State estimation for cox processes with unknown probability law

Let N_i, i?1, be i.i.d. observable Cox processes on a compact metric space E, directed by unobservable random measures M_i. Assume that the probability law of the M_i is completely unknown. Techniques are developed for approximation of state estimators using data from the processes N₁,…,N_n to estimate necessary attributes of the unknown probability law of the time M_i. The techniques are based on representation of the state estimators in terms of reduced Palm distributions of the N_i and on estimation of these Palm distributions. Estimators of Palm distributions are shown to be strongly consistent and asymptotically normal. The difference between the true and the pseudo-state estimators converges to zero in L² at rate n^{?$\text{1}\text{4+δ}$} for each δ > 0.     

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.     

Moser iteration for (quasi)minimizers on metric spaces

We study regularity properties of quasiminimizers of the p-Dirichlet integral on metric measure spaces. We adapt the Moser iteration technique to this setting and show that it can be applied without an underlying differential equation. However, we have been able to run the Moser iteration fully only for minimizers. We prove Caccioppoli inequalities and local boundedness properties for quasisub- and quasisuperminimizers. This is done in metric spaces equipped with a doubling measure and supporting a weak (1, p)-Poincaré inequality. The metric space is not required to be complete. We also provide an example which shows that the dilation constant from the weak Poincaré inequality is essential in the condition on the balls in the Harnack inequality. This fact seems to have been overlooked in the earlier literature on nonlinear potential theory on metric spaces.     

Reflecting Brownian motions and a deletion result for Sobolev spaces of order (1, 2)

Let D be an open set in ^d and E be a relatively closed subset of D having zero Lebesgue measure. A necessary and sufficient integral condition is given for the Sobolev spaces W ^1,2 (D) and W ^1,2(D\E) to be the same. The latter is equivalent to (normally) reflecting Brownian motion (RBM) on being indistinguishable (in distribution) from RBM on . This integral condition is satisfied, for example, when E has zero (d–1)-dimensional Hausdorff measure. Therefore it is possible to delete from D a relatively closed subset E having positive capacity but nevertheless the RBM on is indistinguishable from the RBM on , or equivalently, W ^1,2(D\E)=W^1,2(D). An example of such kind is: D=² and E is the Cantor set. In the proof of above mentioned results, a detailed study of RBMs on general open sets is given. In particular, a semimartingale decomposition and approximation result previously proved in [3] for RBMs on bounded open sets is extended to the case of unbounded open sets.Research supported in part by NSF Grant DMS 86-57483.     

Convergence to equilibrium of critical branching particle systems and superprocesses,and related nonlinear partial differential equations

We consider a class of multitype particle systems in ^d undergoing spatial diffusion and critical stable multitype branching, and their limits known as critical stable multitype Dawson-Watanabe processes, or superprocesses. We show that for large classes of initial states, the particle process and the superprocess converge in distribution towards known equilibrium states as time tends to infinity. As an application we obtain the asymptotic behavior of a system of nonlinear partial differential equations whose solution is related to the distribution of both the particle process and the superprocess.Research partially supported by CONACyT (Mexico), CNRS (France) and BMfWuF (Austria).     

Heat kernel estimates for jump processes of mixed types on metric measure spaces

In this paper, we investigate symmetric jump-type processes on a class of metric measure spaces with jumping intensities comparable to radially symmetric functions on the spaces. The class of metric measure spaces includes the Alfors d-regular sets, which is a class of fractal sets that contains geometrically self-similar sets. A typical example of our jump-type processes is the symmetric jump process with jumping intensity where ν is a probability measure on , c(α, x, y) is a jointly measurable function that is symmetric in (x, y) and is bounded between two positive constants, and c ₀(x, y) is a jointly measurable function that is symmetric in (x, y) and is bounded between γ₁ and γ₂, where either γ₂ ≥ γ₁ > 0 or γ₁ = γ₂ = 0. This example contains mixed symmetric stable processes on as well as mixed relativistic symmetric stable processes on . We establish parabolic Harnack principle and derive sharp two-sided heat kernel estimate for such jump-type processes. Dedicated to Professor Masatoshi Fukushima on the occasion of his 70th birthday. The research of Zhen-Qing Chen is supported in part by NSF Grants DMS-0303310 and DMS-06000206. The research of Takashi Kumagai is supported in part by the Grant-in-Aid for Scientific Research (B) 18340027.     

