## The Annals of Applied Probability

### Control and stopping of a diffusion process on an interval

#### Abstract

Consider a process $X(\cdot) = {X(t), 0 \leq t < \infty}$ which takes values in the interval $I = (0, 1)$, satisfies a stochastic differential equation $$dX(t) = \beta(t)dt + \sigma(t)dW(t), X(0) = x \epsilon I$$ and, when it reaches an endpoint of the interval I, it is absorbed there. Suppose that the parameters $\beta$ and $\sigma$ are selected by a controller at each instant $t \epsilon [0, \infty)$ from a set depending on the current position. Assume also that the controller selects a stopping time $\tau$ for the process and seeks to maximize $\mathbf{E}u(X(\tau))$, where $u: [0, 1] \to \Re$ is a continuous "reward" function. If $\lambda := \inf{x \epsilon I: u(x) = \max u}$ and $\rho := \sup{x \epsilon I: u(x) = \max u}$, then, to the left of $\lambda$, it is best to maximize the mean-variance ratio $(\beta/\sigma^2)$ or to stop, and to the right of $\rho$, it is best to minimize the ratio $(\beta/\sigma^2)$ or to stop. Between $\lambda$ and $\rho$, it is optimal to follow any policy that will bring the process $X(\cdot)$ to a point of maximum for the function $u(\cdot)$ with probability 1, and then stop.

#### Article information

Source
Ann. Appl. Probab., Volume 9, Number 1 (1999), 188-196.

Dates
First available in Project Euclid: 21 August 2002

https://projecteuclid.org/euclid.aoap/1029962601

Digital Object Identifier
doi:10.1214/aoap/1029962601

Mathematical Reviews number (MathSciNet)
MR1682584

Zentralblatt MATH identifier
0938.93067

#### Citation

Karatzas, Ioannis; Sudderth, William D. Control and stopping of a diffusion process on an interval. Ann. Appl. Probab. 9 (1999), no. 1, 188--196. doi:10.1214/aoap/1029962601. https://projecteuclid.org/euclid.aoap/1029962601

#### References

• Dy nkin, E. B. and Yushkevich, A. A. (1969). Markov Processes: Theorems and Problems. Plenum Press, New York.
• EL Karoui, N. (1981). Les aspects probabilistes du contr ole stochastique. Lecture Notes in Math. 876 73-238. Springer, Berlin.
• Fakeev, A. G. (1970). Optimal stopping rules for processes with continuous parameter. Theory Probab. Appl. 15 324-331.
• Fakeev, A. G. (1971). Optimal stopping of a Markov process. Theory Probab. Appl. 16 694-696.
• Feller, W. (1952). The parabolic differential equations and the associated semigroup of transformations. Ann. of Math. 55 468-519.
• Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Springer, New York.
• Pestien, V. C. and Sudderth, W. D. (1985). Continuous-time red-and-black: how to control a diffusion to a goal. Math. Oper. Res. 10 599-611.
• Shiry aev, A. N. (1973). Statistical Sequential Analy sis. Amer. Math. Soc., Providence, RI.
• Shiry aev, A. N. (1978). Optimal Stopping Rules. Springer, New York.