论文标题
叠加和模仿有条件的McKean-Vlasov方程的定理
Superposition and mimicking theorems for conditional McKean-Vlasov equations
论文作者
论文摘要
我们考虑有条件的McKean-Vlasov随机微分方程（SDE），例如在存在共同噪声时具有平均野外相互作用的平均野外游戏和粒子系统的大型系统限制中产生的条件。这些SDE的解决方案的条件时间界限满足了第二阶的非线性随机偏微分方程（SPDE），而条件时间 - 分支机构的定律遵循概率测量空间的Fokker-Planck方程。我们证明了两个叠加原理：第一个确定SPDE的任何解决方案都可以提升为条件McKean-Vlasov SDE的解决方案，并且第二个可以保证在概率度量空间上的Fokker-Planck方程的任何解决方案都可以将其提升为SPDE溶液。我们使用这些结果来获得模仿定理，该定理表明，ITO过程的条件时间界限可以由有条件的McKean-Vlasov SDE的解决方案模仿具有Markovian系数的条件。尤其是在受控的McKean-Vlasov Dynamics的背景下，将开环控制转换为马尔可夫的工具。
We consider conditional McKean-Vlasov stochastic differential equations (SDEs), such as the ones arising in the large-system limit of mean field games and particle systems with mean field interactions when common noise is present. The conditional time-marginals of the solutions to these SDEs satisfy non-linear stochastic partial differential equations (SPDEs) of the second order, whereas the laws of the conditional time-marginals follow Fokker-Planck equations on the space of probability measures. We prove two superposition principles: The first establishes that any solution of the SPDE can be lifted to a solution of the conditional McKean-Vlasov SDE, and the second guarantees that any solution of the Fokker-Planck equation on the space of probability measures can be lifted to a solution of the SPDE. We use these results to obtain a mimicking theorem which shows that the conditional time-marginals of an Ito process can be emulated by those of a solution to a conditional McKean-Vlasov SDE with Markovian coefficients. This yields, in particular, a tool for converting open-loop controls into Markovian ones in the context of controlled McKean-Vlasov dynamics.
