Financial Applications on Fractional Lévy Stochastic Processes

Stochastic Bounds for Lévy Processes

Using the Wiener–Hopf factorization, it is shown that it is possible to bound the path of an arbitrary Lévy process above and below by the paths of two random walks. These walks have the same step distribution, but different random starting points. In principle, this allows one to deduce Lévy process versions of many known results about the large-time behavior of random walks. This is illustrat...

Fractional transport equations for Lévy stable processes.

The influence functional method of Feynman and Vernon is used to obtain a quantum master equation for a system subjected to a Lévy stable random force. The corresponding classical transport equations for the Wigner function are then derived, both in the limits of weak and strong friction. These are fractional extensions of the Klein-Kramers and the Smoluchowski equations. It is shown that the f...

Nonlinear stochastic integrals for hyperfinite Lévy processes

We develop a notion of nonlinear stochastic integrals for hyperfinite Lévy processes, and use it to find exact formulas for expressions which are intuitively of the form Pt s=0 φ(ω, dls, s) and Qt s=0 ψ(ω, dls, s), where l is a Lévy process. These formulas are then applied to geometric Lévy processes, infinitesimal transformations of hyperfinite Lévy processes, and to minimal martingale measure...

Stochastic Processes and their Applications in Financial Pricing

Generalized fractional Lévy processes with fractional Brownian motion limit and applications to stochastic volatility models

Fractional Lévy processes generalize fractional Brownian motion in a natural way. We go a step further and extend the usual fractional Riemann-Liouville kernels to the more general class of regularly varying functions with the corresponding fractional integration parameter. The resulting stochastic processes are called generalized fractional Lévy processes (GFLP). Moreover, we define stochastic...