

A249417


Decimal expansion of E(T_{1,0}), the expected "firstpassage" time required for an OrnsteinUhlenbeck 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
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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, OrnsteinUhlenbeck Process, May 15, 2004. [Cached copy, with permission of the author]
Michael Kopp, Elma Nassar, Etienne Pardoux, Phenotypic lag and population extinction in the movingoptimum model: insights from a smalljumps limit, Journal of Mathematical Biology (2018), Vol. 77, Issue 5, 14311458.
Wikipedia, OrnsteinUhlenbeck 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

JeanFrançois Alcover, Oct 28 2014


STATUS

approved