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.     

A central limit theorem for martingales and an application to branching processes

A functional central limit theorem is obtained for martingales which are not uniformly asymptotically negligible but grow at a geometric rate. The function space is not the usual C[0,1] or D[0,1] but R^N, the space of all real sequences and the metric used leads to a non-separable metric space.The main theorem is applied to a martingale obtained from a supercritical Galton-Watson branching process and as simple corollaries the already known central limit theorems for the Harris and Lotka-Nagaev estimators of the mean of the offspring distribution, are obtained.     

On non-parametric estimation of the Lévy kernel of Markov processes

We consider a recurrent Markov process which is an Itô semi-martingale. The Lévy kernel describes the law of its jumps. Based on observations _X0,_XΔ,…,_XnΔ$X_0,X_{\Delta },\dots ,X_{n\Delta }$, we construct an estimator for the Lévy kernel’s density. We prove its consistency (as nΔ→∞$n\Delta \to \infty$ and Δ→0$\Delta \to 0$) and a central limit theorem. In the positive recurrent case, our estimator is asymptotically normal; in the null recurrent case, it is asymptotically mixed normal. Our estimator’s rate of convergence equals the non-parametric minimax rate of smooth density estimation. The asymptotic bias and variance are analogous to those of the classical Nadaraya–Watson estimator for conditional densities. Asymptotic confidence intervals are provided.     

Filtering of a Markov Jump Process with Counting Observations

This paper concerns the filtering of an R ^d -valued Markov pure jump process when only the total number of jumps are observed. Strong and weak uniqueness for the solutions of the filtering equations are discussed. Accepted 12 November 1999     

The influence of the nonrecent past in prediction for stochastic processes

Consider the stochastic processes X₁, X₂,… and Λ₁, Λ₂,… where the X process can be thought of as observations on the Λ process. We investigate the asymptotic behavior of the conditional distributions of X_t+v given X₁,…, X_t and Λ_t+v given X₁,…, X_t with regard to their dependency on the “early” part of the X process. These distributions arise in various time series and sequential decision theory problems. The results support the intuitively reasonable and often used (as a basic tenet of model building) assumption that only the more recent past is needed for near optimal prediction.     

Queueing systems on a circle

Consider a ring on which customers arrive according to a Poisson process. Arriving customers drop somewhere on the circle and wait there for a server who travels on the ring. Whenever this server encounters a customer, he stops and serves the customer according to an arbitrary service time distribution. After the service is completed, the server removes the client from the circle and resumes his journey.We are interested in the number and the locations of customers that are waiting for service. These locations are modeled as random counting measures on the circle. Two different types of servers are considered: The polling server and the Brownian (or drunken) server. It is shown that under both server motions the system is stable if the traffic intensity is less than 1. Furthermore, several earlier results on the configuration of waiting customers are extended, by combining results from random measure theory, stochastic integration and renewal theory.     

Nonparametric estimation for pure jump Lévy processes based on high frequency data

In this paper, we study nonparametric estimation of the Lévy density for pure jump Lévy processes. We consider n$n$ discrete time observations with step Δ$\Delta$. The asymptotic framework is: n$n$ tends to infinity, Δ=_Δn$\Delta ={\Delta }_n$ tends to zero while n_Δn$n{\Delta }_n$ tends to infinity. First, we use a Fourier approach (“frequency domain”): this allows us to construct an adaptive nonparametric estimator and to provide a bound for the global L²${\mathbb{L}}^2$-risk. Second, we use a direct approach (“time domain”) which allows us to construct an estimator on a given compact interval. We provide a bound for L²${\mathbb{L}}^2$-risk restricted to the compact interval. We discuss rates of convergence and give examples and simulation results for processes fitting in our framework.     

