login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A249417 Decimal expansion of E(T_{1,0}), the expected "first-passage" time required for an Ornstein-Uhlenbeck process to cross the level 1, given that it started at level 0. 10
2, 0, 9, 3, 4, 0, 6, 6, 4, 9, 6, 7, 8, 3, 2, 1, 8, 0, 6, 9, 2, 0, 1, 6, 1, 8, 1, 1, 2, 5, 0, 0, 8, 1, 8, 2, 8, 6, 0, 0, 5, 4, 6, 9, 0, 5, 2, 0, 7, 9, 5, 8, 5, 2, 0, 5, 3, 0, 2, 3, 7, 8, 0, 6, 6, 8, 9, 4, 7, 2, 6, 9, 5, 7, 8, 0, 3, 9, 2, 8, 1, 0, 3, 7, 5, 5, 7, 5, 9, 5, 8, 6, 6, 0, 4, 3, 1, 2, 2, 0, 5, 6, 5 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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 = 1..5000

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

Michael Kopp, Elma Nassar, Etienne Pardoux, Phenotypic lag and population extinction in the moving-optimum model: insights from a small-jumps limit, Journal of Mathematical Biology (2018), Vol. 77, Issue 5, 1431-1458.

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

2.09340664967832180692016181125008182860054690520795852...

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[1, 0]], 10, 103] // First

CROSSREFS

Cf. A249418.

Sequence in context: A199287 A198735 A071120 * A189963 A156649 A197330

Adjacent sequences:  A249414 A249415 A249416 * A249418 A249419 A249420

KEYWORD

nonn,cons

AUTHOR

Jean-Fran├žois Alcover, Oct 28 2014

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified February 18 04:48 EST 2020. Contains 332011 sequences. (Running on oeis4.)