Random recurrence equations and ruin in a Markov-dependent stochastic economic environment

We develop sharp large deviation asymptotics for the probability of ruin in a Markov-dependent stochastic economic environment and study the extremes for some related Markovian processes which arise in financial and insurance mathematics, related to perpetuities and the $\operatorname {ARCH}(1)$ and $\operatorname {GARCH}(1,1)$ 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 "G\"artner--Ellis" assumption on the driving process.Comment: Published in at http://dx.doi.org/10.1214/08-AAP584 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Tail estimates for stochastic fixed point equations via nonlinear renewal theory

This paper introduces a new approach, based on large deviation theory and nonlinear renewal theory, for analyzing solutions to stochastic fixed point equations of the form V D = f(V), where f(v) = A max{v, D} + B for a random triplet (A, B, D) ∈ (0, ∞) × R2. Our main result establishes the tail estimate P {V> u} ∼ Cu−ξ as u → ∞, providing a new, explicit probabilistic characterization for the constant C. Our methods rely on a dual change of measure, which we use to analyze the path properties of the forward iterates of the stochastic fixed point equation. To analyze these forward iterates, we establish several new results in the realm of nonlinear renewal theory for these processes. As a consequence of our techniques, we develop a new characterization of the extremal index, as well as a Lundberg-type upper bound for P {V> u}. Finally, we provide an extension of our main result to random Lipschitz maps of D = f and A max{v, D ∗ } + B ∗ ≤ f(v) ≤ A max{v, D} + B. the form Vn = fn(Vn−1), where fn

Rare event simulation for processes generated via stochastic fixed point equations

In a number of applications, particularly in financial and actuarial mathematics, it is of interest to characterize the tail distribution of a random variable $V$ satisfying the distributional equation $V\stackrel{\mathcal{D}}{=}f(V)$, where $f(v)=A\max\{v,D\}+B$ for $(A,B,D)\in(0,\infty)\times {\mathbb{R}}^2$. This paper is concerned with computational methods for evaluating these tail probabilities. We introduce a novel importance sampling algorithm, involving an exponential shift over a random time interval, for estimating these rare event probabilities. We prove that the proposed estimator is: (i) consistent, (ii) strongly efficient and (iii) optimal within a wide class of dynamic importance sampling estimators. Moreover, using extensions of ideas from nonlinear renewal theory, we provide a precise description of the running time of the algorithm. To establish these results, we develop new techniques concerning the convergence of moments of stopped perpetuity sequences, and the first entrance and last exit times of associated Markov chains on $\mathbb{R}$. We illustrate our methods with a variety of numerical examples which demonstrate the ease and scope of the implementation.Comment: Published in at http://dx.doi.org/10.1214/13-AAP974 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org