Lévy process ≈ Lévy process
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High order schemes for the tempered fractional diffusion equations Open
L,vy flight models whose jumps have infinite moments are mathematically used to describe the superdiffusion in complex systems. Exponentially tempering L,vy measure of L,vy flights leads to the tempered stable L,vy processes which combine …
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Exit identities for Lévy processes observed at Poisson arrival times Open
For a spectrally one-sided Lévy process, we extend various two-sided exit identities to the situation when the process is only observed at arrival epochs of an independent Poisson process. In addition, we consider exit problems of this typ…
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Fractional Brownian motion, the Matérn process, and stochastic modeling of turbulent dispersion Open
Stochastic processes exhibiting power-law slopes in the frequency domain are frequently well modeled by fractional Brownian motion (fBm), with the spectral slope at high frequencies being associated with the degree of small-scale roughness…
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First-passage properties of asymmetric Lévy flights Open
Lévy flights are paradigmatic generalised random walk processes, in which the independent stationary increments—the ‘jump lengths’—are drawn from an -stable jump length distribution with long-tailed, power-law asymptote. As a result, the v…
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Life distribution analysis based on <span>L</span>évy subordinators for degradation with random jumps Open
For a component or a system subject to stochastic degradation with sporadic jumps that occur at random times and have random sizes, we propose to model the cumulative degradation with random jumps using a single stochastic process based on…
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Some Properties of the Kilbas-Saigo Function Open
We characterize the complete monotonicity of the Kilbas-Saigo function on the negative half-line. We also provide the exact asymptotics at −∞, and uniform hyperbolic bounds are derived. The same questions are addressed for the classical Le…
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Dependence calibration and portfolio fit with factor-based subordinators Open
The paper explores the properties of a class of multivariate Lévy processes used for asset returns. We focus on describing both linear and non-linear dependence in an economic sensible and empirically appropriate way. The processes are sub…
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Fractional Brownian Motion Delayed by Tempered and Inverse Tempered Stable Subordinators Open
In recent years subordinated processes have been widely considered in the literature. These processes not only have wide applications but also have interesting theoretical properties. In this paper we consider fractional Brownian motion (F…
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Stability of the solution of stochastic differential equation driven by time-changed Lévy noise Open
This paper studies stabilities of the solution of stochastic differential equations (SDE) driven by time-changed Lévy noise in both probability and moment sense. This provides more flexibility in modeling schemes in application areas inclu…
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Feynman–Kac formulas for regime-switching jump diffusions and their applications Open
This work develops Feynman-Kac formulas for a class of regime-switching jump diffusion processes, in which the jump part is driven by a Poisson random measure associated to a general L\'evy process and the switching part depends on the jum…
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Ornstein-Uhlenbeck Processes Driven by Cylindrical Lévy Processes Open
In this article we introduce a theory of integration for deterministic, operator-valued integrands with respect to cylindrical Lévy processes in separable Banach spaces. Here, a cylindrical Lévy process is understood in the classical frame…
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A generalization of the space-fractional Poisson process and its connection to some Lévy processes Open
The space-fractional Poisson process is a time-changed homogeneous Poisson process where the time change is an independent stable subordinator. In this paper, a further generalization is discussed that preserves the Lévy property. We intro…
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Joint distribution of a spectrally negative Lévy process and its occupation time, with step option pricing in view Open
We study the distribution E x [exp(- q ∫ 0 t 1 ( a , b ) ( X s )d s ); X t ∈ d y ], where -∞ ≤ a < b < ∞, and where q , t > 0 and x ∈ R for a spectrally negative Lévy process X . More precisely, we identify the Laplace transform with respe…
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Regularity of density for SDEs driven by degenerate Lévy noises Open
By using Bismut's approach about the Malliavin calculus with jumps, we study the regularity of the distributional density for SDEs driven by degenerate additive Lévy noises. Under full Hörmander's conditions, we prove the existence of dist…
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Phase-Type Models in Life Insurance: Fitting and Valuation of Equity-Linked Benefits Open
Phase-type (PH) distributions are defined as distributions of lifetimes of finite continuous-time Markov processes. Their traditional applications are in queueing, insurance risk, and reliability, but more recently, also in finance and, th…
