A250719 - OEIS

%I #12 Jun 03 2017 15:34:32

%S 1,4,2,5,2,0,4,5,6,5,5,3,7,7,9,9,7,1,8,9,5,9,7,3,6,6,4,5,6,1,5,1,2,1,

%T 7,1,2,2,0,2,3,0,6,8,5,8,2,4,0,9,6,2,5,8,3,6,3,3,4,3,4,8,1,8,2,0,5,7,

%U 3,9,3,1,9,3,9,7,6,3,3,1,7,2,1,4,3,3,8,0,4,8,8,8,7,6,0,1,0,8,7,2,6,3,8,4

%N Decimal expansion of E(T_{0,2}), the expected "first-passage" time required for an Ornstein-Uhlenbeck process to cross the level 0, given that it started at level 2.

%C 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.

%H G. C. Greubel, <a href="/A250719/b250719.txt">Table of n, a(n) for n = 1..10000</a>

%H Steven R. Finch, <a href="/A249417/a249417.pdf">Ornstein-Uhlenbeck Process</a>, May 15, 2004. [Cached copy, with permission of the author]

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Ornstein%E2%80%93Uhlenbeck_process">Ornstein-Uhlenbeck process</a>

%F E(T_{0,c}) = sqrt(Pi/2)*integrate_{-c..0} (1 + erf(t/sqrt(2)))*exp(t^2/2) dt.

%F E(T_{0,c}) = (1/2)*sum_{k >= 1} (-1)^(k+1)*(sqrt(2)*a)^k/k!*Gamma(k/2).

%F 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.

%e 1.42520456553779971895973664561512171220230685824...

%t 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

%Y Cf. A249417, A249418, A250718.

%K nonn,cons

%O 1,2

%A _Jean-François Alcover_, Nov 27 2014