  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.
