The explicit form of L evy measure for geometric stable (GS) random variables follows from the general L evy{Kchintchine representation of a subordinated innnitely di-visible process. Through this form, asymptotic properties of L evy measure are studied. In particular, logarithmic asymptotics around the origin imply exponential rate of convergence in series representation of GS random variables...

The ever-growing appearance of infinitely divisible laws and related processes in various areas, such as physics, mathematical biology, finance economics, has fuelled an increasing demand for numerical methods sampling sample path generation. In this survey, we review shot noise representation with a view towards generating paths processes. contrast to many conventional methods, the approach re...

An evolutionary programming algorithm with adaptivemutation operators based on L evy prob ability distribution is studied L evy stable distri bution has an in nite second moment Because of this L evy mutation is more likely to generate an o spring that is farther away from its parent than Gaussian mutation which is often used in evolu tionary algorithms Such likelihood depends on a parameter in...

We study the sample path asymptotics of a class of recurrent diiusion processes with random potentials, including the examples of Sinai's simple random walk in random environment and Brox's diiusion process with Brownian potential. The main results consist of several integral criteria which completely characterize all the possible L evy classes, therefore providing a very precise image of the a...

The small ball problem for the integrated process of a real{valued Brownian motion is solved. In sharp contrast to more standard methods, our approach relies on the sample path properties of Brownian motion together with facts about local times and L evy processes.

We study some properties of a new continuous-time model for-nancial time series which is driven by a class of L evy processes instead of a Brownian motion. This model emerged from extensive empirical investigations. We discuss path properties of the driving process and aspects of the valuation of derivatives.

The aim of this paper is to demonstrate how the appropriate numerical statistical and computer techniques can be successfully applied to the modeling of some physical systems We propose to use a fast and accurate method of computer generation of L evy stable random variates

We apply the theory of random walks to quantitatively describe the general problem of how to search eÆciently for randomly located objects that can only be detected in the limited vicinity of a searcher who typically has a nite degree of \free will" to move and search at will. We illustrate L evy ight search processes by comparison to Brownian random walks and discuss experimental observations ...

We study some properties of a new continuous-time model for-nancial time series which is driven by a class of L evy processes instead of a Brownian motion. This model emerged from extensive empirical investigations. We discuss path properties of the driving process and aspects of the valuation of derivatives.

If X is a symmetric L evy process on the line, then there exists a non{decreasing, c adl ag process H such that X(H(x)) = x for all x 0 if and only if X is recurrent and has a non{trivialGaussian component. The minimal such H is a subordinatorK. The law of K is identi ed and shown to be the same as that of a linear time change of the inverse local time at 0 of X. When X is Brownian motion,K is ...