Application of fractional Brownian motion to portfolio optimization: Sharp estimation on almost sure asymptotic behavior of Brownian polymer in fractional Brownian environment
Abstract
In the first part of this thesis 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. The interpretation of the integrals with respect to BH is in the Skorohod sense, not pathwise which is known to lead to arbitrage. We explicitly find the optimal consumption rate and the optimal portfolio in such a market for an agent with logarithmic utility functions. A truly self-financing portfolio is found to lead to a consumption term that is always favorable to the investor. We also present a numerical implementation by Monte-Carlo simulations. The second part of this thesis studies the asymptotic behavior of a one-dimensional directed polymer in a random medium. The latter is represented by a Gaussian field BH on R+ × R which is Riemann-Liouville fractional Brownian motion in time and function-valued in space. The partition function of such a polymer is [special characters omitted]Here b is assumed to be a continuous-time nearest-neighbor random walk on Z with intensity 2κ, defined on a complete probability space independent of BH. The spatial covariance structure of BH is assumed to be homogeneous and periodic with period 2π. For H < ½, we give the existence of the Lyapunov exponent defined as the a.s.-lim t→∞,t ∈N [special characters omitted] log u(t, x), which is deterministic. And moreover, this limit is strictly positive. For H > ½, we give the result on the existence of the a.s.-lim supt →∞,t∈ N [special characters omitted] log u(t, x) and a.s.-lim inf t→∞,t ∈N [special characters omitted] log u(t, x). Both limits are deterministic, finite and strictly positive. As H passes through ½, the exponential behavior of u(t, x) changes abruptly. This can be considered as a phase transition phenomena.
Degree
Ph.D.
Advisors
Viens, Purdue University.
Subject Area
Mathematics
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