A note on weak viability for controlled diffusion

Abstract

In this paper, we study a weak notion of viability for a controlled diffusion. Under a tangential hypothesis, we prove that the weak viability property is satisfied.

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 (X_t^u) be a controlled diffusion taking its values in $\text{R}{}^{\mathrm{d}}$, satisfying the following stochastic differential equation:

$$\text{d}\text{X}{}_{\mathrm{t}}{}^{\mathrm{u}}\text{=b(t,X}{}_{\mathrm{t}}{}^{\mathrm{u}}\text{,u}{}_{\mathrm{t}}\text{)}\text{d}\text{t+σ(t,X}{}_{\mathrm{t}}{}^{\mathrm{u}}\text{,u}{}_{\mathrm{t}}\text{)}\text{d}\text{W}{}_{\mathrm{t}}$$

with initial condition X₀^u=x. In the previous equation, (W_t) is a p-dimensional brownian motion and the functions b and σ taking values, respectively, in $\text{R}{}^{\mathrm{d}}$ and $\text{R}{}^{\mathrm{d\times p}}$ depend on the choice of a control process (u_t) taking its values in a metric space U. One considers a closed susbspace $\text{K⊂}\text{R}{}^{\mathrm{d}}$ 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 (u_t) such that the diffusion X^u satisfies X_t^u∈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 $\text{x∈}\text{R}{}^{\mathrm{d}}$ such that the solution X_t of the equation

$$\text{d}\text{X}{}_{\mathrm{t}}\text{=b(t,X}{}_{\mathrm{t}}\text{)}\text{d}\text{t+σ(t,X}{}_{\mathrm{t}}\text{)}\text{d}\text{W}{}_{\mathrm{t}}$$

is such that , where P^X_t is the distribution of the random variable X_t. 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.