Optimal investment strategies in a certain class of stochastic Merton’s terminal wealth problems

Abstract

The paper studies the classical stochastic Merton’s terminal wealth portfolio problem without transaction costs and also some related problems which may serve as efficient benchmarks for developing relevant analytical techniques as applied to problems with transaction costs. Very few analytical solutions are available for all these types of the Merton’s problem, whereas effective applications of substantially numerical approaches are strongly restricted. Hence, even for quite narrow classes of problems, the development of analytical techniques constructing exact representations for optimal value functions and the corresponding optimal feedback controls is of great theoretical and practical importance. Despite some advancements, sufficiently simple single-input control-affine systems without state constraints and the subject of optimal control synthesis are investigated rather poorly in analytical and qualitative aspects. The present paper is devoted to an entirely analytical construction of the global optimal control synthesis in a certain class of stochastic Merton’s terminal wealth problems. A significant feature of the obtained optimal feedback controls is their capability to be interpreted naturally and adequately from a financial point of view as well as to observe a significant qualitative difference between stochastic and limiting deterministic cases.