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.

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

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: A377647 A275927 A072907 * A211174 A367574 A059833

Adjacent sequences: A250716 A250717 A250718 * A250720 A250721 A250722

Keyword

nonn,cons

Author

Jean-François Alcover, Nov 27 2014

Status

approved