On the interpretation of the Master Equation

Since its introduction by P.L. Lions in his lectures and seminars at the College de France, see Lions [6], and also the very helpful notes of Cardialaguet (2013) on Lions’ lectures, the Master Equation has attracted a lot of interest, and various points of view have been expressed, see for example Carmona and Delarue (2014), Bensoussan et al. (2015), Buckdahn et al. [3]. There are several ways to introduce this type of equation; and in those mentioned works, they involve an argument which is a probability measure, while P.L. Lions has recently proposed the alternative idea of working on the Hilbert space of square integrable random variables. Hence writing the equation is an issue; while another issue is its origin. In this article, we discuss all these various aspects. An important reference is the seminar at College de France delivered by P.L. Lions on November 14, 2014.