Given a spectrally negative Lévy process $X$ drifting to infinity, (inspired on the early ideas of Shiryaev (2002)) we are interested in finding a stopping time that minimises the $L^p$ distance ($p>1$) with $g$, the last time $X$ is negative. The solution is substantially more difficult compared to the case $p=1$, for which it was shown in Baurdoux and Pedraza (2020b) that it is optimal to stop as soon as $X$ exceeds a constant barrier. In the case of $p>1$ treated here, we prove that solving this optimal prediction problem is equivalent to solving an optimal stopping problem in terms of a two-dimensional strong Markov process that incorporates the length of the current excursion away from $0$. We show that an optimal stopping time is now given by the first time that $X$ exceeds a non-increasing and non-negative curve depending on the length of the current excursion away from $0$. We further characterise the optimal boundary and the value function as the unique solution of a non-linear system of integral equations within a subclass of functions. As examples, the case of a Brownian motion with drift and a Brownian motion with drift perturbed by a Poisson process with exponential jumps are considered.
Using a new approach, for spectrally negative Lévy processes we find joint Laplace transforms involving the last exit time (from a semi-infinite interval), the value of the process at the last exit time and the associated occupation time, which generalizes some previous results.
In this paper, we derive the joint Laplace transforms of occupation times until its last passage times as well as its positions. Motivated by Baurdoux [2], the last times before an independent exponential variable are studied. By applying dual arguments, explicit formulas are derived in terms of new analytical identities from Loeffen et al. [12].
We obtain series expansions of the $q$-scale functions of arbitrary spectrally negative Lévy processes, including processes with infinite jump activity, and use these to derive various new examples of explicit $q$-scale functions. Moreover, we study smoothness properties of the $q$-scale functions of spectrally negative Lévy processes with infinite jump activity. This complements previous results of Chan et al. [7] for spectrally negative Lévy processes with Gaussian component or bounded variation.
For refracted spectrally negative Lévy processes, we identify expressions of several quantities related to Laplace transforms on their weighted occupation times until first exit times. Such quantities are expressed in terms of unique solutions to integral equations involving weight functions and scale functions for the associated spectrally negative Lévy processes. Previous results on refracted Lévy processes are recovered.
The joint distribution of the maximum loss and the maximum gain is obtained for a spectrally negative Levy process until the passage time of a given level. Their marginal distributions up to an independent exponential time are also provided. The existing formulas for Brownian motion with drift are recovered using the particular scale functions.Keywords Maximum drawdown Maximum drawup Spectrally negative Reflected process Fluctuation theory
In this paper, we compute the Laplace transform of occupation times (of the negative half-line) of spectrally negative Lévy processes. Our results are extensions of known results for standard Brownian motion and jump-diffusion processes. The results are expressed in terms of the so-called scale functions of the spectrally negative Lévy process and its Laplace exponent. Applications to insurance risk models are also presented.
We study the convergence in distribution of the supremum of the local time and of the favorite site for a transient diffusion in a spectrally negative Lévy potential. To do so, we study the h-valleys of a spectrally negative Lévy process, and we prove in partiular that the renormalized sequence of the h-minima converges to the jumping times sequence of a standard Poisson process.
In this paper we study the draw-down related Parisian ruin problem for spectrally negative Lévy risk processes. We introduce the draw-down Parisian ruin time and solve the corresponding two-sided exit time via excursion theory. We also obtain an expression of the potential measure for the process killed at the draw-down Parisian time. As applications, new results are obtained for spectrally negative Lévy risk process with dividend barrier and Parisian ruin.
Following from recent developments by Hubalek and Kyprianou, the objective of this paper is to provide further methods for constructing new families of scale functions for spectrally negative Lévy processes which are completely explicit. This is the result of an observation in the aforementioned paper which permits feeding the theory of Bernstein functions directly into the Wiener-Hopf factorization for spectrally negative Lévy processes. Many new, concrete examples of scale functions are offered although the methodology in principle delivers still more explicit examples than those listed.
