From rank-based models with common noise to pathwise entropy solutions of SPDEs

We study the mean field limit of a rank-based model with common noise, which arises as an extension to models for the market capitalization of firms in stochastic portfolio theory. We show that, under certain conditions on the drift and diffusion coefficients, the empirical cumulative distribution function converges to the solution of a stochastic PDE. A key step in the proof, which is of independent interest, is to show that any solution to an associated martingale problem is also a pathwise entropy solution to the stochastic PDE, a notion introduced in a recent series of papers (Stoch. Partial Differ. Equ. Anal. Comput. 1 (2013) 664–686; Stoch. Partial Differ. Equ. Anal. Comput. 2 (2014) 517–538; Stoch. Partial Differ. Equ. Anal. Comput. 4 (2016) 635–690; Comm. Pure Appl. Math. 70 (2017) 1562–1597; Stochastic Process. Appl. 127 (2017) 2961–3004).