New Approach to Stochastic Optimal Control

Abstract

This paper provides new insights into the solution of optimal stochastic control problems by means of a system of partial differential equations, which characterize directly the optimal control. This new system is obtained by the application of the stochastic maximum principle at every initial condition, assuming that the optimal controls are smooth enough. The type of problems considered are those where the diffusion coefficient is independent of the control variables, which are supposed to be interior to the control region.

Certainty equivalence principle in stochastic differential games: An inverse problem approach

“…Proof. It is clear that i , as defined in (18), is twice continuously differentiable, for i = 1, … , N. Taking the derivative in (18) with respect to u i , we have…”

“…where C is an arbitrary constant. This is a candidate for robust MPNE, for the game with the utility function i as given in (18). We will use this formula in Section 4.2 below.…”

Euler–Lagrange equations of stochastic differential games: application to a game of a productive asset

“…In Martín-Herrán and Rincón-Zapatero (2005), the EL equations are used to identify games of fishery where the MPNE is Pareto optimal. The subsequent paper Josa-Fombellida and Rincón-Zapatero (2007) focuses on a stochastic control problem with Brownian uncertainty, where the players' decisions cannot affect the size of the uncertainty. 3 The present paper extends this methodology to the game framework.…”

On a PDE Arising in One-Dimensional Stochastic Control Problems

“…Though the initial idea of obtaining a system of PDEs for the optimal control appears in [1] in connection with deterministic control problems, the main antecedents of this paper are: [2] and [3] in deterministic differential games; [4], in stochastic control problems, where the diffusion parameter of the state process is independent of the control variables; [5] in the Merton problem; and [6] in a model of optimal liquidation in illiquid markets. In all these papers, the use of the PDE for optimal control has proved to be useful.…”

“…The relationship between the new PDEs and the HJB equation is shown in Sect. 4, and a sufficient optimality condition is given in terms of a verification theorem in [7]. Section 5 contains applications of the theory to linear models in the dynamics.…”