Differentiability and Convexity of the Solution Maps for Fully Non-linear Parabolic HJB Equations

Abstract

We study the dependence on the terminal condition of solutions to a fully nonlinear parabolic Hamilton--Jacobi--Bellman (HJB) equation, when the terminal condition varies along a finite-dimensional family. This problem is motivated by a dual formulation of semimartingale optimal transport. The equation is posed as a terminal-value Cauchy problem on an unbounded spatial domain. Under structural ellipticity and regularity assumptions on the nonlinear operator, we recall local parabolic Hölder estimates and stability results for bounded uniformly continuous terminal data. We then establish continuous directional differentiability of the solution map with respect to the finite-dimensional parameter, and identify its derivative through the corresponding linearized Cauchy problem. Under an additional convexity assumption on the Hamiltonian and a non-degeneracy condition on the terminal data, we prove local strong convexity of the point-evaluated solution map. The analysis relies on Evans--Krylov theory for fully nonlinear parabolic equations and combines PDE arguments with probabilistic representations. Finally, we extend the differentiability and convexity results to cascades of Cauchy problems in which successive equations are coupled through terminal conditions at several prescribed times. This structure is motivated, for instance, by optimal transport problems with marginal constraints imposed at several maturities.