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

Abstract

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.

Keywords:

• Stochastic optimal control
• game theory
• Brownian motion
• method of similarity solutions

AMS Subject Classification:

• 49N90

Acknowledgements

This research was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). The author also wishes to thank the anonymous reviewers of this paper for their constructive comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by Natural Sciences and Engineering Research Council of Canada [N/A].