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We derive a semi-analytic formula for the transition probability ofthree-dimensional Brownian motion in the positive octant with absorption at theboundaries. Separation of variables in spherical coordinates leads to aneigenvalue problem for the resulting boundary value problem in the two angularcomponents. The main theoretical result is a solution to the original problemexpressed as an expansion into special functions and an eigenvalue which has tobe chosen to allow a matching of the boundary condition. We discuss and testseveral computational methods to solve a finite-dimensional approximation tothis nonlinear eigenvalue problem. Finally, we apply our results to thecomputation of default probabilities and credit valuation adjustments in astructural credit model with mutual liabilities.

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