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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.
We present a numerical method for the Monte Carlo simulation of uncoupled continuous-time random walks with a Levy alpha-stable distribution of jumps in space and a Mittag-Leffler distribution of waiting times, and apply it to the stochastic solution
Intermittent stochastic processes appear in a wide field, such as chemistry, biology, ecology, and computer science. This paper builds up the theory of intermittent continuous time random walk (CTRW) and L{e}vy walk, in which the particles are stocha
We introduce a heterogeneous continuous time random walk (HCTRW) model as a versatile analytical formalism for studying and modeling diffusion processes in heterogeneous structures, such as porous or disordered media, multiscale or crowded environmen
We investigate the effects of markovian resseting events on continuous time random walks where the waiting times and the jump lengths are random variables distributed according to power law probability density functions. We prove the existence of a n
We consider a continuous random walk model for describing normal as well as anomalous diffusion of particles subjected to an external force when these particles diffuse in a uniformly expanding (or contracting) medium. A general equation that relates