Parametric concavity in stochastic dynamic programs

We study a family of dynamic programs that are characterized by a deterministic vector of cost parameters. We show that if the single period cost function is concave with respect to this vector, then the optimal costs of the family of dynamic programs are also concave in the vector of costs. We also establish that the optimal cost inherits other properties, namely, super‐additivity, +8‐star‐shaped, 0‐star‐shaped, concavity‐along‐rays and monotonicity. When the vector of cost parameters evolves as a stochastic process and the single period cost is concave with respect to this vector, we show that the optimal cost is bounded above by the optimal cost for the dynamic program in which these stochastic cost parameters are replaced by their expectations in each period. We provide examples to illustrate how our results can be used to derive bounds which are either easy to compute or have analytical expressions. We also explain why such bounds are useful.