Abstract
We study a singular control problem for a Brownian motion with constant drift and variance reflected at the origin. Exerting control pushes the process towards the origin and generates a concave increasing state-dependent yield which is discounted at a fixed rate. We show that the solution to the problem is sometimes more complicated than anticipated. Indeed, for some parameter values, the optimal policy is a band policy with two reflecting barriers plus a repelling boundary where smooth fit fails. We also show that the apparent anomaly can be understood as involving a switch between two strategies with different risk profiles: The risk-neutral decision maker initially gambles on the more risky strategy and lowers risk if this strategy underperforms.
Acknowledgments
The problem addressed in this paper was presented to me by Larry Shepp on my arrival to Rutgers University as a visiting student from KTH on July 7th, 2003. I am grateful for the highly stimulating (and entertaining!) environment that Larry provided that summer, during which some of the results presented here were obtained. I also wish to thank Naomi and Ofer Zeitouni for helpful discussions in 2003 and 2023. Finally, the presentation of the material in this paper has greatly benefitted from the comments of Luis Alvarez and two anonymous referees.
Citation
Adam Jonsson. "On singular control of reflected diffusions." Electron. Commun. Probab. 29 1 - 14, 2024. https://doi.org/10.1214/24-ECP621
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