Applications of the Gaussian stochastic analysis of fractional Brownian noise to regularity of stochastic heat equations and to portfolio optimization
Two applications of the fractional Brownian motion will be presented. First, we study the time-regularity of the paths of solutions to stochastic heat equations driven by additive infinite-dimensional fractional Brownian noise. Sharp sufficient conditions for almost-sure Hölder continuity, and other, more irregular levels of uniform continuity, are given when the space parameter is fixed. Additionally, a result is included on time-continuity when the solution is understood as a spatially Hölder-continuous-function-valued stochastic process. Tools used for the study include the Brownian representation of fractional Brownian motion, and sharp Gaussian regularity results. ^ Next, we consider the classical Merton problem of finding the optimal consumption rate and the optimal portfolio in a Black-Scholes market driven by fractional Brownian motion BH with Hurst parameter H ∈ (½,1). The interpretation of the integrals with respect to BH is in the Skorohod sense. By using logarithmic utility, we derive formulas for the optimal consumption rate and the optimal portfolio explicitly in the sense that the randomness in these formulas are only given by Wiener integrals with respect to fractional Brownian motion. Therefore, our results stand a good chance of implementation. ^
Major Professor: Frederi G. Viens, Purdue University.