Forward equations for reflected diffusions with jumps

In this paper we obtain the forward equations associated with the evolution of the density, if it exists, of reflected diffusions on the positive orthant with jumps which form a marked point process whose random jump measure possesses a stochastic intensity. These results generalize the so-called generalized Dynkin equations for piecewise deterministic jump processes due to Davis. We then consider the stationary case where the existence of a stochastic intensity is not needed. The techniques are based on local times and the use of random jump measures. We discuss the application of these results to problems arising in queuing and storage processes as well as stationary distributions of diffusions with delayed and jump reflections at the origin.This research was supported in part by the Quebec-France Cooperative Research Program and by the Natural Sciences and Engineering Research Council of Canada under Grant OGP 0042024.     

Drift transforms and Green function estimates for discontinuous processes

In this paper, we consider Girsanov transforms of pure jump type for discontinuous Markov processes. We show that, under some quite natural conditions, the Green functions of the Girsanov transformed process are comparable to those of the original process. As an application of the general results, the drift transform of symmetric stable processes is studied in detail. In particular, we show that the relativistic α-stable process in a bounded C^1,1-smooth open set D can be obtained from symmetric α-stable process in D through a combination of a pure jump Girsanov transform and a Feynman-Kac transform. From this, we deduce that the Green functions for these two processes in D are comparable.     

关于离散型随机变量次序统计量分布矩阵对称性猜想的探索

本文研究了离散型随机变量次序统计量的分布矩阵的对称性 ,获得了二个定理 .定理 1　服从等概率二点分布或等概率三点分布的离散型随机变量的次序统计量的分布矩阵是对称矩阵 .定理 2　取值有限且等概率的离散型随机变量的次序统计量的分布矩阵具有中心对称性 .     

The limit distribution of the maximum increment of a random walk with dependent regularly varying jump sizes

We investigate the maximum increment of a random walk with heavy-tailed jump size distribution. Here heavy-tailedness is understood as regular variation of the finite-dimensional distributions. The jump sizes constitute a strictly stationary sequence. Using a continuous mapping argument acting on the point processes of the normalized jump sizes, we prove that the maximum increment of the random walk converges in distribution to a Fréchet distributed random variable.     

Filtering,Smoothing and M-ary Detection with Discrete Time Poisson Observations

Abstract

In this article, we solve a class of estimation problems, namely, filtering smoothing and detection for a discrete time dynamical system with integer-valued observations. The observation processes we consider are Poisson random variables observed at discrete times. Here, the distribution parameter for each Poisson observation is determined by the state of a Markov chain. By appealing to a duality between forward (in time) filter and its corresponding backward processes, we compute dynamics satisfied by the unnormalized form of the smoother probability. These dynamics can be applied to construct algorithms typically referred to as fixed point smoothers, fixed lag smoothers, and fixed interval smoothers. M-ary detection filters are computed for two scenarios: one for the standard model parameter detection problem and the other for a jump Markov system.     

Continuity of Harmonic Functions for Non-local Markov Generators

We consider a class of pure jump Markov processes in ${\mathbb R}^d$ whose jump kernels are comparable to those of symmetric stable processes. We prove a support theorem, a lower bound on the occupation times of sets, and show that we can approximate resolvents using smooth functions.     

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.     

An inequality connecting entropy distance,Fisher Information and large deviations

In this paper we introduce a new generalisation of the relative Fisher Information for Markov jump processes on a finite or countable state space, and prove an inequality which connects this object with the relative entropy and a large deviation rate functional. In addition to possessing various favourable properties, we show that this generalised Fisher Information converges to the classical Fisher Information in an appropriate limit. We then use this generalised Fisher Information and the aforementioned inequality to qualitatively study coarse-graining problems for jump processes on discrete spaces.     

Convex Ordering for Random Vectors using Predictable Representation

We prove convex ordering results for random vectors admitting a predictable representation in terms of a Brownian motion and a non-necessarily independent jump component. Our method uses forward-backward stochastic calculus and extends the results proved in Klein et al. (Electron J Probab 11(20):27, 2006) in the one-dimensional case. We also study a geometric interpretation of convex ordering for discrete measures in connection with the conditions set on the jump heights and intensities of the considered processes. The work described in this paper was partially supported by a grant from City University of Hong Kong (Project No. 7200108).     

Feynman–Kac formulas for regime-switching jump diffusions and their applications

This work develops Feynman–Kac formulas for a class of regime-switching jump diffusion processes, in which the jump part is driven by a Poisson random measure associated with a general Lévy process and the switching part depends on the jump diffusion processes. Under broad conditions, the connections of such stochastic processes and the corresponding partial integro-differential equations are established. Related initial, terminal and boundary value problems are also treated. Moreover, based on weak convergence of probability measures, it is demonstrated that a sequence of random variables related to the regime-switching jump diffusion process converges in distribution to the arcsine law.     

