After solving for the Ornstein-Uhlenbeck process, i'm still stuck with a stochastic integral that I can't solve.
The Ornstein-Uhlenbeck process:
$X_{t} = X_{0} e^{-\theta t}+ \mu(1-e^{-\theta t}) + \sigma e^{-\theta t} \int_{0}^{t} e^{\theta s}dW_{s} $
Does anyone have an idea on how to solve this stochastic integral? ($\int_{0}^{t} e^{\theta s}dW_{s} $)
*Note that i'm trying to evaluate this expression for a Monte-Carlo simulation. I have already tried discretizing the integral but I would like to improve my results by using the exact solution.
Thanks in advance!
If you only want to simulate that integral, then nothing is mysterious here.
It's just a normal variable of law $ \mathcal{N}\left( 0, \int_{0}^t e^{2 \theta s}.ds \right)$.
If you want to so look at $(W_s)$ as a stochastic process then you can see easily that $ (U_t ):= \left( \int_{0}^t e^{ \theta s}.dW_s \right) $ is just a change time Brownian process. To be more precise,
Letting $g(t)= \int_{0}^t e^{2 \theta s}.ds$.
Roughly speaking, there is a Brownian process $\tilde{W}$ such that:
$(U_t)=\left( \tilde{W}_{g(t)} \right)$