Killed Brownian motion with a prescribed lifetime distribution and models of default
| Autor: | Alexandru Hening, Boris Ettinger, Steven N. Evans |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2011 |
| Předmět: |
Statistics and Probability
inverse first passage time problem Inverse Expected value Lambda Lifetime distribution 01 natural sciences Combinatorics FOS: Economics and business 010104 statistics & probability Feynman–Kac formula killed Brownian motion 91G80 0502 economics and business FOS: Mathematics 0101 mathematics Credit risk Brownian motion stochastic intensity Mathematics 050208 finance 05 social sciences Probability (math.PR) 91G40 Function (mathematics) 16. Peace & justice Risk Management (q-fin.RM) Statistics Probability and Uncertainty First-hitting-time model Cox process Random variable 60J70 Mathematics - Probability Quantitative Finance - Risk Management |
| Zdroj: | Ann. Appl. Probab. 24, no. 1 (2014), 1-33 |
| Popis: | 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 $0t\}}{dt} Comment: Published in at http://dx.doi.org/10.1214/12-AAP902 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org) |
| Databáze: | OpenAIRE |
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