Application of fractional Brownian motion to portfolio optimization: Sharp estimation on almost sure asymptotic behavior of Brownian polymer in fractional Brownian environment

Tao Zhang, Purdue University

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 ut,x=E xbexp 0

    t
BH dr,br+x. 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→∞,tN 1t 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 1t2H log u(t, x) and a.s.-lim inf t→∞,tN 1t2H/logt 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

Frederi G. Viens, Purdue University.

Subject Area

Mathematics

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