Let ${\bar x} > 0 $, $\sigma >0$, $\mu \in {\mathbb R}$, $T > 0 $ and $\alpha \in {\mathbb R}$. Let $\phi(x) := \exp(-x^2/2)/\sqrt{2\pi} $ be a probability density of a standard normal variable. Then consider the following integral:
\begin{equation} F_{\mu\,\sigma}[{\bar x},T]:= \int\limits_T^\infty \frac{1}{\xi^\alpha} \cdot \phi\left(\frac{{\bar x} - \mu \xi}{\sigma \sqrt{\xi}}\right) d\xi \end{equation}
This integral appears in the context of either the first hitting time of a barrier by a Brownian motion or of the law of the supremum of the Brownian motion (see Theorem 46.4 page 348 in
Sato, Ken-Iti, Lévy processes and infinitely divisible distributions, Cambridge Studies in Advanced Mathematics. 68. Cambridge: Cambridge University Press. xii, 486 p. (1999). ZBL0973.60001.).
We have calculated this integral for $\alpha = 3/2 + m$ where $ m\in {\mathbb Z}$. The result reads:
\begin{eqnarray} \left. F_{\mu\,\sigma}[{\bar x},T]:= \frac{e^{\frac{\mu {\bar x}}{\sigma^2}}}{\sqrt{2}} (-1)^m \frac{d^{|m|}}{d \theta^{|m|}} \right\{ \begin{array}{lll} % \left. \frac{e^{-2 \sqrt{\theta b}}}{\sqrt{\theta}} - \frac{1}{2 \sqrt{\theta}} \left( e^{+2\sqrt{\theta b}} erfc[\frac{\sqrt{\theta}}{\sqrt{T}} + \sqrt{b T}] + e^{-2\sqrt{\theta b}} erfc[\frac{\sqrt{\theta}}{\sqrt{T}} - \sqrt{b T}] \right) \right|_{\theta = a} & \mbox{if $m \ge 0 $} \\ % \left. \frac{e^{-2 \sqrt{a \theta}}}{\sqrt{a}} - \frac{1}{2 \sqrt{a}} \left( e^{+2\sqrt{a \theta}} erfc[\frac{\sqrt{a}}{\sqrt{T}} + \sqrt{\theta T}] + e^{-2\sqrt{a \theta}} erfc[\frac{\sqrt{a}}{\sqrt{T}} - \sqrt{\theta T}] \right) \right|_{\theta=b}& \mbox{if $m <0 $} \end{array} \end{eqnarray} where $(a,b) = ({\bar x}^2/(2 \sigma^2),\mu^2/(2 \sigma^2)) $.
The derivation is not complicated. It basically boils down to using this result and then differentiating with respect to parameters of the integrand to induce an arbitrary power function in that integrand. Below is a verification of that result in Mathematica:
In[7307]:= phi[x_] := Exp[-x^2/2]/Sqrt[2 Pi];
sig = RandomReal[{1, 2}];
{mu} = RandomReal[{-2, 2}, 1];
{T, x} = RandomReal[{1, 3}, 2];
m = RandomInteger[{0, 10}];
xi =.;
NIntegrate[
1/xi^((3 + 2 m)/2) phi[(x - mu xi)/(sig Sqrt[xi])], {xi, T, Infinity}]
Exp[(mu x)/sig^2]/Sqrt[2 Pi] NIntegrate[
1/xi^((3 + 2 m)/2) Exp[-(x^2/(2 sig^2 xi)) - (mu^2 xi)/(
2 sig^2)], {xi, T, Infinity}]
{a, b} = {x^2/(2 sig^2), (mu^2) /(2 sig^2)};
Exp[(mu x)/sig^2]/Sqrt[2 Pi] (2) NIntegrate[
xi^(2 m) Exp[-a xi^2 - b/xi^2], {xi, 0, 1/Sqrt[T]}]
th =.;
Exp[(mu x)/sig^2]/
Sqrt[2] (D[
1/Sqrt[th] Exp[-2 Sqrt[th b]] -
1/(2 Sqrt[
th]) (Exp[2 Sqrt[th b]] Erfc[Sqrt[th]/Sqrt[T] + Sqrt[b T]] +
Exp[-2 Sqrt[ th b]] Erfc[Sqrt[th]/Sqrt[T] - Sqrt[b T]]), {th,
m}] /. th :> a) (-1)^m
m = -m;
NIntegrate[
1/xi^((3 + 2 m)/2) phi[(x - mu xi)/(sig Sqrt[xi])], {xi, T, Infinity}]
Exp[(mu x)/sig^2]/
Sqrt[2] (D[
1/Sqrt[a] Exp[-2 Sqrt[a th]] -
1/(2 Sqrt[
a]) (Exp[2 Sqrt[a th]] Erfc[Sqrt[a]/Sqrt[T] + Sqrt[th T]] +
Exp[-2 Sqrt[ a th]] Erfc[
Sqrt[a]/Sqrt[T] - Sqrt[th T]]), {th, -m}] /. th :> b) (-1)^m
Out[7313]= 0.000155233
Out[7314]= 0.000155233
Out[7316]= 0.000155233
Out[7318]= 0.000155233
Out[7320]= 70253.1
Out[7321]= 70253.1
My question would be twofold, firstly how does the result look like when the parameter $\alpha$ is arbitrary? The second question would be more generic. Note that the function $\phi()$ is nothing else but the probability density of a Brownian motion at time one, i.e. $\phi(x) := \phi_{B_1}(x) $. Would it be possible to derive the result if we were to replace the Brownian motion by some other Levy process, for example a Levy stable process or a jump diffusion ?
