On control of solution of stochastic differential equation in Hilbert space. (Ukrainian, English) Zbl 1026.60081

Knopov, P. S.; Shtatland, E. S.

Teor. Jmovirn. Mat. Stat. 65, 53-59 (2001); translation in Theory Probab. Math. Stat. 65, 61-68 (2002).

Summary: Let \(H\) be a real separable Hilbert space with the inner product \((h_1,h_2)\), and the norm \(\|h\|\) and let \(H_{-}\) be the completed Hilbert space with the norm \(\|h\|_{-}=\|W^{-1/2}h\|\), where \(W\) is a self-adjoint positive definite operator on \(H\). Consider a stochastic differential equation of the form \[ d\xi_t=f(t,\xi,u_t(\xi)) dt+dW_t, \quad \xi_0=0, \quad t\in[0,T], \] where \(\{W_t\), \(t\in[0,T]\}\) is a Wiener process with values from \(H_-\), \(u=\{u_t\), \(t\in[0,T]\}\) is a control function with values from the set of possible control functions \(U\), and \(f(t,x,u)\) is an \(H\)-valued drift function. The problem considered is to find a control \(u^{\ast}\in U\) which minimizes \(E\int_0^TC(t,\xi) dt\), where \(C(t,\xi)\) is a risk function. Under some conditions on the functions \(f(t,x,u)\) and \(C(t,\xi)\) the authors prove the existence of the optimal control \(u^{\ast}\in U\).

MSC:

60H20	Stochastic integral equations
93E20	Optimal stochastic control

Keywords:

Wiener process; separable Hilbert space; optimal control