Extremes of Levy processes with light tails

On tails of exit times of multidimensional Lévy processes

Using a very simple argument based on the indepenence of increments and the fact that in a finite dimensional space $R^{d}$ there are not too many directions, we derive a theorem stating that exit time of any (non-constant) Lévy process on $R^{d}$ from a ball has exponentially light tails.

Tail behavior of stopped Lévy processes with Markov modulation

This article concerns the tail probabilities of a light-tailed Markov-modulated Lévy process stopped at a state-dependent Poisson rate. The tails are shown to decay exponentially at rates given by the unique positive and negative roots of the spectral abscissa of a certain matrix-valued function. We illustrate the use of our results with an application to the stationary distribution of wealth in a simple economic model in which agents with constant absolute risk aversion are subject to random mortality and income fluctuation.

First exit times of solutions of stochastic differential equations driven by multiplicative Levy noise with heavy tails

In this paper we study first exit times from a bounded domain of a gradient dynamical system $\dot Y_t=-\nabla U(Y_t)$ perturbed by a small multiplicative Lévy noise with heavy tails. A special attention is paid to the way the multiplicative noise is introduced. In particular we determine the asymptotics of the first exit time of solutions of Itô, Stratonovich and Marcus canonical SDEs.

重尾复合更新过程与Lévy过程在随机时间区间上最大值的渐近性

Maxima over random time intervals for heavy-tailed compound renewal and Lévy processes

本文研究了带负漂移的重尾复合更新过程和Lévy过程在随机时间区间上最大值的次指数尾部渐近性，结果对不依赖未来增量的随机时间一致成立，并与随机游走理论建立联系。

对重尾随机向量进行聚合：从有限和到莱维过程

Aggregating heavy-tailed random vectors: from finite sums to Lévy processes

研究揭示了重尾随机向量聚合的尾部行为，指出其不仅由'一个大跳跃'原则决定，还存在更一般的原理，通过多维正则变化性分析，展示了尾概率的近似方法及莱维过程的多维正则变化性关系。

Stochastic SIR Levy jump model with heavy tailed increments

This paper considers a general stochastic SIR epidemic model driven by a multidimensional Levy jump process with heavy tailed increments and possible correlation between noise components. In this framework, we derive new sufficient conditions for disease extinction and persistence in the mean. Our method differs from previous approaches by the use of Kunita's inequality instead of the Burkholder-Davis-Gundy inequality for continuous processes, and allows for the treatment of infinite Levy measures by the definition of new threshold values. An SIR model driven by a tempered stable process is pr…

Maximum Drawdown and Drawdown Duration of Spectrally Negative Levy Processes Decomposed at Extremes

Path decomposition is performed to characterize the law of the pre/post-supremum, post-infimum and the intermediate processes of a spectrally negative Levy process taken up to an independent exponential time T: As a result, mainly the distributions of the supremum of the post-infimum process and the maximum drawdown of the pre/postsupremum, post-infimum processes and the intermediate processes are obtained together with the law of drawdown durations.

Extreme statistics of superdiffusive Levy flights and every other Levy subordinate Brownian motion

The search for hidden targets is a fundamental problem in many areas of science, engineering, and other fields. Studies of search processes often adopt a probabilistic framework, in which a searcher randomly explores a spatial domain for a randomly located target. There has been significant interest and controversy regarding optimal search strategies, especially for superdiffusive processes. The optimal search strategy is typically defined as the strategy that minimizes the time it takes a given single searcher to find a target, which is called a first hitting time (FHT). However, many systems…

Lévy processes conditioned to stay in a half-space with applications to directional extremes

This paper provides a multivariate extension of Bertoin's pathwise construction of a Lévy process conditioned to stay positive/negative. Thus obtained processes conditioned to stay in half-spaces are closely related to the original process on a compact time interval seen from its directional extremal points. In the case of a correlated Brownian motion the law of the conditioned process is obtained by a linear transformation of a standard Brownian motion and an independent Bessel-3 process. Further motivation is provided by a limit theorem corresponding to zooming in on a Lévy process with a Br…

Long time behavior of Levy-driven Ornstein-Uhlenbeck process with regime-switchin

In this work we investigate the long time behavior of the Ornstein-Uhlenbeck process driven by Levy noise with regime-switching. We provide explicit criteria on the transience and recurrence of this process. Contrasted with the Ornstein-Uhlenbeck process driven simply by Brownian motion, whose stationary distribution must be light-tailed, both the jumps caused by the Levy noise and regime-switching described by Markov chain can derive the heavy-tailed property of the stationary distribution. In this work, the different role played by Levy measure and regime-switching process is clearly charact…

