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!

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


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