The Annals of Applied Probability

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

J.F. Collamore

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Let $\{(X_n, S_n): n = 0, 1, \dots\}$ 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 $\mathbb{R}^d$. We consider $\mathbf{P}\{S_n \in A/\varepsilon, \text{some $n$}\}$ as $\varepsilon \to 0$, where $A \subset \mathbb{R}^d$ is open and the mean drift of $\{S_n\}$ is away from $A$. Our main objective is to study the simulation of $\mathbf{P}\{S_n \in A/\varepsilon, \text{some $n$}\}$ using the Monte Carlo technique of importance sampling. If the set $A$ is convex, then we establish (i) the precise dependence (as $\varepsilon \to 0$) of the estimator variance on the choice of the simulation distribution and (ii) the existence of a unique simulation distribution which is efficient and optimal in the asymptotic sense of D. Siegmund [Ann. Statist. 4 (1976) 673-684]. 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 P. Glasserman and Y. Wang [Ann. Appl. Probab. 7 (1997) 731-746].

Article information

Ann. Appl. Probab., Volume 12, Number 1 (2002), 382-421.

First available in Project Euclid: 12 March 2002

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Zentralblatt MATH identifier

Primary: 65C05: Monte Carlo methods
Secondary: 65U05 60F10: Large deviations 60J15 60K10: Applications (reliability, demand theory, etc.)

Monte Carlo methods rare event simulation hitting probabilities large deviations Harris recurrent Markov chains convex analysis


Collamore, J.F. Importance Sampling Techniques for the Multidimensional Ruin Problem for General Markov Additive Sequences of Random Vectors. Ann. Appl. Probab. 12 (2002), no. 1, 382--421. doi:10.1214/aoap/1015961169.

