Finite state mean field games with Wright–Fisher common noise

Monotone solutions for mean field games master equations: finite state space and optimal stopping

“…We also recall a weak form of Stegall's variational lemma at the end of this section. Let us mention the work [4], which is also concerned with the well-posedness of the master equation in finite state space. In this article the authors derived and studied a particular form of the master equation arising from the presence of a so-called Wright-Fischer common noise in the MFG.…”

“…In this section we consider a problem which arises from the discretization of a continuous problem, mainly to highlight the links between a master equation on O d and on the simplex. For other examples of master equations in finite state space, we refer to [4,11]. Namely we are interested in the case in which G is given by (31…”

Master Equation for the Finite State Space Planning Problem

“…λ > 0, has been treated in [5]. Let us also note that other type of noise have been studied in [2], which contrary to the noise we study here, do not propagate monotonicity but yield some kind of ellipticity for the master equation.…”

Continuous-Time Mean Field Games with Finite State Space and Common Noise

“…To the best of our knowledge, however, to date there has been no extension of these results to settings that include common noise. In the context of finite-state mean field games, we are only aware of two contributions that include common stochasticity (both via the master equation and with a different focus/setting than this paper): [5] analyze the master equation for finite-state mean field games with common noise, and [3] include a common continuous-time Gaussian noise in the aggregate distribution dynamics.…”

“…In Sect. 3 we state the dynamic programming equation and establish a verification theorem for the agent's optimization, given an ex ante aggregate distribution (Theorem 6). Section 4 provides the dynamics of the ex post distribution (Theorem 9) and, on that basis, a system of random forward-backward ODEs for the mean field equilibrium (Definition 10) as well as corresponding existence and uniqueness results (Theorems 13 and 16 ).…”