The Annals of Applied Probability

Killed Brownian motion with a prescribed lifetime distribution and models of default

Boris Ettinger, Steven N. Evans, and Alexandru Hening

Full-text: Open access


The inverse first passage time problem asks whether, for a Brownian motion $B$ and a nonnegative random variable $\zeta$, there exists a time-varying barrier $b$ such that $\mathbb{P}\{B_{s}>b(s),0\leq s\leq t\}=\mathbb{P}\{\zeta>t\}$. We study a “smoothed” version of this problem and ask whether there is a “barrier” $b$ such that $\mathbb{E}[\exp(-\lambda\int_{0}^{t}\psi(B_{s}-b(s))\,ds)]=\mathbb{P}\{\zeta>t\}$, where $\lambda$ is a killing rate parameter, and $\psi:\mathbb{R}\to[0,1]$ is a nonincreasing function. We prove that if $\psi$ is suitably smooth, the function $t\mapsto\mathbb{P}\{\zeta>t\}$ is twice continuously differentiable, and the condition $0<-\frac{d\log\mathbb{P}\{\zeta>t\}}{dt}<\lambda$ holds for the hazard rate of $\zeta$, then there exists a unique continuously differentiable function $b$ solving the smoothed problem. We show how this result leads to flexible models of default for which it is possible to compute expected values of contingent claims.

Article information

Ann. Appl. Probab., Volume 24, Number 1 (2014), 1-33.

First available in Project Euclid: 9 January 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 91G40: Credit risk 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)

Credit risk inverse first passage time problem killed Brownian motion Cox process stochastic intensity Feynman–Kac formula


Ettinger, Boris; Evans, Steven N.; Hening, Alexandru. Killed Brownian motion with a prescribed lifetime distribution and models of default. Ann. Appl. Probab. 24 (2014), no. 1, 1--33. doi:10.1214/12-AAP902.

Export citation


  • [1] Avellaneda, M. and Zhu, J. (2001). Modelling the distance-to-default process of a firm. Risk 14 125–129.
  • [2] Black, F. and Cox, J. C. (1976). Valuing corporate securities: Some effects of bond indenture provisions. J. Finance 31 351–367.
  • [3] Chen, X., Cheng, L., Chadam, J. and Saunders, D. (2011). Existence and uniqueness of solutions to the inverse boundary crossing problem for diffusions. Ann. Appl. Probab. 21 1663–1693.
  • [4] Davis, M. H. A. and Pistorius, M. R. (2011). On an explicit solution to an inverse-first passage problem and quantification of counterparty risk via Bessel bridges. Preprint.
  • [5] Davis, M. H. A. and Pistorius, M. R. (2013). Explicit solution to an inverse first-passage time problem for Lévy processes. Application to counterparty credit risk. Available at arXiv:1306.2719v1.
  • [6] Hull, J. and White, A. (2001). Valuing credit default swaps II: Modeling default correlations. Journal of Derivatives 8 12–22.
  • [7] Hull, J. C. and White, A. (2000). Valuing credit default swaps I: No counterparty default risk. Journal of Derivatives 8 29–40.
  • [8] Iscoe, I. and Kreinin, A. (2002). Default boundary problem. Working paper, Algorithmics Inc. Research Paper Series.
  • [9] Kennedy, C. A. and Carpenter, M. H. (2003). Additive Runge–Kutta schemes for convection–diffusion–reaction equations. Appl. Numer. Math. 44 139–181.
  • [10] Lerche, H. R. (1986). Boundary Crossing of Brownian Motion. Its Relation to the Law of the Iterated Logarithm and to Sequential Analysis. Lecture Notes in Statistics 40. Springer, Berlin.
  • [11] Peskir, G. (2002). Limit at zero of the Brownian first-passage density. Probab. Theory Related Fields 124 100–111.
  • [12] Peskir, G. (2002). On integral equations arising in the first-passage problem for Brownian motion. J. Integral Equations Appl. 14 397–423.
  • [13] Peskir, G. and Shiryaev, A. (2006). Optimal Stopping and Free-Boundary Problems. Birkhäuser, Basel.
  • [14] Valov, A. V. (2009). First passage times: Integral equations, randomization and analytical approximations. Ph.D. thesis, Univ. Toronto.
  • [15] Zucca, C. and Sacerdote, L. (2009). On the inverse first-passage-time problem for a Wiener process. Ann. Appl. Probab. 19 1319–1346.