Markov Additive Processes I. Eigenvalue Properties and Limit Theorems

Ney, Peter; Nummelin, Esa

doi:10.1214/aop/1176992159

Cited by 157 publications

(133 citation statements)

References 17 publications

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“…We will establish our results in some generality, at the level of Markov additive processes in general state space, as studied in a large deviations context by Ney and Nummelin (1987a, b), de Acosta (1988), de Acosta and Ney (1998, and references therein, following along the lines of the seminal papers of Donsker and Varadhan (1975, 1976, 1983. Thus, our results differ from known importance sampling results given e.g.…”

Section: Introductionmentioning

confidence: 63%

“…where π is the stationary measure of the Markov chain {X n }; this is true, for example, if (R) holds and 0 ∈ dom Λ P , or alternatively if (M ) holds and the set dom ψ is open [Ney and Nummelin (1987a), Lemma 3.3 and Lemma 5.2]. From (3.6), we then see that the central tendency of S 1 , S 2 , .…”

Section: Notation Hypotheses and Estimation Resultsmentioning

confidence: 94%

“…For the proofs, see Ney and Nummelin (1987a), Sections 3 and 4, and Iscoe, Ney and Nummelin (1985), Lemma 3.1.…”

Section: Nonnegative Kernels Eigenvalues and Eigenvectorsmentioning

confidence: 99%

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Importance Sampling Techniques for the Multidimensional Ruin Problem for General Markov Additive Sequences of Random Vectors

Collamore¹

2002

Ann. Appl. Probab.

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Let {(X n , S n ) : n = 0, 1, . . .} be a Markov additive process, where {X n } is a Markov chain on a general state space and S n is an additive component on R d . We consider P {S n ∈ A/ , some n} as → 0, where A ⊂ R d is open and the mean drift of {S n } is away from A. Our main objective is to study the simulation of P {S n ∈ A/ , some n} using the Monte Carlo technique of importance sampling. If the set A is convex, then we establish: (i) the precise dependence (as → 0) of the estimator variance on the choice of the simulation distribution; (ii) the existence of a unique simulation distribution which is efficient and optimal in the asymptotic sense of Siegmund (1976). We then extend our techniques to the case where A is not convex. Our results lead to positive conclusions which complement the multidimensional counterexamples of Glasserman and Wang (1997).

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Section: Introductionmentioning

confidence: 63%

Section: Notation Hypotheses and Estimation Resultsmentioning

confidence: 94%

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Importance Sampling Techniques for the Multidimensional Ruin Problem for General Markov Additive Sequences of Random Vectors

Collamore¹

2002

Ann. Appl. Probab.

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“…In the literature, the following stronger condition is often used (see, e.g. [26,27]), but we only need it for some special cases in Section 5.…”

Section: Markov Additive Process and The Wiener-hopf Factorizationmentioning

confidence: 99%

“…If S X is countable, then (3b) is automatically satisfied under assumption (3a). For the general case, useful sufficient conditions are given in Proposition 3.1 of [26].…”

Section: Markov Additive Process and The Wiener-hopf Factorizationmentioning

confidence: 99%

Wiener-Hopf Factorizations for a Multidimensional Markov Additive Process and their Applications to Reflected Processes

Miyazawa

Zwart

2012

Stochastic Systems

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We extend the framework of Neuts' matrix analytic approach to a reflected process generated by a discrete time multidimensional Markov additive process. This Markov additive process has a general background state space and a real vector valued additive component, and generates a multidimensional reflected process. Our major interest is to derive a closed form formula for the stationary distribution of this reflected process. To this end, we introduce a real valued level, and derive new versions of the Wiener-Hopf factorization for the Markov additive process with the multidimensional additive component. In particular, it is represented by moment generating functions, and we consider the domain for it to be valid.Our framework is general enough to include multi-server queues and/or queueing networks as well as non-linear time series which are currently popular in financial and actuarial mathematics. Our results yield structural results for such models. As an illustration, we apply our results to extend existing results on the tail behavior of reflected processes.A major theme of this work is to connect recent work on matrix analytic methods to classical probabilistic studies on Markov additive processes. Indeed, using purely probabilistic methods such as censoring, duality, level crossing and time-reversal (which are known in the matrix analytic methods community but date back to Arjas & Speed [2] and Pitman [29]), we extend and unify existing results in both areas.

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Inequalities between some large deviation rates

Macci

2007

Appl Stoch Models Bus & Ind

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Markov Additive Processes I. Eigenvalue Properties and Limit Theorems

Cited by 157 publications

References 17 publications

Importance Sampling Techniques for the Multidimensional Ruin Problem for General Markov Additive Sequences of Random Vectors

Importance Sampling Techniques for the Multidimensional Ruin Problem for General Markov Additive Sequences of Random Vectors

Wiener-Hopf Factorizations for a Multidimensional Markov Additive Process and their Applications to Reflected Processes

Inequalities between some large deviation rates

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