Dynamic Optimization of Long‐Term Growth Rate for a Portfolio with Transaction Costs and Logarithmic Utility

Abstract

We study the optimal investment policy for an investor who has available one bank account and n risky assets modeled by log-normal diffusions. The objective is to maximize the long-run average growth of wealth for a logarithmic utility function in the presence of proportional transaction costs. This problem is formulated as an ergodic singular stochastic control problem and interpreted as the limit of a discounted control problem for vanishing discount factor. The variational inequalities for the discounted control problem and the limiting ergodic problem are established in the viscosity sense. The ergodic variational inequality is solved by using a numerical algorithm based on policy iterations and multigrid methods. A numerical example is displayed for two risky assets.

Spectral theorem for convex monotone homogeneous maps, and ergodic control

“…In [Ben88], Bensoussan proved uniqueness of the eigenvector (as weak solution of the ergodic HJB equation) under assumptions, which translated in finite dimension imply irreducibility of stochastic matrices P ∈ ∂f (v). Inspired by the results of the present paper, the first author, Sulem and Taksar [AST01] proved uniqueness of the viscosity solution of a special ergodic HJB equation. This yields an example of concrete situation where some non optimal stationary strategies have several final classes, whereas the optimal ones have only one final class (translated to our setting, this means that for some x ∈ R n and P ∈ ∂f (x), P may have several final classes, whereas there exists an eigenvector v such that all elements of ∂f (v) have one final class).…”

Maximization of the portfolio growth rate under fixed and proportional transaction costs

“…They have been widely studied in the context of stochastic control of Markov processes (see [2], [18], [20] and references therein). Financial applications require, however, additional constraints on admissible controls and give rise to a new class of control problems (see [1], [8], [11], [12], [14], [23] for growth-rate optimization problems on finite and infinite time horizons).…”

Growth Optimal Portfolio Selection Under Proportional Transaction Costs with Obligatory Diversification

“…A model with only fixed proportional transaction costs (κ > 0, c ≡ 0) was studied by Morton and Pliska in [10]. Long term growth with only proportional transaction costs was considered in [1]. A simple one asset Black Scholes model with fixed proportional costs plus proportional transaction costs was considered in [5], and the control was restricted to a diversification boundary and the choice of a new portfolio when this boundary was reached.…”

“…space D and a transition matrix P t at time t. Consider also d assets with the ith asset price denoted S i (t) at time t. It is assumed that the evolution of S i (t) is of the form (1) where X i (0) = 0 and X(t) = (X 1 (t), . .…”