## Control and stopping of a diffusion process on an interval.(English)Zbl 0938.93067

Authors’ 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 $dX(t)= \beta(t)dt+ \sigma(t)dW(t),\quad x(0)= x\in I$ 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 $$t\in[0,\infty)$$ 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 $$u:[0,1]\to {\mathfrak R}$$ is a continuous “reward” function. If $$\lambda:= \inf\{x\in I: u(x)= \max u\}$$ and $$\rho:= \sup\{x\in I: u(x)= \max u\}$$, 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”.

### MSC:

 93E20 Optimal stochastic control 62L15 Optimal stopping in statistics 60G40 Stopping times; optimal stopping problems; gambling theory

### Keywords:

optimal stopping; stopping time
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### References:

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