Stochastic non-Markovian differential games and mean-field Langevin dynamics.

Summary This thesis is composed of two independent parts, the first one grouping two distinct problems. In the first part, we first study the Principal-Agent problem in degenerate systems, which arise naturally in partially observed environments where the Agent and the Principal observe only a part of the system. We present an approach based on the stochastic maximum principle, which aims to extend existing work that uses the principle of dynamic programming in non-degenerate systems. First, we solve the Principal problem in an extended contract set given by the first-order condition of the Agent problem in the form of a path-dependent stochastic differential equation (EDSPR). Then we use the sufficient condition of the Agent problem to verify that the obtained optimal contract is implementable. A parallel study is devoted to the existence and uniqueness of the solution of path-dependent EDSPRs in Chapter IV. We extend the decoupling field method to cases where the coefficients of the equations can depend on the trajectory of the forward process. We also prove a stability property for such EDSPRs. Finally, we study the moral hazard problem with several Principals. The Agent can only work for one Principal at a time and thus faces an optimal switching problem. Using the randomization method we show that the Agent's value function and its optimal effort are given by an Itô process. This representation helps us to solve the Principal problem when there are infinitely many Principals in equilibrium according to a mean-field game. We justify the mean-field formulation by a chaos propagation argument.The second part of this thesis consists of chapters V and VI. The motivation of this work is to give a rigorous theoretical foundation for the convergence of gradient descent type algorithms which are often used in the solution of non-convex problems such as the calibration of a neural network. For non-convex problems of the hidden layer neural network type, the key idea is to transform the problem into a convex problem by raising it in the space of measurements. We show that the corresponding energy function admits a unique minimizer which can be characterized by a first order condition using the derivation in the space of measures in the sense of Lions. We then present an analysis of the long term behavior of the Langevin mean-field dynamics, which has a gradient flow structure in the 2-Wasserstein metric. We show that the marginal law flow induced by the mean-field Langevin dynamics converges to a stationary law using La Salle's invariance principle, which is the minimizer of the energy function.In the case of deep neural networks, we model them using a continuous-time optimal control problem. We first give the first order condition using Pontryagin's principle, which will then help us to introduce the system of mean-field Langevin equations, whose invariant measure corresponds to the minimizer of the optimal control problem. Finally, with the reflection coupling method we show that the marginal law of the mean-field Langevin system converges to the invariant measure with an exponential speed.