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
KEYWORD
nonn,cons
AUTHOR
Jean-François Alcover, Oct 28 2014
STATUS
approved