  In this paper we develop a framework for estimating Probability of Default
(PD) based on stochastic models governing an appropriate asset value processes.
In particular, we build upon a L\'evy-driven Ornstein-Uhlenbeck process and
consider a generalized model that incorporates multiple latent variables
affecting the evolution of the process. We obtain an Integral Equation (IE)
formulation for the corresponding PD as a function of the initial position of
the asset value process and the time until maturity, from which we then prove
that the PD function satisfies an appropriate Partial Integro-Differential
Equation (PIDE). These representations allow us to show that appropriate weak
(viscosity) as well as strong solutions exist, and develop subsequent numerical
schemes for the estimation of the PD function. Such a framework is necessary
under the newly introduced International Financial Reporting Standards (IFRS) 9
regulation, which has imposed further requirements on the sophistication and
rigor underlying credit modelling methodologies. We consider special cases of
the generalized model that can be used for applications to credit risk
modelling and provide examples specific to provisioning under IFRS 9, and more.
