On the Singular Control of a Diffusion and Its Running Infimum or Supremum

Giorgio Ferrari and Neofytos Rodosthenous
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Giorgio Ferrari: Center for Mathematical Economics, Bielefeld University
Neofytos Rodosthenous: Center for Mathematical Economics, Bielefeld University

No 745, Center for Mathematical Economics Working Papers from Center for Mathematical Economics, Bielefeld University

Abstract: We study a class of singular stochastic control problems for a one-dimensional diffusion $X$ in which the performance criterion to be optimised depends explicitly on the running infimum $I$ (or supremum $S$) of the controlled process. We introduce two novel integral operators that are consistent with the Hamilton-Jacobi-Bellman equation for the resulting two-dimensional singular control problems. The first operator involves integrals where the integrator is the control process of the two-dimensional process $(X, I)$ or $(X, S)$; the second operator concerns integrals where the integrator is the running infimum or supremum process itself. Using these definitions, we prove a general verification theorem for problems involving two-dimensional state-dependent running costs, costs of controlling the process, costs of increasing the running infimum (or supremum) and exit times. Finally, we apply our results to explicitly solve an optimal dividend problem in which the manager’s time-preferences depend on the company’s historical worst performance.

Keywords: singular stochastic control; one-dimensional diffusions; running infimum; running supremum; free boundary; optimal dividends (search for similar items in EconPapers)
Pages: 28
Date: 2025-08-18
New Economics Papers: this item is included in nep-inv
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Persistent link: https://EconPapers.repec.org/RePEc:bie:wpaper:745

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