PORTFOLIO CHOICE AND THE BAYESIAN-KELLY-CRITERION

We derive optimal gambling and investment policies for cases in which the underlying stochastic process has parameter values that are unobse rved random variables. For the objective of maximizing logarithmic uti lity when the underlying stochastic process is a simple random walk in a random environment, we show that a state-dependent control is optim al, which is a generalization of the celebrated Kelly strategy: the op timal strategy is to bet a fraction of current wealth equal to a linea r function of the posterior mean increment. To approximate more genera l stochastic processes, we consider a continuous-time analog involving Brownian motion. To analyze the continuous-time problem, we study the diffusion limit of random walks in a random environment. We prove tha t they converge weakly to a Kiefer process, or tied-down Brownian shee t. We then find conditions under which the discrete-time process conve rges to a diffusion, and analyze the resulting process. We analyze in detail the case of the natural conjugate prior, where the success prob ability has a beta distribution, and show that the resulting limit dif fusion can be viewed as a rescaled Brownian motion. These results allo w explicit computation of the optimal control policies for the continu ous-time gambling and investment problems without resorting to continu ous-time stochastic-control procedures. Moreover they also allow an ex plicit quantitative evaluation of the financial value of randomness, t he financial gain of perfect information and the financial cost of lea rning in the Bayesian problem.