Two-sided taboo limits for Markov processes and associated perfect simulation (original) (raw)

Perfect simulation and non-monotone Markovian systems

2008

Perfect simulation, or coupling from the past, is an ecient technique for sampling the steady state of monotone discrete time Markov chains. Indeed, one only needs to consider two trajectories corresponding to minimal and maximal state in the system. We show here that even for non-monotone sys- tems one only needs to compute two trajectories: an in- mum and supremum

Semi-Markov approach to continuous time random walk limit processes

The Annals of Probability, 2014

Continuous time random walks (CTRWs) are versatile models for anomalous diffusion processes that have found widespread application in the quantitative sciences. Their scaling limits are typically non-Markovian, and the computation of their finite-dimensional distributions is an important open problem. This paper develops a general semi-Markov theory for CTRW limit processes in R d with infinitely many particle jumps (renewals) in finite time intervals. The particle jumps and waiting times can be coupled and vary with space and time. By augmenting the state space to include the scaling limits of renewal times, a CTRW limit process can be embedded in a Markov process. Explicit analytic expressions for the transition kernels of these Markov processes are then derived, which allow the computation of all finite dimensional distributions for CTRW limits. Two examples illustrate the proposed method. . This reprint differs from the original in pagination and typographic detail. 1 2 M. M. MEERSCHAERT AND P. STRAKA

Stationary Distributions of Continuous-Time Markov Chains: A Review of Theory and Truncation-Based Approximations

SIAM Review

Computing the stationary distributions of a continuous-time Markov chain involves solving a set of linear equations. In most cases of interest, the number of equations is infinite or too large, and cannot be solved analytically or numerically. Several approximation schemes overcome this issue by truncating the state space to a manageable size. In this review, we first give a comprehensive theoretical account of the stationary distributions and their relation to the long-term behaviour of the Markov chain, which is readily accessible to non-experts and free of irreducibility assumptions made in standard texts. We then review truncation-based approximation schemes paying particular attention to their convergence and to the errors they introduce, and we illustrate their performance with an example of a stochastic reaction network of relevance in biology and chemistry. We conclude by elaborating on computational trade-offs associated with error control and some open questions.

On the exact simulation of functionals of stationary Markov chains

Linear Algebra and its Applications, 2004

In performance evaluation domain, simulation is an alternative when numerical analysis fail. To avoid the burn-in time problem, this paper presents an adaptation of the perfect simulation algorithm [10] to finite ergodic Markov chain with arbitrary structure. Simulation algorithms are deduced and provide samplings of functionals of the steady-state without computing the state coupling, it speeds up the algorithm by a significant factor. Based on a sparse representation of the Markov chain, the aliasing technique improves highly the complexity of the simulation. Moreover, with small adaptations, it builds a transition function algorithm that ensures coupling.

Semi-Markov Model for Excursions and Occupation time of Markov Processes

In this paper, we study the excursion time and occupation time of a Markov process below or above a given level by using a simple two states semi-Markov model. In mathematical finance, these results have an important application in the valuation of path dependent options such as Parisian options and cumulative Parisian options. We introduce a new type of Parisian option, single barrier two-sided Parisian option and extend the concept of a ruin probability in ruin theory to a Parisian type of ruin probability.

Exploiting Restricted Transitions in Quasi-Birth-and-Death Processes

2009 Sixth International Conference on the Quantitative Evaluation of Systems, 2009

In this paper we consider Quasi-Birth-and-Death (QBD) processes where the upward (resp. downward) transitions are restricted to occur only from (resp. to) a subset of the phase space. This property is exploited to reduce the computation time to find the matrix R or G of the process. The reduction is done through the definition of a censored process which can be of the M/G/1-or GI/M/1-type. The approach is illustrated through examples that show the applicability and benefits of making use of the additional structure. The examples also show how these special structures arise naturally in the analysis of queuing systems. Even more substantial gains can be realized when we further restrict the class of QBD processes under consideration.

Models of Markov processes with a random transition mechanism

arXiv: Probability, 2015

The paper deals with a certain class of random evolutions. We develop a construction that yields an invariant measure for a continuous-time Markov process with random transitions. The approach is based on a particular way of constructing the combined process, where the generator is defined as a sum of two terms: one responsible for the evolution of the environment and the second representing generators of processes with a given state of environment. (The two operators are not assumed to commute.) The presentation includes fragments of a general theory and pays a particular attention to several types of examples: (1) a queueing system with a random change of parameters (including a Jackson network and, as a special case: a single-server queue with a diffusive behavior of arrival and service rates), (2) a simple-exclusion model in presence of a special `heavy` particle, (3) a diffusion with drift-switching, and (4) a diffusion with a randomly diffusion-type varying diffusion coefficie...

Exact asymptotics for the stationary distribution of a Markov chain: a production model

Queueing Systems, 2009

We derive rough and exact asymptotic expressions for the stationary distribution π of a Markov chain arising in a queueing/production context. The approach we develop can also handle "cascades", which are situations where the fluid limit of the large deviation path from the origin to the increasingly rare event is nonlinear. Our approach considers a process that starts at the rare event. In our production example, we can have two sequences of states that asymptotically lie on the same line, yet π has different asymptotics on the two sequences.