On the convergence of monotone schemes for path-dependent PDE

We propose a reformulation of the convergence theorem of monotone numerical schemes introduced by Zhang and Zhuo for viscosity solutions of path-dependent PDEs, which extends the seminal work of Barles and Souganidis on the viscosity solution of PDE. We prove the convergence theorem under conditions similar to those of the classical theorem in the work of Barles and Souganidis. These conditions are satisfied, to the best of our knowledge, by all classical monotone numerical schemes in the context of stochastic control theory. In particular, the paper provides a unified approach to prove the convergence of numerical schemes for non-Markovian stochastic control problems, second order BSDEs, stochastic differential games etc.Comment: 28 page

Principal-Agent Problem with Common Agency without Communication

In this paper, we consider a problem of contract theory in which several Principals hire a common Agent and we study the model in the continuous time setting. We show that optimal contracts should satisfy some equilibrium conditions and we reduce the optimisation problem of the Principals to a system of coupled Hamilton-Jacobi-Bellman (HJB) equations. We provide conditions ensuring that for risk-neutral Principals, the system of coupled HJB equations admits a solution. Further, we apply our study in a more specific linear-quadratic model where two interacting Principals hire one common Agent. In this continuous time model, we extend the result of Bernheim and Whinston (1986) in which the authors compare the optimal effort of the Agent in a non-cooperative Principals model and that in the aggregate model, by showing that these two optimisations coincide only in the first best case. We also study the sensibility of the optimal effort and the optimal remunerations with respect to appetence parameters and the correlation between the projects

Viscosity Solutions of Fully Nonlinear Elliptic Path Dependent PDEs

This paper introduces a convenient solution space for the uniformly elliptic fully nonlinear path dependent PDEs. It provides a wellposedness result under standard Lipschitz-type assumptions on the nonlinearity and an additional assumption formulated on some partial differential equation defined locally by freezing the path

Large Deviations for Non-Markovian Diffusions and a Path-Dependent Eikonal Equation

This paper provides a large deviation principle for Non-Markovian, Brownian motion driven stochastic differential equations with random coefficients. Similar to Gao and Liu \cite{GL}, this extends the corresponding results collected in Freidlin and Wentzell \cite{FreidlinWentzell}. However, we use a different line of argument, adapting the PDE method of Fleming \cite{Fleming} and Evans and Ishii \cite{EvansIshii} to the path-dependent case, by using backward stochastic differential techniques. Similar to the Markovian case, we obtain a characterization of the action function as the unique bounded solution of a path-dependent version of the Eikonal equation. Finally, we provide an application to the short maturity asymptotics of the implied volatility surface in financial mathematics

Uniform-in-time propagation of chaos for mean field Langevin dynamics

We study the mean field Langevin dynamics and the associated particle system. By assuming the functional convexity of the energy, we obtain the $L^p$-convergence of the marginal distributions towards the unique invariant measure for the mean field dynamics. Furthermore, we prove the uniform-in-time propagation of chaos in both the $L^2$-Wasserstein metric and relative entropy.Comment: 66 pages, 3 figures and 1 table. Contains corrections and enhancements to arXiv:2212.03050v

Comparison of Viscosity Solutions of Semi-linear Path-Dependent PDEs

This paper provides a probabilistic proof of the comparison result for viscosity solutions of path-dependent semilinear PDEs. We consider the notion of viscosity solutions introduced in \cite{EKTZ} which considers as test functions all those smooth processes which are tangent in mean. When restricted to the Markovian case, this definition induces a larger set of test functions, and reduces to the notion of stochastic viscosity solutions analyzed in \cite{BayraktarSirbu1,BayraktarSirbu2}. Our main result takes advantage of this enlargement of the test functions, and provides an easier proof of comparison. This is most remarkable in the context of the linear path-dependent heat equation. As a key ingredient for our methodology, we introduce a notion of punctual differentiation, similar to the corresponding concept in the standard viscosity solutions \cite{CaffarelliCabre}, and we prove that semimartingales are almost everywhere punctually differentiable. This smoothness result can be viewed as the counterpart of the Aleksandroff smoothness result for convex functions. A similar comparison result was established earlier in \cite{EKTZ}. The result of this paper is more general and, more importantly, the arguments that we develop do not rely on any representation of the solution