Before revision:  By rewriting the maximization problem in terms of a convex-analytical support function of a weakly-compact set, crucially leveraging on the recent work by Backhoff and Fontbona arXiv:1405.0251 , the previous formulation let us prove the Hadamard directional differentiability of the value function w.r.t. the drift and interest rate parameters, as well as for volatility matrices under a stability condition on their Kernel, and derive explicit expressions for the directional derivatives.
After revision:  By rewriting the maximization problem in terms of a convex-analytical support function of a weakly-compact set, crucially leveraging on the recent work by Backhoff and Fontbona (SIFIN 2016) , the previous formulation let us prove the Hadamard directional differentiability of the value function w.r.t. the drift and interest rate parameters, as well as for volatility matrices under a stability condition on their Kernel, and derive explicit expressions for the directional derivatives.