On near optimal trajectories for a game associated with the ∞-Laplacian

Abstract

A two-player stochastic differential game representation has recently been obtained for solutions of the equation − ∞ u = h in a C 2 domain with Dirichlet boundary condition, where h is continuous and takes values in R\{0}. Under appropriate assumptions, including smoothness of u, we identify a family of diffusion processes that may arise as the vanishing δ limit law of the state process, when both players play δ-optimally. We also identify the limit law of the state process under a sequence of near saddle points.

A stochastic differential game for the inhomogeneous ∞-Laplace equation

“…Finally, it is natural to ask whether the state process, obtained under δ-optimal play by both players, converges in law as δ tends to zero. Section 5 describes a recently obtained result [3] that addresses this issue.…”

“…A somewhat less ambitious goal, that is the subject of a forthcoming work [3] is the characterization of the limit law of X δ under some choice of a δ-optimal play. The result from [3] states the following. Fix x ∈ G and let X and τ denote such a solution and, respectively, the corresponding exit time from G. Then, given any sequence {δ n } n≥1 , δ n ↓ 0, there exists a sequence of strategy-control pairs (β n , Y n ) ∈ M × Γ, n ≥ 1, with the following properties:…”

“…One may ask whether the law of the process X δ , under an arbitrary δoptimal play (β δ , Y δ ), converges to a limit law as δ → 0; whether this limit law is the same for any choice of such (β δ , Y δ ) pairs; and finally, whether an explicit characterization of this limit law can be provided. A somewhat less ambitious goal, that is the subject of a forthcoming work [3] is the characterization of the limit law of X δ under some choice of a δ-optimal play. The result from [3] states the following.…”

“…A somewhat less ambitious goal, that is the subject of a forthcoming work [3] is the characterization of the limit law of X δ under some choice of a δ-optimal play. The result from [3] states the following. Theorem 5.1.…”

Tug‐of‐war, Market Manipulation, and Option Pricing

“…Here, we also mention the approach based on nonlinear mean value formulas developed in Manfredi et al () and Manfredi, Parviainen, and Rossi (). Continuous time stochastic differential games and infinity harmonic functions were considered in Atar and Budhiraja (), and Atar and Budhiraja (). The equation considered in this paper coincides, modulo the presence of the model related constants and a change of the time direction, with the normalized p ‐Laplace operator considered in Manfredi et al (), in connection with normalized p ‐parabolic equations and tug‐of‐war games, see also Banerjee and Garofalo () and Does ().…”