A note on weak viability for controlled diffusion
Introduction
The aim of this paper is to study sufficient conditions for a viability result for a controlled diffusion. The kind of problem we are looking at is the following: let (Xtu) be a controlled diffusion taking its values in , satisfying the following stochastic differential equation:with initial condition X0u=x. In the previous equation, (Wt) is a p-dimensional brownian motion and the functions b and σ taking values, respectively, in and depend on the choice of a control process (ut) taking its values in a metric space U. One considers a closed susbspace and T>0 a fixed deterministic time. In all generality, the viability problem for K in [0,T] is to check whether there exists a control (ut) such that the diffusion Xu satisfies Xtu∈K for every t∈[0,T].
This problem has been widely studied in the deterministic case and a little bit less in the stochastic case. The major contributions of the 1970s and 1980s are quoted in Aubin and Da Prato (1990) and Gautier and Thibault (1993). A certain renewal of interest for these questions is appeared due to application of this problem to mathematical finance. Let us, in particular mention the nice paper (Buckdahn et al., 1998) where the authors study the viability problem as a standard control problem and (Aubin and Da Prato, 1998) where the authors study the viability problem for differential inclusions.
Some months ago, Michta (1998) has considered a viability problem, where the choice is restricted to the initial condition. He introduced a notion of weak (or approximated) viability that seems quite natural. For ε>0 been given, one looks for an initial condition such that the solution Xt of the equationis such that , where PXt is the distribution of the random variable Xt. Michta has given a sufficient condition in order that this weak viability property be satisfied. In this paper, we are looking for a similar condition for the viability problem for controlled diffusion (0.1). We advise the interested reader, in particular if she or he is not too familiar with probabilistic techniques in stochastic control, to have at least a superficial look at Michta's paper as his case is simpler and allows to write things in a less technical way.
Section snippets
Notations and preliminary results
Let us consider the stochastic differential equationwhere b (resp. σ): (resp. ) are bounded measurable functions, continuous in the second and third parameters, uniformly with respect to time. W is a p-dimensionnal brownian motion. U is a compact metric space and the processes (ut) considered in the previous equation take their values in U.
The systematic study of probabilistic techniques for stochastic control problems
The weak tangential condition
The idea behind viability is that one may chose the control process (ut) in order that the diffusion Xu is constrained to stay in the closed set K (in the case of weak viability this property has to be satisfied at each time with high probability). The sufficient weak tangential condition insures that it is always possible to go “a little bit further” with the constraint satisfied. In our case, as the motion is control dependent, the condition requires the existence of a constant (in time)
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