Viscosity solutions in partially observed control

Abstract

We are concerned with a "simple" partially observed control problem first considered by Beneš and Karatzas in [1], where the relation between Zakai's equation and Mortensen's equation was revealed. In [1] it was pointed out that the dynamic programming techniques do not give us an existence theorem (- as it would be in the completely observed case -), but it was shown that the value function is a lower bound for the solution of Mortensen's equation, if such a solution exists.

In this contribution, we do not attack the existence problem, and our results do not contribute any idea towards such a result, as for as we see.

We consider Mortensen's equation and use the concept of viscosity solution for an attempt towards a better understanding of the important role this equation plays in stochastic control. A viscosity solution is defined to be a subsolution and a supersolution of the problem. We are thus trying to describe the sets of subsolutions and supersolutions and give some control-theoretical interpretation of these sets: Subsolutions may be seen as solutions of a suboptimal problem and supersolutions are related with superoptimal solutions of the control problem. Our results are related to those of Bensoussan in [2], and a lot of techniques are taken from [2] and [3]. The reference on viscosity solutions related to our problem is [6].

This article treats a slimmed problem which was described in more detail in [4]. Some of the technical details which are left out here, can be found there.

This work was supported by the SFB 72 of the DFG at the Universität Bonn, by the British Council during a visit to the Mathematical Institute at the University of Warwick, and by the AFF at the Universität Konstanz