On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions

We introduce a sequence of stopping times that allow us to study an analogue of a life-cycle decomposition for a continuous time Markov process, which is an extension of the well-known splitting technique of Nummelin to the continuous time case. As a consequence, we are able to give deterministic equivalents of additive functionals of the process and to state a generalisation of Chen’s inequality. We apply our results to the problem of non-parametric kernel estimation of the drift of multi-dimensional recurrent, but not necessarily ergodic, diffusion processes.     

Stability of Feynman-Kac formulae with path-dependent potentials

Several particle algorithms admit a Feynman-Kac representation such that the potential function may be expressed as a recursive function which depends on the complete state trajectory. An important example is the mixture Kalman filter, but other models and algorithms of practical interest fall in this category. We study the asymptotic stability of such particle algorithms as time goes to infinity. As a corollary, practical conditions for the stability of the mixture Kalman filter, and a mixture GARCH filter, are derived. Finally, we show that our results can also lead to weaker conditions for the stability of standard particle algorithms for which the potential function depends on the last state only.     

Nonlinear Filtering of Diffusion Processes in Correlated Noise: Analysis by Separation of Variables

Abstract. An approximation to the solution of a stochastic parabolic equation is constructed using the Galerkin approximation followed by the Wiener chaos decomposition. The result is applied to the nonlinear filtering problem for the time-homogeneous diffusion model with correlated noise. An algorithm is proposed for computing recursive approximations of the unnormalized filtering density and filter, and the errors of the approximations are estimated. Unlike most existing algorithms for nonlinear filtering, the real-time part of the algorithm does not require solving partial differential equations or evaluating integrals. The algorithm can be used for both continuous and discrete time observations. \par     

Nonlinear Filtering of Diffusion Processes in Correlated Noise: Analysis by Separation of Variables

   Abstract. An approximation to the solution of a stochastic parabolic equation is constructed using the Galerkin approximation followed by the Wiener chaos decomposition. The result is applied to the nonlinear filtering problem for the time-homogeneous diffusion model with correlated noise. An algorithm is proposed for computing recursive approximations of the unnormalized filtering density and filter, and the errors of the approximations are estimated. Unlike most existing algorithms for nonlinear filtering, the real-time part of the algorithm does not require solving partial differential equations or evaluating integrals. The algorithm can be used for both continuous and discrete time observations. \par     

Particle filters with random resampling times

Particle filters are numerical methods for approximating the solution of the filtering problem which use systems of weighted particles that (typically) evolve according to the law of the signal process. These methods involve a corrective/resampling procedure which eliminates the particles that become redundant and multiplies the ones that contribute most to the resulting approximation. The correction is applied at instances in time called resampling/correction times. Practitioners normally use certain overall characteristics of the approximating system of particles (such as the effective sample size of the system) to determine when to correct the system. As a result, the resampling times are random. However, in the continuous time framework, all existing convergence results apply only to particle filters with deterministic correction times. In this paper, we analyse (continuous time) particle filters where resampling takes place at times that form a sequence of (predictable) stopping times. We prove that, under very general conditions imposed on the sequence of resampling times, the corresponding particle filters converge. The conditions are verified when the resampling times are chosen in accordance to the effective sample size of the system of particles, the coefficient of variation of the particles’ weights and, respectively, the (soft) maximum of the particles’ weights. We also deduce central-limit theorem type results for the approximating particle system with random resampling times.     

First exit time probability for multidimensional diffusions: A PDE-based approach

First exit time distributions for multidimensional processes are key quantities in many areas of risk management and option pricing. The aim of this paper is to provide a flexible, fast and accurate algorithm for computing the probability of the first exit time from a bounded domain for multidimensional diffusions. First, we show that the probability distribution of this stopping time is the unique (weak) solution of a parabolic initial and boundary value problem. Then, we describe the algorithm which is based on a combination of the sparse tensor product finite element spaces and an hp-discontinuous Galerkin method. We illustrate our approach with several examples. We also compare the numerical results to classical Monte Carlo methods.     

