Stochastic calculus for uncoupled continuous-time random walks

TL;DR

This paper develops a stochastic calculus framework for uncoupled continuous-time random walks (CTRWs), including definitions of stochastic integrals, their properties, and applications to anomalous diffusion models with fat-tailed distributions.

Contribution

It introduces a new class of stochastic integrals for CTRWs, proves their martingale properties, and connects them to fractional diffusion equations with explicit quadratic variation expressions.

Findings

Martingale property of zero-mean CTRWs and their stochastic integrals

Numerical validation of relations between CTRW, quadratic variation, and integrals

Analytic expression for quadratic variation in fractional diffusion models

Abstract

The continuous-time random walk (CTRW) is a pure-jump stochastic process with several applications in physics, but also in insurance, finance and economics. A definition is given for a class of stochastic integrals driven by a CTRW, that includes the Ito and Stratonovich cases. An uncoupled CTRW with zero-mean jumps is a martingale. It is proved that, as a consequence of the martingale transform theorem, if the CTRW is a martingale, the Ito integral is a martingale too. It is shown how the definition of the stochastic integrals can be used to easily compute them by Monte Carlo simulation. The relations between a CTRW, its quadratic variation, its Stratonovich integral and its Ito integral are highlighted by numerical calculations when the jumps in space of the CTRW have a symmetric Levy alpha-stable distribution and its waiting times have a one-parameter Mittag-Leffler distribution. Remarkably these distributions have fat tails and an unbounded quadratic variation. In the diffusive limit of vanishing scale parameters, the probability density of this kind of CTRW satisfies the space-time fractional diffusion equation (FDE) or more in general the fractional Fokker-Planck equation, that generalize the standard diffusion equation solved by the probability density of the Wiener process, and thus provides a phenomenologic model of anomalous diffusion. We also provide an analytic expression for the quadratic variation of the stochastic process described by the FDE, and check it by Monte Carlo.