Optimal control of investments with random yield under transaction costs

This paper solves the problem of optimal control of the level of investment in some asset whose price follows a geometric Brownian motion. Each transaction requires both fixed and proportional transaction costs. It is shown that the model can be generalized for a number of different assets. The general form of optimal control is found and a constructive algorithm for identefication of all parameters of the control is presented using quasi-variational inequalities. The algorithm yields a system of six nonlinear equations. It is shown that optimal control, that depends on four parameters in general, depends on two parameters if fixed transaction costs are zero, and depends on three parameters, if proportional transaction costs are zero. Numerical experiment is used to show how optimal control depends on all parameters of the model. Intergal representation of the value function is found, that may help to determine the optimal control. The case when the response to the control takes place after a constant time period is also studied. The property of the optimal control that it depends on only one state variable is proven. This fact is used to solve the problem with standard methods of quasi-variational inequalities.