We first analyse the parameter sensitivity of solutions to parametric linear parabolic partial differential equations. Then, we establish a Schauder-type estimate, which contains quantitative information on the Hölder continuity of solutions. Thanks to a certain compactness given by the estimate to the class of possible solutions, we use a linearization procedure and the iterative method of Banach's fixed-point theorem to show the existence and uniqueness of the PDE of our interest. By studying the nonlocal equation, the results can be used to address an open problem of time-inconsistent stochastic control problems, i.e. the well-posedness of the equilibrium controls or the extended HJB system.

This study is inspired by time-inconsistent stochastic control problems. One can also find its significance and relevance in stochastic differential equation (e.g. approximating backward stochastic Volterra integral equations by a sequence of backward stochastic differential equations) and in behavioural economics and finance (e.g. characterizing the reference dependence in prospect and regret theories), to name a few.