A249418 Decimal expansion of E(T_{0,1}), the expected "first-passage" time required for an Ornstein-Uhlenbeck process to cross the level 0, given that it started at level 1.

9, 0, 1, 9, 0, 8, 0, 1, 2, 6, 5, 2, 8, 0, 6, 5, 0, 0, 6, 3, 9, 4, 3, 1, 2, 0, 8, 4, 4, 3, 7, 7, 6, 7, 4, 2, 8, 4, 3, 4, 1, 9, 2, 6, 0, 6, 1, 9, 5, 7, 8, 9, 5, 3, 9, 6, 3, 1, 9, 6, 5, 0, 2, 5, 3, 0, 0, 6, 9, 3, 5, 3, 5, 4, 6, 4, 0, 8, 0, 8, 6, 6, 5, 7, 5, 1, 5, 8, 5, 3, 5, 4, 7, 8, 8, 9, 8, 3, 1, 1, 1, 4, 2

Offset

0,1

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 = 0..5000

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)*Integral_{t=-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

0.901908012652806500639431208443776742843419260619578953963...

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, 1]], 10, 103] // First

Crossrefs

Cf. A249417.

Sequence in context: A277524 A118811 A200488 * A256036 A065471 A310000

Adjacent sequences: A249415 A249416 A249417 * A249419 A249420 A249421

Keyword

nonn,cons

Author

Jean-François Alcover, Oct 28 2014

Status

approved