Scale functions play a central role in the fluctuation theory of spectrally negative Lévy processes. For spectrally negative compound Poisson processes with positive drift, a new representation of the $q$-scale functions in terms of the characteristics of the process is derived. Moreover, similar representations of the derivatives and the primitives of the $q$-scale functions are presented. The obtained formulae for the derivatives allow for a complete exposure of the smoothness properties of the considered $q$-scale functions. Some explicit examples of $q$-scale functions are given for illust…
For spectrally negative Lévy processes, adapting an approach from \cite{BoLi:sub1} we identify joint Laplace transforms involving local times evaluated at either the first passage times, or independent exponential times, or inverse local times. The Laplace transforms are expressed in terms of the associated scale functions. Connections are made with the permanental process and the Markovian loop soup measure.
In this paper, we identify Laplace transforms of occupation times of intervals until first passage times for spectrally negative Lévy processes. New analytical identities for scale functions are derived and therefore the results are explicitly stated in terms of the scale functions of the process. Applications to option pricing and insurance risk models are also presented.
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 study the scale function of the spectrally negative phase-type Levy process. Its scale function admits an analytical expression and so do a number of its fluctuation identities. Motivated by the fact that the class of phase-type distributions is dense in the class of all positive-valued distributions, we propose a new approach to approximating the scale function and the associated fluctuation identities for a general spectrally negative Levy process. Numerical examples are provided to illustrate the effectiveness of the approximation method.
In this article we get simple explicit formulas for $\Exp\sup_{s\leq t}X(s)$ where $X$ is a spectrally positive or negative Lévy process with infinite variation. As a consequence we derive a generalization of the well-known formula for the supremum distribution of Wiener process that is we obtain $\Prob(\sup_{s\leq t}Z_α(s)\geq u)=α\Prob(Z_α(t)\geq u)$ for $u\geq 0$ where $Z_α$ is a spectrally negative Lévy process with $1<α\leq 2$ which also stems from Kendall's identity for the first crossing time. Our proof uses a formula for the supremum distribution of a spectrally positive Lévy process w…
For spectrally negative Lévy processes, we prove several fluctuation results involving a general draw-down time, which is a downward exit time from a dynamic level that depends on the running maximum of the process. In particular, we find expressions of the Laplace transforms for the two-sided exit problems involving the draw-down time. We also find the Laplace transforms for the hitting time and creeping time over the running-maximum related draw-down level, respectively, and obtain an expression for a draw-down associated potential measure. The results are expressed in terms of scale functio…
For a spectrally negative Lévy process (snLp) $X$, killed according to a rate that is a function $ω$ of its position, we analyse the exit probability of the one-sided upwards-passage problem. When $ω$ is strictly positive, this problem is related to the determination of the Laplace transform of the first passage time upwards for $X$ that has been time-changed by the inverse of the additive functional $\int_0^\cdot ω(X_u)du$. In particular our findings thus shed extra light on related results concerning first passage times upwards (downwards) of spectrally negative positive self-similar Markov …
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.
In this paper we study a spectrally negative Lévy process that is reflected at its draw-down level whenever a draw-down time from the running supremum arrives. Using an excursion-theoretical approach, for such a reflected process we find the Laplace transform of the upper exiting time and an expression of the associated potential measure. When the reflected process is identified as a risk process with capital injections, the expected total amount of discounted capital injections prior to the exiting time and the Laplace transform of the accumulated capital injections until the exiting time are…
We give a review of the state of the art with regard to the theory of scale functions for spectrally negative Levy processes. From this we introduce a general method for generating new families of scale functions. Using this method we introduce a new family of scale functions belonging to the Gaussian Tempered Stable Convolution (GTSC) class. We give particular emphasis to special cases as well as cross-referencing their analytical behaviour against known general considerations.
For a spectrally negative Lévy process $X$, we study the following distribution: $$ \mathbb{E}_x \left[ \mathrm{e}^{- q \int_0^t \mathbf{1}_{(a,b)} (X_s) \mathrm{d}s } ; X_t \in \mathrm{d}y \right], $$ where $-\infty \leq a < b < \infty$, and where $q,t>0$ and $x \in \mathbb{R}$. More precisely, we identify the Laplace transform with respect to $t$ of this measure in terms of the scale functions of the underlying process. Our results are then used to price step options and the particular case of an exponential spectrally negative Lévy jump-diffusion model is discussed.
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.
