OFFSET
1,2
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..10000
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)*integrate_{-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
1.42520456553779971895973664561512171220230685824...
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, 2]], 10, 104] // First
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Jean-François Alcover, Nov 27 2014
STATUS
approved