A stochastic log-Laplace equation

We study a nonlinear stochastic partial differential equation whose solution is the conditional log-Laplace functional of a superprocess in a random environment. We establish its existence and uniqueness by smoothing out the nonlinear term and making use of the particle system representation developed by Kurtz and Xiong [Stochastic Process. Appl. 83 (1999) 103-126]. We also derive the Wong-Zakai type approximation for this equation. As an application, we give a direct proof of the moment formulas of Skoulakis and Adler [Ann. Appl. Probab. 11 (2001) 488-543].Comment: Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000054

Convergence of Particle Filtering Method for Nonlinear Estimation of Vortex Dynamics

In this paper we formulate a numerical approximation method for the nonlinear ¯ltering of vortex dynamics subject to noise using particle ¯lter method. We prove the convergence of this scheme allowing the obser- vation vector to be unbounded.This research is supported by the Army Research Probability and Statistics Program through the grant DODARMY41712Approved for public release; distribution is unlimited

Rates for branching particle approximations of continuous-discrete filters

Herein, we analyze an efficient branching particle method for asymptotic solutions to a class of continuous-discrete filtering problems. Suppose that $t\to X_t$ is a Markov process and we wish to calculate the measure-valued process $t\to\mu_t(\cdot)\doteq P\{X_t\in \cdot|\sigma\{Y_{t_k}, t_k\leq t\}\}$, where $t_k=k\epsilon$ and $Y_{t_k}$ is a distorted, corrupted, partial observation of $X_{t_k}$. Then, one constructs a particle system with observation-dependent branching and $n$ initial particles whose empirical measure at time $t$, $\mu_t^n$, closely approximates $\mu_t$. Each particle evolves independently of the other particles according to the law of the signal between observation times $t_k$, and branches with small probability at an observation time. For filtering problems where $\epsilon$ is very small, using the algorithm considered in this paper requires far fewer computations than other algorithms that branch or interact all particles regardless of the value of $\epsilon$. We analyze the algorithm on L\'{e}vy-stable signals and give rates of convergence for $E^{1/2}\{\|\mu^n_t-\mu_t\|_{\gamma}^2\}$, where $\Vert\cdot\Vert_{\gamma}$ is a Sobolev norm, as well as related convergence results.Comment: Published at http://dx.doi.org/10.1214/105051605000000539 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Problem of nonlinear filtering

Stochastic filtering theory studies the problem of estimating an unobservable `signal' process X given the information obtained by observing an associated process Y (a `noisy' observation) within a certain time window [0; t]. It is possible to explicitly describe the distribution of X given Y in the setting of linear/gaussian systems. Outside the realm of the linear theory, it is known that only a few very exceptional examples have explicitly described posterior distributions. We present in detail a class of nonlinear filters (Benes filters) which allow explicit formulae. Using the explicit expression of the Laplace transform of a functional of Brownian motion we give a direct computation of the unnormalized conditional density of the signal for the Benes filter and obtain the formula for the normalized conditional density of X for two particular filters. In the case in which the signal X is a diffusion process and Y is given by the equation dY t = h(s; X s )ds+dW t ; where W is a Brownian moti..