We provide a description of the excursion measure from a point for a spectrally negative Lévy process. The description is based in two main ingredients. The first is building a spectrally negative Lévy process conditioned to avoid zero and the study of its entrance law at zero. The latter is connected with both the excursion measure from zero of the process reflected in its infimum and reflected in its supremum. This lead us to establish a connection with the excursion measure from the state zero with the excursion measure from zero for the process reflected at the infimum and reflected at the…
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.
In this paper, we consider the mixed ratcheting-periodic dividend strategies for spectrally negative Lévy risk model, in which dividend payments can both be made continuously without falling and discretely at the jump times of an independent Poisson process. The expected net present value(NPV) of dividends paid up to ruin and the Laplace transform of the ruin time are obtained by using Lévy fluctuation theory. All the results are expressed in terms of scale functions. Finally, numerical results for Brownian motion with drift are given.
De Finetti's optimal dividend problem has recently been extended to the case dividend payments can only be made at Poisson arrival times. This paper considers the version with bail-outs where the surplus must be nonnegative uniformly in time. For a general spectrally negative Lévy model, we show the optimality of a Parisian-classical reflection strategy that pays the excess above a given barrier at each Poisson arrival times and also reflects from below at zero in the classical sense.
We study a combination of the refracted and reflected Lévy processes. Given a spectrally negative Lévy process and two boundaries, it is reflected at the lower boundary while, whenever it is above the upper boundary, a linear drift at a constant rate is subtracted from the increments of the process. Using the scale functions, we compute the resolvent measure, the Laplace transform of the occupation times as well as other fluctuation identities that will be useful in applied probability including insurance, queues, and inventory management.
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…
We consider an interesting natural extension to the Parisian ruin problem under the assumption that the risk reserve dynamics are given by a spectrally negative Lévy process. The distinctive feature of this extension is that the distribution of the random implementation delay windows' lengths can depend on the deficit at the epochs when the risk reserve process turns negative, starting a new negative excursion. This includes the possibility of an immediate ruin when the deficit hits a certain subset. In this general setting, we derive a closed-from expression for the Parisian ruin probability …
Path decomposition is performed to analyze the pre-supremum, post-supremum, post-infimum and the intermediate processes of a spectrally negative Levy process taken up to an independent exponential time T as motivated by the aim of finding the joint distribution of the maximum loss and maximum gain. In addition, the joint distribution of the supremum and the infimum before an exponential time is displayed. As an application of path decomposition, the distributions of supremum of the post-infimum process and the maximum loss of the post-supremum process are obtained.
For a spectrally negative positive self-similar Markov process with an a.s. finite overall supremum we provide, in tractable detail, a kind of conditional Wiener-Hopf factorization at the maximum of the absorption time at zero, the conditioning being on the overall supremum and the jump at the overall supremum. In a companion result the Laplace transform of said absorption time (on the event that the process does not go above a given level) is identified under no other assumptions (such as the process admitting a recurrent extension and/or hitting zero continuously), generalizing some existent…
Following Baurdoux and Kyprianou [2] we consider the McKean stochastic game, a game version of the McKean optimal stopping problem (American put), driven by a spectrally negative Levy process. We improve their characterisation of a saddle point for this game when the driving process has a Gaussian component and negative jumps. In particular we show that the exercise region of the minimiser consists of a singleton when the penalty parameter is larger than some threshold and 'thickens' to a full interval when the penalty parameter drops below this threshold. Expressions in terms of scale functio…
Let $a\in (0,\infty)$. For a spectrally negative Lévy process $X$ with infinite variation paths the resolvent of the process killed on hitting the two-point set $V=\{-a,a\}$ is identified. When further $X$ has no diffusion component the Laplace transforms of the entrance laws of the excursion measures of $X$ from $V$ are determined. This is then applied to establishing the Laplace transform of the amount of time that elapses between the last visit of $X$ to a given point $x$, before hitting some other point $y>x$, and the hitting time of $y$. All the expressions are explicit and tractable in t…
