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

Orbit transfer rocket engine integrated control and health monitoring system technology readiness assessment

The objectives of this task were to: (1) estimate the technology readiness of an integrated control and health monitoring (ICHM) system for the Aerojet 7500 lbF Orbit Transfer Vehicle engine preliminary design assuming space based operations; and (2) estimate the remaining cost to advance this technology to a NASA defined 'readiness level 6' by 1996 wherein the technology has been demonstrated with a system validation model in a simulated environment. The work was accomplished through the conduct of four subtasks. In subtask 1 the minimally required functions for the control and monitoring system was specified. The elements required to perform these functions were specified in Subtask 2. In Subtask 3, the technology readiness level of each element was assessed. Finally, in Subtask 4, the development cost and schedule requirements were estimated for bringing each element to 'readiness level 6'

Rare Event Analysis for Minimum Hellinger Distance Estimators via Large Deviation Theory

Hellinger distance has been widely used to derive objective functions that are alternatives to maximum likelihood methods. While the asymptotic distributions of these estimators have been well investigated, the probabilities of rare events induced by them are largely unknown. In this article, we analyze these rare event probabilities using large deviation theory under a potential model misspecification, in both one and higher dimensions. We show that these probabilities decay exponentially, characterizing their decay via a “rate function” which is expressed as a convex conjugate of a limiting cumulant generating function. In the analysis of the lower bound, in particular, certain geometric considerations arise that facilitate an explicit representation, also in the case when the limiting generating function is nondifferentiable. Our analysis involves the modulus of continuity properties of the affinity, which may be of independent interest

Sharp Probability Tail Estimates for Portfolio Credit Risk

Portfolio credit risk is often concerned with the tail distribution of the total loss, defined to be the sum of default losses incurred from a collection of individual loans made out to the obligors. The default for an individual loan occurs when the assets of a company (or individual) fall below a certain threshold. These assets are typically modeled according to a factor model, thereby introducing a strong dependence both among the individual loans, and potentially also among the multivariate vector of common factors. In this paper, we derive sharp tail asymptotics under two regimes: (i) a large loss regime, where the total number of defaults increases asymptotically to infinity; and (ii) a small default regime, where the loss threshold for an individual loan is allowed to tend asymptotically to negative infinity. Extending beyond the well-studied Gaussian distributional assumptions, we establish that—while the thresholds in the large loss regime are characterized by idiosyncratic factors specific to the individual loans—the rate of decay is governed by the common factors. Conversely, in the small default regime, we establish that the tail of the loss distribution is governed by systemic factors. We also discuss estimates for Value-at-Risk, and observe that our results may be extended to cases where the number of factors diverges to infinity

Sharp Probability Tail Estimates for Portfolio Credit Risk

Portfolio credit risk is often concerned with the tail distribution of the total loss, defined to be the sum of default losses incurred from a collection of individual loans made out to the obligors. The default for an individual loan occurs when the assets of a company (or individual) fall below a certain threshold. These assets are typically modeled according to a factor model, thereby introducing a strong dependence both among the individual loans, and potentially also among the multivariate vector of common factors. In this paper, we derive sharp tail asymptotics under two regimes: (i) a large loss regime, where the total number of defaults increases asymptotically to infinity; and (ii) a small default regime, where the loss threshold for an individual loan is allowed to tend asymptotically to negative infinity. Extending beyond the well-studied Gaussian distributional assumptions, we establish that—while the thresholds in the large loss regime are characterized by idiosyncratic factors specific to the individual loans—the rate of decay is governed by the common factors. Conversely, in the small default regime, we establish that the tail of the loss distribution is governed by systemic factors. We also discuss estimates for Value-at-Risk, and observe that our results may be extended to cases where the number of factors diverges to infinity