The correlation function of a queue with Levy and Markov additive input

Let $(Q_t)$ be a stationary workload process, and $r(t)$ the correlation coefficient of $Q_0$ and $Q_t$. In a series of previous papers (i) the transform of $r(\cdot)$ has been derived for the case that the driving process is spectrally-positive (sp) or spectrally-negative (sn) Levy, (ii) it has been shown that for sp-Levy and sn-Levy input $r(\cdot)$ is positive, decreasing, and convex, (iii) in case the driving Levy process is light-tailed (a condition that is automatically fulfilled in the sn case), the decay of the decay rate agrees with that of the tail of the busy period distribution. In…

Asymptotics of densities of first passage times for spectrally negative Lévy processes

We study a first passage time of a Lévy process over a positive constant level. In the spectrally negative case we give conditions for absolutely continuity of the distributions of the first passage times. The tail asymptotics of their densities are also clarified, where the asymptotics depend on tail behaviour of the corresponding Lévy measures. We apply our results to the mathematical finance, in particular, the credit default swap pricing.

分支 Lévy 过程极值的大偏差与几乎必然收敛

Large deviations and almost sure convergence for the extremes of branching Lévy processes

针对空间运动带正则变化尾部的超临界分支 Lévy 过程，本文在已有弱收敛研究基础上，建立其极值的大偏差及最大值的几乎必然收敛结果。

The first passage time problem over a moving boundary for asymptotically stable Lévy processes

We study the asymptotic tail behaviour of the first-passage time over a moving boundary for asymptotically $α$-stable Lévy processes with $α<1$. Our main result states that if the left tail of the Lévy measure is regularly varying with index $- α$ and the moving boundary is equal to $1 - t^γ$ for some $γ<1/α$, then the probability that the process stays below the moving boundary has the same asymptotic polynomial order as in the case of a constant boundary. The same is true for the increasing boundary $1 + t^γ$ with $γ<1/α$ under the assumption of a regularly varying right tail with index $- α…

Metastable Behaviour of Small Noise Levy-Driven Diffusions

We consider a dynamical system in R driven by a vector field -U', where U is a multi-well potential satisfying some regularity conditions. We perturb this dynamical system by a Levy noise of small intensity and such that the heaviest tail of its Levy measure is regularly varying. We show that the perturbed dynamical system exhibits metastable behaviour i.e. on a proper time scale it reminds of a Markov jump process taking values in the local minima of the potential U. Due to the heavy-tail nature of the random perturbation, the results differ strongly from the well studied purely Gaussian case…

Sinai's condition for real valued Lévy processes

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 $\infty$ (resp. at 0) if and only if the underlying Lévy process $ξ$ satisfies Sinai's condition at 0 (resp. at $\infty$). Sinai's condition for real valued Lévy processes is the continuous time analogue of Sinai's condition for random walks. We provide several criteria in terms of the characteristics of $ξ$ to determine whether or not it satisfies Sinai's condition. Some of these criteria are deduced from tail estimates of the Lévy measure of $…

重尾条件下 Lévy 驱动的保险风险模型中的多元破产

Multivariate Ruin in Lévy-driven Insurance Risk Models under Heavy Tails

本文针对重尾索赔下多业务线保险公司的有限时间破产问题，利用多元Lévy过程与子锥正则变化建模，给出了破产概率的精确缩放率及极限行为，并通过模拟验证。

生存概率的渐近性与下尾概率问题

Asymptotics of survival probabilities and lower tail probability problem

本文补遗利用（s）SL过程性质推导小q时Wiener-Hopf因子新公式、生存概率渐近首项，并计算更一般SINH正则过程的下尾概率。

The principle of a single big jump: discrete and continuous time modulated random walks with heavy-tailed increments

We consider a modulated process S which, conditional on a background process X, has independent increments. Assuming that S drifts to -infinity and that its increments (jumps) are heavy-tailed (in a sense made precise in the paper), we exhibit natural conditions under which the asymptotics of the tail distribution of the overall maximum of S can be computed. We present results in discrete and in continuous time. In particular, in the absence of modulation, the process S in continuous time reduces to a Levy process with heavy-tailed Levy measure. A central point of the paper is that we make ful…

Cramér's estimate for stable processes with power drift

We investigate the upper tail probabilities of the all-time maximum of a stable Lévy process with a power negative drift. The asymptotic behaviour is shown to be exponential in the spectrally negative case and polynomial otherwise, with explicit exponents and constants. Analogous results are obtained, at a less precise level, for the fractionally integrated stable Lévy process. We also study the lower tail probabilities of the integrated stable Lévy process in the presence of a power positive drift.

