Stochastic calculus for uncoupled continuous-time random walks

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 Itō 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 Itō 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 Itō integral are highlighted by numerical calculations when the jumps in space of the CTRW have a symmetric Lévy 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.

On the convergence of quadratic variation for compound fractional Poisson processes

“…Some results highlighted here were already published in [8]. However, the main focus of that paper was defining and studying the properties of stochastic integrals driven by pure jump processes.…”

The M-Wright function as a generalization of the Gaussian density for fractional diffusion processes

“…The properties of the corresponding Master Equation (ME) lead to name such diffusion as Erdélyi-Kober fractional diffusion [50,51,52]. Actually, the M-Wright function was also used as a non-Markovian model to generalize the evolution in time of the radius of a premixed flame ball in fractional diffusive media [49,50] and it has emerged to be related to the quadratic variation for compound renewal processes [56] whose convergence is important for stochastic integrals driven by a CTRW [11].…”

Limiting distributions of continuous-time random walks with superheavy-tailed waiting times

“…[2,3]), the probability density P (x, t) of the particle position X(t) depends only on the joint probability density of waiting time and jump length. Unfortunately, even in the simplest (decoupled) case when the joint density is a product of waiting-time density p(τ ) and jump-length density w(ξ), the representation of P (x, t) in terms of special functions is known in a few cases [14][15][16]. In contrast, the long-time behavior of P (x, t) that determines the diffusion and transport properties of walking particles is studied analytically in much more detail [17][18][19][20][21].…”