Traditional diffusion models for random phenomena have paths with Holder
regularity just greater than 1/2 almost surely but there are situations
arising in finance and telecommunications where it is natural to look
for models in which the Holder regularity of the paths can vary.
Such processes are called multifractal and we will construct a class of
such processes on R using ideas from branching processes.
Using connections with multitype branching random walk we will be able
to compute the multifractal spectrum which captures the variability in
the Holder regularity. In addition, if we observe one of our processes
at a fixed resolution then we obtain a finite Markov representation,
which allows efficient simulation.
As an application, we fit the model to some AUD-USD exchange rate data.
Joint work with Geoffrey Decrouez and Ben Hambly