Simulation from a process given by "complicated" SDE

[econmajor]

Actually this is more of a simulation question but since PF doesn't have Simulation category I ask here.
I need to simulate a path from a proces given by this Stochastic DE:
$$ dX_t = -a(X_t-1)dt+b\sqrt{X_t}dB_t $$ where $B_t$ is wiener process/brownian motion and a and b are just some constants. In order to design a simulation scheme to this process I need to find it's distribution. Please help me find the distribution. I don't know whether this is advanced or Intermediate?

 

[andrewkirk]

The SDE is that of the Cox-Ingersoll-Ross model that is used for processes like interest rates. If you look up the wiki page on that model you'll find information about the distribution of a future value $X_t$.

 

[econmajor]

How will a Log Euler scheme for this process look like? I still haven't a proper way to construct a simulation.

 

[andrewkirk]

To simulate a random sequence of projected values of $X_t$ over a period $[0,T]$ with time steps of length $dt\triangleq T/n$, you just start with the initial value $X_0$, generate a set of $n$ independent standard normal pseudo-random numbers $Z_1,...,Z_n$ then apply the above equation $n$ times for $j=1$ to $n$, with $t=t_j$ taking the value $(j-1)dt$ and $dW_{t_j} = Z_j \sqrt{dt}$.

Repeat $m$ times, where $m$ is the number of simulated paths you want, using a different random sequence of $Z_j$ each time.