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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.
4
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
Steven R. Finch, Ornstein-Uhlenbeck Process, May 15, 2004. [Cached copy, with permission of the author]
Eric Weisstein's MathWorld, Digamma Function
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
KEYWORD
nonn,cons
AUTHOR
STATUS
approved