Approximating stochastic integral with Euler scheme

Question

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 $W_Tf(X_0)$?

Answer 1

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:

1. We have some results on the convergence rate of this method (see this)
2. Other schemes exist such as the Milstein scheme which offers better convergence rate ($O(1/n)$-error in $L^p)$.