)$. Finally, assume that $X_t \stackrel{d}{=}X'_t$ for any $t$.

Question: is it true that $$\int_0^T X(t)dW(t) \stackrel{d}{=} \int_0^T X'(t)dW'(t)$$ holds? Intuitively, it seems true to me, but I cannot find a proof for it.

user590533
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  • Yes it's true. Prove the statement for elementary process first, and then approximate $X_t$ and $X_t'$ by elementary process to conclude. – Surb Aug 14 '22 at 09:16
  • I see, it's easier than I thought... Do you have a reference for this fact? – user590533 Aug 14 '22 at 11:11
  • No need reference, you have all elements to prove it your self... – Surb Aug 14 '22 at 11:25

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