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Closed-Form Evaluation of a Normal Quantile Exponential Integral

Article Quant Q&A · Author: user53249
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Summary

The document presents a closed-form expression for an integral whose integrand is an exponential of a scaled inverse standard normal cumulative distribution function, shifted by a constant. The integration variable runs between a lower bound and one, with the inverse cumulative function evaluated at the variable plus a shift.

The stated result expresses the integral as an exponential factor involving the additive constant and half the squared scale parameter, multiplied by the difference between two standard normal cumulative probabilities. Their arguments are the inverse normal quantiles at the shifted integration endpoints, each reduced by the scale parameter. This is a compact evaluation for the specified integrand and parameter setup. The excerpt gives the formula but no derivation, numerical example, or discussion of boundary conditions; users should ensure the shifted arguments lie in the domain of the inverse normal function.

Key ideas

  • The integral contains an exponential of a scaled inverse normal cumulative probability.
  • Its stated closed form uses an exponential prefactor and a difference of normal cumulative probabilities.
  • The two cumulative probability terms correspond to the shifted integration endpoints.
  • The excerpt provides no derivation or numerical illustration.

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# Is there a closed-form solution for the following integral?


# Is there a closed-form solution for the following integral?












The integral under consideration is as follows: $$ F=\int_{a}^{1} \exp\Big\{c\Phi^{-1}(x+b) + d\Big\}\; \mathrm dx, $$ where $0<a, b<1$, and $c>0, d\in\mathbb{R}$ are constants, and the notation $\Phi(\cdot)$ denotes the standard normal distribution function given by $$ \Phi(z) = \mathbb{P}(Z\leq z) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{z}e^{-\frac{u^2}{2}}\; \mathrm du. $$

## Answer by user53249 (score 8)

https://quant.stackexchange.com/a/65661

Thanks to Gordon's help, we have that \begin{eqnarray*} F=exp\Big\{d + \frac{{c}^2}{2}\Big\}\Big[ \Phi\Big(\Phi^{-1}\Big(1+b\Big)-{c}\Big)- \Phi\Big(\Phi^{-1}\Big(a+b\Big)-{c}\Big)\Big] \end{eqnarray*}

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.