Conditional path sampling for stochastic differential equations through drift relaxation

Abstract

We present an algorithm for the efficient sampling of conditional paths of stochastic differential equations (SDEs). While unconditional path sampling of SDEs 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 both the dynamics of the SDE and the initial and endpoint conditions. The dynamics of a SDE are governed by the deterministic term (drift) and the stochastic term (noise). Instead of producing conditional paths directly from the original SDE, one can consider a sequence of SDEs with modified drifts. The modified drifts should be chosen so that it is easier to produce sample paths which satisfy the initial and endpoint conditions. Also, the sequence of modified drifts converges to the drift of the original SDE. We construct a simple Markov Chain Monte Carlo (MCMC) algorithm which samples, in sequence, conditional paths from the modified SDEs, by taking the last sampled path at each level of the sequence as an initial condition for the sampling at the next level in the sequence. The algorithm can be thought of as a stochastic analog of deterministic homotopy methods for solving nonlinear algebraic equations or as a SDE generalization of simulated annealing. The algorithm is particularly suited for filtering/smoothing applications. We show how it can be used to improve the performance of particle filters. Numerical results for filtering of a stochastic differential equation are included.

Forward-reverse expectation-maximization algorithm for Markov chains: convergence and numerical analysis

“…It should be noted here that in recent years the problem of simulating diffusion bridges has attracted much attention. Without pretending to be complete, see, for example, [5], [9], [19], [23], [25], and [26].…”

“…It should be noted here that in the last years the problem of simulating diffusion bridges has attracted much attention. Without pretending to be complete, see for example, Bladt and Sørensen [2014], Delyon and Hu [2006], Milstein and Tretyakov [2004], Stinis [2011], Stuart et al [2004], Schauer et al [2013].…”

Social‐spider optimised particle filtering for tracking of targets with discontinuous measurement data

“…Increasing the number of sample particles could be an improper attempt to resolve this problem, since PF often suffers from a heavy computational cost, especially when the dimension of the model is high [12,13]. To overcome this drawback, the use of an extra step after the resampling step has been suggested in the literatures [14][15][16]. They construct a simple Markov chain Monte Carlo (MCMC) algorithm that samples conditional paths by taking the last sampled path at each level of the sequence as an initial condition for the sampling at the next level of the sequence.…”

Simulation of forward-reverse stochastic representations for conditional diffusions

“…Other notable approaches include those of Milstein and Tretyakov (2004), which treat the case of physically relevant functionals of Wiener integrals with respect to Brownian bridges, and Stinis (2011), who uses an MCMC approach based on successive modifications of the drift of the diffusion process.…”