Euler discretization for Monte Carlo Simulation

[karafrylee]

I tried implementing the Euler discretization to approximate a stochastic process X that satisfies the stochastic differential equation. However, the result I got when compare with the standard Monte Carlo simulation is very far off so I assume I've made an error somewhere in this code. I'm implementing in Java and my Euler discretization equation is

for (int i = 0; i < n; i++) { //n is the no. of runs
for (int j = 0; j < m; j++) { //m is the no. of timesteps
X = X+mu*timestep + sigma*Math.sqrt(timestep)*random.nextGaussian();
}
}

Does anyone know whether I did it wrongly?

 

[trungpham]

As I guess, may be you make a mistake in the expression of X: may be like this
X= X + mu*timestep + sigma*Math.sqrt(j)*random.nextGaussian();
, because as "timestep" is no use in this loop of "j" from 0 to m

 

[Brad Warren]

No, the standard deviation is proportional to the sqrt of the timestep.

The problem is you only have one variable X and you sum all the runs. You need to record the result after each run and start with X at zero again.

 

[trungpham]

Oh, I've got it. Thanks Brad.

 

[karafrylee]

Thanks for your suggestions, I've found out the problem. The standard equation is this:

X = X+mu*X*timestep + sigma*X*Math.sqrt(timestep)*random.nextGaussian();

X should be in ALL the terms, not just the first term. This is a standard Euler equation.