Motivated by classical considerations from risk theory, we investigate boundary crossing problems for refracted Lévy processes. The latter is a Lévy process whose dynamics change by subtracting off a fixed linear drift (of suitable size) whenever the aggregate process is above a pre-specified level. More formally, whenever it exists, a refracted Lévy process is described by the unique strong solution to the stochastic differential equation \[ \D U_t = - δ\mathbf{1}_{\{U_t >b\}}\D t + \D X_t \] where $X=\{X_t :t\geq 0\}$ is a Lévy process with law $\mathbb{P}$ and $b, δ\in \mathbb{R}$ such that the resulting process $U$ may visit the half line $(b,\infty)$ with positive probability. We consider in particular the case that $X$ is spectrally negative and establish a suite of identities for the case of one and two sided exit problems. All identities can be written in terms of the $q$-scale function of the driving Lévy process and its perturbed version describing motion above the level $b$. We remark on a number of applications of the obtained identities to (controlled) insurance risk processes.
Generalizing Kyprianou--Loeffen's refracted Lévy processes, we define a new refracted Lévy process which is a Markov process whose positive and negative motions are Lévy processes different from each other. To construct it we utilize the excursion theory. We study its exit problem and the potential measures of the killed processes. We also discuss approximation problem.
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.
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.
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.
We investigate a refracted Levy process driven by a jump diffusion process, whose jumps have rational Laplace transforms. For such a stochastic process, formulas for the Laplace transform of its occupation times are deduced. To derive the main results, some modifications on our previous approach have been made. In addition, we obtain a very interesting identity, which is conjectured to hold for a general refracted Levy process.
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 refracted jump diffusion process having two-sided jumps with rational Laplace transforms. For such a process, by applying a straightforward but interesting approach, we derive formulas for the Laplace transform of its distribution. Our formulas are presented in an attractive form and the approach is novel. In particular, the idea in the application of an approximating procedure is remarkable. Besides, the results are used to price Variable Annuities with state-dependent fees.
In this paper, we investigate Parisian ruin for a Lévy surplus process with an adaptive premium rate, namely a refracted Lévy process. More general Parisian boundary-crossing problems with a deterministic implementation delay are also considered. Our main contribution is a generalization of the result in Loeffen et al. (2013) for the probability of Parisian ruin of a standard Lévy insurance risk process. Despite the more general setup considered here, our main result is as compact and has a similar structure. Examples are provided.
For given two standard processes with no positive jumps, we construct, using the excursion theory, a Markov process whose positive and negative motions have the same law as the two processes. The resulting process is a generalization of Kyprianou--Loeffen's refracted Lévy processes. We discuss approximation problem for our refracted processes coming from Lévy processes by removing small jumps and taking the limit as the removal level tends to zero. We also discuss conditions for refracted processes to have dual processes.
In this paper, we introduce an insurance ruin model with adaptive premium rate, thereafter refered to as restructuring/refraction, in which classical ruin and bankruptcy are distinguished. In this model, the premium rate is increased as soon as the wealth process falls into the red zone and is brought back to its regular level when the process recovers. The analysis is mainly focused on the time a refracted Lévy risk process spends in the red zone (analogous to the duration of the negative surplus). Building on results from Kyprianou and Loeffen (2010) and Loeffen et al. (2012), we identify th…
A refracted Lévy process is a Lévy process whose dynamics change by subtracting off a fixed linear drift (of suitable size) whenever the aggregate process is above a pre-specified level. More precisely, whenever it exists, a refracted Lévy process is described by the unique strong solution to the stochastic differential equation \[ \ud U_t=-δ\mathbf{1}_{\{U_t>b\}}\ud t +\ud X_t, \] where $X=(X_t, t\ge 0)$ is a Lévy process with law $\p$ and $b,δ\in \R$ such that the resulting process $U$ may visit the half line $(b,\infty)$ with positive probability. In this paper, we consider the case that $X…
In this paper we investigate an optimal dividend problem with transaction costs, where the surplus process is modelled by a refracted Lévy process and the ruin time is considered with Parisian delay. Presence of the transaction costs implies that one need to consider the impulse control problem as a control strategy in such model. An impulse policy $(c_1,c_2)$, which is to reduce the reserves to some fixed level $c_1$ whenever they are above another level $c_2$ is an important strategy for the impulse control problem. Therefore, we give sufficient conditions under which the above described imp…
The growth-fragmentation equation models systems of particles that grow and split as time proceeds. An important question concerns the large time asymptotic of its solutions. Doumic and Escobedo ($2016$) observed that when growth is a linear function of the mass and fragmentations are homogeneous, the so-called Malthusian behaviour fails. In this work we further analyse the critical case by considering a piecewise linear growth, namely \begin{equation} c(x) = \begin{cases} a_{_-} x \quad \quad x < 1 \\ a_{_+} x \quad \quad x \geq 1, \end{cases} \end{equation} with $0 < a_{_+} < a_{_-}$. We giv…
We construct a modified continuous-state branching process whose Malthusian parameter is replaced by another when passing below a certain level. The construction is obtained via a Lamperti-like transform applied to a refracted Lévy process. Infinitesimal generator, probability of vanishing at infinity, of explosion and some path properties are also provided.
