Maximizing the time spent by a diffusion process in an interval (Q2713004)

The author considers the controlled one-dimensional diffusion NEWLINE\[NEWLINE dx_t = a(x_t) dt + b_1(x_t) u_t dt+b_2(x_t) u^2(t) dt + v(x_t)^{1/2} dW_t,NEWLINE\]NEWLINE where \(W\) is a Brownian motion, \(b_2\), \(v\) are suitable positive functions, and control is exercised via the progressively measurable process \(u\). The aim of the paper is to determine controls which maximize the time by the diffusion spent in an interval \([-d,d]\). Using dynamic programming, the author derives first the optimal control under suitable assumptions on the value function of the problem. Moreover, the value function is characterized via the solution of a Kolmogorov backward equation. The paper concludes by some case studies including controlled Wiener, Ornstein-Uhlenbeck, and Bessel processes.