On the Taylor Expansion of Value Functions

We introduce a framework for approximate dynamic programming that we apply to discrete-time chains on ℤ + d with countable action sets. The framework is grounded in the approximation of the (controlled) chain’s generator by that of another Markov process. In simple terms, our approach stipulates applying a second-order Taylor expansion to the value function, replacing the Bellman equation with one in continuous space and time in which the transition matrix is reduced to its first and second moments. In some cases, the resulting equation can be interpreted as a Hamilton–Jacobi–Bellman equation for a Brownian control problem. When tractable, the “Taylored” equation serves as a useful modeling tool. More generally, it is a starting point for approximation algorithms. We develop bounds on the optimality gap—the suboptimality introduced by using the control produced by the Taylored equation. These bounds can be viewed as a conceptual underpinning, analytical rather than relying on weak convergence arguments, for the good performance of controls derived from Brownian approximations. We prove that under suitable conditions and for suitably “large” initial states, (1) the optimality gap is smaller than a 1 – α fraction of the optimal value, with which α ∈ (0, 1) is the discount factor, and (2) the gap can be further expressed as the infinite-horizon discounted value with a “lower-order” per-period reward. Computationally, our framework leads to an “aggregation” approach with performance guarantees. Although the guarantees are grounded in partial differential equation theory, the practical use of this approach requires no knowledge of that theory.