Optimal control of stochastic dynamic systems with past history and Poisson switchings (Q2571528)

An optimal control problem for controlled stochastic hereditary systems described by systems of differential-functional linear equations \[ dx(t)=a(t,x_t,u)\,dt+b(t,x_t,u) \,dw(t)+\int_Z c(t,x_t,z,u)\nu(dz,dt) \] is considered \((t \geq 0)\). Here \(x(t)=x(t,\omega) \in \mathbb R^n\) are trajectories of the process and \(x_t = \{x(t+\theta), -\infty<\theta\leq 0\}\) are initial intervals of the trajectories of \(x(t)\). Vector functionals \(a(t,x_t,u)\), \(b(t,x_t,u)\), \(c(t,x_t,z,u)\) are continuous in all the arguments, \(u \in U\) is an \(l\)-measurable space of piecewise-continuous controls. The control \(u(t) = u(t,x_t)\) is a nonanticipating functional in the second argument. The Wiener process \(w(t)\) and a scalar centered Poisson measure \(\nu(dz,dt)\) are assumed to be independent. The initial condition is \[ x_0(\theta)=\varphi(\theta), -\infty <\theta \leq 0. \] The goal functionals are \[ v_1(\varphi) = \int_0^\infty k(t) g(\varphi(t),\varphi(0))\,dt, \] \[ v_2(t,\varphi) = \int_0^\infty k(s) g(t-s,\varphi(s),\varphi(0))\,ds \] and more complex ones. The Bellman equation is solved for this problem.