A249449 Decimal expansion of Var(T_{0,1}), the variance of the "first-passage" time required for an Ornstein-Uhlenbeck process to cross the level 0, given that it started at level 1.

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

Offset

0,1

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

Table of n, a(n) for n=0..100.

Steven R. Finch, Ornstein-Uhlenbeck Process, May 15, 2004. [Cached copy, with permission of the author]

Eric Weisstein's MathWorld, Digamma Function

Wikipedia, Ornstein-Uhlenbeck process

Formula

Var(T(0,c)) = (1/2)*sum_{k >= 1} ((sqrt(2)*c)^k*Gamma(k/2)*(psi(k/2)+gamma))/k! - E(T(0,c))^2, where 'c' is the start level (c=1), E(T(0,c)) the expectation A249418, and psi the digamma function,

Example

0.8510837032793596531112279067283822476964862279647417977...

Mathematica

digits = 101; Ex[T[0, c_]] := (1/2)*(Pi*Erfi[c/Sqrt[2]] - c^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, c^2/2]); Var[T[0, c_]] := (1/2)*NSum[(-1)^k*((Sqrt[2]*c)^k*Gamma[k/2]*(PolyGamma[k/2] + EulerGamma))/k!, {k, 1, Infinity}, WorkingPrecision -> digits + 5, NSumTerms -> digits] - Ex[T[0, c]]^2 ; RealDigits[Var[T[0, 1]], 10, digits] // First

Crossrefs

Cf. A249417, A249418, A249445.

Sequence in context: A073448 A073745 A086730 * A357466 A334496 A258988

Adjacent sequences: A249446 A249447 A249448 * A249450 A249451 A249452

Keyword

nonn,cons

Author

Jean-François Alcover, Oct 29 2014

Status

approved