Testing numerical solvers with analytic solution to Ornstein-Uhlenbeck SDE?

Question

I have an SDE I want to solve numerically that is fairly close to the Ornstein-Uhlenbeck process: $$ dx_t=θ(μ−x_t)dt+σdW_t $$ which has analytic solution: $$ x_t=x_0e^{−θt}+μ(1−e^{-θt})+\sigma\int_{t_0}^{t}e^{θ(s−t)}dW_s $$ I'm working on implementing some of the numerical solvers from Kloeden and Platen and I'd like to benchmark them against this SDE (as suggested in chapter 4.4), but I'm a tad confused. Doesn't the integral $\int_{t_0}^{t}e^{θ(s−t)}dW_s$ beg the question, in that you need a working numerical solver to calculate it? Any insight would be appreciated.

Answer 1

By solving the SDE as you have, you no longer have an $x_t$ term on the right hand side. You still need to numerically evaluate the stochastic integral, but it doesn't require knowing $x_t$ anymore, as it would if you instead tried to do,

$$ x_t = x_0 + \theta\mu t - \theta\int_0^t x_s\, ds+\sigma W_t $$