Interbank lending with benchmark rates: Pareto optima for a class of singular control games

We analyze a class of stochastic differential games of singular control, motivated by the study of a dynamic model of interbank lending with benchmark rates. We describe Pareto optima for this game and show how they may be achieved through the intervention of a regulator, whose policy is a solution to a singular stochastic control problem. Pareto optima are characterized in terms of the solutions to a new class of Skorokhod problems with piecewise-continuous free boundary. Pareto optimal policies are shown to correspond to the enforcement of endogenous bounds on interbank lending rates. Analytical comparison between Pareto optima and Nash equilibria provides insight into the impact of regulatory intervention on the stability of interbank rates.Comment: 31 pages; 1 figur

Tauberian theorem for value functions

For two-person dynamic zero-sum games (both discrete and continuous settings), we investigate the limit of value functions of finite horizon games with long run average cost as the time horizon tends to infinity and the limit of value functions of $\lambda$-discounted games as the discount tends to zero. We prove that the Dynamic Programming Principle for value functions directly leads to the Tauberian Theorem---that the existence of a uniform limit of the value functions for one of the families implies that the other one also uniformly converges to the same limit. No assumptions on strategies are necessary. To this end, we consider a mapping that takes each payoff to the corresponding value function and preserves the sub- and super- optimality principles (the Dynamic Programming Principle). With their aid, we obtain certain inequalities on asymptotics of sub- and super- solutions, which lead to the Tauberian Theorem. In particular, we consider the case of differential games without relying on the existence of the saddle point; a very simple stochastic game model is also considered

A finite-dimensional approximation for partial differential equations on Wasserstein space

This paper presents a finite-dimensional approximation for a class of partial differential equations on the space of probability measures. These equations are satisfied in the sense of viscosity solutions. The main result states the convergence of the viscosity solutions of the finite-dimensional PDE to the viscosity solutions of the PDE on Wasserstein space, provided that uniqueness holds for the latter, and heavily relies on an adaptation of the Barles & Souganidis monotone scheme to our context, as well as on a key precompactness result for semimartingale measures. We illustrate this result with the example of the Hamilton-Jacobi-Bellman and Bellman-Isaacs equations arising in stochastic control and differential games, and propose an extension to the case of path-dependent PDEs