Lévy process

Abstract: A Historical Introduction.- Probability Concepts.- Markov Processes.- The Ito Calculus and Stochastic Differential Equations.- The Fokker-Planck Equation.- The Fokker-Planck Equation in Several Dimensions.- Small Noise Approximations for Diffusion Processes.- The White Noise Limit.- Beyond the White Noise Limit.- Levy Processes and Financial Applications.- Master Equations and Jump Processes.- The Poisson Representation.- Spatially Distributed Systems.- Bistability, Metastability, and Escape Problems.- Simulation of Stochastic Differential Equations.

Abstract: Preface.- Stochastic Calculus with Levy Processes.- Financial Markets Modelled by Jump Diffusions.- Optimal Stopping of Jump Diffusions.- Backward Stochastic Differential Equations and Risk Measures.- Stochastic Control of Jump Diffusions.- Stochastic Differential Games.- Combined Optimal Stopping and Stochastic Control of Jump Diffusions.- Viscosity Solutions.- Solutions of Selected Exercises.- References.- Notation and Symbols.

Abstract: Three processes reflecting persistence of volatility are initially formulated by evaluating three Levy processes at a time change given by the integral of a mean-reverting square root process. The model for the mean-reverting time change is then generalized to include non-Gaussian models that are solutions to Ornstein-Uhlenbeck equations driven by one-sided discontinuous Levy processes permitting correlation with the stock. Positive stock price processes are obtained by exponentiating and mean correcting these processes, or alternatively by stochastically exponentiating these processes. The characteristic functions for the log price can be used to yield option prices via the fast Fourier transform. In general mean-corrected exponentiation performs better than employing the stochastic exponential. It is observed that the mean-corrected exponential model is not a martingale in the filtration in which it is originally defined. This leads us to formulate and investigate the important property of martingale marginals where we seek martingales in altered filtrations consistent with the one-dimensional marginal distributions of the level of the process at each future date.

Abstract: Translucent materials such as milk, clouds and biological tissues owe their appearance to the way they interact with light, randomly scattering an incident ray many times before it re-emerges. This process — analogous to the brownian motion of particles in a fluid — is called a random walk, a concept central to statistical physics. It is used, for example, to describe the diffusion of heat, light and sound. An extension of this idea is the Levy flight, where a moving entity can occasionally take unusually large steps, thereby transforming a system's behaviour. Levy flights have been recognized in systems as diverse as earthquakes and animal food searches. Barthelemy et al. have now engineered such behaviour into an optical material (titanium dioxide particles in a glass matrix). In the resulting 'Levy glass', rather than regular diffusion, light waves perform a Levy flight, in which photons spread around extremely efficiently. This will be an ideal model for studying Levy flights, and may also lead to novel optical materials. The cover the photons' path, with the light source top right. Photo by Diederik and Leonardo Wiersma An extension of the concept of a random walk is the Levy flight, in which the moving entity can occasionally take unusually large steps. Pierre Barthelemy and colleagues show how such behaviour can be engineered into an optical material. A random walk is a stochastic process in which particles or waves travel along random trajectories. The first application of a random walk was in the description of particle motion in a fluid (brownian motion); now it is a central concept in statistical physics, describing transport phenomena such as heat, sound and light diffusion1. Levy flights are a particular class of generalized random walk in which the step lengths during the walk are described by a ‘heavy-tailed’ probability distribution. They can describe all stochastic processes that are scale invariant2,3. Levy flights have accordingly turned out to be applicable to a diverse range of fields, describing animal foraging patterns4, the distribution of human travel5 and even some aspects of earthquake behaviour6. Transport based on Levy flights has been extensively studied numerically7,8,9, but experimental work has been limited10,11 and, to date, it has not seemed possible to observe and study Levy transport in actual materials. For example, experimental work on heat, sound, and light diffusion is generally limited to normal, brownian, diffusion. Here we show that it is possible to engineer an optical material in which light waves perform a Levy flight. The key parameters that determine the transport behaviour can be easily tuned, making this an ideal experimental system in which to study Levy flights in a controlled way. The development of a material in which the diffusive transport of light is governed by Levy statistics might even permit the development of new optical functionalities that go beyond normal light diffusion.