Journal ArticleDOI
Two renewal theorems for general random walks tending to infinity
Harry Kesten,R. A. Maller +1 more
Reads0
Chats0
TLDR
In this paper, sufficient and necessary conditions for the existence of moments of the first passage time of a random walk are given under the condition that the random walk remains above the level x on K consecutive occasions, which has applications in option pricing.Abstract:
Necessary and sufficient conditions for the existence of moments of the first passage time of a random walk S
n
into [x, ∞) for fixed x≧ 0, and the last exit time of the walk from (−∞, x], are given under the condition that S
n
→∞ a.s. The methods, which are quite different from those applied in the previously studied case of a positive mean for the increments of S
n
, are further developed to obtain the “order of magnitude” as x→∞ of the moments of the first passage and last exit times, when these are finite. A number of other conditions of interest in renewal theory are also discussed, and some results for the first time for which the random walk remains above the level x on K consecutive occasions, which has applications in option pricing, are given.read more
Citations
More filters
Journal ArticleDOI
A continuous-time GARCH process driven by a Lévy process: stationarity and second-order behaviour
TL;DR: In this paper, the authors use a discrete-time analysis, giving necessary and sufficient conditions for the almost-sure convergence of ARCH(1) and GARCH(1,1) discrete time models.
Journal ArticleDOI
Stability of perpetuities
Charles M. Goldie,Ross Maller +1 more
TL;DR: For a series of randomly discounted terms, the authors gave an integral criterion to distinguish between almost-sure absolute convergence and divergence in probability to infinity, these being the only possible forms of asymptotic behavior.
Book ChapterDOI
Stationarity, Mixing, Distributional Properties and Moments of GARCH(p, q)-Processes
TL;DR: In this paper, some of the well known probabilistic properties of GARCH(p, q) processes are collected and analyzed. In particular, the question of strictly and of weakly stationary solutions is addressed.
Journal ArticleDOI
Asymptotic estimates of multi-dimensional stable densities and their applications
TL;DR: In this article, the relation between the upper and lower asymptotic estimates of the density and the fractal dimensions on the sphere of the spectral measure for a multivariate stable distribution is discussed.
Book
Non-homogeneous Random Walks: Lyapunov Function Methods for Near-Critical Stochastic Systems
TL;DR: In this article, the authors present a modern presentation of the "semimartingale" or "Lyapunov function" method applied to near-critical stochastic systems, exemplified by non-homogeneous random walks.
References
More filters
Book ChapterDOI
A Combinatorial Lemma and its Application to Probability Theory
TL;DR: In this article, the authors emphasized the fundamental principle that the expected value of an n-dimensional vector valued random variable is a probability measure defined on euclidean n-space E n.
Book
Nonlinear Renewal Theory in Sequential Analysis
TL;DR: Randomly Stopped Sequences Random Walks The Sequential Probability Ratio Test Nonlinear Renewal Theory Local Limit Theorems Open-Ended Tests Repeated Significance Tests Multiparameter Problems Estimation Following Sequential Testing Sequential Estimation as mentioned in this paper.
Book
Stopped Random Walks: Limit Theorems and Applications
TL;DR: In this paper, limit theorems for stopping random walks with positive drift have been studied in Probability Theory and Regularly Varying Functions (RVF) theory.