We recall four open problems concerning constructing high-order matrix-exponential approximations for the infimum of a spectrally negative Levy process (with applications to first-passage/ruin probabilities, the waiting time distribution in the M/G/1 queue, pricing of barrier options, etc). On the way, we provide a new approximation, for the perturbed Cramer-Lundberg model, and recall a remarkable family of (not minimal order) approximations of Johnson and Taaffe, which fit an arbitrarily high number of moments, greatly generalizing the currently used approximations of Renyi, De Vylder and Whi…
As well known, all functionals of a Markov process may be expressed in terms of the generator operator, modulo some analytic work. In the case of spectrally negative Markov processes however, it is conjectured that everything can be expressed in a more direct way using the $W$ scale function which intervenes in the two-sided first passage problem, modulo performing various integrals. This conjecture arises from work on Levy processes \cite{AKP,Pispot,APP,Iva,IP, ivanovs2013potential,AIZ,APY}, where the $W$ scale function has explicit Laplace transform, and is therefore easily computable; furth…
We consider a company that receives capital injections so as to avoid ruin. Differently from the classical bail-out settings where the underlying process is restricted to stay at or above zero, we study the case bail-out can only be made at independent Poisson times. Namely, we study a version of the reflected process that is pushed up to zero only on Poisson observation times at which the process is below zero. We also study the case with additional classical reflection above so as to model a company that pays dividends according to a barrier strategy. Focusing on the spectrally negative Lévy…
Scale functions play a central role in the fluctuation theory of spectrally negative Lévy processes and often appear in the context of martingale relations. These relations are often complicated to establish requiring excursion theory in favour of Itô calculus. The reason for the latter is that standard Itô calculus is only applicable to functions with a sufficient degree of smoothness and knowledge of the precise degree of smoothness of scale functions is seemingly incomplete. The aim of this article is to offer new results concerning properties of scale functions in relation to the smoothnes…
Last passage times arise in a number of areas of applied probability, including risk theory and degradation models. Such times are obviously not stopping times since they depend on the whole path of the underlying process. We consider the problem of finding a stopping time that minimises the $L^1$-distance to the last time a spectrally negative Lévy process $X$ is below zero. Examples of related problems in a finite horizon setting for processes with continuous paths are Du Toit et al. (2008) and Glover and Hulley (2014), where the last zero is predicted for a Brownian motion with drift, and f…
For a refracted Lévy process driven by a spectrally negative Lévy process, we use a different approach to derive expressions for its q-potential measures without killing. Unlike previous methods whose derivations depend on scale functions which are defined only for spectrally negative Lévy processes, our approach is free of scale functions. This makes it possible to extend the result here to a quite general refracted Lévy process by applying the approach presented below.
In this paper we solve the exit problems for (reflected) spectrally negative Lévy processes, which are exponentially killed with a killing intensity dependent on the present state of the process and analyze respective resolvents. All identities are given in terms of new generalizations of scale functions. For the particular cases $ω(x)=q$ and $ω(x)=q \mathbf{1}_{(a,b)}(x)$, we obtain results for the classical exit problems and the Laplace transforms of the occupation times in a given interval, until first passage times, respectively. Our results can also be applied to find the bankruptcy proba…
The existence of moments of first downward passage times of a spectrally negative Lévy process is governed by the general dynamics of the Lévy process, i.e. whether the Lévy process is drifting to $+\infty$, $-\infty$ or oscillates. Whenever the Lévy process drifts to $+\infty$, we prove that the $κ$-th moment of the first passage time (conditioned to be finite) exists if and only if the $(κ+1)$-th moment of the Lévy jump measure exists. This generalises a result shown earlier by Delbaen for Cramér-Lundberg risk processes \cite{Delbaen1990}. Whenever the Lévy process drifts to $-\infty$, we pr…
In this article we derive formulas for the probability $P(\sup_{t\leq T} X(t)>u)$ $T>0$ and $P(\sup_{t<\infty} X(t)>u)$ where $X$ is a spectrally positive Lévy process with infinite variation. The formulas are generalizations of the well-known Takács formulas for stochastic processes with non-negative and interchangeable increments. Moreover, we find the joint distribution of $\inf_{t\leq T} Y(t)$ and $Y(T)$ where $Y$ is a spectrally negative Lévy process.
In this paper, we introduce the concept of \emph{Poissonian occupation times} below level $0$ of spectrally negative Lévy processes. In this case, occupation time is accumulated only when the process is observed to be negative at arrival epochs of an independent Poisson process. Our results extend some well known continuously observed quantities involving occupation times of spectrally negative Lévy processes. As an application, we establish a link between Poissonian occupation times and insurance risk models with Parisian implementation delays.
