Linear Stochastic Control

Abstract

Let us next consider stochastic ‘control’ problems for the linear system:

$$x\left( {t;\omega } \right) = \int\limits_0^t A \left( s \right)x\left( {s;\omega } \right)ds + \int\limits_0^t B \left( s \right)u\left( s \right)ds + \int\limits_0^t F \left( s \right)dW\left( {s;\omega } \right),$$

$$0 \leqslant t \leqslant 1$$

((7.1))

$$Y\left( {t;\omega } \right) = \int\limits_0^t C \left( s \right)x\left( {s;\omega } \right)ds + \int\limits_0^t G \left( s \right)dW\left( {s;\omega } \right),0 \leqslant t \leqslant 1$$

where u(.) denotes the ‘control’ function to be determined, and where we assume that all the coefficients are continuous on [0,1], and G(s)G(s)*>0 0≤s≤1 and of course W(s;ω) is a Wiener process as before.