Optimal Controls for Zakai Equations

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This chapter is an application of previous one. In Sect. 5.5, we have introduced the (controlled) Zakai equation, which is a stochastic linear parabolic equation with a Brownian adapted control process. By using the results in Chap. 5, we will study control problems for Zakai equations related to partially observable diffusions. The control problem for partially observable diffusions turns out to be a completely observable control problem on a Hilbert space, by using the unnormalized conditional probability density given by the Zakai equation (cf. Bensoussan A, Stochastic control of partially observable systems. Cambridge University Press, Cambridge/New York, 1992; Lions, J Commun PDE 8:1101–1134, 1983, I, II; Gozzi and Świech, J Funct Analy 172:466–510, 2000). Section 6.1 is devoted to the analysis of controlled Zakai equations. In Sect. 6.2, we formulate control problems for a system governed by Zakai equations, in the same way as in Chap. 2. When a control process \(\gamma (\cdot )\) is chosen, the cost on a time internal [T 0, t] is given by \(\int _{T_{0}}^{t}r(u^{\gamma (\cdot )}(s),\gamma (s))\,ds + F(u^{\gamma (\cdot )}(t))\), where \(u^{\gamma (\cdot )}(\cdot )\) is the response of \(\gamma (\cdot )\). By taking a suitable control process, we want to minimize (or maximize) the expectation of the cost. In Sect. 6.3 we formulate the DPP via the semigroup constructed from the value function, whose generator is related to the HJB equation on a Hilbert space. The viscosity solution of HJB equation is introduced following Gozzi and Świech (J Funct Analy 172:466–510, 2000) in Sect. 6.4. Example 6.1 makes explicitly the connection between controlled Zakai equations and control of partially observable diffusions.