The book discusses a class of discrete time stochastic growth processes for which the growth
rate is proportional to the exponential of a Gaussian Markov process. These growth processes
appear naturally in problems of mathematical finance as discrete time approximations of
stochastic volatility models and stochastic interest rates models such as the Black-Derman-Toy
and Black-Karasinski models. These processes can be mapped to interacting one-dimensional
lattice gases with long-range interactions. The book gives a detailed discussion of these
statistical mechanics models including new results not available in the literature and their
implication for the stochastic growth models. The statistical mechanics analogy is used to
understand observed non-analytic dependence of the Lyapunov exponents of the stochastic growth
processes considered which is related to phase transitions in the lattice gas system. The
theoretical results are applied to simulations of financial models and are illustrated with
Mathematica code. The book includes a general introduction to exponential stochastic growth
with examples from biology population dynamics and finance. The presentation does not assume
knowledge of mathematical finance. The new results on lattice gases can be read independently
of the rest of the book. The book should be useful to practitioners and academics studying the
simulation and application of stochastic growth models.
