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30
Iterated random functions
 SIAM Review
, 1999
"... Abstract. Iterated random functions are used to draw pictures or simulate large Ising models, among other applications. They offer a method for studying the steady state distribution of a Markov chain, and give useful bounds on rates of convergence in a variety of examples. The present paper surveys ..."
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Cited by 131 (1 self)
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Abstract. Iterated random functions are used to draw pictures or simulate large Ising models, among other applications. They offer a method for studying the steady state distribution of a Markov chain, and give useful bounds on rates of convergence in a variety of examples. The present paper surveys the field and presents some new examples. There is a simple unifying idea: the iterates of random Lipschitz functions converge if the functions are contracting on the average. 1. Introduction. The
Levy Integrals and the Stationarity of generalised OrnsteinUhlenbeck processes
"... The generalised OrnsteinUhlenbeck process constructed from a bivariate Lévy process (ξt, ηt)t≥0 is defined as Vt = e −ξt ( ∫ t ..."
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Cited by 21 (9 self)
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The generalised OrnsteinUhlenbeck process constructed from a bivariate Lévy process (ξt, ηt)t≥0 is defined as Vt = e −ξt ( ∫ t
Elementary fixed points of the BRW smoothing transforms with infinite number of summands
, 2003
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Integrated insurance risk models with exponential Lévy investment
 Insurance Math. Econ
, 2008
"... We consider an insurance risk model for the cashflow of an insurance company, which invests its reser.v.e into a portfolio consisting of risky and riskless assets. The price of the risky asset is modeled by an exponential Lévy process. We derive the integrated risk process and the corresponding disc ..."
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Cited by 13 (4 self)
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We consider an insurance risk model for the cashflow of an insurance company, which invests its reser.v.e into a portfolio consisting of risky and riskless assets. The price of the risky asset is modeled by an exponential Lévy process. We derive the integrated risk process and the corresponding discounted net loss process. We calculate certain quantities as characteristic functions and moments. We also show under weak conditions stationarity of the discounted net loss process and derive the left and right tail behaviour of the model. Our results show that the model carries a high risk, which may originate either from large insurance claims or from the risky investment.
On distributional properties of perpetuities
, 2008
"... SUMMARY. We study probability distributions of convergent random series of a special structure, called perpetuities. By giving a new argument, we prove that such distributions are of pure type: degenerate, absolutely continuous, or continuously singular. We further provide necessary and sufficient c ..."
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Cited by 13 (3 self)
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SUMMARY. We study probability distributions of convergent random series of a special structure, called perpetuities. By giving a new argument, we prove that such distributions are of pure type: degenerate, absolutely continuous, or continuously singular. We further provide necessary and sufficient criteria for the finiteness of pmoments, p> 0 as well as exponential moments. In particular, a formula for the abscissa of convergence of the moment generating function is provided. The results are illustrated with a number of examples at the end of the article.
Density Approximation and Exact Simulation of Random Variables that are Solutions of FixedPoint Equations
 Adv. Appl. Probab
, 2002
"... An algorithm is developed for the exact simulation from distributions that are defined as fixedpoints of maps between spaces of probability measures. The fixedpoints of the class of maps under consideration include examples of limit distributions of random variables studied in the probabilistic an ..."
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Cited by 10 (6 self)
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An algorithm is developed for the exact simulation from distributions that are defined as fixedpoints of maps between spaces of probability measures. The fixedpoints of the class of maps under consideration include examples of limit distributions of random variables studied in the probabilistic analysis of algorithms. Approximating sequences for the densities of the fixedpoints with explicit error bounds are constructed. The sampling algorithm relies on a modified rejection method. AMS subject classifications. Primary: 65C10; secondary: 65C05, 68U20, 11K45.
The tail of the stationary distribution of a random coefficient AR(q) model
, 2001
"... We investigate a stationary random coefficient autoregressive process. Using renewal type arguments tailormade for such processes we show that the stationary distribution has a powerlaw tail. When the model is normal, we show that the model is in distribution equivalent to an autoregressive proces ..."
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Cited by 8 (3 self)
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We investigate a stationary random coefficient autoregressive process. Using renewal type arguments tailormade for such processes we show that the stationary distribution has a powerlaw tail. When the model is normal, we show that the model is in distribution equivalent to an autoregressive process with ARCH errors. Hence we obtain the tail behaviour of any such model of arbitrary order.
RENEWAL THEORY FOR FUNCTIONALS OF A MARKOV CHAIN WITH COMPACT STATE SPACE
, 2003
"... Motivated by multivariate random recurrence equations we prove a new analogue of the Key Renewal Theorem for functionals of a Markov chain with compact state space in the spirit of Kesten [Ann. Probab. 2 (1974) 355–386]. Compactness of the state space and a certain continuity condition allows us to ..."
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Cited by 6 (4 self)
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Motivated by multivariate random recurrence equations we prove a new analogue of the Key Renewal Theorem for functionals of a Markov chain with compact state space in the spirit of Kesten [Ann. Probab. 2 (1974) 355–386]. Compactness of the state space and a certain continuity condition allows us to simplify Kesten’s proof considerably.
RANDOM RECURRENCE EQUATIONS AND RUIN IN A MARKOVDEPENDENT STOCHASTIC ECONOMIC ENVIRONMENT
, 2009
"... We develop sharp large deviation asymptotics for the probability of ruin in a Markovdependent stochastic economic environment and study the extremes for some related Markovian processes which arise in financial and insurance mathematics, related to perpetuities and the ARCH(1) and GARCH(1, 1) time ..."
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Cited by 6 (1 self)
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We develop sharp large deviation asymptotics for the probability of ruin in a Markovdependent stochastic economic environment and study the extremes for some related Markovian processes which arise in financial and insurance mathematics, related to perpetuities and the ARCH(1) and GARCH(1, 1) time series models. Our results build upon work of Goldie [Ann. Appl. Probab. 1 (1991) 126–166], who has developed tail asymptotics applicable for independent sequences of random variables subject to a random recurrence equation. In contrast, we adopt a general approach based on the theory of Harris recurrent Markov chains and the associated theory of nonnegative operators, and meanwhile develop certain recurrence properties for these operators under a nonstandard “Gärtner–Ellis” assumption on the driving process.