The Tax Identity For Markov Additive Risk Processes

Taxed risk processes, i.e. processes which change their drift when reaching new maxima, represent a certain type of generalizations of Lévy and of Markov additive processes (MAP), since the times at which their Markovian mechanism changes are allowed to depend on the current position. In this paper we study generalizations of the tax identity of Albrecher and Hipp (2007) from the classical risk model to more general risk processes driven by spectrally-negative MAPs. We use the Sparre Andersen risk processes with phase-type interarrivals to illustrate the ideas in their simplest form.     

Monotone matrices and monotone Markov processes

A stochastic matrix is “monotone” [4] if its row-vectors are stochastically increasing. Closure properties, characterizations and the availability of a second maximal eigenvalue are developed. Such monotonicity is present in a variety of processes in discrete and continous time. In particular, birth-death processes are monotone. Conditions for the sequential monotonicity of a process are given and related inequalities presented.     

随机单调Markov积分半群

讨论了Markov积分半群的单调性和转移函数的单调性的等价性,并得到最小的Q半群是单调的充要条件.     

Pathwise Numerical Approximations of SPDEs with Additive Noise under Non-global Lipschitz Coefficients

We consider the pathwise numerical approximation of nonlinear parabolic stochastic partial differential equations (SPDEs) driven by additive white noise under local assumptions on the coefficients only. We avoid the standard global Lipschitz assumption in the literature on the coefficients by first showing convergence under global Lipschitz coefficients but with a strong error criteria and then by applying a localization technique for one sample path on a bounded set.     

Self-Intersections of Markov Processes

We describe explicit conditions on the transition density functionsof a time-homogeneous continuous. Markov process so that almostevery path has multiple points. The application to two- andthree-dimensional diffusions is exhibited.     

On the Pathwise Comparison of Jump Processes Driven by Stochastic Intensities

In this paper we present a pathwise comparison theorem for jump processes governed by stochastic intensities and taking values in an arbitrary partially ordered Polish space. This generalizes recent results for the real-valued case. The proof given here is based on competing risk arguments rather than on thinning and allows to avoid additional domination conditions. For real valued processes we obtain an almost sure pathwise representation of the comparison result based on an i.i.d. sequence of (0, 1)-uniformly distributed random variables. For Markov chains our conditions coincide with the classical comparison conditions.     

Lumpability and Commutativity of Markov Processes

Abstract

We introduce the concepts of lumpability and commutativity of a continuous time discrete state space Markov process, and provide a necessary and sufficient condition for a lumpable Markov process to be commutative. Under suitable conditions we recover some of the basic quantities of the original Markov process from the jump chain of the lumped Markov process.     

马尔可夫过程的与时间有关的函数

设（Ｘｔ）是有转移函数的马尔可夫过程,其中Ｘｔ取值于状态空间（Ｅｔ,ξ,ｔ≥０。设ｆｔ是（Ｅｔ,ξ）到状态空间（Ｅｔ,ξ是的可测变换。本文给出了使（ｆｔ（Ｘｔ）仍是有转移函数的马尔可夫过程的充分条件,对于有函数的马尔可夫过程族,也讨论了类似的问题。     

Gilt-Edged Switching and Markov Decision Processes

A model for switching between gilt-edged securities is developed using a modified version of Howard's Markov decision algorithm. The model makes use of empirical observations of the behaviour of relative price movements. It produced some interesting results in the theory of Markov decision processes, and empirical tests of methods of implementation, which allow for constraints not included in the formal model, showed very promising results.     

Measure-Valued Markov Processes and Stochastic Flows

We consider a new class of Markov processes in the space of measures with constant mass. We present the construction of such processes in terms of probabilities that control the motion of individual particles. We study additive functionals of such processes and give examples related to stochastic flows with interaction.     

Existence and Uniqueness of Fixed Points for Markov Operators and Markov Processes

This paper concerns a Markov operator T on a space L₁, and aMarkov process P which defines a Markov operator on a spaceM of finite signed measures. For T, the paper presents necessaryand sufficient conditions for:

a the existence of invariant probabilitydensities (IPDs)

b the existence of strictly positive IPDs,and

c the existence and uniqueness of IPDs.

Similar resultson invariant probability measures for P are presented. The basicapproach is to pose a fixed-point problem as the problem ofsolving a certain linear equation in a suitable Banach space,and then obtain necessary and sufficient conditions for thisequation to have a solution. 1991 Mathematics Subject Classification:60J05, 47B65, 47N30.     

Markov Processes with Equal Capacities

Let X and be transient standard Markov processes in weak duality with respect to a -finite measure m. Let (Y, , ) be a second dual pair with the same state space E as (X, , m). Let Cap ^X and Cap ^Y be the 0-order capacities associated with (X, , m) and (Y, , ), and let V and denote the potential kernels for Y and . Assume that singletons are polar with respect to both X and Y, and that semipolar sets are of capacity zero for both dual pairs. We show that if Cap ^X (B)=Cap ^Y (B) for every Borel subset of E then there is a strictly increasing continuous additive functional D=(D _t) _t0 of (X, , m) such that

On weakly mixing Markov Processes

A result of England and Martin on weak mixing (see [6]) is extended to Markov Processes in a strengthened form, and also to continuous time Markov Processes.