Stochastic Optimal Control in Infinite Dimension

Dynamic Programming for Optimal Control of Stochastic McKean--Vlasov Dynamics

“…Indeed, comparison principles for viscosity solutions in the Wasserstein space, or more generally in metric spaces, are difficult to obtain as we have to deal with locally non compact spaces (see e.g. [2], [26], [24]), and instead by working in separable Hilbert spaces, one can essentially reduce to the case of Euclidian spaces by projection, and then take advantage of the results developed for viscosity solutions, in particular here, for second order Hamilton-Jacobi-Bellman equations, see [34], [23]. We shall assume that the σ-algebra G is countably generated upto null sets, which ensures that the Hilbert space L 2 (G; R d ) is separable, see [22], p. 92.…”

“…In view of our definition of viscosity solution, we have to show a comparison principle for viscosity solutions to the lifted Bellman equation (4.10). We use the comparison principle proved in Theorem 3.50 in [23] and only need to check that the hypotheses of this theorem are satisfied in our context for the lifted Hamiltonian H defined in (4.11). Notice that the Bellman equation ( We conclude this section with a verification theorem, which gives an analytic feedback form of the optimal control when there is a smooth solution to the Bellman equation (4.7) in the Wasserstein space.…”

“…By adapting standard arguments to our context, we prove the viscosity property of the value function to the Bellman equation from the dynamic programming principle. To complete our PDE characterization of the value function with a uniqueness result, it is convenient to work in the lifted Hilbert space of square integrable random variables instead of the Wasserstein metric space of probability measures, in order to rely on the general results for viscosity solutions of second order Hamilton-Jacobi-Bellman equations in separable Hilbert spaces, see [33], [34], [23]. We also state a verification theorem which is useful for getting an analytic feedback form of the optimal control when there is a smooth solution to the Bellman equation.…”

Necessary and sufficient conditions of risk‐sensitive optimal control and differential games for stochastic differential delayed equations

“…One of the remarkable features of SDDEs is that they allow to model situations where the system depends on current and past history of the state and control variables. Hence, applications of SDDEs can be found in a wide variety of fields, such as economics and portfolio optimization, physics, and power and communication systems …”

“…One is the stochastic maximum principle, in which, unlike the nondelay case, the adjoint equation (the backward SDE) involves the current and future evolution of the solution, which is known as an anticipated backward SDE (ABSDE) . Another approach is dynamic programming, where the Hamilton‐Jacobi‐Bellman (HJB) equation becomes infinite‐dimensional . In some special cases of the SDDE, the corresponding HJB equation reduces to a finite‐dimensional equation .…”