Optimal Control and Zero-Sum Stochastic Differential Game Problems of Mean-Field Type

Abstract

We establish existence of nearly-optimal controls, conditions for existence of an optimal control and a saddle-point for respectively a control problem and zero-sum differential game associated with payoff functionals of mean-field type, under dynamics driven by weak solutions of stochastic differential equations of mean-field type.

Appendix

For the sake of completeness, we display a poof of the fact that the set of probability measures ${\mathcal {P}}(\Omega )$ endowed with the total variation metric $D_T$ defined on $(\Omega ,\mathcal {F}_T)$ by

$$\begin{aligned} D_T(P,Q):=2\sup _{A\in \mathcal {F}_T}|P(A)-Q(A)| \end{aligned}$$

(5.20)

is complete. Indeed, let $(Q_n)_{n\ge 0}$ be a Cauchy sequence for $D_T$. Then , for each set $A\in \mathcal {F}_T$, the sequence $(Q_n(A))_{n\ge 0}$ is a Cauchy sequence in $\mathbb {R}$ and thus is a convergent sequence. By the Vitali-Hahn-Saks-Nikodym Theorem, the set-function Q defined on $(\Omega ,\mathcal {F}_T)$ by

$$\begin{aligned} Q(A):=\lim _{n\rightarrow \infty } Q_n(A),\quad A\in \mathcal {F}_T, \end{aligned}$$

is indeed a probability measure.

We will now show that $D_T(Q_n,Q)\rightarrow 0$. Given $\varepsilon >0$, there exists an integer $n_0$ such that if $m,n>n_0$ and $A\in \mathcal {F}_T$, such that

$$\begin{aligned} |Q_n(A)-Q_m(A)|< \varepsilon . \end{aligned}$$

Sending m to infinity, we obtain

$$\begin{aligned} |Q_n(A)-Q(A)|\le \varepsilon . \end{aligned}$$

Now taking the supremum over all $A\in \mathcal {F}_T$, we finally get that $D_T(Q_n,Q)\rightarrow 0$, as $n\rightarrow \infty$.