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”.


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