Optimal Control of Diffusion Processes with Terminal Constraint in Law

Abstract

Stochastic optimal control problems with constraints on the probability distribution of the final output are considered. Necessary conditions for optimality in the form of a coupled system of partial differential equations involving a forward Fokker–Planck equation and a backward Hamilton–Jacobi–Bellman equation are proved using convex duality techniques.

Appendix

Since it appears twice in our article and in particular in the proof of Theorem 3.2, we recall the statement of the von Neumann theorem we are using. The statement and proof can be found in Appendix of [36] and in a slightly different setting, in [40].

Theorem A.1

(Von Neumann) Let ${\mathbb {A}}$ and ${\mathbb {B}}$ be convex sets of some vector spaces and suppose that ${\mathbb {B}}$ is endowed with some Hausdorff topology. Let ${\mathcal {L}}$ be a function satisfying:

$$\begin{aligned}&a \rightarrow {\mathcal {L}}(a,b) \hbox { is concave in }{\mathbb {A}} \hbox { for every }b \in {\mathbb {B}}, \\&b \rightarrow {\mathcal {L}}(a,b) \hbox { is convex in }{\mathbb {B}} \hbox { for every }a \in {\mathbb {A}}. \end{aligned}$$

Suppose also that there exist $a_* \in {\mathbb {A}}$ and $C_* > \sup _{a \in {\mathbb {A}} } \inf _{b \in {\mathbb {B}}} {\mathcal {L}}(a,b)$ such that:

$$\begin{aligned}&{\mathbb {B}}_*:= \left\{ b \in {\mathbb {B}}, {\mathcal {L}}(a_*,b) \le C_* \right\} \hbox { is not empty and compact in } {\mathbb {B}}, \\&b \rightarrow {\mathcal {L}}(a,b) \hbox { is lower-semicontinuous in } {\mathbb {B}}_* \hbox { for every } a \in {\mathbb {A}}. \end{aligned}$$

Then,

$$\begin{aligned} \min _{b\in {\mathbb {B}} } \sup _{a \in {\mathbb {A}} } {\mathcal {L}}(a,b) = \sup _{ a \in {\mathbb {A}} } \inf _{b \in {\mathbb {B}} } {\mathcal {L}}(a,b). \end{aligned}$$

Remark A.1

The fact that the infimum in the “$\inf \sup$” problem is in fact a minimum is part of the theorem.