In this paper, we obtain analytical expression for the distribution of the occupation time in the red (below level $0$) up to an (independent) exponential horizon for spectrally negative Lévy risk processes and refracted spectrally negative Lévy risk processes. This result improves the existing literature in which only the Laplace transforms are known. Due to the close connection between occupation time and many other quantities, we provide a few applications of our results including future drawdown, inverse occupation time, Parisian ruin with exponential delay, and the last time at running ma…
In this paper, we study de Finetti's optimal dividend problem with capital injection under the assumption that the dividend strategies are absolutely continuous. In many previous studies, the process before being controlled was assumed to be a spectrally one-sided Lévy process, however in this paper we use a Lévy process that may have both positive and negative jumps. In the main theorem, we show that a refraction--reflection strategy is an optimal strategy. We also mention the existence and uniqueness of solutions of the stochastic differential equations that define refracted Lévy processes.
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…
A level-dependent Lévy process solves the stochastic differential equation $dU(t) = dX(t)-ϕ(U(t)) dt$, where $X$ is a spectrally negative Lévy process. A special case is a multi-refracted Lévy process with $ϕ_k(x)=\sum_{j=1}^kδ_j1_{\{x\geq b_j\}}$. A general rate function $ϕ$ that is non-decreasing and continuously differentiable is also considered. We discuss solutions of the above stochastic differential equation and investigate the so-called scale functions, which are counterparts of the scale functions from the theory of Lévy processes. We show how fluctuation identities for $U$ can be exp…
In this paper we study a spectrally negative Lévy process which is refracted at its running maximum and at the same time reflected from below at a certain level. Such a process can for instance be used to model an insurance surplus process subject to tax payments according to a loss-carry-forward scheme together with the flow of minimal capital injections required to keep the surplus process non-negative. We characterize the first passage time over an arbitrary level and the cumulative amount of injected capital up to this time by their joint Laplace transform, and show that it satisfies a sim…
We revisit the classical singular control problem of minimizing running and controlling costs. The problem arises in inventory control, as well as in healthcare management and mathematical finance. Existing studies have shown the optimality of a barrier strategy when driven by the Brownian motion or Lévy processes with one-side jumps. Under the assumption that the running cost function is convex, we show the optimality of a barrier strategy for a general class of Lévy processes.
We consider the multi-refraction strategies in two equivalent versions of the optimal dividend problem in the dual (spectrally positive Lévy) model. The first problem is a variant of the bail-out case where both dividend payments and capital injections must be absolutely continuous with respect to the Lebesgue measure. The second is an extension of Avanzi et al. [4] where a strategy is a combination of two absolutely continuous dividend payments with different upper bounds and different transaction costs. In both problems, it is shown to be optimal to refract the process at two thresholds, wit…
We develop a general construction for nonlinear Lévy processes with given characteristics. More precisely, given a set $Θ$ of Lévy triplets, we construct a sublinear expectation on Skorohod space under which the canonical process has stationary independent increments and a nonlinear generator corresponding to the supremum of all generators of classical Lévy processes with triplets in $Θ$. The nonlinear Lévy process yields a tractable model for Knightian uncertainty about the distribution of jumps for which expectations of Markovian functionals can be calculated by means of a partial integro-di…
We present the Levy's distributional property for symmetric Levy processes with generating triplet $(0, 0,ν)$ or $(σ>0, γ, ν)$ where $ν$ is a symmetric measure on $R\backslash\{0\}$. This generalizes the classical Levy's theorem about Brownian motions with drift.
Semi-Levy process is an additive process with periodically stationary increments. In particular, it is a generalization of Levy process. The dichotomy of recurrence and transience of Levy processes is well known, but this is not necessarily true for general additive processes. In this paper, we prove the recurrence and transience dichotomy of semi-Levy processes. For the proof, we introduce a concept of semi-random walk and discuss its recurrence and transience properties. An example of semi-Levy process constructed from two independent Levy processes is investigated. Finally, we prove the law…
We show on- and off-diagonal upper estimates for the transition densities of symmetric Levy and Levy-type processes. To get the an-diagonal estimates we prove a Nash type inequality for the related Dirichlet form. For the off-diagonal estimates we assume that the characteristic function of a Levy (type) process is analytic, which allows to apply the complex analysis technique.
