Variance of Itô integral on general time interval

Question

I have a question concerning the variance of the Itô integral on general time intervals, i.e. I want to calculate \begin{align*}\operatorname{Var}\left(\int_s^T f(t)dW_t\right),\end{align*} where $f: [0, T]\to \mathbb{R}$ is deterministic, $W_t$ is the standard Wiener-process and $0<s<T$. For $s=0$ it is a standard fact that this equals $\int_0^Tf(t)^2dt$. I suspect that this should also hold for $s>0$ as it is true for step processes. Is this true or am I missing something?

Answer 1

For any $s\in[0,T)$, the variance in question is $$\int_s^T f(t)^2\,dt.$$ This is shown just as in the case $s=0$.

Alternatively, the case $s\in(0,T)$ can be reduced to the case $s=0$ by noting that

• the random function $[0,T-s]\ni u\mapsto W^s_u:=W_{s+u}-W_s$ is a standard Wiener process on the interval $[0,T-s]$;

• $\int_s^T f(t)\,dW_t=\int_0^{T-s} f(s+u)\,dW^s_u$;

• the variance of $\int_0^{T-s} f(s+u)\,dW^s_u$ is $\int_0^{T-s} f(s+u)^2\,du=\int_s^T f(t)^2\,dt$.