Symmetric jump processes and their heat kernel estimates

We survey the recent development of the DeGiorgi-Nash-Moser-Aronson type theory for a class of symmetric jump processes (or equivalently, a class of symmetric integro-differential operators). We focus on the sharp two-sided estimates for the transition density functions (or heat kernels) of the processes, a priori Hölder estimate and parabolic Harnack inequalities for their parabolic functions. In contrast to the second order elliptic differential operator case, the methods to establish these properties for symmetric integro-differential operators are mainly probabilistic.     

Harnack Inequalities for Jump Processes

We consider a class of pure jump Markov processes in R ^d whose jump kernels are comparable to those of symmetric stable processes. We establish a Harnack inequality for nonnegative functions that are harmonic with respect to these processes. We also establish regularity for the solutions to certain integral equations.     

On option pricing under a completely random measure via a generalized Esscher transform

In this paper, we develop an option valuation model when the price dynamics of the underlying risky asset is governed by the exponential of a pure jump process specified by a shifted kernel-biased completely random measure. The class of kernel-biased completely random measures is a rich class of jump-type processes introduced in [James, L.F., 2005. Bayesian Poisson process partition calculus with an application to Bayesian Lévy moving averages. Ann. Statist. 33, 1771–1799; James, L.F., 2006. Poisson calculus for spatial neutral to the right processes. Ann. Statist. 34, 416–440] and it provides a great deal of flexibility to incorporate both finite and infinite jump activities. It includes a general class of processes, namely, the generalized Gamma process, which in its turn includes the stable process, the Gamma process and the inverse Gaussian process as particular cases. The kernel-biased representation is a nice representation form and can describe different types of finite and infinite jump activities by choosing different mixing kernel functions. We employ a dynamic version of the Esscher transform, which resembles an exponential change of measures or a disintegration formula based on the Laplace functional used by James, to determine an equivalent martingale measure in the incomplete market. Closed-form option pricing formulae are obtained in some parametric cases, which provide practitioners with a convenient way to evaluate option prices.     

Log-Convexity of Counting Processes Evaluated at a Random end of Observation Time with Applications to Queueing Models

We consider a counting processes with independent inter-arrival times evaluated at a random end of observation time T, independent of the process. For instance, this situation can arise in a queueing model when we evaluate the number of arrivals after a random period which can depend on the process of service times. Provided that T has log-convex density, we give conditions for the inter-arrival times in the counting process so that the observed number of arrivals inherits this property. For exponential inter-arrival times (pure-birth processes) we provide necessary and sufficient conditions. As an application, we give conditions such that the stationary number of customers waiting in a queue is a log-convex random variable. We also study bounds in the approximation of log-convex discrete random variables by a geometric distribution.     

等概离散无记忆信源率失真函数的计算方法

研究了等概信源的率失真函数计算问题,得到了等概信源在对称失真与列对称失真下率失真函数的计算方法,并举例说明了其具体应用.     

Quenched invariance principle for simple random walk on discrete point processes

We consider the simple random walk on random graphs generated by discrete point processes. This random walk moves on graphs whose vertex set is a random subset of a cubic lattice and whose edges are lines between any consecutive vertices on lines parallel to each coordinate axis. Under the assumption that the discrete point processes are finitely dependent and stationary, we prove that the quenched invariance principle holds, i.e., for almost every configuration of the point process, the path distribution of the walk converges weakly to that of a Brownian motion.     

Strong approximations of stochastic differential equations with jumps

This paper is a survey of strong discrete time approximations of jump-diffusion processes described by stochastic differential equations (SDEs). It also presents new results on strong discrete time approximations for the specific case of pure jump SDEs.     

Jump diffusion processes and their applications in insurance and finance

For insurance risks, jump processes such as homogeneous/non-homogeneous compound Poisson processes and compound Cox processes have been used to model aggregate losses. If we consider the economic assumption of a positive interest to aggregate losses, Lévy processes have proven to be useful. Also in financial modelling, it has been observed that diffusion models are not robust enough to capture the appearance of jumps in underlying asset prices and interest rates. As a result, jump diffusion processes, which are, simply speaking, combinations of compound Poisson processes with Brownian motion, have gained popularity for modelling in insurance and finance. In this paper, considering a jump diffusion process, we obtain the explicit expression of the joint Laplace transform of the distribution of a jump diffusion process and its integrated process, assuming that jump size follows the mixture of two exponential distributions, which is a special case of phase-type distributions. Based on this Laplace transform, we derive the moments of the aggregate accumulated claim amounts of insurance risk. For a financial application, we concern non-defaultable zero-coupon bond pricing. We also provide several numerical examples for the moments of aggregate accumulated claims and default-free zero-coupon bond prices.     

Mean first passage times of two-dimensional processes with jumps

In this paper, we incorporate a jump component into the model based on a two-dimensional degenerate diffusion process for the remaining lifetime of machines in the recent paper [Lefebvre, M., 2010. Mean first-passage time to zero for wear processes. Stochastic Models 26, 46-53] by the second author. We calculate explicitly the expected value of first passage times associated to the two-dimensional process when the jump component is taken to be a compound Poisson process with exponential jumps and random proportion of jumps.