具有可测漂移的随机微分方程随机控制的最大值原理
Maximum principle for stochastic control of SDEs with measurable drifts
摘要：
本文研究了具有非规则漂移系数的随机微分方程驱动系统的随机最优控制问题。建立了随机最大值原理的充要条件。为了实现这一点，我们首先推导出一个明确的表示的一阶变分过程（在Sobolev意义上）的控制扩散。由于漂移系数是不光滑的，因此用状态过程的局部时表示。然后，我们构造了一个序列的最优控制问题的光滑系数的逼近论点。最后，我们利用Ekeland变分原理得到一个近似伴随过程，由此我们通过极限推导出最大值原理。
英文摘要：
In this paper, we consider stochastic optimal control of systems driven by stochastic differential equations with irregular drift coefficient. We establish a necessary and sufficient stochastic maximum principle. To achieve this, we first derive an explicit representation of the first variation process (in Sobolev sense ) of the controlled diffusion. Since the drift coefficient is not smooth, the representation is given in terms of the local time of the state process. Then we construct a sequence of optimal control problems with smooth coefficients by an approximation argument. Finally, we use Ekeland's variational principle to obtain an approximating adjoint process from which we derive the maximum principle by passing to the limit.
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