## The Annals of Applied Probability

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

#### Abstract

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

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

Dates
First available in Project Euclid: 9 January 2014

https://projecteuclid.org/euclid.aoap/1389278717

Digital Object Identifier
doi:10.1214/12-AAP902

Mathematical Reviews number (MathSciNet)
MR3161639

Zentralblatt MATH identifier
1328.60188

#### Citation

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. https://projecteuclid.org/euclid.aoap/1389278717

#### References

• [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.