A bank salvage model by impulse stochastic controls

TL;DR

This paper develops a stochastic impulse control model for bank salvage, focusing on minimizing costs through capital injections, with a rigorous mathematical analysis of the associated quasi-variational inequality and solution properties.

Contribution

It introduces a novel impulse control framework for bank salvage with a unique viscosity solution and regularity results for the associated QVI.

Findings

Existence and uniqueness of viscosity solutions to the QVI.

Lipschitz continuity in space and Holder continuity in time.

Smooth-fit property under mild assumptions.

Abstract

The present paper is devoted to the study of a bank salvage model with finite time horizon and subjected to stochastic impulse controls. In our model, the bank's default time is a completely inaccessible random quantity generating its own filtration, then reflecting the unpredictability of the event itself. In this framework the main goal is to minimize the total cost of the central controller who can inject capital to save the bank from default. We address the latter task showing that the corresponding quasi-variational inequality (QVI) admits a unique viscosity solution, Lipschitz continuous in space and Holder continuous in time. Furthermore, under mild assumptions on the dynamics the smooth-fit $W^{(1,2),p}_{loc}$ property is achieved for any $1<p<+\infty$.