# interchange stochastic and deterministic integration

If $f$ is a function in $L^2([0,1]^m)$, W is one-dimensional Brownian motion, $a,b \in [0,1]$, are the following two integrals equal?

$$\int_0^1\int_0^{t_{m-1}}\cdots \int_0^{t_2} \left(\int_a^bf(t_1, t_2, \cdots, t_{m-1}, s)ds\right) dW_{t_1}dW_{t_2}\cdots dW_{t_{m-1}}$$

$$\int_a^b\left(\int_0^1\int_0^{t_{m-1}}\cdots \int_0^{t_2}f(t_1, t_2, \cdots, t_{m-1}, s)dW_{t_1}dW_{t_2}\cdots dW_{t_{m-1}}\right)ds$$

I know we can use Fubini's theorem to interchage integral order in deterministic cases, but since here we have mixed deterministic and stochastic integral, I don't know how to make justification.

Thank you for help.

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Do it when $f$ is the characteristic function of $S=S_1\times\dots \times S_p$, where $S_i$ are measurable subsets of $[0,1]$. Then using Itô's isometry, to an arbitrary measurable $S$, then to simple functions and finally for $f$ square integrable. – Davide Giraudo Nov 29 '13 at 12:04
@DavideGiraudo: thanks, I manage to get that the two integrals are both linear continuous mappings from $L^2([0,1]^m)$ to the space of square integrable r.v.s, since they are identical for all simple functions, they are identical for any function in $L^2([0,1]^m)$ – Petite Etincelle Nov 29 '13 at 12:56
OP: Close the question? Or post an answer? – Did Dec 25 '13 at 13:13
@Did: Davide has given an answer in his comment, please close the question, thanks – Petite Etincelle Dec 26 '13 at 14:53
OP: YOU are supposed to close the question, not others. – Did Dec 26 '13 at 17:34