[irchans]

It seems to me that if we let

h(b) = E[ e^{-bT}],

Then

h(b) = E[ 1 - b*T + (b*T)^2/2 - (b*T)^3/6 + ... ]
= 1 - b*E[T] + b^2*E[T^2]/2 - b^3*E[T^3]/6 + ....

So

h'(b) = E[T] + b*E[T^2] - b^2*E[T^3]/2 + ...
h''(b) = E[T^2] - b*E[T^3] + ...
h''(0) = E[T^2].


Jason said that

E[e^{-bT}]
= f(b)e^{-ga}/(f(b)cosh(f(b)a) - g sinh(f(b)a))

where

f(b) = sqrt{g^2 + 2b/s^2}.

I used Mathematica to do the algebra and got

E[T^2] = (-2 + e^(4*a*g) + 2*a^2*g^2 + e^(2*a*g)*(1 - 6*a*g))/(2*g^4*s^4).

Hmm.