The OEIS is supported by the many generous donors to the OEIS Foundation.

 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-François 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 | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified August 12 03:07 EDT 2024. Contains 375085 sequences. (Running on oeis4.)