Nonlinear Filtering Equation of a Jump Process with Counting Observations

We study the filtering problem of an R ^d -valued pure jump process when the observations is a counting process. We assume that the dynamic of the state and the observations may be strongly dependent and that the two processes may jump together. Weak and pathwise uniqueness of solution of the Kushner–Stratonovich equation are discussed.     

Markov Chain Approximations for Symmetric Jump Processes

Markov chain approximations of reversible jump processes are investigated. Tightness results and a central limit theorem are established. Moreover, given the generator of a reversible jump process with state space ℝ ^d , the approximating Markov chains are constructed explicitly. As a byproduct we obtain a definition of the Sobolev space H ^α/2(ℝ ^d ), α∈(0,2), that is equivalent to the standard one.      

Existence of Optimal Controls for Partially Observed Jump Processes

A partially observable control problem for an R ^d -valued jump process with counting observations is studied. The state and the observations may be strongly dependent and, in particular, the two processes may jump together. An equivalent separated problem is introduced and the existence of an optimal control for the separated problem is obtained in the class of relaxed and generalized controls. Equivalence between the initial problem and the relaxed generalized separated control problem is discussed.     

Existence and characterization of product-form invariant distributions for state-dependent stochastic networks in the heavy-traffic diffusion limit

We consider state-dependent stochastic networks in the heavy-traffic diffusion limit represented by reflected jump-diffusions in the orthant ℝ₊ ⁿ with state-dependent reflection directions upon hitting boundary faces. Jumps are allowed in each coordinate by means of independent Poisson random measures with jump amplitudes depending on the state of the process immediately before each jump. For this class of reflected jump-diffusion processes sufficient conditions for the existence of a product-form stationary density and an ergodic characterization of the stationary distribution are provided. Moreover, such stationary density is characterized in terms of semi-martingale local times at the boundaries and it is shown to be continuous and bounded. A central role is played by a previously established semi-martingale local time representation of the regulator processes. F.J. Piera’s research supported in part by CONICYT, Chile, FONDECYT Project 1070797. R.R. Mazumdar’s research supported in part by NSF, USA, Grant 0087404 through Networking Research Program, and a Discovery Grant from NSERC, Canada.     

Towards a Characterization of Markov Processes Enjoying the Time-Inversion Property

We give a necessary and sufficient condition for a homogeneous Markov process taking values in ℝ ⁿ to enjoy the time-inversion property of degree α. The condition sets the shape for the semigroup densities of the process and allows to further extend the class of known processes satisfying the time-inversion property. As an application we recover the result of Watanabe (Z. Wahrscheinlichkeitstheor. Verwandte Geb. 31:115–124, 1975) for continuous and conservative Markov processes on ℝ₊. As new examples we generalize Dunkl processes and construct a matrix-valued process with jumps related to the Wishart process by a skew-product representation.      

Counting Observations: A Note on State Estimation Sensitivity with an L 1 -Bound

Let (X _t , Y _t ) be a pure jump Markov process: the state X _t takes real values and the observation Y _t is a counting process. The two processes are allowed to have common jump times. Let ϕ(X(⋅)) be a functional of the state trajectory restricted to the time interval [0, T] . If we change the infinitesimal parameters and/ or the initial distribution, then we introduce an error in computing the conditional law of ϕ(X(⋅)) given the observation up to time T . In this paper we give an explicit L ¹ -bound for this error. Accepted 9 March 2001. Online publication 20 June 2001.     

Diffusion approximations for open queueing networks with service interruptions

This paper establishes functional central limit theorems describing the heavy-traffic behavior of open single-class queueing networks with service interruptions. In particular, each station has a single server which is alternatively up and down. There are two treatments of the up and down times. The first treatment corresponds to fixed up and down times and leads to a reflected Brownian motion, just as when there are no service interruptions, but with different parameters. To represent long rare interruptions, the second treatment has growing up and down times with the up and down times being of ordern andn ^1/2, respectively, when the traffic intensities are of order 1-n^–1/2. In this case we establish convergence in the SkorohodM ₁ topology to a multidimensional reflection of multidimensional Brownian motion plus a multidimensional jump process.     

A posteriori error analysis for a class of integral equations and variational inequalities

We consider elliptic and parabolic variational equations and inequalities governed by integro-differential operators of order ${2s \in (0,2]}We consider elliptic and parabolic variational equations and inequalities governed by integro-differential operators of order 2s ? (0,2]{2s \in (0,2]}. Our main motivation is the pricing of European or American options under Lévy processes, in particular pure jump processes or jump diffusion processes with tempered stable processes. The problem is discretized using piecewise linear finite elements in space and the implicit Euler method in time. We construct a residual-type a posteriori error estimator which gives a computable upper bound for the actual error in H ^s -norm. The estimator is localized in the sense that the residuals are restricted to the discrete non-contact region. Numerical experiments illustrate the accuracy of the space and time estimators, and show that they can be used to measure local errors and drive adaptive algorithms.     

Remark on pathwise uniqueness of stochastic differential equations driven by Lévy processes

Consider real-valued processes determined by stochastic differential equations driven by Lévy processes. The jump parts of the driving Lévy process are not always α-stable ones, nor symmetric ones. In the present article, we shall study the pathwise uniqueness of the solutions to the stochastic differential equations under the conditions on the coefficients that the diffusion and the jump terms are Hölder continuous, while the drift one is monotonic. Our approach is based on Gronwall’s inequality.     

