On a Numerical Solution of Structural model for the Probability of Defaultunder a Regime-Switching Synchronous-Jump Tempered Stable LévyModel with Desingularized Meshfree Collocation method

In the paper [Hainaut, D. and Colwell, D.B., A structural model for credit risk with switchingprocesses and synchronous jumps, The European Journal of Finance44(33) (4238):3262-3284],the authors exploit a synchronous-jump regime-switching model to compute the default probabilityof a publicly traded company. Here, we first generalize the proposed L\'evy model to more generalsetting of tempered stable processes recently introduced into the finance literature. Based on thesingularity of the resulting partial integro-differential operator, we propose a general frameworkbased on strictly positive-definite functions to de-singularize the operator. We then analyze anefficient meshfree collocation method based on radial basis functions to approximate the solution ofthe corresponding system of partial integro-differential equations arising from the structural creditrisk model. We show that under some regularity assumptions, our proposed method naturallyde-sinularizes the problem in the tempered stable case. Numerical results of applying the methodon some standard examples from the literature confirms the accuracy of our theoretical results andnumerical algorithm.