Control and Stopping of a Diffusion Process on an Interval

• Published in: The Annals of applied probability 1999-02, Vol.9 (1), p.188-196
• Main Authors: Karatzas, Ioannis, Sudderth, William D.
• Format: Article
• Language: English
• Subjects: 60D60; 60G40; 62L15; 93E20; Markov processes; Mathematical functions; one-dimensional diffusions; optimal stopping; Stochastic control; Stopping distances
• ISSN: 1050-5164; 2168-8737
• DOI: 10.1214/aoap/1029962601

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$dX(t) = \beta(t)dt + \sigma(t) dW(t), \quad X(0) = x \in I$and, when it reaches an endpoint of the interval I, it is absorbed there. Suppose that the parameters β and σ are selected by a controller at each instant t ∈ [0, ∞) from a set depending on the current position. Assume also that the controller selects a stopping time τ for the process and seeks to maximize Eu(X(τ)), where u:[0, 1] → R is a continuous "reward" function. If λ:=inf{x ∈ I: u(x) = max u} and$\rho:= \text{sup} \{x \in I: u(x)= \text{max}\,u\}$, then, to the left of λ, it is best to maximize the mean-variance ratio (β/σ2) or to stop, and to the right of ρ, it is best to minimize the ratio (β/σ2) or to stop. Between λ and ρ, it is optimal to follow any policy that will bring the process X(·) to a point of maximum for the function u(·) with probability 1, and then stop.