Quasi-stationary distributions for Lévy processes (original) (raw)

Parisian quasi-stationary distributions for asymmetric Lévy processes

Statistics & Probability Letters

In recent years there has been some focus on quasi-stationary behaviour of an onedimensional Lévy process X, where we ask for the law P(Xt ∈ dy|τ − 0 > t) for t → ∞ and τ − 0 = inf{t ≥ 0 : Xt < 0}. In this paper we address the same question for so-called Parisian ruin time τ θ , that happens when process stays below zero longer than independent exponential random variable with intensity θ.

B No 5 SCALING LIMITS OF OVERSHOOTING LÉVY WALKS ∗

2012

In this paper, we obtain the scaling limits of one-dimensional overshooting Lévy walks. We also find the limiting processes for extensions of Lévy walks, in which the waiting times and jumps are related by powerlaw, exponential and logarithmic dependence. We find that limiting processes of overshooting Lévy walk are characterized by infinite mean-squaredisplacement. It also occurs that introducing different dependence between waiting times and jumps of Lévy walks results in subdiffusive properties.

Lévy walks with rests: Long-time analysis

Physical Review E, 2022

In this paper we analyze the asymptotic behavior of Lévy walks with rests. Applying recent results in the field of functional convergence of continuous-time random walks we find the corresponding limiting processes. Depending on the parameters of the model, we show that in the limit we can obtain standard Lévy walk or the process describing competition between subdiffusion and Lévy flights. Some other more complicated limit forms are also possible to obtain. Finally we present some numerical results, which confirm our findings.

On doubly reflected completely asymmetric Lévy processes

Stochastic processes and their applications, 2003

Consider a completely asymmetric Lévy process X and let Z be X reflected at 0 and at a > 0. In applied probability (e.g. ), the process Z turns up in the study of the virtual waiting time in a M/G/1-queue with finite buffer a or the water level in a finite dam of size a. We find an expression for the resolvent density of Z. We show Z is positive recurrent and determine the invariant measure. Using the regenerative property of Z, we determine the asymptotic law of t −1 t 0 f (Z s )ds for an appropriate class of functions f . Finally, the long time average of the local time of Z in x ∈ [0, a] is studied. . Primary 60J30. Secondary 28D10.

On the drawdown of completely asymmetric Levy processes

The drawdown process Y=barX−XY=\bar{X} - XY=barX−X of a completely asymmetric L\'{e}vy process XXX is given by XXX reflected at its running supremum barX\bar{X}barX.In this paper we explicitly express the law of the sextuple (taua,barGtaua,underlineXtaua,barXtaua,Ytaua−,Ytaua−a)(\tau_a,\bar{G}_{\tau_a},\underline{X}_{\tau_a},\bar{X}_{\tau_a},Y_{\tau_a-},Y_{\tau_a}-a)(taua,barGtaua,underlineXtaua,barXtaua,Ytaua−,Ytaua−a) in terms of the scale function and the L\'evy measure of XXX, where taua\tau_ataua denotes the first-passage time of YYY over the level a>0a>0a>0, barGtaua\bar{G}_{\tau_a}barGtaua is the time of the last supremum of XXX prior to taua\tau_ataua and underlineX\underline{X}underlineX is the running infimum of XXX. We also explicitly identify the distribution of the drawup hatYtaua\hat{Y}_{\tau_a}hatYtaua at the moment taua\tau_ataua, where hatY=X−underlineX\hat{Y} = X-\underline{X}hatY=X−underlineX, and derive the probability of a large drawdown preceding a small rally. These results are applied to the Carr & Wu \cite{CarrWu} model for S&P 500.

Submitted to the Annals of Probability SAMPLE PATH LARGE DEVIATIONS FOR LÉVY PROCESSES AND RANDOM WALKS WITH REGULARLY VARYING INCREMENTS By

2018

Let X be a Lévy process with regularly varying Lévy measure ν. We obtain sample-path large deviations for scaled processes X̄n(t) , X(nt)/n and obtain a similar result for random walks with regularly varying increments. Our results yield detailed asymptotic estimates in scenarios where multiple big jumps in the increment are required to make a rare event happen; we illustrate this through detailed conditional limit theorems. In addition, we investigate connections with the classical large deviations framework. In that setting, we show that a weak large deviation principle (with logarithmic speed) holds, but a full large deviation principle does not hold.

Conditioned limit theorems for random walks with negative drift

Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, 1980

In this paper we will solve a problem posed by Iglehart. In (1975) he conjectured that if Sn is a random walk with negative mean and finite variance then there is a constant c~ so that (St,.j/c~nl/2[N>n) converges weakly to a process which he called the Brownian excursion. It will be shown that his conjecture is false or, more precisely, that if ES~ =-a<0, ES~ < oo, and there is a slowly varying function L so that P(SI > x)~ x-q L(x) as x-~ oo then (SE,,.j/nlS,,>O) and (St,,jnlN>n) converge weakly to nondegenerate limits. The limit processes have sample paths which have a single jump (with d.f. (1-(x/a)-q) +) and are otherwise linear with slope-a. The jump occurs at a uniformly distributed time in the first case and at t = 0 in the second.

On the asymptotic behaviour of Lévy processes, Part I: Subexponential and exponential processes

Stochastic Processes and their Applications, 2009

We study tail probabilities of the suprema of Lévy processes with subexponential or exponential marginal distributions over compact intervals. Several of the processes for which the asymptotics are studied here for the first time have recently become important to model financial time series. Hence our results should be important, for example, in the assessment of financial risk.

Sinaıˇ's condition for real valued Lévy processes

Annales De L Institut Henri Poincare-probabilites Et Statistiques, 2007

We prove that the upward ladder height subordinator H associated to a real valued Lévy process ξ has Laplace exponent ϕ that varies regularly at ∞ (resp. at 0) if and only if the underlying Lévy process ξ satisfies Sinaǐ's condition at 0 (resp. at ∞). Sinaǐ's condition for real valued Lévy processes is the continuous time analogue of Sinaǐ's condition for random walks. We provide several criteria in terms of the characteristics of ξ to determine whether or not it satisfies Sinaǐ's condition. Some of these criteria are deduced from tail estimates of the Lévy measure of H, here obtained, and which are analogous to the estimates of the tail distribution of the ladder height random variable of a random walk which are due to Veraverbeke and Grübel.