A250721 Decimal expansion of Var(T_{0,2}), the variance of the "first-passage" time required for an Ornstein-Uhlenbeck process to cross the level 0, given that it started at level 2.

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

Offset

1,3

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=1..102.

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

1.066945439352308129579312628531952087804791333966...

Mathematica

digits = 102; 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 -> 2*digits] - Ex[T[0, c]]^2 ; RealDigits[Var[T[0, 2]], 10, digits] // First

Crossrefs

Cf. A249418, A249445, A249449, A250720.

Sequence in context: A155880 A217852 A021603 * A065480 A198116 A200023

Adjacent sequences: A250718 A250719 A250720 * A250722 A250723 A250724

Keyword

nonn,cons

Author

Jean-François Alcover, Nov 27 2014

Status

approved