# Conditional path sampling of stochastic differential equations by drift relaxation - Mathematics > Numerical Analysis

Conditional path sampling of stochastic differential equations by drift relaxation - Mathematics > Numerical Analysis - Download this document for free, or read online. Document in PDF available to download.

Abstract: We present an algorithm for the efficient sampling of conditional paths ofstochastic differential equations SDEs. While unconditional path sampling ofSDEs is straightforward, albeit expensive for high dimensional systems of SDEs,conditional path sampling can be difficult even for low dimensional systems.This is because we need to produce sample paths of the SDE which respect boththe dynamics of the SDE and the initial and endpoint conditions. The dynamicsof a SDE are governed by the deterministic term drift and the stochastic termnoise. Instead of producing conditional paths directly from the original SDE,one can consider a sequence of SDEs with modified drifts. The modified driftsshould be chosen so that it is easier to produce sample paths which satisfy theinitial and endpoint conditions. Also, the sequence of modified driftsconverges to the drift of the original SDE. We construct a simple Markov ChainMonte Carlo MCMC algorithm which samples, in sequence, conditional paths fromthe modified SDEs, by taking the last sampled path at each level of thesequence as an initial condition for the sampling at the next level in thesequence. The algorithm can be thought of as a stochastic analog ofdeterministic homotopy methods for solving nonlinear algebraic equations or asa SDE generalization of simulated annealing. The algorithm is particularlysuited for filtering-smoothing applications. We show how it can be used toimprove the performance of particle filters. Numerical results for filtering ofa stochastic differential equation are included.

Author: ** Panagiotis Stinis**

Source: https://arxiv.org/