Regularity Properties for General HJB Equations: A Backward Stochastic Differential Equation Method
Regularity Properties for General HJB Equations: A Backward Stochastic Differential Equation Method
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DOI：
10.1137/110828629
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发表时间：
2012-06
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影响因子：
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通讯作者：
R. Buckdahn;Jianhui Huang;Juan Li
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文献类型：
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作者：
R. Buckdahn;Jianhui Huang;Juan Li
在这项工作中，我们研究了一大类有障碍物或无障碍物的 Hamilton-Jacobi-Bellman (HJB) 方程的正则性质，这些方程可以以随机控制系统的形式进行随机解释，其中非线性成本函数借助后向随机微分方程 (BSDE) 或反射 BSDE 来定义。更准确地说，我们证明，首先，HJB 方程在时间间隔 (0,T) 上的唯一粘度解 V (t, x)，有或没有障碍物，并且在时间 T 处具有终端条件，在 (t, x ) 中联合 Lipschitz 运行 (0,T) 的任何紧凑子区间。其次，对于 V 在没有障碍物或有上部障碍物的情况下求解 HJB 方程的情况，在适当的假设下显示 V (t, x )i 在 (t, x) 中是联合半凹的。这些结果扩展了 Buckdahn、Cannarsa 和 Quincampoix 的早期结果（Nonlinear Differential Equations Appl., 17 (2010), pp. 715-728）。我们的方法将他们的时间变化想法嵌入到 BSDE 分析中。我们还提供了一个基本的反例，它表明，一般来说，对于 V 求解具有较低障碍的 HJB 方程的情况，半凹性不成立。
In this work we investigate regularity properties of a large class of Hamilton-Jacobi- Bellman (HJB) equations with or without obstacles, which can be stochastically interpreted in the form of a stochastic control system in which nonlinear cost functional is defined with the help of a backward stochastic differential equation (BSDE) or a reflected BSDE. More precisely, we prove that, first, the unique viscosity solution V (t, x) of an HJB equation over the time interval (0,T), with or without an obstacle, and with terminal condition at time T , is jointly Lipschitz in (t, x )f ort running any compact subinterval of (0,T). Second, for the case that V solves an HJB equation without an obstacle or with an upper obstacle it is shown under appropriate assumptions that V (t, x )i s jointly semiconcave in (t, x). These results extend earlier ones by Buckdahn, Cannarsa, and Quincampoix (Nonlinear Differential Equations Appl., 17 (2010), pp. 715-728). Our approach embeds their idea of time change into a BSDE analysis. We also provide an elementary counterexample which shows that, in general, for the case that V solves an HJB equation with a lower obstacle the semiconcavity doesn't hold true.