A note on refracted Lévy processes without positive jumps

For a refracted spectrally negative Levy process, we find some new and fantastic formulas for its q-potential measures without killing. Unlike previous results, which are written in terms of the known q-scale functions, our formulas are free of the q-scale functions. This makes our results become extremely important since it is likely that our formulas also hold for a general refracted Levy process.

On Lévy processes conditioned to stay positive

We construct the law of Lévy processes conditioned to stay positive under general hypotheses. We obtain a Williams type path decomposition at the minimum of these processes. This result is then applied to prove the weak convergence of the law of Lévy processes conditioned to stay positive as their initial state tends to 0. We describe an absolute continuity relationship between the limit law and the measure of the excursions away from 0 of the underlying Lévy process reflected at its minimum. Then, when the Lévy process creeps upwards, we study the lower tail at 0 of the law of the height this…

Hitting times of points and intervals for symmetric Lévy processes

For one-dimensional symmetric Lévy processes, which hit every point with positive probability, we give sharp bounds for the tail function of the first hitting time of B which is either a single point or an interval. The estimates are obtained under some weak type scaling assumptions on the characteristic exponent of the process. We apply these results to prove optimal estimates of the transition density of the process killed after hitting B.

Excursions away from a regular point for one-dimensional symmetric Levy processes without Gaussian part

The characteristic measure of excursions away from a regular point is studied for a class of symmetric Lévy processes without Gaussian part. It is proved that the harmonic transform of the killed process enjoys Feller property. The result is applied to prove extremeness of the excursion measure and to prove several sample path behaviors of the excursion and the $ h $-path processes.

带稳定分支的临界分支Lévy过程中最大位移的尾概率

Tail probability of maximal displacement in critical branching Lévy process with stable branching

该研究在更弱的后代分布尾假设下，推导了带稳定分支的临界分支Lévy过程最大位移的尾概率，推广了分支布朗运动、分支随机游走的相关经典结果。

Hitting probabilities for Lévy processes on the real line

We prove sharp two-sided estimates on the tail probability of the first hitting time of bounded interval as well as its asymptotic behaviour for general non-symmetric processes which satisfy an integral condition \[ \int_0^{\infty} \frac{dξ}{1+\operatorname{Re} ψ(ξ)}<\infty. \] To this end, we first prove and then apply the global scale invariant Harnack inequality. Results are obtained under certain conditions on the characteristic exponent. We provide a wide class of Lévy processs which satisfy these assumptions.

Scaling limit results for the sum of many inverse Lévy subordinators

The first passage time process of a Lévy subordinator with heavy-tailed Lévy measure has long-range dependent paths. The random fluctuations that appear under two natural schemes of summation and time scaling of such stochastic processes are shown to converge weakly. The limit process is fractional Brownian motion in one case and a non-Gaussian and non-stable process in the other case. The latter appears to be of independent interest as a random process that arises under the influence of coexisting Gaussian and stable domains of attraction and is known from other applications to provide a brid…

Mittag-Leffler Lévy Processes

In this article, we introduce Mittag-Leffler Lévy process and provide two alternative representations of this process. First, in terms of Laplace transform of the marginal densities and next as a subordinated stochastic process. Both these representations are useful in analyzing the properties of the process. Since integer order moments for this process are not finite, we obtain fractional order moments. Its density function and corresponding Lévy measure density is also obtained. Further, heavy tailed behavior of densities and stochastic self-similarity of the process is demonstrated. Our res…

由下不完全伽马从属过程及其变体控制的带跳Lévy过程

Lévy processes with jumps governed by lower incomplete gamma subordinator and its variations

本文研究三类由独立Lévy从属过程时间变换的Lévy过程，推导其分布性质、讨论长程相关性，给出保险领域应用及从属过程模拟样本路径。

临界分支带杀Lévy过程的灭绝时间与最大位移的尾分布

Tails of extinction time and maximal displacement for critical branching killed Lévy process

