Note on the Tail Behavior of Random Walk Maxima with Heavy Tails and Negative Drift

2003 ◽  
Vol 7 (3) ◽  
pp. 57-61 ◽  
Author(s):  
Rob Kaas ◽  
Qihe Tang
Keyword(s):  
Random Walk ◽  
Heavy Tails ◽  
Tail Behavior ◽  
Negative Drift

Heavy Tails of Discounted Aggregate Claims in the Continuous-Time Renewal Model

2007 ◽  
Vol 44 (02) ◽  
pp. 285-294 ◽  
Author(s):  
Qihe Tang
Keyword(s):  
Asymptotic Formula ◽  
Continuous Time ◽  
Heavy Tails ◽  
Tail Behavior ◽  
Renewal Model ◽  
Discounted Aggregate Claims ◽  
Aggregate Claims ◽  
Tail Asymptotic

We study the tail behavior of discounted aggregate claims in a continuous-time renewal model. For the case of Pareto-type claims, we establish a tail asymptotic formula, which holds uniformly in time.

scholarly journals On the Maximum of a Random Walk with Small Negative Drift

1983 ◽  
Vol 11 (3) ◽  
pp. 491-505 ◽  
Author(s):  
Michael J. Klass
Keyword(s):  
Random Walk ◽  
Negative Drift

scholarly journals Pricing of Catastrophe Insurance Options Under Immediate Loss Reestimation

2008 ◽  
Vol 45 (03) ◽  
pp. 831-845 ◽  
Author(s):  
Francesca Biagini ◽  
Yuliya Bregman ◽  
Thilo Meyer-Brandis
Keyword(s):  
Fourier Transform ◽  
Option Pricing ◽  
Heavy Tails ◽  
Initial Estimate ◽  
Tail Behavior ◽  
Catastrophe Insurance ◽  
Catastrophe Modeling ◽  
Loss Distributions ◽  
Loss Index

We specify a model for a catastrophe loss index, where the initial estimate of each catastrophe loss is reestimated immediately by a positive martingale starting from the random time of loss occurrence. We consider the pricing of catastrophe insurance options written on the loss index and obtain option pricing formulae by applying Fourier transform techniques. An important advantage is that our methodology works for loss distributions with heavy tails, which is the appropriate tail behavior for catastrophe modeling. We also discuss the case when the reestimation factors are given by positive affine martingales and provide a characterization of positive affine local martingales.

scholarly journals Asymptotic Probabilities of an Exceedance Over Renewal Thresholds with an Application to Risk Theory

2005 ◽  
Vol 42 (01) ◽  
pp. 153-162 ◽  
Author(s):  
Christian Y. Robert
Keyword(s):  
Random Walk ◽  
Stopping Time ◽  
Random Variables ◽  
Risk Theory ◽  
Heavy Tailed Distribution ◽  
Negative Drift ◽  
Heavy Tailed ◽  
Independent And Identically Distributed

Let (Y n , N n ) n≥1 be independent and identically distributed bivariate random variables such that the N n are positive with finite mean ν and the Y n have a common heavy-tailed distribution F. We consider the process (Z n ) n≥1 defined by Z n = Y n - Σ n-1, where It is shown that the probability that the maximum M = max n≥1 Z n exceeds x is approximately as x → ∞, where F' := 1 - F. Then we study the integrated tail of the maximum of a random walk with long-tailed increments and negative drift over the interval [0, σ], defined by some stopping time σ, in the case in which the randomly stopped sum is negative. Finally, an application to risk theory is considered.

A local limit theorem for random walk maxima with heavy tails

2002 ◽  
Vol 56 (4) ◽  
pp. 399-404 ◽  
Author(s):  
Søren Asmussen ◽  
Vladimir Kalashnikov ◽  
Dimitrios Konstantinides ◽  
Claudia Klüppelberg ◽  
Gurami Tsitsiashvili
Keyword(s):  
Random Walk ◽  
Limit Theorem ◽  
Heavy Tails ◽  
Local Limit Theorem ◽  
Local Limit

On the maximum and absorption time of left-continuous random walk

1978 ◽  
Vol 15 (2) ◽  
pp. 292-299 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Random Walk ◽  
Limit Theorems ◽  
Initial Position ◽  
Absorption Time ◽  
Conditional Limit Theorems ◽  
Negative Drift ◽  
Continuous Random Walk ◽  
Conditional Limit ◽  
Transformed Distribution

In a recent paper Green (1976) obtained some conditional limit theorems for the absorption time of left-continuous random walk. His methods require that in the driftless case the increment distribution has exponentially decreasing tails and that the same is true for a transformed distribution in the case of negative drift.Here we take a different approach which will produce Green's results under minimal conditions. Limit theorems are given for the maximum as the initial position of the random walk tends to infinity.

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