2006/05/13 by Mark M. Meerschaert, Enrico Scalas · 1 citation
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Anomalous diffusion #Applied mathematics #Brownian motion #Complex Systems and Time Series Analysis #Computer science #Continuous-time random walk #Economics #Finance #Fractional Differential Equations Solutions #Futures contract #Limit (mathematics) #Mathematical analysis #Mathematical finance #Mathematics #Physics #Random walk #Statistical physics #Statistics #Stochastic differential equation #Stochastic process #physics.data-an #physics.soc-ph #q-fin.ST #stochastic dynamics and bifurcation
paper · pdf · doi:10.1016/j.physa.2006.04.034
published as Physica A, vol. 370, 114-118, 2006 · 7 pages, 2 figures. Paper presented at the Econophysics Colloquium, Canberra, Australia, November 2005
openalex publication_date 2006/05/13 · arxiv created 2006/08/29 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Continuous time random walks (CTRWs) are used in physics to model anomalous diffusion, by incorporating a random waiting time between particle jumps. In finance, the particle jumps are log-returns and the waiting times measure delay between transactions. These two random variables (log-return and waiting time) are typically not independent. For these coupled CTRW models, we can now compute the limiting stochastic process (just like Brownian motion is the limit of a simple random walk), even in the case of heavy tailed (power-law) price jumps and/or waiting times. The probability density functions for this limit process solve fractional partial differential equations. In some cases, these equations can be explicitly solved to yield descriptions of long-term price changes, based on a high-resolution model of individual trades that includes the statistical dependence between waiting times and the subsequent log-returns. In the heavy tailed case, this involves operator stable space-time random vectors that generalize the familiar stable models. In this paper, we will review the fundamental theory and present two applications with tick-by-tick stock and futures data.