Which Levy processes satisfy Hunt's hypothesis (H) is a long-standing open problem in probabilistic potential theory. The study of this problem for one-dimensional Levy processes suggests us to consider (H) from the point of view of the sum of Levy processes. In this paper, we present theorems and examples on the validity of (H) for the sum of two independent Levy processes. We also give a novel condition on the Levy measure which implies (H) for a large class of one-dimensional Levy processes.
In this paper we introduce a new class of Lévy processes which we call hypergeometric-stable Lévy processes, because they are obtained from symmetric stable processes through several transformations and where the Gauss hypergeometric function plays an essential role. We characterize the Lévy measure of this class and obtain several useful properties such as the Wiener Hopf factorization, the characteristic exponent and some associated exit problems.
In this paper, we study the existence of the transition densities of one-dimensional Lévy processes. Compared with past results, our results contain the Lévy processes whose Lévy symbols have logarithm behavior at infinity. Our results contain the Lévy symbol induced by the following Laplace exponent $ψ(ξ) := (\ln(1 + \ln(1 + \ln(\cdots \ln(1 + |ξ|)))))^ε$ ($n$ times), $0 < ε< 1$, $2 \le n$. We also show that $ψ(ξ)$ is a Lévy symbol with transition density.
A functional representation of free Lévy processes is established via an ensemble of unitarily invariant Hermitian matrix-valued Lévy processes. This is accomplished by proving functional asymptotics of their empirical spectral processes towards the law of a free Lévy processes. This result recovers a functional version of Wigner's theorem and introduces a functional version of Marchenko-Pastur's theorem providing the free Poisson process as the noncommutative limit process.
The inversion of a Levy measure was first introduced (under a different name) in Sato 2007. We generalize the definition and give some properties. We then use inversions to derive a relationship between weak convergence of a Levy process to an infinite variance stable distribution when time approaches zero and weak convergence of a different Levy process as time approaches infinity. This allows us to get self contained conditions for a Levy process to converge to an infinite variance stable distribution as time approaches zero. We formulate our results both for general Levy processes and for t…
We give an interpretation of the bilateral exit problem for Lévy processes via the study of an elementary Markov chain. We exhibit a strong connection between this problem and Krein's theory on strings. For instance, for symmetric Lévy processes with bounded variations, the Lévy exponent is the correspondant spectral density and the Wiener-Hopf factorization turns out to be a version of Krein's entropy formula.
We review the probabilistic properties of Ornstein-Uhlenbeck processes in Hilbert spaces driven by Lévy processes. The emphasis is on the different contexts in which these processes arise, such as stochastic partial differential equations, continuous-state branching processes, generalised Mehler semigroups and operator self-decomposable distributions. We also examine generalisations to the case where the driving noise is cylindrical.
We generalise multivariate subordination of Lévy processes as introduced by Barndorff-Nielsen, Pedersen, and Sato to Hilbert space valued Lévy processes. The processes are explicitly characterised and conditions for integrability and martingale properties are derived under various assumptions of the Lévy process and subordinator. As an application of our theory we construct explicitly some Hilbert space valued versions of Lévy processes which are popular in the univariate and multivariate case. In particular, we define a normal inverse Gaussian Lévy process in Hilbert space as a subordination …
We prove two martingale identities which involve exit times of Levy-driven Ornstein--Uhlenbeck processes. Using these identities we find an explicit formula for the Laplace transform of the exit time under the assumption that positive jumps of the Levy process are exponentially distributed.
For a certain class of asymmetric Lévy processes where the origin is regular for itself, the renormalized zero resolvent is proved to be harmonic for the killed process upon hitting zero.
Combinatorial Levy processes evolve on general state spaces of countable combinatorial structures. In this setting, the usual Levy process properties of stationary, independent increments are defined in an unconventional way in terms of the symmetric difference operation on sets. In discrete time, the description of combinatorial Levy processes gives rise to the notion of combinatorial random walks. These processes behave differently than random walks and Levy processes on other state spaces. Standard examples include processes on sets, graphs, and n-ary relations, but the framework permits fa…
Kuznetsov et al. (2011) and Kuznetsov and Pardo (2013) introduced the family of Hypergeometric Lévy processes. They appear naturally in the study of fluctuations of stable processes when one analyses stable processes through the theory of positive self-similar Markov processes. Hypergeometric Lévy processes are defined through their characteristic exponent, which, as a complex-valued function, has four independent parameters. Kyprianou et al. (2014) showed that the definition of a Hypergeometric Lévy process could be taken to include a greater range of the aforesaid parameters than originally …
We completely describe the size and large intersection properties of the Holder singularity sets of Levy processes. We also study the set of times at which a given function cannot be a modulus of continuity of a Levy process. The Holder singularity sets of the sample paths of certain random wavelet series are investigated as well.
