Continuous-time random walk

In mathematics, a continuous-time random walk (CTRW) is a generalization of a random walk where the wandering particle waits for a random time between jumps. It is a stochastic jump process with arbitrary distributions of jump lengths and waiting times.^[1]^[2]^[3] More generally it can be seen to be a special case of a Markov renewal process.

Motivation

CTRW was introduced by Montroll and Weiss ^[4] as a generalization of physical diffusion process to effectively describe anomalous diffusion, i.e., the super- and sub-diffusive cases. An equivalent formulation of the CTRW is given by generalized master equations.^[5] A connection between CTRWs and diffusion equations with fractional time derivatives has been established.^[6] Similarly, time-space fractional diffusion equations can be considered as CTRWs with continuously distributed jumps or continuum approximations of CTRWs on lattices. ^[7]

Formulation

A simple formulation of a CTRW is to consider the stochastic process $X(t)$ defined by

X(t)=X_{0}+\sum _{{i=1}}^{{N(t)}}\Delta X_{i},

whose increments $\Delta X_{i}$ are iid random variables taking values in a domain $\Omega$ and $N(t)$ is the number of jumps in the interval $(0,t)$ . The probability for the process taking the value $X$ at time $t$ is then given by

P(X,t)=\sum _{{n=0}}^{\infty }P(n,t)P_{n}(X).

Here $P_{n}(X)$ is the probability for the process taking the value $X$ after $n$ jumps, and $P(n,t)$ is the probability of having $n$ jumps after time $t$ .

Montroll-Weiss formula

We denote by $\tau$ the waiting time in between two jumps of $N(t)$ and by $\psi (\tau )$ its distribution. The Laplace transform of $\psi (\tau )$ is defined by

{\tilde {\psi }}(s)=\int _{0}^{\infty }d\tau \,e^{-\tau s}\psi (\tau ).

Similarly, the characteristic function of the jump distribution $f(\Delta X)$ is given by its Fourier transform:

{\hat {f}}(k)=\int _{\Omega }d(\Delta X)\,e^{ik\Delta X}f(\Delta X).

One can show that the Laplace-Fourier transform of the probability $P(X,t)$ is given by

{\hat {{\tilde {P}}}}(k,s)={\frac {1-{\tilde {\psi }}(s)}{s}}{\frac {1}{1-{\tilde {\psi }}(s){\hat {f}}(k)}}.

The above is called Montroll-Weiss formula.

Examples

The Wiener process is the standard example of a continuous time random walk in which the waiting times are exponential and the jumps are continuous and normally distributed.

References

↑ Klages, Rainer; Radons, Guenther; Sokolov, Igor M. Anomalous Transport: Foundations and Applications.
↑ Paul, Wolfgang; Baschnagel, Jörg (2013-07-11). Stochastic Processes: From Physics to Finance. Springer Science & Business Media. pp. 72–. ISBN 9783319003276. Retrieved 25 July 2014.
↑ Slanina, Frantisek (2013-12-05). Essentials of Econophysics Modelling. OUP Oxford. pp. 89–. ISBN 9780191009075. Retrieved 25 July 2014.
↑ Elliott W. Montroll; George H. Weiss (1965). "Random Walks on Lattices. II". J. Math. Phys. 6: 167. doi:10.1063/1.1704269.
↑ . M. Kenkre; E. W. Montroll; M. F. Shlesinger (1973). "Generalized master equations for continuous-time random walks". Journal of Statistical Physics. 9 (1): 45–50. doi:10.1007/BF01016796.
↑ Hilfer, R.; Anton, L. (1995). "Fractional master equations and fractal time random walks". Phys. Rev. E. 51 (2): R848––R851. doi:10.1103/PhysRevE.51.R848.
↑ Gorenflo, Rudolf; Mainardi, Francesco; Vivoli, Alessandro (2005). "Continuous-time random walk and parametric subordination in fractional diffusion". Chaos, Solitons \& Fractals. Elsevier. 34 (1): 87–103. doi:10.1016/j.chaos.2007.01.052.

Stochastic processes

Discrete time	Bernoulli process Branching process Chinese restaurant process Galton–Watson process Independent and identically distributed random variables Markov chain Moran process Random walk Loop-erased Self-avoiding

Continuous time	Bessel process Birth–death process Brownian motion Bridge Excursion Fractional Geometric Meander Cauchy process Contact process Continuous-time random walk Cox process Diffusion process Empirical process Feller process Fleming–Viot process Gamma process Hunt process Interacting particle systems Itô diffusion Itô process Jump diffusion Jump process Lévy process Local time Markov additive process McKean–Vlasov process Ornstein–Uhlenbeck process Poisson process Compound Non-homogeneous Point process Schramm–Loewner evolution Semimartingale Sigma-martingale Stable process Superprocess Telegraph process Variance gamma process Wiener process Wiener sausage

Both	Branching process Galves–Löcherbach model Gaussian process Hidden Markov model (HMM) Markov process Martingale Differences Local Sub- Super- Random dynamical system Regenerative process Renewal process Stochastic chains with memory of variable length White noise

Fields and other	Dirichlet process Gaussian random field Gibbs measure Hopfield model Ising model Potts model Boolean network Markov random field Percolation Pitman–Yor process Point process Cox Poisson Random field Random graph

Time series models	Autoregressive conditional heteroskedasticity (ARCH) model Autoregressive integrated moving average (ARIMA) model Autoregressive (AR) model Autoregressive–moving-average (ARMA) model Generalized autoregressive conditional heteroskedasticity (GARCH) model Moving-average (MA) model

Financial models	Black–Derman–Toy Black–Karasinski Black–Scholes Chen Constant elasticity of variance (CEV) Cox–Ingersoll–Ross (CIR) Garman–Kohlhagen Heath–Jarrow–Morton (HJM) Heston Ho–Lee Hull–White LIBOR market Rendleman–Bartter SABR volatility Vašíček Wilkie

Actuarial models	Bühlmann Cramér–Lundberg Risk process Sparre–Anderson

Queueing models	Bulk Fluid Generalized queueing network M/G/1 M/M/1 M/M/c

Properties	Càdlàg paths Continuous Continuous paths Ergodic Exchangeable Feller-continuous Gauss–Markov Markov Mixing Piecewise deterministic Predictable Progressively measurable Self-similar Stationary Time-reversible

Limit theorems	Central limit theorem Donsker's theorem Doob's martingale convergence theorems Ergodic theorem Fisher–Tippett–Gnedenko theorem Large deviation principle Law of large numbers (weak/strong) Law of the iterated logarithm Maximal ergodic theorem Sanov's theorem

Inequalities	Burkholder–Davis–Gundy Doob's martingale Kunita–Watanabe

Tools	Cameron–Martin formula Convergence of random variables Doléans-Dade exponential Doob decomposition theorem Doob–Meyer decomposition theorem Doob's optional stopping theorem Dynkin's formula Feynman–Kac formula Filtration Girsanov theorem Infinitesimal generator Itô integral Itô's lemma Kolmogorov continuity theorem Kolmogorov extension theorem Lévy–Prokhorov metric Malliavin calculus Martingale representation theorem Optional stopping theorem Prokhorov's theorem Quadratic variation Reflection principle Skorokhod integral Skorokhod's representation theorem Skorokhod space Snell envelope Stochastic differential equation Tanaka Stopping time Stratonovich integral Uniform integrability Usual hypotheses Wiener space Classical Abstract

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