Numerical methods to solve PDE models for pricing business companies in different regimes and implementation in GPUs

In this paper we propose appropriate numerical methods for companies valuation models proposed in [2]. Moreover, small modifications of these models allow to price the debt of the company and obtain the credit spread. The models are formulated in terms of final-boundary value problems associated to Kolmogorov type equations, which in some cases include an additional unilateral constraint on the solution. We also analyze the required boundary conditions so that the final-boundary value problem is well posed. This allows us to remove one of the unnecessary boundary conditions proposed in [2]. The numerical methods are mainly based on the use of characteristics (also known as semilagrangian) schemes in the direction without diffusion combined with implicit second order finite differences schemes in the direction where diffusion is present in the equation. This particular choice of numerical methods allows to develop an original parallelization strategy, which results to be specially highly efficient when using GPUs technologies.