2007/07/08 by Rainer Buckdahn, Juan Li, Buckdahn, Rainer +1 · 1 citation
Economics, Econometrics and Finance · Engineering · Mathematics · #60H10 #90C39 #93E05 #FOS: Mathematics #Mathematical Biology Tumor Growth #Optimization and Control (math.OC) #Probability (math.PR) #Stability and Controllability of Differential Equations #Stochastic processes and financial applications #math.OC #math.PR #msc:60H10 #msc:90C39 #msc:93E05
paper · pdf · doi:10.48550/arxiv.0707.1133
This paper was submitted. Moreover, lectures about this paper were given by Rainer Buckdahn in the summer school of Yantai in P. R. China in July, 2007
openalex publication_date 2007/07/08 · arxiv created 2008/09/30 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we first investigate zero-sum two-player stochastic differential games with reflection with the help of theory of Reflected Backward Stochastic Differential Equations (RBSDEs). We will establish the dynamic programming principle for the upper and the lower value functions of this kind of stochastic differential games with reflection in a straight-forward way. Then the upper and the lower value functions are proved to be the unique viscosity solutions of the associated upper and the lower Hamilton-Jacobi-Bellman-Isaacs equations with obstacles, respectively. The method differs heavily from those used for control problems with reflection, it has its own techniques and its own interest. On the other hand, we also prove a new estimate for RBSDEs being sharper than that in El Karoui, Kapoudjian, Pardoux, Peng and Quenez [7], which turns out to be very useful because it allows to estimate the Lp-distance of the solutions of two different RBSDEs by the p-th power of the distance of the initial values of the driving forward equations. We also show that the unique viscosity solution of the approximating Isaacs equation which is constructed by the penalization method converges to the viscosity solution of the Isaacs equation with obstacle.