Optimal control and dependence modeling of insurance portfolios with Lévy dynamics

In this paper we are interested in optimizing proportional reinsurance and investment policies in a multidimensional Lévy-driven insurance model. The criterion is that of maximizing exponential utility. Solving the classical Hamilton-Jacobi-Bellman equation yields that the optimal retention level keeps a constant amount of claims regardless of time and the company’s wealth level.A special feature of our construction is to allow for dependencies of the risk reserves in different business lines. Dependence is modeled via an Archimedean Lévy copula. We derive a sufficient and necessary condition for an Archimedean Lévy generator to create a multidimensional positive Lévy copula in arbitrary dimension.Based on these results we identify structure conditions for the generator and the Lévy measure of an Archimedean Lévy copula under which an insurance company reinsures a larger fraction of claims from one business line than from another.     

Symmetrization of Lévy processes and applications

It is shown that many of the classical generalized isoperimetric inequalities for the Laplacian when viewed in terms of Brownian motion extend to a wide class of Lévy processes. The results are derived from the multiple integral inequalities of Brascamp, Lieb and Luttinger but the probabilistic structure of the processes plays a crucial role in the proofs.     

The Segal-Bargmann transform for Lévy white noise functionals associated with non-integrable Lévy processes

By using a method of truncation, we derive the closed form of the Segal-Bargmann transform of Lévy white noise functionals associated with a Lévy process with the Lévy spectrum without the moment condition. Besides, a sufficient and necessary condition to the existence of Lévy stochastic integrals is obtained.     

A Stochastic Process Generated by the Lévy Laplacian

In this paper we give a stochastic process generated by the Lévy Laplacian in the white noise analysis with a characterization of the Laplacian.     

A computational analysis for mean exit time under non-Gaussian Lévy noises

Complex dynamical systems are often subject to non-Gaussian random fluctuations. The exit phenomenon, i.e., escaping from a bounded domain in state space, is an impact of randomness on the evolution of these dynamical systems. The existing work is about asymptotic estimate on mean exit time when the noise intensity is sufficiently small. In the present paper, however, the authors analyze mean exit time for arbitrary noise intensity, via numerical investigation. The mean exit time for a dynamical system, driven by a non-Gaussian, discontinuous (with jumps), α-stable Lévy motion, is described by a differential equation with nonlocal interactions. A numerical approach for solving this nonlocal problem is proposed. A computational analysis is conducted to investigate the relative importance of jump measure, diffusion coefficient and non-Gaussianity in affecting mean exit time.     

Lévy random bridges and the modelling of financial information

The information-based asset-pricing framework of Brody-Hughston-Macrina (BHM) is extended to include a wider class of models for market information. To model the information flow, we introduce a class of processes called Lévy random bridges (LRBs), generalising the Brownian bridge and gamma bridge information processes of BHM. Given its terminal value at T, an LRB has the law of a Lévy bridge. We consider an asset that generates a cash-flow X_T at T. The information about X_T is modelled by an LRB with terminal value X_T. The price process of the asset is worked out, along with the prices of options.     

Invariant Measures and Symmetry Property of Lévy Type Operators

Second order elliptic integro-differential operators (Lévy type operators) are investigated. The notion of regular (infinitesimal) invariant probability measures for such operators is posed. Sufficient conditions for the existence of such regular infinitesimal invariant probability measures are obtained and the symmetrization problem is discussed.     

Ratio Limit Theorems for Random Walks and Lévy Processes

In this paper, we study quasi-symmetric random walks and Lévy processes, a property first introduced by C.J. Stone, discuss the -invariant Radon measures for random walks and Lévy processes, and formulate some nice ratio limit theorems which are closely related to -invariant Radon measures. Mathematics Subject Classifications (2000) 60G51, 60G50.Research supported in part by NSFC 10271109.     