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  • ASMUSSEN, S. (1989). Risk theory in a Markovian environment. Scand. Actuar. J. 69-100.
  • ASMUSSEN, S. (1999). Stochastic simulation with a view towards stochastic processes. Univ. Aarhus.
  • ASMUSSEN, S. and RUBINSTEIN, R. Y. (1995). Steady-state rare events simulation in queueing models and its complexity properties. In Advances in Queueing: Models, Methods and Problems (J. Dshalalow, ed.) 429-461. CRC Press, Boca Raton, FL.
  • ATHREYA, K. B. and NEY, P. (1978). A new approach to the limit theory of recurrent Markov chains. Trans. Amer. Math. Soc. 245 493-501.
  • BOROVKOV, A. A. (1997). Limit theorems for time and place of the first boundary passage by a multidimensional random walk. Doklady 55 254-256.
  • BOROVKOV, A. A. and MOGULSKII, A. A. (1996). Large deviations for stationary Markov chains in a quarter plane. In Probability Theory and Mathematical Statistics. World Scientific, River Edge, NJ.
  • BROCKWELL, P. J. and DAVIS, R. A. (1991). Time Series: Theory and Methods, 2nd ed. Springer, New York.
  • BUCKLEW, J. (1998). The blind simulation problem and regenerative processes. IEEE Trans. Inform. Theory 44 2877-2891.
  • BUCKLEW, J. A., NEY, P. and SADOWSKY, J. S. (1990). Monte Carlo simulation and large deviations theory for uniformly recurrent Markov chains. J. Appl. Probab. 27 44-59.
  • CHEN, J.-C., LU, D., SADOWSKY, J. S. and YAO, K. (1993). On importance sampling in digital communications. I. Fundamentals. II. Trellis-coded modulation. IEEE J. Selected Areas Comm. 11 289-308.
  • COLLAMORE, J. F. (1996a). Hitting probabilities and large deviations. Ann. Probab. 24 2065-2078.
  • COLLAMORE, J. F. (1996b). Large deviation techniques for the study of the hitting probabilities of rare sets. Ph.D. dissertation, Univ. Wisconsin-Madison.
  • COLLAMORE, J. F. (1998). First passage times of general sequences of random vectors: a large deviations approach. Stochastic Process. Appl. 78 97-130.
  • COTTRELL, M., FORT, J.-C. and MALGOUYRES, G. (1983). Large deviations and rare events in the study of stochastic algorithms. IEEE Trans. Automat. Control 28 907-920.
  • CRAMÉR, H. (1930). On the mathematical theory of risk. In Scandinavian Jubilee Volume. Stockholm.
  • DE ACOSTA, A. (1988). Large deviations for vector-valued functionals of a Markov chain: lower bounds. Ann. Probab. 16 925-960.
  • DE ACOSTA, A. and NEY, P. (1998). Large deviation lower bounds for arbitrary additive functionals of a Markov chain. Ann. Probab. 26 1660-1682.
  • DEMBO, A. and ZEITOUNI, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Springer, New York.
  • DONSKER, M. D. and VARADHAN, S. R. S. (1975, 1976, 1983). Asymptotic evaluation of certain Markov process expectations for large time, I, II, III, IV. Comm. Pure Appl. Math. 28 1-47; 28 279-301; 29 389-461; 36 183-212.
  • FREIDLIN, M. I. and WENTZELL, A. D. (1984). Random Perturbations of Dynamical Systems. Springer, New York.
  • GLASSERMAN, P. and KOU, S. (1995). Analysis of an importance sampling estimator for tandem queues. ACM Trans. Modeling Comp. Simul. 4 22-42.
  • GLASSERMAN, P. and WANG, Y. (1997). Counterexamples in importance sampling for large deviations probabilities. Ann. Appl. Probab. 7 731-746.
  • HAMMERSLEY, J. M. and HANDSCOMB, D. C. (1964). Monte Carlo Methods. Chapman and Hall, London.
  • ISCOE, I., NEY, P. and NUMMELIN, E. (1985). Large deviations of uniformly recurrent Markov additive processes. Adv. in Appl. Math. 6 373-412.
  • LEHTONEN, T. and NYRHINEN, H. (1992a). On asymptotically efficient simulation of ruin probabilities in a Markovian environment. Scand. Actuar. J. 60-75.
  • LEHTONEN, T. and NYRHINEN, H. (1992b). Simulating level-crossing probabilities by importance sampling. Adv. Appl. Probab. 24 858-874.
  • LUNDBERG, F. (1909). Zur Theorie der Rückversicherung. In Verhandl. Kongr. Versicherungsmath. Wien.
  • MEYN, S. P. and TWEEDIE, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.
  • NEY, P. (1983). Dominating points and the asymptotics of large deviations for random walk on Rd. Ann. Probab. 11 158-167.
  • NEY, P. and NUMMELIN, E. (1984). Some limit theorems for Markov additive processes. In Proceedings of the Symposium on Semi-Markov Processes, Brussels.
  • NEY, P. and NUMMELIN, E. (1987a). Markov additive processes. I. Eigenvalue properties and limit theorems. Ann. Probab. 15 561-592.
  • NEY, P. and NUMMELIN, E. (1987b). Markov additive processes. II. Large deviations. Ann. Probab. 15 593-609.
  • NUMMELIN, E. (1978). A splitting technique for Harris recurrent Markov chains. Z. Wahrsch. Verw. Gebiete 43 309-318.
  • NUMMELIN, E. (1984). General Irreducible Markov Chains and Non-negative Operators. Cambridge Univ. Press.
  • REVUZ, D. (1975). Markov Chains. North-Holland, Amsterdam.
  • ROCKAFELLAR, R. T. (1970). Convex Analysis. Princeton Univ. Press.
  • SADOWSKY, J. S. (1993). On the optimality and stability of exponential twisting in Monte Carlo estimation. IEEE Trans. Inform. Theory 39 119-128.
  • SADOWSKY, J. S. (1996). On Monte Carlo estimation of large deviations probabilities. Ann. Appl. Probab. 6 399-422.
  • SIEGMUND, D. (1976). Importance sampling in the Monte Carlo study of sequential tests. Ann. Statist. 4 673-684.