Computing Sensitivities for Distortion Risk MeasuresJose Blanchet and Peter W. Glynn Naval Research Logistics (2022). Perpetuities (i.e., random variables of the form $ D={\int}_0^{\infty }{e}^{-\Gamma \left(t-\right)}d\Lambda (t) $ play an important role in many application settings. We develop approximations for the distribution of $ D $ when the “accumulated short rate process”, $ \Gamma $ , is small. We provide: (1) characterizations for the distribution of $ D $ when $ \Gamma $ and $ \Lambda $ are driven by Markov processes; (2) general sufficient conditions under which weak convergence results can be derived for $ D $ , and (3) Edgeworth expansions for the distribution of $ D $ in the iid case and the case in which $ \Lambda $ is a Levy process and the interest rate is a function of an ergodic Markov process. |