# how to sample a conditioned diffusion

there are several reasons why we could be interested in sampling conditioned diffusions:

• if we observed a diffusion at discrete time and want to do some kind of inference on the parameters of the diffusion, we often need to interpolate in between (missing data)
• physicists/chemist often study the behaviour of a particle in a double well potential: how do you simulate one transition between the two meta-stable positions ?

To be concrete, let us say that we want to sample/study (compute some reasonable approximation of path functionals, say) the conditioned SDE $$dX_t = -\nabla U(X_t) \ dt + dW_t, \qquad X_0=x_{-}, \; X_{T}=x^+.$$ If $T$ is sufficiently small, since this is easy to write the density of the conditioned SDE with respect to the Brownian bridge measure, under mild assumptions on the potential $U$ we could hope that an accept-reject approach might work: this is not the case we are interested. Instead, we suppose that $T$ is large in some sense so that a Brownian bridge is very unlikely to look like a path of this conditioned SDE: the accept reject method does not work.

Question: what is the right way to attack this problem ?

• we could start from any path, and at each iteration choose random times $t$ and $t+\Delta t$ and update in between by the usual accept-reject method: if $\Delta t$ is small enough, this might be not too bad.
• we could run a MCMC on the (discretized) path space - articles have been written, this works reasonably well in theory, does not work in practice
• we could try a h-transform: we do not know or want to approximate the transition densities of the conditioned diffusion

Since this situation looks sufficiently general, and should have showed up in different places, I was wondering if there were fundamentally different (i.e. better) approaches to tackle the curse of dimensionality (ie: $T$ is large) inherent to this problem.

## 2 Answers

You might take a look at http://cims.nyu.edu/~eve2/rare_events.htm

• thanks for these references - these are the kind of approaches I had in mind. – Alekk May 23 '10 at 21:09

Hi Alekk I don't know if you can sample from it but I think that Fabrice Baudoin has written papers on what he calls "Conditional SDEs" which looks related to your problem.

Ohterwise there is litterature about what is called (final)-"augmentation of filtration" (I m sure you know about it). When there is sufficient regularity conditions you can view it as a Girsanov transform. So you can evaluate the girsanov transform explictly that results only in a chagne in the drift in the original SDE and then sample from this new SDE.

Hope this helps

• Thank you The Bridge - these approaches are very related to what we usually call a Doob H-transform. As you said, a conditioned SDE can be viewed as an un-conditioned SDE, after a suitable change in the drift. Nevertheless, this is often quite difficult to compute this new drift. Basically, you need to have a very good approximation of the Markov transition densities, which is quite often not very efficient from a numerical point of view. The SDE for the Brownian Bridge is a typical example where all the computations can be carried out explicitly. It is generally much more involved, though. – Alekk May 23 '10 at 21:13
• Hi Alekk Would you please be kind enough to give an explicit example where h-transform is not computable because of the too complex probability transition semi-group with large T. Regards – The Bridge May 24 '10 at 9:58