Success or failure of a firm under different financing policies: A dynamic stochastic model

The value of a firm evolves in time, according to a standard Ito's stochastic differential equation, and under different types of financial support. In particular both the cases of legal and illegal financing are considered. The ruin/success probabilities for the firm are evaluated in closed form via special functions in the various settings. Some functionals of the process, which are relevant from the viewpoint of the financiers, (via Kac's lemma) are shown to obey a partial differential equation, which is studied both with analytic tools (second-order ordinary differential equations of hypergeometric type) and with numerical ones. The case of insufficient payments is analyzed via stochastic calculus and a new model for loan renegotiation is offered.