Consider the problem to explicitly calculate the law of the first passage time T(a) of a general Levy process Z above a positive level a. In this paper it is shown that the law of T(a) can be approximated arbitrarily closely by the laws of T^n(a), the corresponding first passages time for X^n, where (X^n)_n is a sequence of Levy processes whose positive jumps follow a phase-type distribution. Subsequently, explicit expressions are derived for the laws of T^n(a) and the upward ladder process of X^n.
The derivation is based on an embedding of X^n into a class of Markov additive processes and on the solution of the fundamental (matrix) Wiener-Hopf factorisation for this class.
This Wiener-Hopf factorisation can be computed explicitly by solving iteratively a certain fixed point equation. It is shown that, typically, this iteration converges geometrically fast.
Consider a sequence (Z_n,Z_n^M) of bivariate Lévy processes, such that Z_n is a spectrally positive Lévy process with finite variation, and Z_n^M is the counting process of marks in {0,1} carried by the jumps of Z_n. The study of these processes is justified by their interpretation as contour processes of a sequence of splitting trees with mutations at birth. Indeed, this paper is the first part of a work aiming to establish an invariance principle for the genealogies of such populations enriched with their mutational histories. To this aim, we define a bivariate subordinator that we call the …
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 $…
We prove that when a sequence of Lévy processes $X^{(n)}$ or a normed sequence of random walks $S^{(n)}$ converges a.s. on the Skorokhod space toward a Lévy process $X$, the sequence $L^{(n)}$ of local times at the supremum of $X^{(n)}$ converges uniformly on compact sets in probability toward the local time at the supremum of $X$. A consequence of this result is that the sequence of (quadrivariate) ladder processes (both ascending and descending) converges jointly in law towards the ladder processes of $X$. As an application, we show that in general, the sequence $S^{(n)}$ conditioned to stay…
In this paper we study the mean of the first exit time from a bounded interval of various Lévy processes. We establish sharp two-sided estimates of the mean for Lévy processes under certain condition on their characteristic exponents. We also study the cumulative distribution function of the supremum and infimum processes. Finally, we establish integral conditions that assure that the renewal function of the ladder height process is comparable with the linear one.
We study subexponential tail asymptotics for the distribution of the maximum $M_t:=\sup_{u\in[0,t]}X_u$ of a process $X_t$ with negative drift for the entire range of $t>0$. We consider compound renewal processes with linear drift and Lévy processes. For both we also formulate and prove the principle of a single big jump for their maxima. The class of compound renewal processes particularly includes Cramér-Lundberg risk process.
We show that if a Lévy process creeps then, as a function of $u$, the renewal function $V(t,u)$ of the bivariate ascending ladder process $(L^{-1},H)$ is absolutely continuous on $[0,\infty)$ and left differentiable on $(0,\infty)$, and the left derivative at $u$ is proportional to the (improper) distribution function of the time at which the process creeps over level $u$, where the constant of proportionality is $\rmd_H^{-1}$, the reciprocal of the (positive) drift of $H$. This yields the (missing) term due to creeping in the recent quintuple law of Doney and Kyprianou (2006). As an applicati…
In this paper we study processes which are constructed by a convolution of a deterministic kernel with a martingale. A special emphasis is put on the case where the driving martingale is a centred Lévy process, which covers the popular class of fractional Lévy processes. As a main result we show that, under appropriate assumptions on the kernel and the martingale, the maximum process of the corresponding `convoluted martingale' is $p$-integrable and we derive maximal inequalities in terms of the kernel and of the moments of the driving martingale.
