We obtain a new fluctuation identity for a general Lévy process giving a quintuple law describing the time of first passage, the time of the last maximum before first passage, the overshoot, the undershoot and the undershoot of the last maximum. With the help of this identity, we revisit the results of Klüppelberg, Kyprianou and Maller [Ann. Appl. Probab. 14 (2004) 1766--1801] concerning asymptotic overshoot distribution of a particular class of Lévy processes with semi-heavy tails and refine some of their main conclusions. In particular, we explain how different types of first passage contribute to the form of the asymptotic overshoot distribution established in the aforementioned paper. Applications in insurance mathematics are noted with emphasis on the case that the underlying Lévy process is spectrally one sided.
Let be $(X_t, t\geq 0)$ be a Lévy process which is the sum of a Brownian motion with drift and a compound Poisson process. We consider the first passage time $τ_x$ at a fixed level $x>0$ by $(X_t, t\geq 0)$ and $K_x:= X_{τ_x}-x$ the overshoot and $L_x:= x-X_{τ_x^-}$ the undershoot. We first study the regularity of the density of the first passage time. Secondly, we calculate the joint law of $(τ_x, K_x, L_x).$
Recent models of the insurance risk process use a Lévy process to generalise the traditional Cramér-Lundberg compound Poisson model. This paper is concerned with the behaviour of the distributions of the overshoot and undershoots of a high level, for a Lévy process which drifts to $-\infty$ and satisfies a Cramér or a convolution equivalent condition. We derive these asymptotics under minimal conditions in the Cramér case, and compare them with known results for the convolution equivalent case, drawing attention to the striking and unexpected fact that they become identical when certain parame…
Let $τ(x)$ be the epoch of first entry into the interval $(x,\infty)$, $x>0$, of the reflected process $Y$ of a Lévy process $X$, and define the overshoot $Z(x) = Y(τ(x))-x$ and undershoot $z(x) = x - Y(τ(x)-)$ of $Y$ at the first-passage time over the level $x$. In this paper we establish, separately under the Cramér and positive drift assumptions, the existence of the weak limit of $(z(x), Z(x))$ as $x$ tends to infinity and provide explicit formulae for their joint CDFs in terms of the Lévy measure of $X$ and the renewal measure of the dual of $X$. We apply our results to analyse the behavi…
Let (X_t, t>=0) be a Levy process started at 0, with Levy measure nu and T_x the first hitting time of level x>0: T_x:=inf{t>=0; X_t>x}. Let $F(theta, mu, rho,.) be the joint Laplace transform of (T_x, K_x, L_x): F(theta,mu,rho,x) :=E(e^(-theta T_x - mu K_x ρL_x) 1_(T_x<+infinity)), where theta>=0, mu>=0, rho>=0, x>=0, K_x:=X_(T_x)-x and L_x:=x-X_(T_(x^-)). If we assume that nu has finite exponential moments we exhibit an asymptotic expansion for F(theta,mu,rho,x), as x -> +infinity. A limit theorem involving a normalization of the triplet (T_x,K_x,L_x) as x -> +infinity, may be deduced. At la…
In this note we apply the recently established Wiener-Hopf Monte Carlo (WHMC) simulation technique for Levy processes from Kuznetsov et al. [17] to path functionals, in particular first passage times, overshoots, undershoots and the last maximum before the passage time. Such functionals have many applications, for instance in finance (the pricing of exotic options in a Levy model) and insurance (ruin time, debt at ruin and related quantities for a Levy insurance risk process). The technique works for any Levy process whose running infimum and supremum evaluated at an independent exponential ti…
Let (X_t, t >=0) be a Levy process started at 0, with Levy measure nu, and T_x the first hitting time of level x>0: T_x := inf{t>=0; X_t>x}. Let F(theta,mu,rho,.) be the joint Laplace transform of (T_x, K_x, L_x): F(theta,mu,rho,x) := E (e^{-theta T_x - mu K_x - rho L_x} 1_{T_x<+infinity}), where theta>=0, mu>=0, rho>=0, x>0, K_x := X_{T_x} - x and L_x := x - X_{T_{x^-}}. If nu(R) < + \infinity and integral_1^{+\infty} e^{sy} nu (dy) < +infinity for some s>0, then we prove that F(theta,mu,rho,.) is the unique solution of an integral equation and has a subexponential decay at infinity when thet…
For a stable process, we give an explicit formula for the potential measure of the process killed outside a bounded interval and the joint law of the overshoot, undershoot and undershoot from the maximum at exit from a bounded interval. We obtain the equivalent quantities for a stable process reflected in its infimum. The results are obtained by exploiting a simple connection with the Lamperti representation and exit problems of stable processes.
