Differentiability and Convexity of Solution Maps for Fully Nonlinear Parabolic HJB Equations

This study focuses on the dependence of solutions to fully nonlinear parabolic Hamilton-Jacobi-Bellman (HJB) equations on finite-dimensional terminal conditions, motivated by the dual formulation of optimal transport for semimartingales. Based on Evans-Krylov theory and combining PDE arguments with probabilistic representation methods, the study proves continuous directional differentiability and local strong convexity under point evaluation of the solution map, and extends the results to cascaded Cauchy problems with multiple coupled terminal times.