On a linearly quadratic problem of optimal control of a stochastic Volterra equation (Q757821)

We consider the optimal control problem of the linear stochastic Volterra equation \[ (1)\quad \xi (t)=\eta (t)\quad +\int^{t}_{0}a_ 0(t,s)u(s)ds\quad +\int^{t}_{0}a_ 1(t,s)\xi (s)ds \] with the quadratic performance criterion \[ (2)\quad J(u)\quad =\quad M[\xi '(T)H\xi (T)\quad +\int^{T}_{0}(\xi '(s)F(s)\xi (s)\quad +\quad u'(s)N(s)u(s))ds]. \] This problem is the natural generalization to integral equations of the results of the theory of optimal control of stochastic differential equations. For F(s)\(\equiv 0\) the problem (1), (2) has been solved by the author [J. Appl. Math. Mech. 49, 704-712 (1985); translation from Prikl. Mat. Mekh. 49, 923-934 (1985; Zbl 0615.93073); Probl. Control Inf. Theory 13, 141-152 (1984; Zbl 0566.93071)]. In this paper we show that some refinement of the solution method allows us to solve the problem also in the general case.