Max-plus probabilistic approach to problems related to stochasitic control

将传统的加法和乘法分别用极大值和实数加法代替，得到了极大加法代数。将极大代数应用于连续时间和连续状态控制问题，得到了随机控制和确定性控制中动态规划偏微分方程解的结构和值函数的刻划的新结果。这些结果是基于最大加法得出的。我们还用极大概率的观点建立了数学金融中的最优消费和最优投资问题，刻画了最优值。

Max-plus algebra is obtained by replacing the conventional addition and multiplication with maximum and the addition in reals, respectively. Using max-plus algebra in continuous-time and continuum-state control problems, we obtained new results on structures of solutions and characterizations of value functions for dynamic programming partial differential equations in stochastic and deterministic controls. These results are based on max-plus methodology. We also formulated optimal consumption and investment problems in mathematical finance by max-plus probabilistic point of view and characterized optimal values.