Germano, Guido; Politi, Mauro; Scalas, Enrico; Schilling, René L; (2009) Stochastic calculus for uncoupled continuous-time random walks. Physical Review E , 79 (6) , Article 066102. 10.1103/PhysRevE.79.066102.

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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 It¯o 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 It¯o 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 It¯o integral are highlighted by numerical calculations when the jumps in space of the CTRW have a symmetric L´evy α-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 FokkerPlanck 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.

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