16-FEB-2025
We develop a rigorous treatment of Feller processes as pseudo-differential operators and apply the resulting symbol calculus to stochastic optimal control. Starting from the Courège–Lévy–Khintchine representation theorem, we define the symbol $q(x,\xi)$ of a Feller generator as the position-dependent analogue of the Lévy exponent, and prove that $\xi \mapsto q(x,\xi)$ is a continuous negative definite function for each $x$. We then introduce controlled Feller processes, in which the full Lévy characteristics $(b(x,u), a(x,u), \nu(x,u,\cdot))$ depend on a control parameter $u$, and establish three principal results: (i) the HJB Hamiltonian $\mathcal{H}$ equals the infimum over $u$ of the controlled symbol evaluated at the gradient of the value function (gradient-symbol identity); (ii) the optimal symbol $q^*(x,\xi) = \inf_{u \in U} q^u(x,\xi)$ preserves the Lévy–Khintchine structure whenever $U$ is convex and the infimum is attained; (iii) a conservativeness criterion for the optimally controlled process stated directly in terms of symbol growth. We conclude by showing that the Blumenthal–Getoor index of $q^*$ governs the local Sobolev regularity of the value function, providing a spatial profile of HJB regularity through the optimal symbol.