Wiener’s test for super-Brownian motion and the Brownian snake

We prove a Wiener-type criterion for super-Brownian motion and the Brownian snake.If F is a Borel subset of ℝ ^d and x ∈ ℝ ^d , we provide a necessary and sufficientcondition for super-Brownian motion started at δ _x to immediately hit the set F. Equivalently, this condition is necessary and sufficient for the hitting time of F by theBrownian snake with initial point x to be 0. A key ingredient of the proof isan estimate showing that the hitting probability of F is comparable, up to multiplicative constants,to the relevant capacity of F. This estimate, which is of independent interest, refines previous results due to Perkins and Dynkin. An important role is played by additivefunctionals of the Brownian snake, which are investigated here via the potentialtheory of symmetric Markov processes. As a direct application of our probabilisticresults, we obtain a necessary and sufficient condition for the existence in a domain D of a positivesolution of the equation Δ; u = u ² which explodes at a given point of ∂ D. Received: 5 January 1996 / In revised form: 30 October 1996     

Continuity of eigenvalues of subordinate processes in domains

Let X={X_t,t≥0} be a symmetric Markov process in a state space E and D an open set of E. Let S⁽ⁿ⁾={S⁽ⁿ⁾_t, t ≥ 0} be a subordinator with Laplace exponent ϕ_n and S={S_t,t≥0} a subordinator with Laplace exponent ϕ. Suppose that X is independent of S and S⁽ⁿ⁾. In this paper we consider the subordinate processes and and their subprocesses and X^ϕ,D killed upon leaving D. Suppose that the spectra of the semigroups of and X^ϕ,D are all discrete, with being the eigenvalues of the generator of and being the eigenvalues of the generator of X^ϕ,D. We show that, if lim_n→∞ϕ_n(λ)=ϕ(λ) for every λ>0, then The research of this author is supported in part by NSF Grant DMS-0303310. The research of this author is supported in part by a joint US-Croatia grant INT 0302167.     

Classical limit theorems for measure-valued Markov processes

Laws of large numbers, central limit theorems, and laws of the iterated logarithm are obtained for discrete and continuous time Markov processes whose state space is a set of measures. These results apply to each measure-valued stochastic process itself and not simply to its real-valued functionals.     

Rescaled contact processes converge to super-Brownian motion in two or more dimensions

We show that in dimensions two or more a sequence of long range contact processes suitably rescaled in space and time converges to a super-Brownian motion with drift. As a consequence of this result we can improve the results of Bramson, Durrett, and Swindle (1989) by replacing their order of magnitude estimates of how close the critical value is to 1 with sharp asymptotics. Received: 2 February 1998 / Revised version: 28 August 1998     

Filtering with Discrete State Observations

The problem of estimating a finite state Markov chain observed via a process on the same state space is discussed. Optimal solutions are given for both the ``weak' and ``strong' formulations of the problem. The ``weak' formulation proceeds using a reference probability and a measure change for the Markov chain. The ``strong' formulation considers an observation process related to perturbations of the counting processes associated with the Markov chain. In this case the ``small noise' convergence is investigated. Accepted 7 April 1998     

Large deviations w.r.t. quasi-every starting point for symmetric right processes on general state spaces

Summary In the work of Donsker and Varadhan, Fukushima and Takeda and that of Deuschel and Stroock it has been shown, that the lower bound for the large deviations of the empirical distribution of an ergodic symmetric Markov process is given in terms of its Dirichlet form. We give a short proof generalizing this principle to general state spaces that include, in particular, infinite dimensional and non0metrizable examples. Our result holds w.r.t. quasi-every starting point of the Markov process. Moreover we show the corresponding weak upper bound w.r.t. quasi-every starting point.This research was supported by the Graduiertenkolleg Algebraische, analytische und geometrische Methoden und ihre Wechselwirkung in der modernen Mathematik, Bonn