A250721 - OEIS

%I #10 Jun 03 2017 15:34:38

%S 1,0,6,6,9,4,5,4,3,9,3,5,2,3,0,8,1,2,9,5,7,9,3,1,2,6,2,8,5,3,1,9,5,2,

%T 0,8,7,8,0,4,7,9,1,3,3,3,9,6,6,1,7,7,4,7,9,4,5,3,0,6,6,3,6,8,9,6,3,9,

%U 8,4,0,8,8,9,7,8,3,8,6,9,6,7,5,2,2,2,5,7,8,5,2,7,0,7,4,8,4,0,6,8,8,1

%N Decimal expansion of Var(T_{0,2}), the variance of the "first-passage" time required for an Ornstein-Uhlenbeck process to cross the level 0, given that it started at level 2.

%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 1.066945439352308129579312628531952087804791333966...

%t digits = 102; 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 -> 2*digits] - Ex[T[0, c]]^2 ; RealDigits[Var[T[0, 2]], 10, digits] // First

%Y Cf. A249418, A249445, A249449, A250720.

%K nonn,cons

%O 1,3

%A _Jean-François Alcover_, Nov 27 2014