# Simulation of pinned diffusion process

Suppose I have a stochastic differential equation (in the Ito sense): $$dX_t = \mu(X_t)\,dt + \sigma(X_t)\, dW_t$$ in $\mathbb{R}^n$, where I know that $X_0=a$ and $X_T=b$. In other words, the process has been "pinned" at fixed times $0$ and $T$.

I want to know how to simulate such an equation (i.e. produce trajectories numerically).

I've seen some questions ( 1, 2, 3, 4, 5 ) on the "Brownian Bridge", which is a special case of this.

Edit (081617): it appears that this process is also referred to as an Ito bridge or as a diffusion bridge. It turns out this is not as easy as I'd hoped. A promising paper (found with these better search terms) is Simulation of multivariate diffusion bridges by Bladt et al. Any help/suggestions are still appreciated!

• Have you tried Euler Maruyama method to simulate it ? Aug 16, 2017 at 19:32
• @Khosrotash How can I apply Euler-Maruyama to a pinned diffusion? Aug 16, 2017 at 19:41
• Do you mean $$@ t=0 \to x(0)=a \\@t=T \to x(T)=b$$ and $a,b$ are assumed ? Aug 16, 2017 at 19:44
• @Khosrotash Yes indeed, all are fixed or known in advance. Aug 16, 2017 at 19:52