Growth Optimal Portfolio Selection Under Proportional Transaction Costs with Obligatory Diversification

Abstract

A continuous time long run growth optimal or optimal logarithmic utility portfolio with proportional transaction costs consisting of a fixed proportional cost and a cost proportional to the volume of transaction is considered. The asset prices are modeled as exponent of diffusion with jumps whose parameters depend on a finite state Markov process of economic factors. An obligatory portfolio diversification is introduced, accordingly to which it is required to invest at least a fixed small portion of our wealth in each asset.

Impulse Control Maximizing Average Cost per Unit Time: A Nonuniformly Ergodic Case

“…By (7) also lim α→0 b α = 0 for any fixed R. By (A1) and [11, Proposition 2.1] taking into account integrability of K(X t ) we obtain that c R → 0 as R → ∞. Consequently, q(X t ) is integrable and q(x) = E x T 0 (f (X t ) − µ(f ))dt + q(X T ) , from which it follows that Z s = s 0 (f (X t ) − µ(f ))dt + q(X s ) is a martingale and we immediately have (5).…”

“…as T → ∞ provided that lim T →∞ sup x∈Γ P x {A T } = 0. It remains to show (5). For α > 0 and T ≥ 0 we have q α (x) = E x T 0 e −αt (f (X t ) − µ(f ))dt + e −αT q α (X T ) .…”

“…Controlling random systems by impulses, i.e., discrete interventions, is often the only feasible strategy from an application point of view and, therefore, the literature is extensive. For applications in finance, the reader is referred to [5,10] and references therein. Intensive studies of impulse control of diffusions and diffusions with jumps are presented in [4].…”

Optimal relaxed portfolio strategies for growth rate maximization problems with transaction costs

“…Unfortunately, this strategy is still not realizable in practice, since it involves continuous trading at the boundaries. One way out is to introduce artificial fixed transaction costs, which punish high frequent trading, see [Morton & Pliska, 1995], [Korn, 1998], [Øksendal & Sulem, 2002], [Irle & Sass, 2006a, Irle & Sass, 2006b, [Tamura, 2006, Tamura, 2008, and [Duncan et al, 2011]. The advantage of using fixed transaction costs that are proportional to the investors wealth is that the optimal strategies turn out to be realizable and easy to describe: They are determined by four parameters a < α ≤ β < b.…”

On an approximation of average cost per unit time impulse control of Markov processes

“…This kind of cost functional is also important in mathematics of finance under the name of Kelly criterion -see [16]. In particular impulse control appear in the case of growth optimal portfolio under proportional transaction costs in [10] and [7] and references there in. Average cost per unit time impulse control in a general (locally compact separable) state space was considered in [21].…”