We provide short and simple proofs of the continuous time ballot theorem for processes with cyclically interchangeable increments and Kendall's identity for spectrally positive Lévy processes. We obtain the later result as a direct consequence of the former. The ballot theorem is extended to processes having possible negative jumps. Then we prove through straightforward arguments based on the law of bridges and Kendall's identity, Theorem 2.4 in \cite{mpp} which gives an expression for the law of the supremum of spectrally positive Lévy processes. An analogous formula is obtained for the supre…
We consider the classical optimal dividend control problem which was proposed by de Finetti [Trans. XVth Internat. Congress Actuaries 2 (1957) 433--443]. Recently Avram, Palmowski and Pistorius [Ann. Appl. Probab. 17 (2007) 156--180] studied the case when the risk process is modeled by a general spectrally negative Lévy process. We draw upon their results and give sufficient conditions under which the optimal strategy is of barrier type, thereby helping to explain the fact that this particular strategy is not optimal in general. As a consequence, we are able to extend considerably the class of…
First passage problems for spectrally negative Lévy processes with possible absorbtion or/and reflection at boundaries have been widely applied in mathematical finance, risk, queueing, and inventory/storage theory. Historically, such problems were tackled by taking Laplace transform of the associated Kolmogorov integro-differential equations involving the generator operator. In the last years there appeared an alternative approach based on the solution of two fundamental "two-sided exit" problems from an interval (TSE). A spectrally one-sided process will exit smoothly on one side on an interv…
Given a spectrally negative Lévy process, we predict, in a $L_1$ sense, the last passage time of the process below zero before an independent exponential time. This optimal prediction problem generalises Baurdoux and Pedraza (2020) where the infinite horizon problem is solved. Using a similar argument as that in Urusov (2005), we show that this optimal prediction problem is equivalent to solving an optimal prediction problem in a finite horizon setting. Surprisingly (unlike the infinite horizon problem) an optimal stopping time is based on a curve that is killed at the moment the mean of the e…
In this paper, we unify two popular approaches for the definition of actuarial ruin with implementation delays, also known as Parisian ruin. Our new definition of ruin includes both deterministic delays and exponentially distributed delays: ruin is declared the first time an excursion in the red zone lasts longer than an implementation delay with a deterministic and a stochastic component. For this Parisian ruin with mixed delays, we identify the joint distribution of the time of ruin and the deficit at ruin, therefore providing generalizations of many results previously obtained, such as in \…
The purpose of this review article is to give an up to date account of the theory and application of scale functions for spectrally negative Levy processes. Our review also includes the first extensive overview of how to work numerically with scale functions. Aside from being well acquainted with the general theory of probability, the reader is assumed to have some elementary knowledge of Levy processes, in particular a reasonable understanding of the Levy-Khintchine formula and its relationship to the Levy-Ito decomposition. We shall also touch on more general topics such as excursion theory …
In this paper we analyze so-called Parisian ruin probability that happens when surplus process stays below zero longer than fixed amount of time $ζ>0$. We focus on general spectrally negative Lévy insurance risk process. For this class of processes we identify expression for ruin probability in terms of some other quantities that could be possibly calculated explicitly in many models. We find its Cramér-type and convolution-equivalent asymptotics when reserves tends to infinity. Finally, we analyze few explicit examples.
In the setting of the classical Cramer-Lundberg risk insurance model, Albrecher and Hipp (2007) introduced the idea of tax payments. More precisely, if $X = \{X_t : t\geq 0\}$ represents the Cramer-Lundberg process and, for all $t\geq 0$, $S_t = \sup_{s\leq t}X_s$, then Albrecher and Hipp (2007) study $X_t - γS_t$, $t\geq 0$, where $γ\in(0,1)$ is the rate at which tax is paid. This model has been generalised to the setting that $X$ is a spectrally negative Lévy process by Albrecher et al. \cite{albr_ren_zhou}. Finally Kyprianou and Zhou (2009) extend this model further by allowing the rate at …
We study the almost sure asymptotic behavior of the supremum of the local time for a transient diffusion in a spectrally negative Lévy environment. More precisely, we provide the proper renormalizations for the extremely large and the extremely small values of the supremum of the local time. In particular, in the sub-ballistic regime, we link the almost sure behavior of the supremum of the local time with the left asymptotic tail of the exponential functional of the environment conditioned to stay positive, which allows to use the properties of such exponential functionals to characterize the …
