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.

中文翻译：

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基于去奇异化Meshfree配置方法的体制切换同步跳跃调和稳定Lévy模型下违约概率结构模型的数值解

在论文 [Hainaut, D. 和 Colwell, DB，具有转换过程和同步跳跃的信用风险结构模型，欧洲金融杂志 44(33) (4238):3262-3284] 中，作者利用了同步跳跃机制- 转换模型来计算上市公司的违约概率。在这里，我们首先将提出的 L\'evy 模型推广到最近引入金融文献的缓和稳定过程的更一般设置。基于所得偏积分微分算子的奇异性，我们提出了一个基于严格正定函数的通用框架来对算子进行去奇异化。然后，我们分析了一种基于径向基函数的高效无网格配置方法，以逼近由结构信用风险模型产生的偏积分微分方程组的相应系统的解。我们表明，在一些规律性假设下，我们提出的方法自然地将问题在调和稳定情况下进行了去奇异化。将该方法应用于文献中一些标准实例的数值结果证实了我们的理论结果和数值算法的准确性。