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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

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  • $\begingroup$ (Not a complete answer.) I'd simulate $W$ 10,000 times as well. By convergence I guess you mean that of the distribution. If you fix $W$, then in general you wouldn't get the full picture. $\endgroup$ – AddSup Apr 6 at 15:20
  • $\begingroup$ Do you mean for each 10000 paths I simulate 10000 $W$? So the resulting random variable can take 100 million values ? $\endgroup$ – user3503589 Apr 6 at 15:22
  • $\begingroup$ I thought that by "the same sample from a normally distributed random variable" you meant you simulate one path for $W$. Maybe I misinterpreted it. I meant to simulate a new path for $W$ for each of the 10,000 paths for $X$, which will result in 10,000 values for the integral. $\endgroup$ – AddSup Apr 7 at 5:36
  • $\begingroup$ In the possible solution I did mean 1 Brownian math. But since at each time $t_i$ in the grid, $X$ can take 10000 values, so if I use 10000 possible Brownian paths, $X_{t_{i+1}}$ could take would 10000^2 values. How do you intend to update $X_{t_i}$ to $X_{t_{i+1}}$ so that you get only 10000 possible values? $\endgroup$ – user3503589 Apr 7 at 10:55

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