Expected Value of product of Ito's Integral

Question

Any idea on how to compute the expected value of product of Ito's Integral with two different upper limit?

For example: $$\mathbb{E}\left[\int_0^r f(t)\,dB(t) \int_0^s f(t)\,dB(t)\right]$$

I only know how to compute when the upper limit r and s are the same...but don't know how when r and s are different...help.

Answer 1

You can split the integrals up into parts over their domain. The part where they overlap can use the usual formula, and the variables are independent on the part where they don't overlap, so those expectations are products of the expectation of the factors.

Answer 2

Let $g(t)=f(t)\mathbf 1_{[0,r]}(t)$ and $h(t)=f(t)\mathbf 1_{[0,s]}(t)$ then $$\int_0^r f(t)\,\mathrm dB(t)=\int_0^\infty g(t)\,\mathrm dB(t),\qquad \int_0^r f(t)\,\mathrm dB(t)= \int_0^\infty h(t)\,\mathrm dB(t)$$ hence $$\mathbb{E}\left[\int_0^r f(t)\,\mathrm dB(t)\cdot \int_0^s f(t)\,\mathrm dB(t)\right]=\int_0^\infty g(t)h(t)\mathrm dt=\int_0^{\min(r,s)}f^2(t)\mathrm dt.$$