2013-10-17 : Sylvain Le Corff and Yohan Petetin

Post date: 23-Sep-2013 11:41:18

Continuous-time importance sampling for Jump diffusions

This talk introduces a new algorithm to sample from continuous-time jump diffusions and to estimate expectations of functionals of such diffusions. Recently, new exact algorithms have been proposed to draw samples from finite-dimensional distributions of diffusion processes and jump diffusions without any discretization step. These algorithms are based on a rejection sampling procedure and draw skeletons at some random time steps. However, these exact methods rely on strong assumptions such as the reducibility to a unit-volatility jump diffusion using the Lamperti transform. While this assumption can be proved easily for scalar diffusions, much stronger conditions are required in the multidimensional case.

In this contribution, we introduce a new algorithm to compute unbiased estimates of expectations of functionals of jump diffusions which can be used under weaker assumptions. This Jump Continuous Importance Sampling (JCIS) algorithm draws weighted skeletons using an importance sampling mechanism recently introduced for diffusion processes. In this case, the sampled paths are not distributed as the diffusion process but the weighted samples can be used to produce unbiased Monte Carlo estimates. The JCIS algorithm is compared to several other algorithms (Euler scheme with thinned jumps, Multilevel Monte Carlo path simulation, Jump Exact algorithm) using different models (Merton model, Sinus model, Double Jump model).

* 16 h 15 : Yohan Petetin

Single- and multiple-object filtering for Markov models with jumps

The objective of this communication is to propose algorithms for the single- and multiple-object filtering problems in hidden Markov models which involve a discrete process which models regime switching or jumps.

First, we focus on hidden Markov models with Markovian jumps and we recall how to use sequential Monte Carlo (MC) methods for the statistical filtering problem. More precisely, we focus on Rao-Blackwellised Particle filtering for linear and Gaussian Jump Markov State Space Systems (JMSS), for the approximation of the optimal Bayesian estimate (in the minimum mean square error sense).

Next, we generalise the previous filtering problem to multiple object filtering: we now look for estimating the state vectors and the jumps associated to a random number of objects which can appear or disappear over time, in a clutter environment and in the presence of misdetections. We thus focus on the Probability Hypothesis Density filter and we discuss on the MC implementation of this filter.

Finally, the last is part is devoted to alternative estimation techniques for linear and Gaussian JMSS. Rather than approximating the estimate in the classical JMSS model, we build a class of dynamical models with jumps which share with JMSS some physical properties of interest, and in which the optimal Bayesian estimate (in the minimum mean square error sense) can be computed exactly (i.e., without numerical or MC approximations) and at a computational cost linear in the number of observations. We show that these models may provide an alternative to MC methods in the JMSS context.

Bibliography

· A. Doucet, N. J. Gordon, and V. Krishnamurthy. Particle filters for state estimation of jump Markov linear systems. IEEE Transactions on Signal Processing, 49(3), 613–24, March 2001.

· Y. Petetin, M. Morelande and F. Desbouvries, " Marginalized particle PHD filters for multiple object Bayesian filtering", IEEE Transactions on Aerospace and Electronic Systems, Accepted for publication, 2013.

· Y. Petetin and F. Desbouvries, « Un nouvel algorithme de filtrage dans les modèles de Markov à saut linéaires et Gaussiens », Actes du 24ème colloque Gretsi, Brest, France, september 3-6 2013.