For a Lévy process $ξ=(ξ_t)_{t\geq0}$ drifting to $-\infty$, we define the so-called exponential functional as follows \[{\rm{I}}_ξ=\int_0^{\infty}e^{ξ_t} dt.\] Under mild conditions on $ξ$, we show that the following factorization of exponential functionals \[{\rm{I}}_ξ\stackrel{d}={\rm{I}}_{H^-} \times {\rm{I}}_{Y}\] holds, where, $\times $ stands for the product of independent random variables, $H^-$ is the descending ladder height process of $ξ$ and $Y$ is a spectrally positive Lévy process with a negative mean constructed from its ascending ladder height process. As a by-product, we gener…
We consider a branching Markov process in continuous time in which the particles evolve independently as spectrally negative Lévy processes. When the branching mechanism is critical or subcritical, the process will eventually die and we may define its overall maximum, i.e. the maximum location ever reached by a particule. The purpose of this paper is to give asymptotic estimates for the survival function of this maximum. In particular, we show that in the critical case the asymptotics is polynomial when the underlying Lévy process oscillates or drifts towards $+\infty$, and is exponential when…
Let us consider a real Lévy process X whose transition probabilities are absolutely continuous and have bounded densities. Then the law of the past supremum of X before any deterministic time t is absolutely continuous on (0,\infty). We show that its density f_t(x) is continuous on (0,\infty) if and only if the potential density h' of the upward ladder height process is continuous on (0,\infty). Then we prove that f_t behaves at 0 as h'. We also describe the asymptotic behaviour of f_t, when t tends to infinity. Then some related results are obtained for the density of the meander and this of …
A maximal inequality is an inequality which involves the (absolute) supremum $\sup_{s\leq t}|X_{s}|$ or the running maximum $\sup_{s\leq t}X_{s}$ of a stochastic process $(X_t)_{t\geq 0}$. We discuss maximal inequalities for several classes of stochastic processes with values in an Euclidean space: Martingales, Lévy processes, Lévy-type - including Feller processes, (compound) pseudo Poisson processes, stable-like processes and solutions to SDEs driven by a Lévy process -, strong Markov processes and Gaussian processes. Using the Burkholder-Davis-Gundy inequalities we als discuss some relation…
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-…
We formulate the insurance risk process in a general Levy process setting, and give general theorems for the ruin probability and the asymptotic distribution of the overshoot of the process above a high level, when the process drifts to -\infty a.s. and the positive tail of the Levy measure, or of the ladder height measure, is subexponential or, more generally, convolution equivalent. Results of Asmussen and Kluppelberg [Stochastic Process. Appl. 64 (1996) 103-125] and Bertoin and Doney [Adv. in Appl. Probab. 28 (1996) 207-226] for ruin probabilities and the overshoot in random walk and compou…
Distributional identities for a Lévy process $X_t$, its quadratic variation process $V_t$ and its maximal jump processes, are derived, and used to make "small time" (as $t\downarrow0$) asymptotic comparisons between them. The representations are constructed using properties of the underlying Poisson point process of the jumps of $X$. Apart from providing insight into the connections between $X$, $V$, and their maximal jump processes, they enable investigation of a great variety of limiting behaviours. As an application, we study "self-normalised" versions of $X_t$, that is, $X_t$ after divisio…
A Lévy process is said to creep through a curve if, at its first passage time across this curve, the process reaches it with positive probability. We first study this property for bivariate subordinators. Given the graph $\{(t,f(t)):t\ge0\}$ of any continuous, non increasing function $f$ such that $f(0)>0$, we give an expression of the probability that a bivariate subordinator $(Y,Z)$ issued from 0 creeps through this graph in terms of its renewal function and the drifts of the components $Y$ and $Z$. We apply this result to the creeping probability of any real Lévy process through the graph o…
We study the small-time asymptotics of sample paths of Lévy processes and Lévy-type processes. Namely, we investigate under which conditions the limit $$\limsup_{t \to 0} \frac{1}{f(t)} |X_t-X_0|$$ is finite resp.\ infinite with probability $1$. We establish integral criteria in terms of the infinitesimal characteristics and the symbol of the process. Our results apply to a wide class of processes, including solutions to Lévy-driven SDEs and stable-like processes. For the particular case of Lévy processes, we recover and extend earlier results from the literature. Moreover, we present a new ma…
The running infimum of a Levy process relative to its point of issue is know to have the same range that of the negative of a certain subordinator. Conditioning a Levy process issued from a strictly positive value to stay positive may therefore be seen as implicitly conditioning its descending ladder heigh subordinator to remain in a strip. Motivated by this observation, we consider the general problem of conditioning a subordinator to remain in a strip. Thereafter we consider more general contexts in which subordinators embedded in the path decompositions of Markov processes are conditioned t…
In this article, we give a new proof of the Itô formula for some integral processes related to the space-time Lévy white noise introduced in Balan (2015) as an alternative for the Gaussian white noise perturbing an SPDE. We discuss two applications of this result, which are useful in the study of SPDEs driven by a space-time Lévy noise with finite variance: a maximal inequality for the $p$-th moment of the stochastic integral, and the Itô representation theorem leading to a chaos expansion similar to the Gaussian case.
