  Brownian motion is a well-known model for normal diffusion, but not all
physical phenomena behave according to a Brownian motion. Many phenomena
exhibit irregular diffusive behavior, called anomalous diffusion. Examples of
anomalous diffusion have been observed in physics, hydrology, biology, and
finance, among many other fields. Continuous-time random walks (CTRWs),
introduced by Montroll and Weiss, serve as models for anomalous diffusion.
CTRWs generalize the usual random walk model by allowing random waiting times
between successive random jumps. Under certain conditions on the jumps and
waiting times, scaled CTRWs can be shown to converge in distribution to a limit
process M(t) in the cadlag space D[0,infinity) with the Skorohod J_1 or M_1
topology. An interesting question is whether stochastic integrals driven by the
scaled CTRWs X^n(t) converge in distribution to a stochastic integral driven by
the CTRW limit process M(t). We prove weak convergence of the stochastic
integrals driven by CTRWs for certain classes of CTRWs, when the CTRW limit
process is an alpha-stable Levy motion and when the CTRW limit process is a
time-changed Brownian motion.
