Stochastic Integral Help

Question

Let W(t) be a Brownian Motion.

Show that the integral: $$ \int_t^T W(s)ds $$

can be written in terms of the stochastic integral:

$$ \int_t^T (T-s)dW(S) $$

Is there an error with this question? I can only show that it can be written as:

$$ \int_0^T (T-s)dW(S) $$

Cheers

Answer 1

By Itô's formula, $$ \mathrm d(W(s)s)=W(s)\,\mathrm ds+s\,\mathrm dW(s), $$ so integrating between $t$ and $T$ yields $$ T\,W(T)-t\,W(t)=\int_t^TW(s)\,\mathrm ds+\int_t^Ts\,\mathrm dW(s), $$ whereby $$ \int_t^TW(s)\,\mathrm ds=T\int_t^T\,\mathrm dW(s)+(T-t)W(t)-\int_t^Ts\,\mathrm dW(s) =(T-t)W(t)+\int_t^T(T-s)\,\mathrm dW(s). $$