We consider a process $Z$ on the real line composed from a Lévy process and its exponentially tilted version killed with arbitrary rates and give an expression for the joint law of $Z$ seen from its supremum, the supremum $\overline Z$ and the time $T$ at which the supremum occurs. In fact, it is closely related to the laws of the original and the tilted Lévy processes conditioned to stay negative and positive. The result is used to derive a new representation of stationary particle systems driven by Lévy processes. In particular, this implies that a max-stable process arising from Lévy proces…
We study stationary max-stable processes $\{η(t)\colon t\in\mathbb R\}$ admitting a representation of the form $η(t)=\max_{i\in\mathbb N}(U_i+ Y_i(t))$, where $\sum_{i=1}^{\infty} δ_{U_i}$ is a Poisson point process on $\mathbb R$ with intensity ${\rm e}^{-u} {\rm d} u$, and $Y_1,Y_2,\ldots$ are i.i.d.\ copies of a process $\{Y(t)\colon t\in\mathbb R\}$ obtained by running a Lévy process for positive $t$ and a dual Lévy process for negative $t$. We give a general construction of such Lévy-Brown-Resnick processes, where the restrictions of $Y$ to the positive and negative half-axes are Lévy pro…
For a positive self-similar Markov process, X, we construct a local time for the random set, $Θ$, of times where the process reaches its past supremum. Using this local time we describe an exit system for the excursions of X out of its past supremum. Next, we define and study the ladder process (R,H) associated to a positive self-similar Markov process X, namely a bivariate Markov process with a scaling property whose coordinates are the right inverse of the local time of the random set $Θ$ and the process X sampled on the local time scale. The process (R,H) is described in terms of a ladder p…
We consider solutions of Lévy-driven stochastic differential equations of the form $\mathrm{d} X_t=σ(X_{t-})\mathrm{d} L_t$, $X_0=x$ where the function $σ$ is twice continuously differentiable and maximal of linear growth and the driving Lévy process $L=(L_t)_{t\geq0}$ is either vector or matrix-valued. While the almost sure short-time behavior of Lévy processes is well-known and can be characterized in terms of the characteristic triplet, there is no complete characterization of the behavior of the process $X$. Using methods from stochastic calculus, we derive limiting results for stochastic …
Bañuelos and Bogdan (2004) and Bogdan, Palmowski and Wang (2016) analyse the asymptotic tail distribution of the first time a stable (Lévy) process in dimension $d\geq 2$ exists a cone. We use these results to develop the notion of a stable process conditioned to remain in a cone as well as the the notion of a stable process conditioned to absorb continuously at the apex of a cone (without leaving the cone). As self-similar Markov processes we examine some of their fundamental properties through the lens of its Lamperti-Kiu decomposition. In particular we are interested to understand the under…
We study a general perturbed risk process with cumulative claims modelled by a subordinator with finite expectation, with the perturbation being a spectrally negative Levy process with zero expectation. We derive a Pollaczek-Hinchin type formula for the survival probability of that risk process, and give an interpretation of the formula based on the decomposition of the dual risk process at modified ladder epochs.
