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

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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
 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 (list; constant; graph; refs; listen; history; text; internal format)
 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

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 7 08:03 EDT 2024. Contains 375008 sequences. (Running on oeis4.)