布朗运动在八分区中的转移概率及其在默认建模中的应用

Q3 Mathematics

Applied Mathematical Finance Pub Date : 2017-12-01 DOI:10.1080/1350486X.2018.1481439

Vadim Kaushansky, A. Lipton, C. Reisinger

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引用次数: 4

摘要

导出了边界有吸收的正八邻域中三维布朗运动跃迁概率的半解析公式。在球坐标系中分离变量会导致两个角分量边值问题的特征值问题。主要的理论结果是原始问题的解，表示为特殊函数的展开和必须选择的特征值，以允许边界条件的匹配。我们讨论并测试了几种计算方法来解决这个非线性特征值问题的有限维近似。最后，我们将研究结果应用于具有相互负债的结构性信用模型中违约概率和信用估值调整的计算。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Transition Probability of Brownian Motion in the Octant and its Application to Default Modelling

ABSTRACT We derive a semi-analytical formula for the transition probability of three-dimensional Brownian motion in the positive octant with absorption at the boundaries. Separation of variables in spherical coordinates leads to an eigenvalue problem for the resulting boundary value problem in the two angular components. The main theoretical result is a solution to the original problem expressed as an expansion into special functions and an eigenvalue which has to be chosen to allow a matching of the boundary condition. We discuss and test several computational methods to solve a finite-dimensional approximation to this nonlinear eigenvalue problem. Finally, we apply our results to the computation of default probabilities and credit valuation adjustments in a structural credit model with mutual liabilities.

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来源期刊

Applied Mathematical Finance Economics, Econometrics and Finance-Finance

CiteScore

2.30

自引率

0.00%

发文量

期刊介绍： The journal encourages the confident use of applied mathematics and mathematical modelling in finance. The journal publishes papers on the following: •modelling of financial and economic primitives (interest rates, asset prices etc); •modelling market behaviour; •modelling market imperfections; •pricing of financial derivative securities; •hedging strategies; •numerical methods; •financial engineering.