The thermodynamic limit of polygonal models

We discuss the basic models of polygonal Markov fields with a two-dimensional continuous parameter % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgI% GioNWaajaadsfaoiabgkOimlabl2riHoaaCaaabeqaaiaaikdaaaaa% aa!3DEB!\[x \in T \subset \mathbb{R}^2 \], which were introduced by Arak and studied later by Arak and surgailis. There are two types of polygonal models, either with a given initial distribution of the lines or of the points (vertices) of a random polygonal graph. The main result of this paper is proof of the existence of a thermodynamic limit (as % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaW2ajmaqca% WGubacciqceaQae8xKH0QdcqWIDesOdaahaaqabeaacaaIYaaaaaaa% !3C7D!\[T \uparrow \mathbb{R}^2 \]) for a class of polygonal models with a small contour length'. It is based on the study of Kirkwood-Salzburg-type equations for the correlation functions. We also discuss some examples of consistent polygonal models for which the existence of the thermodynamic limit is trivial.     

Weak convergence to a Markov process: The martingale approach

Summary In this article, we obtain some sufficient conditions for weak convergence of a sequence of processes {X _n } toX, whenX arises as a solution to a well posed martingale problem. These conditions are tailored for application to the case when the state space for the processesX _n ,X is infinite dimensional. The usefulness of these conditions is illustrated by deriving Donsker's invariance principle for Hilbert space valued random variables. Also, continuous dependence of Hilbert space valued diffusions on diffusion and drift coefficients is proved.Research supported by National Board for Higher Mathematics, Bombay, IndiaPart of the work was done at University of California, Santa Barbara, USA     

A new adaptive Runge–Kutta method for stochastic differential equations

In this paper, we will present a new adaptive time stepping algorithm for strong approximation of stochastic ordinary differential equations. We will employ two different error estimation criteria for drift and diffusion terms of the equation, both of them based on forward and backward moves along the same time step. We will use step size selection mechanisms suitable for each of the two main regimes in the solution behavior, which correspond to domination of the drift-based local error estimator or diffusion-based one. Numerical experiments will show the effectiveness of this approach in the pathwise approximation of several standard test problems.     

Long time asymptotics for constrained diffusions in polyhedral domains

We study long time asymptotic properties of constrained diffusions that arise in the heavy traffic analysis of multiclass queueing networks. We first consider the classical diffusion model with constant coefficients, namely a semimartingale reflecting Brownian motion (SRBM) in a d$d$-dimensional positive orthant. Under a natural stability condition on a related deterministic dynamical system [P. Dupuis, R.J. Williams, Lyapunov functions for semimartingale reflecting brownian motions, Annals of Probability 22 (2) (1994) 680–702] showed that an SRBM is ergodic. We strengthen this result by establishing geometric ergodicity for the process. As consequences of geometric ergodicity we obtain finiteness of the moment generating function of the invariant measure in a neighborhood of zero, uniform time estimates for polynomial moments of all orders, and functional central limit results. Similar long time properties are obtained for a broad family of constrained diffusion models with state dependent coefficients under a natural condition on the drift vector field. Such models arise from heavy traffic analysis of queueing networks with state dependent arrival and service rates.     

Representations of the optimal filter in the context of nonlinear filtering of random fields with fractional noise

The problem of nonlinear filtering of multiparameter random fields, observed in the presence of a long-range dependent spatial noise, is considered. When the observation noise is modelled by a persistent fractional Wiener sheet, several pathwise representations of the optimal filter are derived. The representations involve series of multiple stochastic integrals of different types and are particularly important since the evolution equations, satisfied by the best mean-square estimate of the signal random field, have a complicated analytical structure and fail to be proper (measure-valued) stochastic partial differential equations. Several of the above optimal filter representations involve a new family of strong martingale transforms associated to the multiparameter fractional Brownian sheet; the latter martingale family is of independent interest in fractional stochastic calculus of multiparameter random fields.     

