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A249445
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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.
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5
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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FORMULA
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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,
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EXAMPLE
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5.8420278024190413735330352325420327086898351632336...
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MATHEMATICA
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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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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