A249445 Decimal expansion of Var(T_{1,0}), the variance of the "first-passage" time required for an Ornstein-Uhlenbeck process to cross the level 1, given that it started at level 0.

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

Offset

1,1

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=1..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(a,0)) = E(T(a,0))^2 -(1/2)*sum_{k >= 1} ((sqrt(2)*a)^k*Gamma(k/2)*(psi(k/2)+gamma))/k!, where 'a' is the hit level (a=1), E(T(a,0)) the expectation A249417, and psi the digamma function,

Example

5.8420278024190413735330352325420327086898351632336...

Mathematica

digits = 100; Ex[T[a_, 0]] := (1/2)*(HypergeometricPFQ[{1, 1}, {3/2, 2}, a^2/2]*a^2 + Pi*Erfi[a/Sqrt[2]]); Var[T[a_, 0]] := Ex[T[a, 0]]^2 - (1/2)*NSum[((Sqrt[2]*a)^k*Gamma[k/2]*(PolyGamma[k/2] + EulerGamma))/k!, {k, 1, Infinity}, WorkingPrecision -> digits+5, NSumTerms -> digits]; RealDigits[Var[T[1, 0]], 10, digits] // First

Crossrefs

Cf. A249417, A249418.

Sequence in context: A248007 A212194 A212162 * A199450 A366072 A019649

Adjacent sequences: A249442 A249443 A249444 * A249446 A249447 A249448

Keyword

nonn,cons

Author

Jean-François Alcover, Oct 29 2014

Status

approved