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Letbe a one-dimensional controlled diffusion process, andbe a deterministic process. Various models are considered for. The optimizer tries to bring the ratioto a certain value  d  as soon as possible. There is also a second optimizer whose aim is to prevent the ratiofrom reaching  d  as long as possible. Explicit solutions are obtained. The particular case when the control variables are assumed to be constants and the general case are treated.

Minimizing or maximizing the first-passage time to a time-dependent boundary