Performance Bounds and Suboptimal Policies for Multi-Period Investment

We consider dynamic trading of a portfolio of assets in discrete periods over a finite time horizon, with arbitrarytime-varying distribution of asset returns. The goal is to maximize the total expected revenue from the portfolio,while respecting constraints on the portfolio such as a required terminal portfolio and leverage and risk limits. Therevenue takes into account the gross cash generated in trades, transaction costs, and costs associated with thepositions, such as fees for holding short positions. Our model has the form of a stochastic control problem with lineardynamics and convex cost function and constraints. While this problem can be tractably solved in several special cases,such as when all costs are convex quadratic, or when there are no transaction costs, our focus is on the more generalcase, with nonquadratic cost terms and transaction costs.We show how to use linear matrix inequality techniques and semidefinite programming to produce a quadratic bound onthe value function, which in turn gives a bound on the optimal performance. This performance bound can be used to judgethe performance obtained by any suboptimal policy. As a by–product of the performance bound computation, we obtain anapproximate dynamic programming policy that requires the solution of a convex optimization problem, often a quadraticprogram, to determine the trades to carry out in each step. While we have no theoretical guarantee that the performanceof our suboptimal policy is always near the performance bound (which would imply that it is nearly optimal) we observethat in numerical examples the two values are typically close.