## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 1, Number 1 (1991), 167-172.

### Forcing a Stochastic Process to Stay in or to Leave a Given Region

#### Abstract

Systems defined by $dx(t) = a\lbrack x(t), t \rbrack dt + B\lbrack x(t), t \rbrack u(t) dt + N^{1/2}\lbrack x(t), t \rbrack dW(t)$, where $x(t)$ is the state variable, $u(t)$ is the control variable, $a$ is a vector function, $B$ and $N$ are matrices and $W(t)$ is a Brownian motion process, are considered. The aim is to minimize the expected value of a cost function with quadratic control costs on the way and terminal cost function $K(T)$, where $T = \inf\{s: x(s) \in D \mid x(t) = x\}, D$ being a given region in $\mathbb{R}^n$. The function $K$ is taken to be 0 if $T \geq (\leq) \tau$, where $\tau$ is a positive constant and $+\infty$ if $T < (>) \tau$ when the aim is to force $x(t)$ to stay in (resp., to leave) the region $C$, the complement of $D$. A particular one-dimensional problem is solved explicitly and a risk-sensitive version of the general problem is also considered.

#### Article information

**Source**

Ann. Appl. Probab., Volume 1, Number 1 (1991), 167-172.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1177005986

**Digital Object Identifier**

doi:10.1214/aoap/1177005986

**Mathematical Reviews number (MathSciNet)**

MR1097469

**Zentralblatt MATH identifier**

0728.93078

**JSTOR**

links.jstor.org

**Subjects**

Primary: 93E20: Optimal stochastic control

Secondary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx]

**Keywords**

Stochastic control optimal control Brownian motion risk sensitivity

#### Citation

Lefebvre, Mario. Forcing a Stochastic Process to Stay in or to Leave a Given Region. Ann. Appl. Probab. 1 (1991), no. 1, 167--172. doi:10.1214/aoap/1177005986. https://projecteuclid.org/euclid.aoap/1177005986