2013/05/22 by Mihai Ŝırbu, Mihai Sîrbu, Sîrbu, Mihai · 3 citations
Computer Science · Decision Sciences · Economics, Econometrics and Finance · Mathematics · #49L20 #49L25 #91A05 #91A15 #Artificial Intelligence in Games #Economic theories and models #FOS: Mathematics #Mathematical Biology Tumor Growth #Optimization and Control (math.OC) #Risk and Portfolio Optimization #Sports Analytics and Performance #Stochastic processes and financial applications #math.OC #msc:49L20 #msc:49L25 #msc:91A05 #msc:91A15
paper · pdf · doi:10.48550/arxiv.1305.5083
to appear in SIAM Journal on Control and Optimization
openalex publication_date 2013/05/22 · arxiv created 2014/02/18 · arxiv updated 2014/02/20 · openalex created_date 2022/08/22 · openalex updated_date 2026/07/28
We develop here the Stochastic Perron Method in the framework of two-player zero-sum differential games. We consider the formulation of the game where both players play, symmetrically, feed-back strategies (as in [CR09] or [PZ12]) as opposed to the Elliott-Kalton formulation prevalent in the literature. The class of feed-back strategies we use is carefully chosen so that the state equation admits strong solutions and the technicalities involved in the Stochastic Perron Method carry through in a rather simple way. More precisely, we define the game over elementary strategies, which are well motivated by intuition. Within this framework, the Stochastic Perron Method produces a viscosity sub-solution of the upper Isaacs equation dominating the upper value of the game, and a viscosity super-solution of the upper Isaacs equation lying below the upper value of the game. Using a viscosity comparison result we obtain that the upper value is the unique and continuous viscosity solution of the upper Isaacs equation. An identical statement holds for the lower value and lower Isaacs equation. A version of the Dynamic Programming Principle is obtained as a by-product. If the Isaacs condition holds, the game has a value over elementary (pure) strategies.