Let $X\left(t\right)$ be a one-dimensional controlled diffusion process, and $Y\left(t\right)$ be a deterministic process. Various models are considered for $\left(\dot{X}\right(t),\dot{Y}(t\left)\right)$. The optimizer tries to bring the ratio $X\left(t\right)/Y\left(t\right)$ to a certain value d as soon as possible. There is also a second optimizer whose aim is to prevent the ratio $X\left(t\right)/Y\left(t\right)$ from 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.

中文翻译：

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最小化或最大化第一次通过时间到与时间相关的边界

让$X\left(吨\right)$是一维受控扩散过程，并且$是\left(吨\right)$是一个确定性的过程。考虑了各种模型$\left(\dot{X}\right(吨),\dot{是}(吨\left)\right)$. 优化器试图带来比率$X\left(吨\right)/是\left(吨\right)$尽快达到某个值d。还有第二个优化器，其目的是防止比率$X\left(吨\right)/是\left(吨\right)$从达到d尽可能长的时间。得到明确的解决方案。假设控制变量为常数时的特殊情况，处理一般情况。