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$L^{p}$-Wasserstein distance for stochastic differential equations driven by Lévy processes Open
Coupling by reflection mixed with synchronous coupling is constructed for a\nclass of stochastic differential equations (SDEs) driven by L\\'{e}vy noises. As\nan application, we establish the exponential contractivity of the associated\nse…
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Asymptotics for ruin probabilities in Lévy-driven risk models with heavy-tailed claims Open
Consider a bivariate Lévy-driven risk model in which the loss process of an insurance company and the investment return process are two independent Lévy processes. Under the assumptions that the loss process has a Lévy measure of consisten…
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Lévy processes and stochastic integrals in the sense of generalized convolutions Open
In this paper, we present a comprehensive theory of generalized and weak\ngeneralized convolutions, illustrate it by a large number of examples, and\ndiscuss the related infinitely divisible distributions. We consider L\\'{e}vy\nand additi…
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Subgeometric rates of convergence for Markov processes under subordination Open
We are interested in the rate of convergence of a subordinate Markov process to its invariant measure. Given a subordinator and the corresponding Bernstein function (Laplace exponent), we characterize the convergence rate of the subordinat…
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Hardy-Stein identities and square functions for semigroups Open
We prove a Hardy-Stein type identity for the semigroups of symmetric, pure-jump L\'evy processes. Combined with the Burkholder-Gundy inequalities, it gives the $L^p$ two-way boundedness, for $1<p<\infty$, of the corresponding Littlewood-Pa…
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Mixed Periodic-Classical Barrier Strategies for Lévy Risk Processes Open
Given a spectrally-negative Lévy process and independent Poisson observation times, we consider a periodic barrier strategy that pushes the process down to a certain level whenever the observed value is above it. We also consider the versi…
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Asymptotics for ruin probabilities of a non-standard renewal risk model with dependence structures and exponential Lévy process investment returns Open
Consider a non-standard renewal risk model with dependence structures, where claim sizes follow a one-sided linear process with independent and identically distributed step sizes, the step sizes and inter-arrival times respectively form a …
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Multilevel Monte Carlo for exponential Lévy models Open
We apply the multilevel Monte Carlo method for option pricing problems using exponential Lévy models with a uniform timestep discretisation. For lookback and barrier options, we derive estimates of the convergence rate of the error introdu…
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Continuous-state branching processes with competition: duality and\n reflection at Infinity Open
The boundary behavior of continuous-state branching processes with quadratic\ncompetition is studied in whole generality. We first observe that despite\ncompetition, explosion can occur for certain branching mechanisms. We obtain a\nnecess…
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Distributional representations and dominance of a Lévy process over its maximal jump processes Open
Distributional identities for a Lévy process XtXt, its quadratic variation process VtVt and its maximal jump processes, are derived, and used to make “small time” (as t↓0t↓0) asymptotic comparisons between them. The representations are con…
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Efficient Computation of Various Valuation Adjustments Under Local Lévy Models Open
Various valuation adjustments (XVAs) can be written in terms of nonlinear partial integro-differential equations equivalent to forward-backward SDEs (FBSDEs). In this paper we develop a Fourier-based method for solving FBSDEs in order to e…
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Detecting Stock Crashes Using Levy Distribution Open
In this paper we study the possibility of construction indicators-precursors relying on one of the most power-law tailed distributions – Levy’s stable distribution. Here, we apply Levy’s parameters for 29 stock indices for the period from …
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Entrance and exit at infinity for stable jump diffusions Open
In his seminal work from the 1950s, William Feller classified all one-dimensional diffusions on $-\\infty \\leq a<b\\leq \\infty $ in terms of their ability to access the boundary (Feller’s test for explosions) and to enter the interior fr…
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Stochastic integration with respect to cylindrical Lévy processes Open
A cylindrical Lévy process does not enjoy a cylindrical version of thesemimartingale decomposition which results in the need to develop a completely novel approach to stochastic integration. In this work, we introduce a stochastic integral…
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A Feynman–Kac-type formula for Lévy processes with discontinuous killing rates Open
The roots of the present paper go back to the author’s dissertation [17], which was financially supported by the DFG through project EB66/11-1.