3,000 research outputs found
A Viscosity Solution Theory of Stochastic Hamilton-Jacobi-Bellman equations in the Wasserstein Space
This paper is devoted to a viscosity solution theory of the stochastic
Hamilton-Jacobi-Bellman equation in the Wasserstein spaces for the mean-field
type control problem which allows for random coefficients and may thus be
non-Markovian. The value function of the control problem is proven to be the
unique viscosity solution. The major challenge lies in the mixture of the lack
of local compactness of the Wasserstein spaces and the non-Markovian setting
with random coefficients and various techniques are used, including Ito
processes parameterized by random measures, the conditional law invariance of
the value function, a novel tailor-made compact subset of measure-valued
processes, finite dimensional approximations via stochastic n-player
differential games with common noises, and so on.Comment: 41 page
Nash equilibria for non zero-sum ergodic stochastic differential games
In this paper we consider non zero-sum games where multiple players control
the drift of a process, and their payoffs depend on its ergodic behaviour. We
establish their connection with systems of Ergodic BSDEs, and prove the
existence of a Nash equilibrium under the generalised Isaac's conditions. We
also study the case of interacting players of different type
Reinforcement learning with restrictions on the action set
Consider a 2-player normal-form game repeated over time. We introduce an
adaptive learning procedure, where the players only observe their own realized
payoff at each stage. We assume that agents do not know their own payoff
function, and have no information on the other player. Furthermore, we assume
that they have restrictions on their own action set such that, at each stage,
their choice is limited to a subset of their action set. We prove that the
empirical distributions of play converge to the set of Nash equilibria for
zero-sum and potential games, and games where one player has two actions.Comment: 28 page
Mean field games with controlled jump-diffusion dynamics: Existence results and an illiquid interbank market model
We study a family of mean field games with a state variable evolving as a
multivariate jump diffusion process. The jump component is driven by a Poisson
process with a time-dependent intensity function. All coefficients, i.e. drift,
volatility and jump size, are controlled. Under fairly general conditions, we
establish existence of a solution in a relaxed version of the mean field game
and give conditions under which the optimal strategies are in fact Markovian,
hence extending to a jump-diffusion setting previous results established in
[30]. The proofs rely upon the notions of relaxed controls and martingale
problems. Finally, to complement the abstract existence results, we study a
simple illiquid inter-bank market model, where the banks can change their
reserves only at the jump times of some exogenous Poisson processes with a
common constant intensity, and provide some numerical results.Comment: 37 pages, 6 figure
Stochastic Target Games and Dynamic Programming via Regularized Viscosity Solutions
We study a class of stochastic target games where one player tries to find a
strategy such that the state process almost-surely reaches a given target, no
matter which action is chosen by the opponent. Our main result is a geometric
dynamic programming principle which allows us to characterize the value
function as the viscosity solution of a non-linear partial differential
equation. Because abstract mea-surable selection arguments cannot be used in
this context, the main obstacle is the construction of measurable
almost-optimal strategies. We propose a novel approach where smooth
supersolutions are used to define almost-optimal strategies of Markovian type,
similarly as in ver-ification arguments for classical solutions of
Hamilton--Jacobi--Bellman equations. The smooth supersolutions are constructed
by an exten-sion of Krylov's method of shaken coefficients. We apply our
results to a problem of option pricing under model uncertainty with different
interest rates for borrowing and lending.Comment: To appear in MO
- …