A249418 - OEIS

%I #17 Jun 11 2024 01:37:03

%S 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,

%T 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,

%U 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

%N 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.

%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="/A249418/b249418.txt">Table of n, a(n) for n = 0..5000</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)*Integral_{t=-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 0.901908012652806500639431208443776742843419260619578953963...

%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, 1]], 10, 103] // First

%Y Cf. A249417.

%K nonn,cons

%O 0,1

%A _Jean-François Alcover_, Oct 28 2014