On some functionals of the first passage times in jump models of stochastic volatility

Abstract

We compute some functionals related to the generalized joint Laplace transforms of the first times at which two-dimensional jump processes exit half strips. It is assumed that the state space components are driven by Cox processes with both independent and common (positive) exponential jump components. The method of proof is based on the solutions of the equivalent partial integro-differential boundary-value problems for the associated value functions. The results are illustrated on several two-dimensional jump models of stochastic volatility which are based on non-affine analogs of certain mean-reverting or diverting diffusion processes representing closed-form solutions of the appropriate stochastic differential equations.     

ExponentialL 2-convergence andL 2-spectral gap for Markov processes

This paper deals with the exponentialL ²-convergence for jump processes. We introduce some reduction methods and improve some previous results. Then we prove that for birth-death processes, exponentialL ²-convergence coincides indeed with exponential ergodicity which is widely studied in the Markov chain theory. Research supported in part by the National Natural Science Foundation of China and Fok Ying-Tung Educational Foundation     

Functional Inequalities for the Decay of Sub-Markov Semigroups

A general functional inequality is introduced to describe various decays of semigroups. Our main result generalizes the classical one on the equivalence of the L ²-exponential decay of a sub-Markov semigroup and the Poincaré inequality for the associated Dirichlet form. Conditions for the general inequality to hold are presented. The corresponding isoperimetric inequality is studied in the context of diffusion and jump processes. In particular, Cheeger's inequality for the principal eigenvalue is generalized. Moreover, our results are illustrated by examples of diffusion and jump processes.     

Boundary trace of reflecting Brownian motions

We establish a uniform dimensional result for normally reflected Brownian motion (RBM) in a large class of non-smooth domains. Hausdorff dimensions for the boundary occupation time and the boundary trace of RBM are determined. Extensions to stable-like jump processes and to symmetric reflecting diffusions are also given.Mathematics Subject Classification (2000):Primary 60G17, 60J60, Secondary 28A80, 30C35, 60G52, 60J50     

A mixed nonconforming finite element for linear elasticity

This article considers a mixed finite element method for linear elasticity. It is based on a modified mixed formulation that enforces the continuity of the stress weakly by adding a jump term of the approximated stress on interior edges. The symmetric stress are approximated by nonconforming linear elements and the displacement by piecewise constants. We establish ??(h) error bound in the (broken) L² norm for the divergence of the stress and ??(h) error bound in the L² norm for both the displacement and the stress tensor. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005.     

An Extension of Itô's Formula for Anticipating Processes

In this paper we introduce a class of square integrable processes, denoted by L^F, defined in the canonical probability space of the Brownian motion, which contains both the adapted processes and the processes in the Sobolev space L^2,2. The processes in the class L^F satisfy that for any time t, they are twice weakly differentiable in the sense of the stochastic calculus of variations in points (r, s) such that r s t. On the other hand, processes belonging to the class L^F are Skorohod integrable, and the indefinite Skorohod integral has properties similar to those of the Ito integral. In particular we prove a change-of-variable formula that extends the classical Itô formula. Those results are generalization of similar properties proved by Nualart and Pardoux⁽⁷⁾ for processes in L^2,2.     

An Ito formula for domain-valued processes driven by stochastic flows

 We consider a natural class of stochastic processes taking values in the space of smoothly bounded domains in ℝ ⁿ with compact closure. These processes are generated by stochastic flows on ℝ ⁿ which are obtained as the solutions of stochastic differential equations on ℝ ⁿ . We establish an Ito formula for smooth domain functionals, applied to processes in this class. Received: 2 March 2001 / Revised version: 10 January 2002 / Published online: 22 August 2002     

The Coding Complexity of Lévy Processes

We introduce a new coding scheme for general real-valued Lévy processes and control its performance with respect to L _p [0,1]-norm distortion under different complexity constraints. We also establish lower bounds that prove the optimality of our coding scheme in many cases.      

Hölder Continuity of Harmonic Functions with Respect to Operators of Variable Order

ABSTRACT

We consider a class of integrodifferential operators and their corresponding harmonic functions. Under mild assumptions on the family of jump measures we prove a priori estimates and establish Hölder continuity of bounded functions that are harmonic in a domain.     

L p Estimates for Maxwell's Equations with Source Term

The goal of this paper is to establish interior and global L ^p -type estimates for the solutions of Maxwell's equations with source term in a domain filled with two different materials separated by a ² interface. The usual elliptic estimates cannot be applied directly, due to the singularity of the dielectric permittivity. A special curl-div decomposition is introduced for the electric field to reduce the problem to an elliptic equation in divergence form with jump coefficients. The potential analysis and the jump condition lead to the interior L ^p estimates which are superior to the straightforward Nash-Moser estimates. The reduction procedure is expected to be useful for future numerical simulation. Because of the natural physical requirements, the boundary condition is nonlocal and involves a first order pseudo-differential operator, the boundary estimate is established by delicate new maximum principles and Riesz convexity arguments. These estimates are then employed to solve a nonlinear optics problem that arises in the modeling of surface enhanced second-harmonic generation of nonlinear diffractive optics in periodic structures (gratings).     

Stochastic Differentiation – A Generalized Approach

The space (D ^*) of Wiener distributions allows a natural Pettis-type stochastic calculus. For a certain class of generalized multiparameter processes X: R ^N (D ^*) we prove several differentiation rules (Itô formulas); these processes can be anticipating. We then apply these rules to some examples of square integrable Wiener functionals and look at the integral versions of the resulting formulas.