Approximation of quantum Lévy processes by quantum random walks

Every quantum Lévy process with a bounded stochastic generator is shown to arise as a strong limit of a family of suitably scaled quantum random walks.     

Fluctuation theory for level-dependent Lévy risk processes

A level-dependent Lévy process solves the stochastic differential equation $dU\left(t\right)=dX\left(t\right)??\left(U\left(t\right)\right)dt$, where $X$ is a spectrally negative Lévy process. A special case is a multi-refracted Lévy process with ${?}_k\left(x\right)={\sum }_{j=1}^k{\delta }_j{\mathbf{1}}_{\{x\ge b_j\}}$. A general rate function $?$ that is non-decreasing and locally Lipschitz continuous 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 expressed via these scale functions. We demonstrate that the derivatives of the scale functions are solutions of Volterra integral equations.     

Pricing credit default swaps with a random recovery rate by a double inverse Fourier transform

We evaluate the par spread for a single-name credit default swap with a random recovery rate. It is carried out under the framework of a structural default model in which the asset-value process is of infinite activity but finite variation. The recovery rate is assumed to depend on the undershoot of the asset value below the default threshold when default occurs. The key part is to evaluate a generalized expected discounted penalty function, which is a special case of the so-called Gerber–Shiu function in actuarial ruin theory. We first obtain its double Laplace transform in time and in spatial variable, and then implement a numerical Fourier inversion integration. Numerical experiments show that our algorithm gives accurate results within reasonable time and different shapes of spread curve can be obtained.     

Quadratic–exponential growth BSDEs with jumps and their Malliavin’s differentiability

We investigate a class of quadratic–exponential growth BSDEs with jumps. The quadratic structure introduced by Barrieu & El Karoui (2013) yields the universal bounds on the possible solutions. With local Lipschitz continuity and the so-called $A_{\Gamma }$-condition for the comparison principle to hold, we prove the existence of a unique solution under the general quadratic–exponential structure. We have also shown that the strong convergence occurs under more general (not necessarily monotone) sequence of drivers, which is then applied to give the sufficient conditions for the Malliavin’s differentiability.     

Lévy noise perturbation for an epidemic model with impact of media coverage

ABSTRACT

This work is devoted to study the existence and uniqueness of global positive solution for a stochastic epidemic model with media coverage driven by Lévy noise. We also investigate the dynamic properties of the solution around both disease-free and endemic equilibria points of the deterministic model. Numerical simulations are presented to confirm the theoretical results.     

Malliavin smoothness on the Lévy space with Hölder continuous or BV functionals

We consider Malliavin smoothness of random variables $f\left(X_1\right)$, where $X$ is a pure jump Lévy process and the function $f$ is either bounded and Hölder continuous or of bounded variation. We show that Malliavin differentiability and fractional differentiability of $f\left(X_1\right)$ depend both on the regularity of $f$ and the Blumenthal–Getoor index of the Lévy measure.     

Strong and Weak Disorder for Lévy Directed Polymers in Random Environment

Abstract

We consider Lévy directed polymers in the Poisson random environment. We give conditions for strong or weak disorder in terms of the Lévy exponent of symmetric Lévy process.     

Pricing and hedging of variable annuities with state-dependent fees

We investigate the problem of pricing and hedging variable annuity contracts for which the fee deducted from the policyholder’s account depends on the account value. It is believed that state-dependent fees are beneficial to policyholders and insurers since they reduce policyholders’ incentives to lapse the policies and match the costs incurred by policyholders with the pay-offs received from embedded guarantees. We consider an incomplete financial market which consists of two risky assets modelled with a two-dimensional Lévy process. One of the assets is a security which can be traded by the insurer, and the second asset is a security which is the underlying fund for the variable annuity contract. In our model we derive an equation from which the fee for the guaranteed benefit can be calculated and we characterize a strategy which allows the insurer to hedge the benefit. To solve the pricing and hedging problem in an incomplete financial market we apply a quadratic objective.     

Lévy–Khinchin Formula and Existence of Densities for Convolution Semigroups on Symmetric Spaces

We prove the existence of a smooth density for a convolution semigroup on a symmetric space and obtain its spherical representation.      

Uniform asymptotics for ruin probabilities in a dependent renewal risk model with stochastic return on investments

In this paper, an insurer is allowed to make risk-free and risky investments, and the price process of the investment portfolio is described as an exponential Lévy process. We study the asymptotic tail behavior for a non-standard renewal risk model with dependence structures. The claim sizes are assumed to follow a one-sided linear process with independent and identically distributed step sizes, and the step sizes and inter-arrival times form a sequence of independent and identically distributed random pairs with a dependence structure. When the step-size distribution is heavy tailed, we obtain some uniform asymptotics for the finite-and infinite-time ruin probabilities.     

Non-homogeneous space-time fractional Poisson processes

The space-time fractional Poisson process (STFPP), defined by Orsingher and Poilto (2012), is a generalization of the time fractional Poisson process (TFPP) and the space fractional Poisson process (SFPP). We study the fractional generalization of the non-homogeneous Poisson process and call it the non-homogeneous space-time fractional Poisson process (NHSTFPP). We compute their pmf and generating function and investigate the associated differential equation. The limit theorems for the NHSTFPP process are studied. We study the distributional properties, the asymptotic expansion of the correlation function of the non-homogeneous time fractional Poisson process (NHTFPP) and subsequently investigate the long-range dependence (LRD) property of a special NHTFPP. We investigate the limit theorem for the fractional non-homogeneous Poisson process (FNHPP) studied by Leonenko et al. (2014). Finally, we present some simulated sample paths of the NHSTFPP process.     

Local Malliavin calculus for Lévy processes and applications

The Malliavin derivative operator for the Poisson process introduced by Carlen and Pardoux [Differential calculus and integration by parts on a Poisson space, in Stochastics, Algebra and Analysis in Classical and Quantum Dynamics, S. Albeverio et al. (eds), Kluwer, Dordrecht, 1990, pp. 63–73] is extended to Lévy processes. It is a true derivative operator (in the sense that it satisfies the chain rule), and we deduce a sufficient condition for the absolute continuity of functionals of the Lévy process. As an application, we analyse the absolute continuity of the law of the solution of some stochastic differential equations with jumps.     

On the pathwise uniqueness of solutions of stochastic differential equations driven by symmetric stable Lévy processes1

We investigate a sufficient condition for pathwise uniqueness property for 1D stochastic differential equation driven by symmetric α-stable Lévy process, where α ∈ (1, 2).