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Register a new userOn near optimal trajectories for a game associated with the infty-Laplacian
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A two-player stochastic differential game representation has recently been obtained for solutions of the equation -Delta_infty u=h in a calC^2 domain with Dirichlet boundary condition, where h is continuous and takes values in RRsetminus{0}. Under appropriate assumptions, including smoothness of u, the vanishing delta limit law of the state process, when both players play delta-optimally, is identified as a diffusion process with coefficients given explicitly in terms of derivatives of the function u.
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Given a bounded $mathcaligr{C}^2$ domain $Gsubset{mathbb{R}}^m$, functions $ginmathcaligr{C}(partial G,{mathbb{R}})$ and $hinmathcaligr {C}(bar{G},{mathbb{R}}setminus{0})$, let $u$ denote the unique viscosity solution to the equation $-2Delta_{infty}u=h$ in $G$ with boundary data $g$. We provide a representation for $u$ as the value of a two-player zero-sum stochastic differential game.
We introduce a new non-zero-sum game of optimal stopping with asymmetric information. Given a stochastic process modelling the value of an asset, one player has full access to the information and observes the process completely, while the other player can access it only periodically at independent Poisson arrival times. The first one to stop receives a reward, different for each player, while the other one gets nothing. We study how each player balances the maximisation of gains against the maximisation of the likelihood of stopping before the opponent. In such a setup, driven by a Levy process with positive jumps, we not only prove the existence, but also explicitly construct a Nash equilibrium with values of the game written in terms of the scale function. Numerical illustrations with put-option payoffs are also provided to study the behaviour of the players strategies as well as the quantification of the value of information.
In this paper we prove a Holder regularity estimate for viscosity solutions of inhomogeneous equations governed by the infinite Laplace operator relative to a frame of vector fields.
In this paper, we establish several different characterizations of the vanishing mean oscillation space associated with Neumann Laplacian $Delta_N$, written ${rm VMO}_{Delta_N}(mathbb{R}^n)$. We first describe it with the classical ${rm VMO}(mathbb{R}^n)$ and certain ${rm VMO}$ on the half-spaces. Then we demonstrate that ${rm VMO}_{Delta_N}(mathbb{R}^n)$ is actually ${rm BMO}_{Delta_N}(mathbb{R}^n)$-closure of the space of the smooth functions with compact supports. Beyond that, it can be characterized in terms of compact commutators of Riesz transforms and fractional integral operators associated to the Neumann Laplacian. Additionally, by means of the functional analysis, we obtain the duality between certain ${rm VMO}$ and the corresponding Hardy spaces on the half-spaces. Finally, we present an useful approximation for ${rm BMO}$ functions on the space of homogeneous type, which can be applied to our argument and otherwhere.
This paper investigates a partially observable queueing system with $N$ nodes in which each node has a dedicated arrival stream. There is an extra arrival stream to balance the load of the system by routing its customers to the shortest queue. In addition, a reward-cost structure is considered to analyze customers strategic behaviours. The equilibrium and socially optimal strategies are derived for the partially observable mean field limit model. Then, we show that the strategies obtained from the mean field model are good approximations to the model with finite $N$ nodes. Finally, numerical experiments are provided to compare the equilibrium and socially optimal behaviours, including joining probabilities and social benefits for different system parameters.
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