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A250719 Decimal expansion of E(T_{0,2}), the expected "first-passage" time required for an Ornstein-Uhlenbeck process to cross the level 0, given that it started at level 2. 2
1, 4, 2, 5, 2, 0, 4, 5, 6, 5, 5, 3, 7, 7, 9, 9, 7, 1, 8, 9, 5, 9, 7, 3, 6, 6, 4, 5, 6, 1, 5, 1, 2, 1, 7, 1, 2, 2, 0, 2, 3, 0, 6, 8, 5, 8, 2, 4, 0, 9, 6, 2, 5, 8, 3, 6, 3, 3, 4, 3, 4, 8, 1, 8, 2, 0, 5, 7, 3, 9, 3, 1, 9, 3, 9, 7, 6, 3, 3, 1, 7, 2, 1, 4, 3, 3, 8, 0, 4, 8, 8, 8, 7, 6, 0, 1, 0, 8, 7, 2, 6, 3, 8, 4 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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 = 1..10000

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

Wikipedia, Ornstein-Uhlenbeck process

FORMULA

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

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

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

EXAMPLE

1.42520456553779971895973664561512171220230685824...

MATHEMATICA

Ex[T[0, c_]] := (1/2)*(Pi*Erfi[c/Sqrt[2]] - c^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, c^2/2]); RealDigits[Ex[T[0, 2]], 10, 104] // First

CROSSREFS

Cf. A249417, A249418, A250718.

Sequence in context: A267184 A275927 A072907 * A211174 A059833 A123152

Adjacent sequences: A250716 A250717 A250718 * A250720 A250721 A250722

KEYWORD

nonn,cons

AUTHOR

Jean-François Alcover, Nov 27 2014

STATUS

approved

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Last modified November 30 21:57 EST 2022. Contains 358453 sequences. (Running on oeis4.)