Parameter sensitivity analysis for mean field games of production

We study a mean field game system introduced by Chan and Sircar (AMO, 2015) to model production of an exhaustible resource. In particular, we study the sensitivity of the solution with respect to a parameter $\varepsilon$, which measures the degree to which producers are interchangeable. We prove that on some interval $[0,\varepsilon_0]$, where $\varepsilon_0 > 0$, the solution is infinitely differentiable with respect to $\varepsilon$. The result is based on a set of new a priori estimates for forward-backward systems of linear partial differential equations.