本文研究临界分支带杀Lévy过程的灭绝时间与最大位移尾渐近行为，推导了存活概率与尾概率的衰减速率，并得到相关标度极限的表示结果。

Passage time and fluctuation calculations for subexponential Lévy processes

We consider the passage time problem for Lévy processes, emphasising heavy tailed cases. Results are obtained under quite mild assumptions, namely, drift to $-\infty$ a.s. of the process, possibly at a linear rate (the finite mean case), but possibly much faster (the infinite mean case), together with subexponential growth on the positive side. Local and functional versions of limit distributions are derived for the passage time itself, as well as for the position of the process just prior to passage, and the overshoot of a high level. A significant connection is made with extreme value theory…

一维Lévy驱动SDE的有限爆炸及其在SPDE中的应用

Finite Explosion for One-dimensional Lévy Driven SDEs and its Application to SPDEs

本文研究纯跳Lévy过程驱动的一维SDE的有限时间爆炸，提出三组条件刻画爆炸行为，并应用于半线性抛物型SPDE，证明其局部弱解几乎必然有限寿命。

On ergodic properties of some Levy-type processes

In this note we prove some sufficient conditions for ergodicity of a Levy-type process, such that on the test functions the generator of the respective semigroup is of the form $$ Lf(x) = a(x)f'(x) + \int_{\mathbb{R}}{ \left( f(x+u)-f(x)- \nabla f(x)\cdot u \mathbb{I}_{|u|\leq 1} \right) ν(x,du)}, \quad f\in C_{\infty}^{2}(\mathbb{R}). $$ Here $ν(x,du)$ is a Levy-type kernel and $a(\cdot): \mathbb{R}\to \mathbb{R}$. We consider the case when the tails are of polynomial decay as well as the case when the decay is (sub)-exponential. For the proof the Foster-Lyapunov approach is used.

Moderate deviations and laws of the iterated logarithm for the local times of additive Lévy processes and additive random walks

We study the upper tail behaviors of the local times of the additive Lévy processes and additive random walks. The limit forms we establish are the moderate deviations and the laws of the iterated logarithm for the L_2-norms of the local times and for the local times at a fixed site.

Cramér's Estimate for the Reflected Process Revisited

The reflected process of a random walk or Lévy process arises in many areas of applied probability, and a question of particular interest is how the tail of the distribution of the heights of the excursions away from zero behaves asymptotically. The Lévy analogue of this is the tail behaviour of the characteristic measure of the height of an excursion. Apparently the only case where this is known is when Cramér's condition hold. Here we establish the asymptotic behaviour for a large class of Lévy processes which have exponential moments but do not satisfy Cramér's condition. Our proof also app…

Asymptotics for first-passage times of Lévy processes and random walks

We study the exact asymptotics for the distribution of the first time $τ_x$ a Lévy process $X_t$ crosses a negative level $-x$. We prove that $\mathbf P(τ_x>t)\sim V(x)\mathbf P(X_t\ge 0)/t$ as $t\to\infty$ for a certain function $V(x)$. Using known results for the large deviations of random walks we obtain asymptotics for $\mathbf P(τ_x>t)$ explicitly in both light and heavy tailed cases. We also apply our results to find asymptotics for the distribution of the busy period in an M/G/1 queue.

First exit times for Lévy-driven diffusions with exponentially light jumps

We consider a dynamical system described by the differential equation $\dot{Y}_t=-U'(Y_t)$ with a unique stable point at the origin. We perturb the system by the Lévy noise of intensity $\varepsilon$ to obtain the stochastic differential equation $dX^{\varepsilon}_t=-U'(X^{\varepsilon}_{t-}) dt+\varepsilon dL_t.$ The process $L$ is a symmetric Lévy process whose jump measure $ν$ has exponentially light tails, $ν([u,\infty))\sim\exp(-u^α)$, $α>0$, $u\to \infty$. We study the first exit problem for the trajectories of the solutions of the stochastic differential equation from the interval $(-1,1…

分支Lévy过程加性鞅的矩及其应用

Moments of additive martingales of branching Lévy processes and applications

本文研究超临界分支Lévy过程中Biggins鞅及其极限的矩和尾部行为，并据此建立其差的中心极限定理。

Small-time expansions for the transition distributions of Lévy processes

Let $X$ be a Lévy process with absolutely continuous Lévy measure $ν$. Small time polynomial expansions of order $n$ in $t$ are obtained for the tails $P(X_{t}\geq{}y)$ of the process, assuming smoothness conditions on the Lévy density away from the origin. By imposing additional regularity conditions on the transition density $p_{t}$ of $X_{t}$, an explicit expression for the remainder of the approximation is also given. As a byproduct, polynomial expansions of order $n$ in $t$ are derived for the transition densities of the process. The conditions imposed on $p_{t}$ require that its derivati…

