derive integration by parts for a stochastic integral

Question

The question is to show the following identity:

$\int_{0}^{T}tdW(t) = TW(T)-\int_{0}^{T}W(t)dt$

This can be done quite easily with ito's however the question explicitly says to show the identity by taking the limit of the forward euler method.

So from what I've understood one should be able to do the following:

$\int_{0}^{T}tdW(t)=\sum_{n=1}^N t_n(W(t_n+1)-W(t_n))$.

However I'm stuck after this simple step. I thought about using similar steps that are used when showing:

$\int_T W(t)dW(t) = \frac{W(T)^2}{2}-\frac{T}{2}$.

However I was unsure. Did try to google the problem but pretty much all the solutions used ito's.

Answer 1

Using summation by parts, as suggested, it actually becomes quite clear.

$$\sum_{n=1}^{N}t_n(W(t_{n+1})-W(t_n))=t_N W(t_N)-t_1W(t_1)-\sum_{n=1}^N W(t_n)(t_{n+1}-t_n)$$

With $t_N=T$ and $t_1=0$. This gives:

$$\int_0^TtdW(t)= TW(T)-\sum_{n=1}^N W(t_n)(t_{n+1}-t_n)$$ Taking $n \rightarrow \infty$ gives the Riemann integral. Thus the results ends up being:

$$\int_0^TtdW(t)= TW(T)-\int_0^TW(t)dt$$

Unless I've made some kind of mistake this should be the correct answer

Answer 2

Hint: Use the summation by parts formula (also known as Abel's summation formula):

$$\sum_{k=m}^n f_k (g_{k+1}-g_k) = (f_{n+1} g_{n+1}-f_m g_m) - \sum_{k=m}^n g_{k+1} (f_{k+1}-f_k).$$