ERC ELISA Online Seminar</p>

Title: On the optimal rate for the convergence problem in mean-field control

Abstract: In this talk I will present a recent joint work with François Delarue and Joe Jackson about the convergence problem in mean-field control theory. The goal is to obtain optimal rates for the convergence of the value functions associated to the different problems, in general situations where no structural condition (convexity, displacement convexity) ensures the uniqueness and stability of optimal solutions for the limiting problem. In particular we don’t expect propagation of chaos for the particle system and the value function associated to the limiting problem might not be differentiable. Our main result is to derive sharp rates of convergence in two distinct regimes. When the data is sufficiently regular we obtain rates proportional to N^(-1/2), N being the number of particles. When the data is merely Lipschitz and semi-concave with respect to the first Wasserstein distance we obtain rates proportional to N^{-2/3d}, close to the optimal rates for uncontrolled particle systems. Although one inequality between the value functions follows by classical control theoretic argument, the other one proves more difficult. Our strategy consists then in several mollifying arguments to produce approximations of the value function for the mean-field problem which are almost classical sub-solutions to the dynamic programming equation and conclude by a comparison argument. We also provide some examples to show that our results are sharp.

Title: Approximately optimal distributed stochastic controls beyond the mean field setting

Abstract: In this talk I will present a recent joint work with Daniel Lacker, in which we study a class of high-dimensional stochastic optimal control problems. We consider both the full-information problem, in which each agent observes the states of all other agents, and the distributed problem, in which each agent observes only its own state. Our main results are sharp non-asymptotic bounds on the gap between these two problems, measured both in terms of their value functions and optimal states. Along the way, we develop theory for distributed optimal stochastic control in parallel with the classical setting, by characterizing optimizers in terms of an associated stochastic maximum principle and a Hamilton-Jacobi-type equation. Our original motivation for this project comes from mean field control, and by specializing our results to the mean field setting we obtain the optimal rate of convergence for displacement convex data. We also apply our results to a control problem with "heterogeneous doubly stochastic interactions" and derive conditions under which the usual mean field limit is a good approximation, despite the asymmetry in the model.

Abstract: We present a model of a financial market where a large number of firms determine their dynamic emission strategies under climate transition risk in the presence of both green-minded and neutral investors. The firms aim to achieve a trade-off between financial and environmental performance, while interacting through the stochastic discount factor, determined in equilibrium by the investors’ allocations. We formalize the problem in the setting of mean-field games and prove the existence and uniqueness of a Nash equilibrium for firms. We then present a convergent numerical algorithm for computing this equilibrium and illustrate the impact of climate transition risk and the presence of green-minded investors on the market decarbonization dynamics and share prices. We show that uncertainty about future climate risks and policies leads to higher overall emissions and higher spreads between share prices of green and brown companies. This effect is partially reversed in the presence of environmentally concerned investors, whose impact on the cost of capital spurs companies to reduce emissions.. Joint work with Peter Tankov. See arXiv submission.

Title: McKean--Vlasov equations with rough common noise and quenched propagation of chaos

Abstract: We show well-posedness and propagation of chaos for McKean–Vlasov equations with rough common noise and progressively measurable coefficients. Our results are valid under minimal regularity assumptions on the coefficients, in agreement with the respective requirements of Itô and rough path theory. To achieve these goals, we work in framework of rough stochastic differential equations. Joint work with Peter Friz and Antoine Hocquet

ERC ELISA Online Probability Seminar