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%I #12 Jun 03 2017 15:34:07
%S 8,5,1,0,8,3,7,0,3,2,7,9,3,5,9,6,5,3,1,1,1,2,2,7,9,0,6,7,2,8,3,8,2,2,
%T 4,7,6,9,6,4,8,6,2,2,7,9,6,4,7,4,1,7,9,7,7,1,9,5,9,7,0,3,2,9,8,0,1,5,
%U 5,6,0,1,9,8,3,0,0,2,2,5,8,1,0,0,6,2,1,1,5,2,3,6,3,9,2,2,1,3,7,4,6
%N Decimal expansion of Var(T_{0,1}), the variance of the "first-passage" time required for an Ornstein-Uhlenbeck process to cross the level 0, given that it started at level 1.
%C Following Steven Finch, it is assumed that the values of the parameters of the stochastic differential equation dX_t = -rho (X_t - mu) dt + sigma dW_t, satisfied by the process, are mu = 0, rho = 1 and sigma^2 = 2.
%H Steven R. Finch, <a href="/A249417/a249417.pdf">Ornstein-Uhlenbeck Process</a>, May 15, 2004. [Cached copy, with permission of the author]
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/DigammaFunction.html">Digamma Function</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Ornstein%E2%80%93Uhlenbeck_process">Ornstein-Uhlenbeck process</a>
%F Var(T(0,c)) = (1/2)*sum_{k >= 1} ((sqrt(2)*c)^k*Gamma(k/2)*(psi(k/2)+gamma))/k! - E(T(0,c))^2, where 'c' is the start level (c=1), E(T(0,c)) the expectation A249418, and psi the digamma function,
%e 0.8510837032793596531112279067283822476964862279647417977...
%t digits = 101; Ex[T[0, c_]] := (1/2)*(Pi*Erfi[c/Sqrt[2]] - c^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, c^2/2]); Var[T[0, c_]] := (1/2)*NSum[(-1)^k*((Sqrt[2]*c)^k*Gamma[k/2]*(PolyGamma[k/2] + EulerGamma))/k!, {k, 1, Infinity}, WorkingPrecision -> digits + 5, NSumTerms -> digits] - Ex[T[0, c]]^2 ; RealDigits[Var[T[0, 1]], 10, digits] // First
%Y Cf. A249417, A249418, A249445.
%K nonn,cons
%O 0,1
%A _Jean-François Alcover_, Oct 29 2014
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