Approximating stochastic integral with Euler scheme I am a bit confused on how to numerically calculate stochastic integral:
$$\int_0^Tf(X_s)dW_s$$
Suppose that I dont want to make partition of $(0,T)$.
Then, how should integral be approximated $W_Tf(X_T)$ or $W_Tf(X_0)$?
 A: We will denote $\{Y_t\}_{t\geq 0}$ by $Y_t = \int_0^t f(X_s)dW_s$. The Euler-maruyama scheme consists of numerically (and recursively) estimate $Y$ at different point times. Assume that we have a tenor $\{t_1,...,t_n\}$ where $t_n=T$. We have that by linearity of the stochastic integral:
\begin{align}
Y_{t_{i+1}}-Y_{t_i} = \int_{t_i}^{t_{i+1}}f(X_s)dW_s
\end{align}
The above quantity can be estimated if we come back to definition of the stochastic integral:
\begin{align}
Y_{t_i+1}-Y_{t_i} &\approx f(X_{t_i})\left(W_{t_{i+1}}-W_{t_i}\right) \\
&= f(X_{t_i}).\sqrt{t_{i+1}-t_{i}}\mathcal{N}(0,1)
\end{align}
Overall we have that, 
\begin{align}
\forall i\in\{1,\dots, n\} \quad Y_{t_i+1} = Y_{t_i} + f(X_{t_i}).\sqrt{t_{i+1}-t_{i}}\mathcal{N}(0,1)
\end{align}
Note that in the above, I assumed that $f(X_t)$ is known at each point of time. 
Some remarks: 


*

*We have some results on the convergence rate of this method (see this)

*Other schemes exist such as the Milstein scheme which offers better convergence rate ($O(1/n)$-error in $L^p)$.

