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 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.

12 pages, 3 figures, submitted to Phys. Rev. E

References in corpus (10)

• Coupled continuous time random walks in finance
• Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation
• Polymer translocation through a nanopore - a showcase of anomalous diffusion
• Driven polymer translocation through a nanopore: a manifestation of anomalous diffusion
• Non-Markovian diffusion equations and processes: analysis and simulations
• The continuous time random walk formalism in financial markets
• A fractional generalization of the Poisson processes
• Closed-form solutions for continuous time random walks on finite chains
• Path probability density functions for semi-Markovian random walks
• Stochastic integration based on simple, symmetric random walks

Cited by in corpus (25)

• Paradoxical diffusion: Discriminating between normal and anomalous random walks
• Full characterization of the fractional Poisson process
• First-passage and first-exit times of a Bessel-like stochastic process
• Subordination Pathways to Fractional Diffusion
• Variable Order Fractional Fokker-Planck Equations derived from Continuous Time Random Walks
• Asymptotic solutions of decoupled continuous-time random walks with superheavy-tailed waiting time and heavy-tailed jump length distributions
• Time evolution towards q-Gaussian stationary states through unified Ito-Stratonovich stochastic equation
• Anomalous diffusion originated by two Markovian hopping-trap mechanisms
• Semi-Markov Graph Dynamics
• Mittag-Leffler Functions and Their Applications
• Covariant Formulation of Non-linear Langevin Theory with Multiplicative Gaussian White Noises
• Describe NMR relaxation by effective diffusion equation
• Describe NMR relaxation by anomalous rotational or translational diffusion
• Stochastic solutions of generalized time-fractional evolution equations
• Random numbers from the tails of probability distributions using the transformation method
• General PFG signal attenuation expressions for anisotropic anomalous diffusion by modified-Bloch equations
• General pulsed-field gradient signal attenuation expression based on a fractional integral modified-Bloch equation
• Limiting distributions of continuous-time random walks with superheavy-tailed waiting times
• Describing NMR chemical exchange by effective phase diffusion approach
• Parrondo-like behavior in continuous-time random walks with memory
• Exit times in non-Markovian drifting continuous-time random walk processes
• Effect of phase and time coupling on NMR relaxation rate by random walk in phase space
• Weak convergence of stochastic integrals driven by continuous-time random walks
• Continuous-time statistics and generalized relaxation equations
• Riemann-Liouville Fractional Cosine Functions