  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.