Understanding the space-time features of how a Lévy process crosses a constant barrier for the first time, and indeed the last time, is a problem which is central to many models in applied probability such as queueing theory, financial and actuarial mathematics, optimal stopping problems, the theory of branching processes to name but a few. In \cite{KD} a new quintuple law was established for a general Lévy process at first passage below a fixed level. In this article we use the quintuple law to establish a family of related joint laws, which we call $n$-tuple laws, for Lévy processes, Lévy pr…
We consider the spectrally negative Levy processes and determine the joint laws for the quantities such as the first and last passage times over a fixed level, the overshoots and undershoots at first passage, the minimum, the maximum and the duration of negative values. We apply our results to insurance risk theory to find an explicit expression for the generalized expected discounted penalty function in terms of scale functions. Further, a new expression for the generalized Dickson's formula is provided.
For a spectrally positive and strictly stable process with index in (1,2), a series representation is obtained for the joint distribution of the "first passage triple" that consists of the time of first passage and the undershoot and the overshoot at first passage. The result leads to several corollaries, including 1) the joint law of the first passage triple and the pre-passage running supremum, and 2) at a fixed time point, the joint law of the process' value, running supremum, and the time of the running supremum. The representation can be decomposed as a sum of strictly positive functions …
For a (killed) spectrally negative Lévy process we provide an analytic expression for the distribution of its overshoot over a fixed level in terms of the infinitesimal generator and the scale function of the process. Our identity involves an auxiliary function and the simplicity of the identity depends very much on the choice of this function. In particular, for specific choices one recovers various previous established formulas in the literature. We review several applications and also show that one can get in a similar way identities of overshoots for reflected and refracted spectrally nega…
We develop criteria for recurrence and transience of one-dimensional Markov processes which have jumps and oscillate between $+\infty$ and $-\infty$. The conditions are based on a Markov chain which only consists of jumps (overshoots) of the process into complementary parts of the state space. In particular we show that a stable-like process with generator $-(-Δ)^{α(x)/2}$ such that $α(x)=α$ for $x<-R$ and $α(x)=β$ for $x>R$ for some $R>0$ and $α,β\in(0,2)$ is transient if and only if $α+β<2$, otherwise it is recurrent. As a special case this yields a new proof for the recurrence, point recurr…
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…
For the sum process $X=X^1+X^2$ of a bivariate Lévy process $(X^1,X^2)$ with possibly dependent components, we derive a quintuple law describing the first upwards passage event of $X$ over a fixed barrier, caused by a jump, by the joint distribution of five quantities: the time relative to the time of the previous maximum, the time of the previous maximum, the overshoot, the undershoot and the undershoot of the previous maximum. The dependence between the jumps of $X^1$ and $X^2$ is modeled by a Lévy copula. We calculate these quantities for some examples, where we pay particular attention to …
Using Lamperti's relationship between Lévy processes and positive self-similar Markov processes (pssMp), we study the weak convergence of the law $\mathbb{P}_x$ of a pssMp starting at $x>0$, in the Skorohod space of càdlàg paths, when $x$ tends to 0. To do so, we first give conditions which allow us to construct a càdlàg Markov process $X^{(0)}$, starting from 0, which stays positive and verifies the scaling property. Then we establish necessary and sufficient conditions for the laws $\mathbb{P}_x$ to converge weakly to the law of $X^{(0)}$ as $x$ goes to 0. In particular, this answers a quest…
In this paper a class of Ornstein--Uhlenbeck processes driven by compound Poisson processes is considered. The jumps arrive with exponential waiting times and are allowed to be two-sided. The jumps are assumed to form an iid sequence with distribution a mixture (not necessarily convex) of exponential distributions, independent of everything else. The fact that downward jumps are allowed makes passage of a given lower level possible both by continuity and by a jump. The time of this passage and the possible undershoot (in the jump case) is considered. By finding partial eigenfunctions for the i…
For a stopped diffusion process in a multidimensional time-dependent domain $\D$, we propose and analyse a new procedure consisting in simulating the process with an Euler scheme with step size $Δ$ and stopping it at discrete times $(iΔ)_{i\in\N^*}$ in a modified domain, whose boundary has been appropriately shifted. The shift is locally in the direction of the inward normal $n(t,x)$ at any point $(t,x)$ on the parabolic boundary of $\D$, and its amplitude is equal to $0.5826 (...) |n^*σ|(t,x)\sqrt Δ$ where $σ$ stands for the diffusion coefficient of the process. The procedure is thus extremel…
Autoregressive processes are intensively studied in statistics and other fields of applied stochastics. For many applications the overshoot and the threshold-time are of special interest. When the upward innovations are in the class of phasetype distributions we determine the joint distribution of this two quantities and apply this result to problems of optimal stopping. Using a principle of continuous fit this leads to explicit solutions.