A spectrally positive additive Lévy field is a multidimensional field obtained as the sum $\mathbf{X}_{\rm t}={\rm X}^{(1)}_{t_1}+{\rm X}^{(2)}_{t_2}+\dots+{\rm X}^{(d)}_{t_d}$, ${\rm t}=(t_1,\dots,t_d)\in\mathbb{R}_+^d$, where ${\rm X}^{(j)}={}^t (X^{1,j},\dots,X^{d,j})$, $j=1,\dots,d$, are $d$ independent $\mathbb{R}^d$-valued Lévy processes issued from 0, such that $X^{i,j}$ is non decreasing for $i\neq j$ and $X^{j,j}$ is spectrally positive. It can also be expressed as $\mathbf{X}_{\rm t}=\mathbb{X}_{\rm t}{\bf 1}$, where ${\bf 1}={}^t(1,1,\dots,1)$ and $\mathbb{X}_{\rm t}=(X^{i,j}_{t_j})…
We investigate statistics of lead changes of the maxima of two discrete-time random walks in one dimension. We show that the average number of lead changes grows as $π^{-1}\ln(t)$ in the long-time limit. We present theoretical and numerical evidence that this asymptotic behavior is universal. Specifically, this behavior is independent of the jump distribution: the same asymptotic underlies standard Brownian motion and symmetric Levy flights. We also show that the probability to have at most n lead changes behaves as $t^{-1/4}[\ln t]^n$ for Brownian motion and as $t^{-β(μ)}[\ln t]^n$ for symmet…
Consider a population where individuals give birth at constant rate during their lifetimes to i.i.d. copies of themselves. Individuals bear clonally inherited types, but (neutral) mutations may happen at the birth events. The smallest subtree containing the genealogy of all the extant individuals at a fixed time τ, is called the coalescent point process. We enrich this process with the history of the mutations that appeared over time, and call it the marked coalescent point process. With the help of limit theorems for Lévy processes with marked jumps established in a previous work (arXiv:1305.…
For a strictly stationary sequence of random variables we derive functional convergence of the joint partial sum and partial maxima process under joint regular variation with index $α\in (0,2)$ and weak dependence conditions. The limiting process consists of an $α$--stable Lévy process and an extremal process. We also describe the dependence between these two components of the limit. The convergence takes place in the space of $\mathbb{R}^{2}$--valued càdlàg functions on $[0,1]$, with the Skorohod weak $M_{1}$ topology. We further show that this topology in general can not be replaced by the s…
We study a generalized risk process $X(t)=Y(t)-C(t)$, $t\in[0,τ]$, where $Y$ is a Lévy process, $C$ an independent subordinator and $τ$ an independent exponential time. Dropping the standard assumptions on the finite expectations of the processes $Y$ and $C$ and the net profit condition, we derive a Pollaczek-Khinchine type formula for the supremum of the dual process $\widehat{X}=-X$ on $[0,τ]$ which generalizes the results obtained in \cite{HPSV1}. We also discuss which assumptions are necessary for deriving this formula, specially from the point of view of the ladder process.
We present an $L_{p}$-theory ($p\geq 2$) for time-fractional stochastic partial differential equations driven by Lévy processes of the type $$ \partial^α_{t}u=\sum_{i,j=1}^d a^{ij}u_{x^{i}x^{j}} +f+\sum_{k=1}^{\infty}\partial^β_{t}\int_{0}^{t} (\sum_{i=1}^dμ^{ik} u_{x^i} +g^k) dZ^k_{s} $$ given with nonzero intial data. Here $\partial^α_t$ and $\partial^β_t$ are the Caputo fractional derivatives, $α\in (0,2), β\in (0,α+1/p)$, and $\{Z^k_t:k=1,2,\cdots\}$ is a sequence of independent Lévy processes. The coefficients are random functions depending on $(t,x)$. We prove the uniqueness and existenc…
This article treats both discrete time and continuous time stopping problems for general Markov processes on the real line with general linear costs. Using an auxiliary function of maximum representation type, conditions are given to guarantee the optimal stopping time to be of threshold type. The optimal threshold is then characterized as the root of that function. For random walks our results condense in the fact that all combinations of concave increasing pay-off functions and convex cost functions lead to a one-sided solution. For Lévy processes an explicit way to obtain the auxiliary func…
In his 1972 paper, John Lamperti characterized all positive self-similar Markov processes as time-changes of exponentials of Levy processes. In the past decade the problem of classifying all non-negative self-similar Markov processes that do not necessarily have zero as a trap has been solved gradually via connections to ladder height processes and excursion theory. Motivated by a recent article of Chaumont, Rivero, Panti we classify via jump-type SDEs the symmetric real-valued self-similar Markov processes that only decrease the absolute value by jumps and leave zero continuously. Our constru…
In this article we study a class of stochastic functional differential equations driven by Lévy processes (in particular, $α$-stable processes), and obtain the existence and uniqueness of Markov solutions in small time intervals. This corresponds to the local solvability to a class of quasi-linear partial integro-differential equations. Moreover, in the constant diffusion coefficient case, without any assumptions on the Lévy generator, we also show the existence of a unique maximal weak solution for a class of semi-linear partial integro-differential equation systems under bounded Lipschitz as…
This paper considers a continuous time analogue of the classical autoregressive moving average processes, Lévy-driven CARMA processes. First we describe limiting properties of the periodogram by means of the so-called truncated Fourier transform if observations are available continuously. The obtained results are in accordance with their counterparts from the discrete-time case. Then we discuss the numerical approximation of the truncated Fourier transform based on non-equidistant high frequency data. In order to ensure convergence of the numerical approximation to the true value of the trunca…
The Lamperti--Kiu transformation for real-valued self-similar Markov processes (rssMp) states that, associated to each rssMp via a space-time transformation, there is a Markov additive process (MAP). In the case that the rssMp is taken to be an $α$-stable process with $α\in(0,2)$, Chaumont et al. (2013) and Kuznetsov et al. (2014) have computed explicitly the characteristics of the matrix exponent of the semi-group of the embedded MAP, which we henceforth refer to as the {\it Lamperti-stable MAP}. Specifically, the matrix exponent of the Lamperti-stable MAP's transition semi-group can be writt…
We find a wide class of Levy-Loewner evolutions for which the value of integral means beta-spectrum $β(q)$ at $q=2$ is the maximal real eigenvalue of a three-diagonal matrix. The second moments of derivatives of corresponding conformal mappings are expressed through solutions of matrix Fuchsian systems with three singular points.
