This dissertation consists of two different research topics: Lyapunov exponents and portfolio optimization. The second chapter states the study of Lyapunov exponents under stochastic Anderson model. The stochastic Anderson model in discrete or continuous space is defined for a class of non-Gaussian space-time potentials as solutions to the multiplicative stochastic heat equation for some diffusivity parameter and inverse-temperature parameter. The relation with the corresponding polymer model in a random environment is given. The large time exponential behavior of the solution is studied via its almost sure Lyapunov exponent, which is proved to exist, and is estimated as a function of the diffusivity parameter and the inverse-temperature parameter: positivity and non-trivial upper bounds are established, generalizing and improving existing results.

The third chapter of this dissertation presents how to evaluate (approximately) optimal self-financing strategy and optimal trading frequency for a portfolio with a risky asset and a risk-free asset. The objective is to maximize the expected future utility of the terminal wealth in a stochastic volatility setting, when transaction costs are incurred at each discrete trading time. A HARA utility function is used, allowing a simple approximation of the optimization problem, which is implementable forward in time. For each of various transaction cost rates, we find the optimal trading frequency, i.e. the one that attains the maximum of the expected utility at time zero. We study the relation between transaction cost rate and optimal trading frequency. The numerical method used is based on a stochastic volatility particle filtering algorithm, combined with a Monte-Carlo method. The filtering algorithm updates the estimate of the volatility distribution forward in time, as new stock observations arrive; these updates are used at each of these discrete times to compute the new portfolio allocation.