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%I #11 Jun 03 2017 15:34:02
%S 5,8,4,2,0,2,7,8,0,2,4,1,9,0,4,1,3,7,3,5,3,3,0,3,5,2,3,2,5,4,2,0,3,2,
%T 7,0,8,6,8,9,8,3,5,1,6,3,2,3,3,6,0,9,8,7,9,3,4,8,9,6,6,7,6,4,4,1,8,5,
%U 9,3,9,6,9,1,4,2,8,7,7,4,9,2,9,2,7,3,7,1,6,5,4,2,5,6,5,5,6,9,3,6
%N Decimal expansion of Var(T_{1,0}), the variance of the "first-passage" time required for an Ornstein-Uhlenbeck process to cross the level 1, given that it started at level 0.
%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(a,0)) = E(T(a,0))^2 -(1/2)*sum_{k >= 1} ((sqrt(2)*a)^k*Gamma(k/2)*(psi(k/2)+gamma))/k!, where 'a' is the hit level (a=1), E(T(a,0)) the expectation A249417, and psi the digamma function,
%e 5.8420278024190413735330352325420327086898351632336...
%t digits = 100; Ex[T[a_, 0]] := (1/2)*(HypergeometricPFQ[{1, 1}, {3/2, 2}, a^2/2]*a^2 + Pi*Erfi[a/Sqrt[2]]); Var[T[a_, 0]] := Ex[T[a, 0]]^2 - (1/2)*NSum[((Sqrt[2]*a)^k*Gamma[k/2]*(PolyGamma[k/2] + EulerGamma))/k!, {k, 1, Infinity}, WorkingPrecision -> digits+5, NSumTerms -> digits]; RealDigits[Var[T[1, 0]], 10, digits] // First
%Y Cf. A249417, A249418.
%K nonn,cons
%O 1,1
%A _Jean-François Alcover_, Oct 29 2014
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