Large deviations of uniformly recurrent Markov additive processes

Iscoe, Ian; Ney, Peter; Nummelin, Esa

doi:10.1016/0196-8858(85)90017-x

Cited by 81 publications

(86 citation statements)

References 31 publications

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“…For further applications to Markov and semi-Markov processes, see e.g. Iscoe, Ney and Nummelin (1985), Ney and Nummelin (1987a), and Meyn and Tweedie (1993).…”

Section: Notation Hypotheses and Estimation Resultsmentioning

confidence: 99%

“…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%

“…Proof (i) Following Iscoe, Ney and Nummelin (1985), Lemma 3.4, introduce the generating function Nummelin (1984), Proposition 4.7 (i)]. Note by the construction of K M Q that individual terms and number of nonzero terms in the summand on the right of (4.21) are finite; consequently,…”

Section: Proof Of Theorem 31: Lower Boundmentioning

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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“…For further applications to Markov and semi-Markov processes, see e.g. Iscoe, Ney and Nummelin (1985), Ney and Nummelin (1987a), and Meyn and Tweedie (1993).…”

Section: Notation Hypotheses and Estimation Resultsmentioning

confidence: 99%

“…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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“…The paper [21] also shows that the rate function of the large deviation principle for {S n /n} is the convex conjugate of H, i.e.,…”

Section: Ldp For a Uniformly Recurrent Markov Chainmentioning

confidence: 99%

“…Fix any α ∈ R d . Then by [21], the non-negative kernel exp{hα, g(y)i}p(x, dy) admits a unique real eigenvalue exp{H(α)} and a unique (up to a multiplicative constant) eigenfunction r(x; α) in the sense that, for every x ∈ S,…”

Section: Ldp For a Uniformly Recurrent Markov Chainmentioning

confidence: 99%

Adaptive Importance Sampling for Uniformly Recurrent Markov Chains

Dupuis¹,

Wang²

2003

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Importance sampling is a variance reduction technique for efficient estimation of rare-event probabilities by Monte Carlo. In standard importance sampling schemes, the system is simulated using an a priori fixed change of measure suggested by a large deviation lower bound analysis. Recent work, however, has suggested that such schemes do not work well in many situations. In this paper, we consider adaptive importance sampling in the setting of uniformly recurrent Markov chains. By "adaptive," we mean that the change of measure depends on the history of the samples. Based on a control-theoretic approach to large deviations, the existence of asymptotically optimal adaptive schemes is demonstrated in great generality. In this framework, the difference between a static change of measure and an adaptive change of measure amounts to the difference between an open-loop control and a feed-back control. The implementation of the adaptive schemes is carried out with the help of a limiting Bellman equation. Also presented are numerical examples contrasting the adaptive and standard schemes.

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