In the paper [Hainaut, D. and Colwell, D.B., {\rm A structural model for credit risk with switching processes and synchronous jumps}, The European Journal of Finance 22(11) (2016): 1040-1062], the authors exploit a synchronous-jump regime-switching model to compute the default probability of a publicly traded company. Here, we first generalize the proposed L\'{e}vy model to more general setting of tempered stable processes recently introduced into the finance literature. Based on the singularity of the resulting partial integro-differential operator, we propose a general framework based on strictly positive-definite functions to de-singularize the operator. We then analyze an efficient meshfree collocation method based on radial basis functions to approximate the solution of the corresponding system of partial integro-differential equations arising from the structural credit risk model. We show that under some regularity assumptions, our proposed method naturally de-sinularizes the problem in the tempered stable case. Numerical results of applying the method on some standard examples from the literature confirms the accuracy of our theoretical results and numerical algorithm.

翻译：在论文[Hainaut, D. and Colwell, D.B., 和 D.B., rm A 用于转换过程和同步跳跃的信用风险的结构模型 中,《欧洲金融杂志》22(11)(2016):1040-1062],作者们利用同步跳式套管转换模型来计算上市公司的默认概率。在这里,我们首先将拟议的L\'{e}vy模型概括为最近引入金融文献的温和稳定流程的更一般设置。根据由此产生的部分食堂差异操作器的独一性,我们提出了一个基于严格肯定性功能的总体框架,使操作器去分类化。然后,我们根据辐射性功能分析了一种高效的无间隔合用法,以近似于结构信用风险模型所产生的部分内分化方等方的对应系统的解决办法。我们根据一些常规性假设,我们拟议的方法自然地淡化了内脏稳定的操作器中的问题。在应用某种标准的理论性模型和数字学分析结果方面的结果证实了我们数字学的精确性。