In this paper we prove the uniform boundary Harnack principle in general open sets for harmonic functions with respect to a large class of rotationally symmetric purely discontinuous Lévy processes.
We investigate the high resolution coding problem for general real-valued Lévy processes under L^p[0,1]-norm distortion. Tight asymptotic formulas are found under mild regularity assumptions.
We find necessary and sufficient conditions for almost sure finiteness of integral functionals of spectrally positive Lévy processes. Via Lamperti type transforms, these results can be applied to obtain new integral tests on extinction and explosion behaviors for a class of continuous-state nonlinear branching processes.
In this paper we prove a criterion of convergence in distribution in Skorokhod space. We apply this criterion to some special Levy processes and obtain almost-sure versions of limit theorems for these processes.
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…
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…
The law of a positive infinitely divisible process with no drift is characterized by its Lévy measure on the paths space. Based on recent results of the two authors, it is shown that even for simple examples of such processes, the knowledge of their Lévy measures allows to obtain remarkable distributional identities.
We ask for necessary and sufficient conditions for almost sure finiteness of the perpetual integrals of a Levy process. Zero-one laws are already known for Brownian motion with drift and spectrally one-sided Levy processes. Under the assumption that local times exist, we use fluctuation theory and Jeulin's lemma to prove the zero-one law.
Motivated by the recent results of Nualart and Xu \cite{Nualart} concerning limits laws for occupation times of one dimensional symmetric stable processes, this paper proves a decomposition for functionals of one dimensional symmetric Lévy processes under certain conditions on the characteristic exponent and computes the moments of the decomposition.
In this paper we consider the argmin process of random walks and Lévy processes. We prove that they enjoy the Markov property, and provide their transition kernels in some special cases.
Exponential functionals of Lévy processes appear as stationary distributions of generalized Ornstein-Uhlenbeck (GOU) processes. In this paper we obtain the infinitesimal generator of the GOU process and show that it is a Feller process. Further we use these results to investigate properties of the mapping Φ, which maps two independent Lévy processes to their corresponding exponential functional, where one of the processes is assumed to be fixed. We show that in many cases this mapping is injective and give the inverse mapping in terms of (Lévy) characteristics. Also, continuity of Φis treated …
In this paper a Malliavin calculus for Lévy processes based on a family of true derivative operators is developed. The starting point is an extension to Lévy processes of the pioneering paper by Carlen and Pardoux [8] for the Poisson process, and our approach includes also the classical Malliavin derivative for Gaussian processes. We obtain a sufficient condition for the absolute continuity of functionals of the Lévy process. As an application, we analyze the absolute continuity of the law of the solution of some stochastic differential equations.
We present a new approach to fluctuation identities for reflected Lévy processes with one-sided jumps. This approach is based on a number of easy to understand observations and does not involve excursion theory or Itô calculus. It also leads to more general results.
We show the existence of Lévy-type stochastic processes in one space dimension with characteristic triplets that are either discontinuous at thresholds, or are stable-like with stability index functions for which the closures of the discontinuity sets are countable. For this purpose, we formulate the problem in terms of a Skorokhod-space martingale problem associated with non-local operators with discontinuous coefficients. These operators are approximated along a sequence of smooth non-local operators giving rise to Feller processes with uniformly controlled symbols. They converge uniformly o…
In this paper we study the convergence of solutions for (possibly degenerate) stochastic differential equations driven by Lévy processes, when the coefficients converge in some appropriate sense. First, we prove, by means of a superposition principle, a limit theorem of stochastic differential equations driven by Lévy processes. Then we apply the result to a type of nonlinear filtering problems and obtain the convergence of the nonlinear filterings.
Markov-modulated Lévy processes with two different regimes of restarting are studied. These regimes correspond to the completely renewed process and to the process of Markov modulation, accompanied by jumps. We give explicit expressions for the Lévy-Khintchine exponent in the case of a two-state underlying Markov chain. For the renewal case, the limit distributions (as $t\to\infty$) are obtained. In the case of processes with jumps, we present some results for the exponential functional.