The {\em drawdown} process $Y$ of a completely asymmetric Lévy process $X$ is equal to $X$ reflected at its running supremum $\bar{X}$: $Y = \bar{X} - X$. In this paper we explicitly express in terms of the scale function and the Lévy measure of $X$ the law of the sextuple of the first-passage time of $Y$ over the level $a>0$, the time $\bar{G}_{τ_a}$ of the last supremum of $X$ prior to $τ_a$, the infimum $\unl X_{τ_a}$ and supremum $\ovl X_{τ_a}$ of $X$ at $τ_a$ and the undershoot $a - Y_{τ_a-}$ and overshoot $Y_{τ_a}-a$ of $Y$ at $τ_a$. As application we obtain explicit expressions for the …
Conditioning Markov processes to avoid a set is a classical problem that has been studied in many settings. In the present article we study the question if a Levy process can be conditioned to avoid an interval and, if so, the path behavior of the conditioned process. For Levy processes with finite second moments we show that conditioning is possible and identify the conditioned process as an h-transform of the original killed process. The h-transform is explicit in terms of successive overshoot distributions and is used to prove that the conditioned process diverges to plus infinity and minus…
Several two-boundary problems are solved for a special Lévy process: the Poisson process with an exponential component. The jumps of this process are controlled by a homogeneous Poisson process, the positive jump size distribution is arbitrary, while the distribution of the negative jumps is exponential. Closed form expressions are obtained for the integral transforms of the joint distribution of the first exit time from an interval and the value of the overshoot through boundaries at the first exit time. Also the joint distribution of the first entry time into the interval and the value of th…
We investigate the behavior of Lévy processes with convolution equivalent Lévy measures, up to the time of first passage over a high level u. Such problems arise naturally in the context of insurance risk where u is the initial reserve. We obtain a precise asymptotic estimate on the probability of first passage occurring by time T. This result is then used to study the process conditioned on first passage by time T. The existence of a limiting process as $u\to\infty$ is demonstrated, which leads to precise estimates for the probability of other events relating to first passage, such as the ove…
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…
Let $N$ be a positive integer, $c$ be a positive constant and $(U_n)_{n\ge 1}$ be a sequence of independent identically distributed pseudo-random variables. We assume that the $U_n$'s take their values in the discrete set $\{-N,-N+1,...,N-1,N\}$ and that their common pseudo-distribution is characterized by the \textit{(positive or negative) real} numbers \[\mathbb{P}\{U_n=k\}=δ_{k0}+(-1)^{k-1} c\binom{2N}{k+N}\] for any $k\in\{-N,-N+1,...,N-1,N\}$. Let us finally introduce $(S_n)_{n\ge 0}$ the associated pseudo-random walk defined on $\mathbb{Z}$ by $S_0=0$ and $S_n=\sum_{j=1}^n U_j$ for $n\ge…
We provide asymptotic results and develop high frequency statistical procedures for time-changed Lévy processes sampled at random instants. The sampling times are given by first hitting times of symmetric barriers whose distance with respect to the starting point is equal to $\varepsilon$. This setting can be seen as a first step towards a model for tick-by-tick financial data allowing for large jumps. For a wide class of Lévy processes, we introduce a renormalization depending on $\varepsilon$, under which the Lévy process converges in law to an $α$-stable process as $\varepsilon$ goes to $0$…
The purpose of this article is to provide, with the help of a fluctuation identity, a generic link between a number of known identities for the first passage time and overshoot above/below a fixed level of a Levy process and the solution of Gerber and Shiu [Astin Bull. 24 (1994) 195-220], Boyarchenko and Levendorskii [Working paper series EERS 98/02 (1998), Unpublished manuscript (1999), SIAM J. Control Optim. 40 (2002) 1663-1696], Chan [Original unpublished manuscript (2000)], Avram, Chan and Usabel [Stochastic Process. Appl. 100 (2002) 75-107], Mordecki [Finance Stoch. 6 (2002) 473-493], Asm…
We characterize the small-time asymptotic behavior of the exit probability of a Lévy process out of a two-sided interval and of the law of its overshoot, conditionally on the terminal value of the process. The asymptotic expansions are given in the form of a first-order term and a precise computable error bound. As an important application of these formulas, we develop a novel adaptive discretization scheme for the Monte Carlo computation of functionals of killed Lévy processes with controlled bias. The considered functionals appear in several domains of mathematical finance (e.g., structural …
