Transition probability of Brownian motion in the octant and its application to default modelling

Journal article

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.

Authors: Kaushansky, V; Lipton, A; Reisinger, C

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Routledge
• Journal: Applied Mathematical Finance
• Volume: 25
• Issue: 5-6
• Pages: 434-465
• Publication date: 2018-06-18
• Acceptance date: 2018-05-22
• DOI: 10.1080/1350486X.2018.1481439
• EISSN: 1466-4313
• ISSN: 1350-486X
• Keywords: three-dimensional Brownian motion; nonlinear eigen-value problem; mutual liabilities; transition probability; structural default model