Portfolio choice and the Bayesian kelly criterion

The authors derive optimal gambling and investment policies for cases in which the underlying stochastic process has parameter values that are unobserved random variables. For the objective of maximizing logarithmic utility when the underlying stochastic process is a simple random walk in a random environment, they show that a state-dependent control is optimal, which is a generalization of the celebrated Kelly strategy: the optimal strategy is to bet a fraction of current wealth equal to a linear function of the posterior mean increment. To approximate more general stochastic processes, the authors consider a continuous-time analog involving Brownian motion. To analyze the continuous-time problem, they study the diffusion limit of random walks in a random environment. The authors prove that they converge weakly to a Kiefer process, or tied-down Brownian sheet. They then find conditions under which the discrete-time process converges to a diffusion, and analyze the resulting process. The authors analyze in detail the case of the natural conjugate prior, where the success probability has a beta distribution, and show that the resulting limit diffusion can be viewed as a rescaled Brownian motion. These results allow explicit computation of the optimal control policies for the continuous-time gambling and investment problems without resorting to continuous-time stochastic-control procedures. Moreover they also allow an explicit quantitative evaluation of the financial value of randomness, the financial gain of perfect information and the financial cost of learning in the Bayesian problem.