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A250720 Decimal expansion of Var(T_{2,0}), the variance of the "first-passage" time required for an Ornstein-Uhlenbeck process to cross the level 2, given that it started at level 0. 1
1, 0, 5, 2, 7, 5, 2, 0, 3, 5, 4, 8, 8, 1, 8, 3, 9, 1, 5, 1, 2, 2, 7, 2, 5, 3, 2, 7, 7, 2, 1, 2, 3, 2, 1, 6, 5, 4, 8, 6, 7, 0, 0, 2, 5, 1, 9, 5, 2, 5, 2, 4, 8, 1, 2, 9, 0, 4, 3, 7, 2, 6, 0, 2, 1, 6, 6, 5, 5, 3, 1, 0, 7, 2, 7, 1, 0, 2, 8, 7, 9, 5, 5, 7, 3, 0, 2, 9, 5, 0, 7, 1, 5, 7, 4, 1, 1, 7, 5, 4, 2, 6, 6, 4 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

3,3

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=3..106.

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(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,

EXAMPLE

105.27520354881839151227253277212321654867002519525248129...

MATHEMATICA

digits = 104; 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 -> 2*digits]; RealDigits[Var[T[2, 0]], 10, digits] // First

CROSSREFS

Cf. A249417, A249445, A249449, A250721.

Sequence in context: A088520 A084340 A065925 * A080350 A204899 A253545

Adjacent sequences:  A250717 A250718 A250719 * A250721 A250722 A250723

KEYWORD

nonn,cons

AUTHOR

Jean-François Alcover, Nov 27 2014

STATUS

approved

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Last modified August 11 06:25 EDT 2020. Contains 336422 sequences. (Running on oeis4.)