A Semi-Markov Dynamic Capital Injection Problem for Distressed Banks

Luca Di Persio (), Luca Prezioso and Yilun Jiang
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Luca Di Persio: Department of Computer Science, College of Mathematics, University of Verona, via Ca’ Vignal 2, 37129 Verona, Italy
Luca Prezioso: Department of Mathematics, University of Trento, Via Sommarive, 14, 38123 Trento, Italy
Yilun Jiang: Department of Mathematics, The Pennsylvania State University, University Park, State College, PA 16802, USA

Risks, 2023, vol. 11, issue 4, 1-16

Abstract: Our study investigates the optimal dividend strategy for a bank, taking into account the potential for government capital injections. We explore different types of government interventions, such as liberal, transparent, or uncertain strategies, and consider both single and multiple types of interventions. Our approach differs from others as it focuses on interventions that aim to maintain the overall stability of the financial system, rather than just addressing banks that have already sought government assistance or are in dire need of it. Specifically, we focus on situations where the government is more likely to assist banks that have not requested its intervention or that are not too difficult to save. To accomplish this, we conduct a comprehensive examination of all possible scenarios involving a single, one-time capital injection and derive explicit solutions for the associated optimal control problem. Furthermore, we expand the model to include semi-Markov dynamic capital injection processes and show that the optimal control is the unique viscosity solution of a Hamilton–Jacobi–Bellman equation. The government’s strategy also takes into account the bank’s solvency and any past government interventions.

Keywords: semi-Markov process; capital injection; dividend payment; stochastic control; jump-diffusion; HJB equation (search for similar items in EconPapers)
JEL-codes: C G0 G1 G2 G3 K2 M2 M4 (search for similar items in EconPapers)
Date: 2023
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