We propose a new approach to studying classical solutions of the Bellman equation and Master equation for mean field type control problems, using a novel form of the "lifting" idea introduced by P.-L. Lions. Rather than studying the usual system of Hamilton-Jacobi/Fokker-Planck PDEs using analytic techniques, we instead study a stochastic control problem on a specially constructed Hilbert space, which is reminiscent of a tangent space on the Wasserstein space in optimal transport. On this Hilbert space we can use classical control theory techniques, despite the fact that it is infinite dimensional. A consequence of our construction is that the mean field type control problem appears as a special case. Thus we preserve the advantages of the lifiting procedure, while removing some of the difficulties. Our approach extends previous work by two of the coauthors, which dealt with a deterministic control problem for which the Hilbert space could be generic.
Optimal control of stochastic Volterra integral equations with completely monotone kernels and stochastic differential equations on Hilbert spaces with unbounded control and diffusion operators
Sapienza Università di Roma（罗马大学）；Technische Universität Berlin（柏林工业大学）
A note on convergence in mean for $d$-dimensional arrays of random vectors in Hilbert spaces under the Cesàro uniform integrability
本文证明了在Cesàro一致可积条件下，希尔伯特空间中$d$维随机向量阵列的$p$，$0<p\le 1$阶平均收敛性.在$0<p<1$的情况下，我们的$L_p$收敛是有效的，无论任何依赖结构。在$p=1$的情况下，假设基础随机向量是成对独立的。的平均收敛结果建立最大部分和，而以前的贡献，到目前为止被认为是部分和。推广了文献中的一些结果。各种性质的Cesàro一致可积的$d$维阵列的随机向量，如经典的等价标准和de La Vallée Poussin定理也详细。