We introduce a general distributional framework that results in a unifying description and characterization of a rich variety of continuous-time stochastic processes. The cornerstone of our approach is an innovation model that is driven by some generalized white noise process, which may be Gaussian or not (e.g., Laplace, impulsive Poisson or alpha stable). This allows for a conceptual decoupling between the correlation properties of the process, which are imposed by the whitening operator L, and its sparsity pattern which is determined by the type of noise excitation. The latter is fully speci…
We consider a one-dimensional continuous time random walk (CTRW) on a fixed time interval $T$ where at each time step the walker waits a random time $τ$, before performing a jump drawn from a symmetric continuous probability distribution function (PDF) $f(η)$, of Lévy index $0 < μ\leq 2$. Our study includes the case where the waiting time PDF $Ψ(τ)$ has a power law tail, $Ψ(τ) \propto τ^{-1 - γ}$, with $0< γ< 1$, such that the average time between two consecutive jumps is infinite. The random motion is sub-diffusive if $γ< μ/2$ (and super-diffusive if $γ> μ/2$). We investigate the joint PDF of…
Let $Z=(Z_t)_{t\geq0}$ be an additive process with a bounded triplet $(0,0,Λ_t)_{t\geq0}$. Suppose that for any Schwartz function $φ$ on $\mathbb{R}^d$ whose Fourier transform is in $C_c^{\infty}(B_{c_s} \setminus B_{c_s^{-1}} )$, there exist positive constants $N_0$, $N_1$, and $N_2$ such that \begin{equation*} \int_{\mathbb{R}^d}|\mathbb{E}[φ(x+r^{-1}Z_t)]|dx\leq N_0 e^{- \frac{N_1 t}{s(r)}},\quad \forall (r,t)\in(0,1)\times[0,T], \end{equation*} and $$ \|ψ^μ(r^{-1}D)φ\|_{L_1(\mathbb{R}^d)}\leq \frac{N_2}{s(r)},\quad \forall r\in(0,1). $$ where $s$ is a scaling function (Definition 2.4), $c_…
In the context of order statistics of discrete time random walks (RW), we investigate the statistics of the gap, $G_n$, and the number of time steps, $L_n$, between the two highest positions of a Markovian one-dimensional random walker, starting from $x_0 = 0$, after $n$ time steps (taking the $x$-axis vertical). The jumps $η_i = x_i - x_{i-1}$ are independent and identically distributed random variables drawn from a symmetric probability distribution function (PDF), $f(η)$, the Fourier transform of which has the small $k$ behavior $1 - \hat f(k) \propto |k|^μ$, with $0 < μ\leq 2$. For $μ=2$, …
We present an exact solution for the probability density function $P(τ=t_{\min}-t_{\max}|T)$ of the time-difference between the minimum and the maximum of a one-dimensional Brownian motion of duration $T$. We then generalise our results to a Brownian bridge, i.e. a periodic Brownian motion of period $T$. We demonstrate that these results can be directly applied to study the position-difference between the minimal and the maximal height of a fluctuating $(1+1)$-dimensional Kardar-Parisi-Zhang interface on a substrate of size $L$, in its stationary state. We show that the Brownian motion result …
