Impulsive Control of Portfolios

Abstract

In the paper a general model of a market with asset prices and economical factors of Markovian structure is considered. The problem is to find optimal portfolio strategies maximizing a discounted infinite horizon reward functional consisting of an integral term measuring quality of the portfolio at each moment and a discrete term measuring the reward from consumption. There are general transaction costs which, in particular, cover fixed plus proportional costs. It is shown, under general conditions, that there exists an optimal impulse strategy and the value function is a solution to the Bellman equation which corresponds to suitable quasi-variational inequalities.

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

“…Models with economic factors have been gaining popularity in financial mathematics recently although it has been noted that they add substantially to the complicacy of mathematical methods required for their analysis as compared to models without factors (see eg. [3], [4], [13], [23]). A main advantage of models with economic factors lies in the fact that economic factors can influence market trends therefore change the long-term behaviour of prices.…”

Growth-optimal portfolios under transaction costs

“…This type of transaction costs prevents continuous trading in continuous-time models (see e.g. [26]) and emulates existing market mechanisms. The framework of this paper covers more general transaction costs structures as well (see Section 6).…”

“…However, a unified framework has only recenly been introduced and has attracted a lot of interest (see eg. [5], [6], [12], [26], [25], [31]). Existing literature concentrates mainly on continuous-time diffusion models.…”

“…Fleming and Sheu [12] allow both processes to have dependent Brownian motions but their diffusions are of a special form. Palczewski and Stettner [26], though, assume only that asset prices and economic factors follow one general continuous-time Markov process and prove results concerning optimal portfolio selection for infinite time disounted performance functional under transaction costs.…”

On Exact Controllability of First-Order Impulsive Differential Equations

“…In 16 the authors investigated the optimal harvesting policy for an ecosystem with impulsive harvest. For some recent references on different control strategies, including impulsive control, we refer the reader to 13,[19][20][21][22][23][24][25][26] We say that system 1.1 -1.2 is exactly controllable in the time T > 0 if for any x 0 , x 1 there exists a control u such that a solution x of 1.1 -1.2 satisfies x 0 x 0 , x T x 1 . Of course, we specify below the space of solutions and controls.…”