Conditions for interchanging Ito and Riemannian integrals

59 Views Asked by At

Let $X_{s,\ r}$ be an adapted real-valued stochastic process, $s,\ r \geq 0$ and $W_s$ be the standard one-dimensional Wiener process (Brownian motion). Under which condition on $X_{s,\ r}$ does the following hold? For a fixed $t\geq 0$, $$\int_0^t\int_0^t X_{s,\ r} dW_s dr = \int_0^t\int_0^t X_{s,\ r} dr dW_s .$$

Thank you!

————————————————

Edit: Mark Veraar’s comment answers this question.