A Maximum Principle for SDEs of Mean-Field Type
A Maximum Principle for SDEs of Mean-Field Type
复制标题
DOI：
10.1007/s00245-010-9123-8
复制
发表时间：
2011-06
影响因子：
1.8
通讯作者：
Daniel Andersson;Boualem Djehiche
中科院分区：
文献类型：
--
作者：
Daniel Andersson;Boualem Djehiche
研究了一类平均场型随机微分方程的最优控制问题，其中允许系数依赖于律的某些泛函以及过程的状态。此外，成本功能也是平均场型，这使得控制问题的时间不一致的意义上，贝尔曼最优性原则不成立。在凸作用空间的假设下，导出了局部形式的极大值原理，给出了最优性的必要条件。这些也被证明是足够的额外的假设。这个最大值原理不同于经典的最大值原理，经典的最大值原理中伴随方程是一个线性后向场，因为这里的伴随方程是一个线性平均场后向场。作为一个例子，我们将结果应用到均值-方差投资组合选择问题。
We study the optimal control of a stochastic differential equation (SDE) of mean-field type, where the coefficients are allowed to depend on some functional of the law as well as the state of the process. Moreover the cost functional is also of mean-field type, which makes the control problem time inconsistent in the sense that the Bellman optimality principle does not hold. Under the assumption of a convex action space a maximum principle of local form is derived, specifying the necessary conditions for optimality. These are also shown to be sufficient under additional assumptions. This maximum principle differs from the classical one, where the adjoint equation is a linear backward SDE, since here the adjoint equation turns out to be a linear mean-field backward SDE. As an illustration, we apply the result to the mean-variance portfolio selection problem.