Filtering and stochastic control: a historical perspective

We attempt to give a historical account of the main ideas leading to the development of nonlinear filtering and stochastic control as we know it today. We present a development of linear filtering theory, beginning with Wiener-Kolmogoroff filtering and ending with Kalman filtering. The linear-quadratic-Gaussian problem of stochastic control is considered and states that for this problem the optimal stochastic control can be constructed by solving separately a state estimation problem and a deterministic optimal control problem. Many of the ideas presented here generalize to the nonlinear situation. A reasonably detailed discussion of nonlinear filtering, again from the innovations viewpoint, is given. Finally, we deal with optimal stochastic control. The general method of discussing these problems is dynamic programming.