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A250718 Decimal expansion of E(T_{2,0}), the expected "first-passage" time required for an Ornstein-Uhlenbeck process to cross the level 2, given that it started at level 0. 2
1, 0, 4, 2, 8, 4, 0, 9, 3, 9, 7, 9, 9, 5, 9, 4, 9, 0, 0, 4, 1, 5, 5, 3, 6, 6, 3, 0, 1, 1, 0, 1, 3, 5, 6, 4, 3, 1, 9, 8, 7, 4, 9, 9, 3, 2, 4, 3, 8, 6, 4, 6, 0, 5, 6, 7, 4, 7, 0, 3, 2, 3, 9, 5, 7, 0, 4, 1, 2, 4, 7, 9, 3, 3, 0, 2, 6, 2, 5, 1, 8, 3, 7, 9, 1, 4, 0, 5, 7, 7, 2, 7, 9, 8, 7, 0, 4, 5, 1, 6 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

2,3

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

G. C. Greubel, Table of n, a(n) for n = 2..10000

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

Wikipedia, Ornstein-Uhlenbeck process

FORMULA

E(T_{a,0}) = sqrt(Pi/2)*integrate_{0..a} (1 + erf(t/sqrt(2)))*exp(t^2/2) dt.

E(T_{a,0}) = (1/2)*sum_{k >= 1} (sqrt(2)*a)^k/k!*Gamma(k/2).

E(T_{a,0}) = (1/2)*(Pi*erfi(a/sqrt(2)) + a^2 * 2F2(1,1; 3/2,2; a^2/2)), where erfi is the imaginary error function, and 2F2 the hypergeometric function.

EXAMPLE

10.42840939799594900415536630110135643198749932438646...

MATHEMATICA

Ex[T[a_, 0]] := (1/2)*(Pi*Erfi[a/Sqrt[2]] + a^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, a^2/2]); RealDigits[Ex[T[2, 0]], 10, 100] // First

CROSSREFS

Cf. A249417, A249418, A250719.

Sequence in context: A026192 A026142 A095399 * A068504 A080965 A083703

Adjacent sequences:  A250715 A250716 A250717 * A250719 A250720 A250721

KEYWORD

nonn,cons

AUTHOR

Jean-Fran├žois Alcover, Nov 27 2014

STATUS

approved

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Last modified August 26 03:34 EDT 2019. Contains 326324 sequences. (Running on oeis4.)