# 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 *B ^{H}* with Hurst parameter

*H*> ½. The interpretation of the integrals with respect to

*B*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

^{H}*B*on

_{H}**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**exp 0

^{ x}b- t

*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

*B*. The spatial covariance structure of

_{H}*B*is assumed to be homogeneous and periodic with period 2π. For

_{H}*H*< ½, we give the existence of the Lyapunov exponent defined as the a.s.-lim

_{ t}_{→∞,}

_{t}_{ ∈}

**1t log**

_{N}*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 sup

_{t}_{ →∞,}

_{t}_{∈}

**1t**

_{ N}^{2H}log

*u*(

*t, x*) and a.s.-lim inf

_{ t}_{→∞,}

_{t}_{ ∈}

**1t**

_{N}^{2H}/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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