Lévy Processes and Stochastic Calculus

Abstract

Lévy processes form a wide and rich class of random process, and have many applications ranging from physics to finance. Stochastic calculus is the mathematics of systems interacting with random noise. Here, the author ties these two subjects together, beginning with an introduction to the general theory of Lévy processes, then leading on to develop the stochastic calculus for Lévy processes in a direct and accessible way. This fully revised edition now features a number of new topics. These include: regular variation and subexponential distributions; necessary and sufficient conditions for Lévy processes to have finite moments; characterisation of Lévy processes with finite variation; Kunita's estimates for moments of Lévy type stochastic integrals; new proofs of Ito representation and martingale representation theorems for general Lévy processes; multiple Wiener-Lévy integrals and chaos decomposition; an introduction to Malliavin calculus; an introduction to stability theory for Lévy-driven SDEs.

A concave—convex elliptic problem involving the fractional Laplacian

“…Observe that α = 2 corresponds to the standard local Laplacian. This type of diffusion operators arises in several areas such as physics, probability and finance (see, for example, [6,8,23,44]). In particular, the fractional Laplacian can be understood as the infinitesimal generator of a stable Lévy process [8].…”

A fast and accurate numerical method for symmetric Lévy processes based on the Fourier transform and sinc-Gauss sampling formula

“…Then, the corresponding Lévy process becomes symmetric. As examples of the Lévy process with such measure, we can give the variance gamma (VG) process (Applebaum, 2009), also known as the symmetric Laplace motion (Kozubowski et al, 2006), with γ = 1 and µ(y) = e −y , and the normal inverse Gaussian (NIG) process (Applebaum, 2009) with γ = 2 and µ(y) = y K 1 (y)/π, where K 1 is the modified Bessel function of the second kind. Then, applying (1.8) to the operator A in (1.4) and taking its adjoint, we have a special form of equation (1.6) as…”

“…Example 2 (Normal inverse Gaussian (NIG) process (Applebaum, 2009)) Setting γ = 2 and µ(y) = y K 1 (y)/π in (1.8), we have the Lévy measure…”

“…A Lévy process {X t } t 0 is essentially a stochastic process with stationary and independent increments (Applebaum, 2009), which is used to describe uncertain phenomena in various fields. Let us consider a brief non-rigorous review of the scalar Lévy process {X t } t 0 .…”

Existence and stability of solutions to non-Lipschitz stochastic differential equations driven by Lévy noise

“…A number of meritorious results concerning SDEs with non-Gaussian Lévy noise has been presented in existing literatures [7,8,[15][16][17]. Among them, conditions which can guarantee the existence and uniqueness of the solutions to the SDEs with non-Gaussian Lévy noise are to be assumed as the one of the most basic and important Lipschitz condition.…”