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Abstract

In this paper, we study a Hamilton-Jacobi-Bellman (HJB) equation set on the Wasserstein space $\mathcal{P}_2(\mathbb{R}^d)$, with a second order term arising from a purely common noise. We do not assume that the Hamiltonian is convex in the momentum variable, which means that we cannot rely on representation formulas coming from mean field control. In this setting, Gangbo, Mayorga, and \'Swi\k{e}ch showed via viscosity solutions methods that the HJB equation on $\mathcal{P}_2(\mathbb{R}^d)$ can be approximated by a sequence of finite-dimensional HJB equations. Our main contribution is to quantify this convergence result. The proof involves a doubling of variables argument, which leverages the Hilbertian approach of P.L. Lions for HJB equations in the Wasserstein space, rather than working with smooth metrics which have been used to obtain similar results in the presence of idiosyncratic noise. In dimension one, our doubling of variables argument is made relatively simply by the rigid structure of one-dimensional optimal transport, but in higher dimension the argument is significantly more complicated, and relies on some estimates concerning the "simultaneous quantization" of probability measures.