Canonical Lévy process and Malliavin calculus

A suitable canonical Lévy process is constructed in order to study a Malliavin calculus based on a chaotic representation property of Lévy processes proved by Itô using multiple two-parameter integrals. In this setup, the two-parameter derivative _Dt,x$D_{t,x}$ is studied, depending on whether x=0$x=0$ or x≠0$x\ne 0$; in the first case, we prove a chain rule; in the second case, a formula by trajectories.     

Large Deviations of Local Times of Lévy Processes

For X(t) a real-valued symmetric Lévy process, its characteristic function is E(e ^iX(t))=exp(–t()). Assume that is regularly varying at infinity with index 1<2. Let L ^x _t denote the local time of X(t) and L* _t =sup _xR L ^x _t . Estimates are obtained for P(L ⁰ _t y) and P(L* _t y) as y and t fixed.     

Continuity Conditions for a Class of Gaussian Chaos Processes Related to Continuous Additive Functionals of Lévy Processes

Let be sequences of real numbers which are symmetric in k. Let be independent sequences of independent normal random variables with mean zero and variance one. For each fixed choice of we consider

Mortality modelling with Lévy processes

This paper addresses the modelling of human mortality by the aid of doubly stochastic processes with an intensity driven by a positive Lévy process. We focus on intensities having a mean reverting stochastic component. Furthermore, driving Lévy processes are pure jump processes belonging to the class of α-stable subordinators. In this setting, expressions of survival probabilities are inferred, the pricing is discussed and numerical applications to actuarial valuations are proposed.     

Some independence results related to the arc-sine law

We provide proofs for recent results of Getoor and Sharpe on the distribution of local times on rays for certain planar Lévy processes which were invalidated by an appeal to an incorrect assertion. Our arguments rely on independence properties related to the arc-sine law.     

Donsker's Delta Function of Lévy Process

Let X={X(t):tR} be a Lévy process and a non-decreasing, right continuous, bounded function with (–)=0 (((1+u ²)/u ²)d(u) is the Lévy measure). In this paper we define the Donsker delta function (X(t)–a), t>0 and aR, as a generalized Lévy functional under the condition that (0)–(0–)>0. This leads us to define F(X(t)) for any tempered distribution F, and as an application, we derive an Itô formula for F(X(t)) when has jumps at 0 and 1.     

s-Stable Laws in Insurance and Finance and Generalization to Nilpotent Lie Groups

s-stable laws on Hilbert spaces, associated with some nonlinear transformations, were introduced by Jurek.^{(16, 18)} Here, we interpret certain s-stable motions as limits of total amount of claims processes (up to a deterministic reserve) of a portfolio of (nontraded) excess-of-loss reinsurance contracts and show that they lead to Erlang's model. We also give explicit formulas for the price of perpetual American options in case the logarithm of the price of the underlying asset is an s-stable motion. Furthermore, we generalize the concept of s-stability to simply connected nilpotent Lie groups. For step 2-nilpotent Lie groups we characterize the Lévy measure and the s-domain of attraction of nongaussian s-stable convolution semigroups.     

The ergodicity of stochastic generalized porous media equations with lévy jump

In this article,we first prove the existence and uniqueness of the solution to the stochastic generalized porous medium equation perturbed by Lévy process,and then show the exponential convergence of(pt)t≥0 to equilibrium uniform on any bounded subset in H.     

On the Lévy Measures for Symmetric Stable Semigroups on the Torus T∞

We investigate analytical properties of the Lévy measures fir symmetric stable semigroups on the compact Lie-projective group T. We apply these properties to describe the domain of the fractional powers of Laplacians on T. Among the analytical tools involved are the intrinsic metric and the scale of Hölder continuous functions w.r.t. this metric.     

Sur les petites déviations d'un processus de Lévy

Nous caractérisons les processus de Lévy sur partant de 0 qui peuvent être approchés par la fonction nulle pour la topologie localement uniforme. Ceci nous permet de déterminer le support d'un processus de Lévy quelconque dans l'espace de Skorohod, et de retrouver un théorème de Tortrat sur le support des lois indéfiniment divisibles.