50 research outputs found
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 -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