The Merton Problem

Robert Merton opened a new chapter in finance with his two papers (Merton, 1969; Merton, 1971), reprinted in his book (Merton, 1992), on dynamic asset allocation. Aside from taking a decisive step away from Markowitz-style single-period models, these papers made the key link with stochastic control theory. All theories of optimal investment in continuous time developed since then, including the theory described in this book, are direct descendants of these early papers, and it is essential to be familiar with the basic ideas, which we present below in a stripped-down setting: one risky asset, no consumption, finite time horizon. The intention is to give a quick account, with a minimum of notational distractions, of the solution techniques, all of which will reappear in much more general settings later in the book…