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.
Viscosity solutions in partially observed control
1986-01-01
10 pages
Article/Chapter (Book)
Electronic Resource
English
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