This paper studies optimal control for an infinite horizon cash management problem where the cash fund fluctuates as a Brownian motion. Holding-penalty costs are assumed to be a quadratic function of the cash level and there are fixed and proportional transaction costs. Using the "impulse technique", we prove that optimal control exists and takes the form of a control band policy. (2000): 93E20, 49K45
Mathematics Subject Classification
The aim of this work is to investigate a portfolio optimization problem in presence of fixed transaction costs. We consider an economy with two assets: one risky, modeled by a geometric Brownian motion, and one risk-free which grows at a certain fixed rate. The agent is fully described by his/her utility function and the objective is to maximize the expected utility from the liquidation of wealth at a terminal date. We deal with different forms of utility functions (power, logarithmic and exponential utility), describing in each case how the fixed transaction costs influence the agent's behavior. We show when it is optimal to recalibrate his/her portfolio and which are the best adjusted portfolios. We also analyze how the optimal strategy is influenced by the risk-aversion, as well as other model parameters.
In this paper we consider the optimal impulse control of a system which evolves randomly in accordance with a homogeneous diffusion process in ℜ 1. Whenever the system is controlled a cost is incurred which has a fixed component and a component which increases with the magnitude of the control applied. In addition to these controlling costs there are holding or carrying costs which are a positive function of the state of the system. Our objective is to minimize the expected discounted value of all costs over an infinite planning horizon. Under general assumptions on the cost functions we show that the value function is a weak solution of a quasi-variational inequality and we deduce from this solution the existence of an optimal impulse policy. The computation of the value function is performed by means of the Finite Element Method on suitable truncated domains, whose convergence is discussed. Copyright Springer-Verlag Berlin/Heidelberg 2006Impulse control, stochastic cash management, quasi-variational inequalities, finite element approximation,
We consider a passive investor who divides his capital between two assets: a risk-free money market instrument and an index fund, or ETF, tracking a broad market index. We model the evolution of the market index by a lognormal diffusion. The agent faces both fixed and proportional transaction costs and solvency constraints. The objective is to maximize the expected utility from the portfolio liquidation at a fixed horizon but if the portfolio reaches a pre-set target value then the position in the risky asset is liquidated. The model is formulated as a parabolic impulse control problem and we characterize the value function as the unique constrained viscosity solution of the associated quasi-variational inequality. We show the existence of an impulse policy which is arbitrarily close to the optimal one by reducing the model to a sequence of iterated optimal stopping problems. The value function and the quasi-optimal policy are computed numerically by an iterative finite element discretization technique. We present extended numerical results in the case of a CRRA utility function, showing the non-stationary shape of the optimal strategy and how it varies with respect to the model parameters. The numerical experiments reveal that, even with small transaction costs and distant horizons, the optimal strategy is essentially a buy-and-hold trading strategy where the agent recalibrates his portfolio very few times.
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