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Generalized solutions of HJB equations applied to stochastic control on Hilbert space. (English) Zbl 1018.93030

A stochastic control problem for a controlled stochastic differential equation (SDE) in Hilbert space is considered.
The existence of Borel measurable optimal controls for the infinite-dimensional nonstationary stochastic control problem in Hilbert \(H\) space, using the theory of measurable selections and the general theory of existence and uniqueness of generalized solutions, is proved.
For the solution of the nonstationary control problems, the author uses some delicate arguments to prove a verification theorem asserting that the generalized solution of the HJB equation is indeed the value function. He deals with a parabolic problem in infinite-dimensional Hilbert space \(H,\) while for the stationary control problem an associated elliptic-type problem in \(H\) needs to be solved.
In fact, the existence of a solution to the elliptic problem implies the existence of its Ito differential along any solution trajectory of the original SDE. In the case of a nonstationary problem, the generalized solutions are not regular enough to admit the Ito differential. The author uses some delicate arguments using regularized problems for which the Ito differential exists.
These results are based on some recent results on the existence of generalized solutions to HJB equations in the Hilbert space \(L_2(H,\mu),\) where \(\mu\) is a suitable reference measure on \(H\).

MSC:

93E20 Optimal stochastic control
49J27 Existence theories for problems in abstract spaces
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