How can I simulate the Stochastic integral $\int_0^1 X_sdW_s$ where $X$ is strong solution of of an SDE driven by a Brownian motion independent of $W$(the integrator above). I have already computed $X$ along $10000$ paths an over an equally spaced time grid(containing $500$ points) on the interval $[0,1]$
What recipe should I use to approximate $\int_0^1 X_sdW_s$. If $X$ were deterministic I could simply sample normally distributed random variables and approximate by $\sum_{i=1}^{N-1} X_{t_i}N(0,t_{i+1}-t_i)$ and repeat the procedure 1000 times to get the random variable $\int_0^1 X_sdW_s$
But how do I handle that $X_{t_i}$ can take a 1000 values in the case $X$ is a stochastic process instead of deterministic function? What would be the correct way to approximate the stochastic integral in a mathematically coherent way?
**Possible Solution: Could I approximate $\int_0^1 X_sdW_s(\omega)= \sum_{i=1}^{N-1}X_{t_i}(\omega)N(0,t_{i+1}-t_i)(\omega_i)$. In this sum we multiply $X_{t_i}(\omega)$ for every one of 10000 paths by the same sample from a normally distributed random variable(of course we multiply with a different independent Normal sample as $i$ varies). This would give me $\int_0^1 X_sdW_s$( as a random variable with 10000 possible values). **
I am not sure if this approximation converges in any sense to