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A250721 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. 1
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, 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..102.

Steven R. Finch, Ornstein-Uhlenbeck Process, May 15, 2004. [Cached copy, with permission of the author]

Eric Weisstein's MathWorld, Digamma Function

Wikipedia, Ornstein-Uhlenbeck process

FORMULA

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,

EXAMPLE

1.066945439352308129579312628531952087804791333966...

MATHEMATICA

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

CROSSREFS

Cf. A249418, A249445, A249449, A250720.

Sequence in context: A155880 A217852 A021603 * A065480 A198116 A200023

Adjacent sequences:  A250718 A250719 A250720 * A250722 A250723 A250724

KEYWORD

nonn,cons

AUTHOR

Jean-Fran├žois Alcover, Nov 27 2014

STATUS

approved

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Last modified September 26 15:09 EDT 2022. Contains 357000 sequences. (Running on oeis4.)