Cramer's estimate for a reflected Levy process

The natural analogue for a Levy process of Cramer's estimate for a reflected random walk is a statement about the exponential rate of decay of the tail of the characteristic measure of the height of an excursion above the minimum. We establish this estimate for any Levy process with finite negative mean which satisfies Cramer's condition, and give an explicit formula for the limiting constant. Just as in the random walk case, this leads to a Poisson limit theorem for the number of ``high excursions.''

Occupation times of alternating renewal processes with Lévy applications

This paper presents a set of results relating to the occupation time $α(t)$ of a process $X(\cdot)$. The first set of results concerns exact characterizations of $α(t)$ for $t\geq0$, e.g., in terms of its transform up to an exponentially distributed epoch. In addition we establish a central limit theorem (entailing that a centered and normalized version of $α(t)/t$ converges to a zero-mean Normal random variable as $t\rightarrow\infty$) and the tail asymptotics of $P(α(t)/t\geq q)$. We apply our findings to spectrally positive Lévy processes reflected at the infimum and establish various new o…

Estimates on the tail probabilities of subordinators and applications to general time fractional equations

In this paper, we study estimates on tail probabilities $\mathbb{P}(S_r \ge t)$ of several classes of subordinators under mild assumptions on the tail of its Lévy measure. As an application of that result, we obtain two-sided estimates for fundamental solutions of general homogeneous time fraction equations including those with Dirichlet boundary conditions.

On Future Drawdowns of Lévy processes

For a given Lévy process $X=(X_t)_{t\in\mathbb{R}_+}$ and for fixed $s\in \mathbb{R}_{+}\cup\{\infty\}$ and $t\in\mathbb{R}_+$ we analyse the {\it future drawdown extremes} that are defined as follows: \begin{eqnarray*} \overline D^*_{t,s} = \sup_{0\leq u\leq t} \inf_{u\leq w < t+s}(X_w-X_u), \qquad\qquad \underline D^*_{t,s} = \inf_{0\leq u\leq t} \inf_{u\leq w < t+s}(X_w-X_u). \end{eqnarray*} The path-functionals $\overline D^*_{t,s}$ and $\underline D^*_{t,s}$ are of interest in various areas of application, including financial mathematics and queueing theory. In the case that $X$ has a str…

Zooming-in on a Lévy process: Failure to observe threshold exceedance over a dense grid

For a Lévy process $X$ on a finite time interval consider the probability that it exceeds some fixed threshold $x>0$ while staying below $x$ at the points of a regular grid. We establish exact asymptotic behavior of this probability as the number of grid points tends to infinity. We assume that $X$ has a zooming-in limit, which necessarily is $1/α$-self-similar Lévy process with $α\in(0,2]$, and restrict to $α>1$. Moreover, the moments of the difference of the supremum and the maximum over the grid points are analyzed and their asymptotic behavior is derived. It is also shown that the zooming-…

Density and tails of unimodal convolution semigroups

We give sharp bounds for the isotropic unimodal probability convolution semigroups when their Lévy-Khintchine exponent has Matuszewska indices strictly between 0 and 2.

Fluctuation theory for Lévy processes with completely monotone jumps

We study the Wiener-Hopf factorization for Lévy processes $X_t$ with completely monotone jumps. Extending previous results of L.C.G. Rogers, we prove that the space-time Wiener-Hopf factors are complete Bernstein functions of both the spatial and the temporal variable. As a corollary, we prove complete monotonicity of: (a) the tail of the distribution function of the supremum of $X_t$ up to an independent exponential time; (b) the Laplace transform of the supremum of $X_t$ up to a fixed time $T$, as a function of $T$. The proof involves a detailed analysis of the holomorphic extension of the c…

通过特征函数模拟 Lévy 过程、其极值以及极值的到达时间

Simulation of a Lévy process, its extremum, and hitting time of the extremum via characteristic functions

提出利用特征函数、条件特征函数与共形变形技术模拟 Lévy 过程及其极值、到达时间的通用框架，并通过预计算与插值高效处理分布尾部和中心区域。