Reflected BSDEs with nonpositive jumps,and controller-and-stopper games

We study a class of reflected backward stochastic differential equations with nonpositive jumps and upper barrier. Existence and uniqueness of a minimal solution are proved by a double penalization approach under regularity assumptions on the obstacle. In a suitable regime switching diffusion framework, we show the connection between our class of BSDEs and fully nonlinear variational inequalities. Our BSDE representation provides in particular a Feynman–Kac type formula for PDEs associated to general zero-sum stochastic differential controller-and-stopper games, where control affects both drift and diffusion term, and the diffusion coefficient can be degenerate. Moreover, we state a dual game formula of this BSDE minimal solution involving equivalent change of probability measures, and discount processes. This gives in particular a new representation for zero-sum stochastic differential controller-and-stopper games.     

An adaptive extrapolation discontinuous Galerkin method for the valuation of Asian options

We consider the approximation of the optimal stopping problem associated with ultradiffusion processes in the context of mathematical finance and the valuation of Asian options. In particular, the value function is characterized as the solution of an ultraparabolic variational inequality. Employing the penalty method and a regularization of the state space, we develop higher-order adaptive approximation schemes which utilize the extrapolation discontinuous Galerkin method in temporal space. Numerical examples are provided in order to demonstrate the approach.     

Slow and fast scales for superprocess limits of age-structured populations

A superprocess limit for an interacting birth-death particle system modeling a population with trait and physical age-structures is established. Traits of newborn offspring are inherited from the parents except when mutations occur, while ages are set to zero. Because of interactions between individuals, standard approaches based on the Laplace transform do not hold. We use a martingale problem approach and a separation of the slow (trait) and fast (age) scales. While the trait marginals converge in a pathwise sense to a superprocess, the age distributions, on another time scale, average to equilibria that depend on traits. The convergence of the whole process depending on trait and age, only holds for finite-dimensional time-marginals. We apply our results to the study of examples illustrating different cases of trade-off between competition and senescence.     

Limit theorems for hybridization reactions on oligonucleotide microarrays

We derive herein the limiting laws for certain stationary distributions of birth-and-death processes related to the classical model of chemical adsorption-desorption reactions due to Langmuir. The model has been recently considered in the context of a hybridization reaction on an oligonucleotide DNA-microarray. Our results imply that the truncated-gamma- and beta-type distributions can be used as approximations to the observed distributions of the fluorescence readings of the oligo-probes on a microarray. These findings might be useful in developing new model-based, probe-specific methods of extracting target concentrations from array fluorescence readings.     

Study of dependence for some stochastic processes: Symbolic Markov copulae

We study dependence between components of multivariate (nice Feller) Markov processes: what conditions need to be satisfied by a multivariate Markov process so that its components are Markovian with respect to the filtration of the entire process and such that they follow prescribed laws? To answer this question, we introduce a symbolic approach, which is rooted in the concept of pseudo-differential operator (PDO). We investigate connections between dependence, in the sense described above, and the PDOs. In particular, we study the problem of constructing a multivariate nice Feller process with given marginal laws in terms of symbols of the related PDOs. This approach leads to relatively simple conditions, which provide solutions to this problem.     

Inclusion-exclusion and point processes

A combinatorial (inclusion-exclusion) approach to the construction of point processes starting from densities is proposed. A formal sufficient crifficient criterion is derived and then applied with positive results to systems of functions having a special product form. Thus, a new class of point processes is derived to play a role within classical Gibbs processes.     

Adaptive finite element methods for Cahn–Hilliard equations

We develop a method for adaptive mesh refinement for steady state problems that arise in the numerical solution of Cahn–Hilliard equations with an obstacle free energy. The problem is discretized in time by the backward-Euler method and in space by linear finite elements. The adaptive mesh refinement is performed using residual based a posteriori estimates; the time step is adapted using a heuristic criterion. We describe the space–time adaptive algorithm and present numerical experiments in two and three space dimensions that demonstrate the usefulness of our approach.