Hi, I have the following expected value to compute


$E[ \int\_{o}^{T} f(t) dt \int\_{o}^{T} H(s) dW(s)]$,


where $f(t)$ and $H(s)$ are two stochastic processes adapted to the filtration generated by the Brownian motion W.


I think that this expected value could be equal to zero, but I really don't know how to give this proof.


Thank you in advance for any kind of advice or references.


Imma



Compute the expected value of the product between a Lebesgue–Stieltjes type integral and an Ito integral
You can write $h(s)=\int\_0^Tf(t)dtH(s)$ then your expectation could be written as $E[ \int\_{o}^{T} h(s) dW(s)]$. This integral is $0$ if you can prove $(\int\_{o}^{t} h(s) dW(s)) $ to be a martingale, for example $E(\int\_{o}^{T} h^2(s) ds)<+\infty$. 


As you noticed in your comment, this procedure is correct if we can suppose that the process $h(s)$ is adapted, which is the case if $f$ is deterministic. Otherwise we can the decompose the integral as the in the following


$$\int\_0^T(\int\_0^Tf(t)dt)H(s)dW\_s =\int\_0^T(\int\_0^sf(t)dt)H(s)dW\_s+\int\_0^T(\int\_s^Tf(t)dt)H(s)dW\_s$$


then with interverting of the order of integration in the second integral we can write
$$\int\_0^T(\int\_s^Tf(t)dt)H(s)dW\_s=\int\_0^T(\int\_0^tH(s)dW\_s)f(t)dt$$


The process apperaing in the integrals are now adapted, and you can add the condition so that Fubini works and to justify the martangality of the integrals. Hope thsi help.


