A bank salvage model by impulse stochastic controls

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 capitals to save the bank from default. We address the latter task showing that the corresponding quasivariational inequality (QVI) admits a unique viscosity solution, Lipschitz continuous in space and Hölder 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 < +∞.