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A250718
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Decimal expansion of E(T_{2,0}), the expected "first-passage" time required for an Ornstein-Uhlenbeck process to cross the level 2, given that it started at level 0.
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2
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1, 0, 4, 2, 8, 4, 0, 9, 3, 9, 7, 9, 9, 5, 9, 4, 9, 0, 0, 4, 1, 5, 5, 3, 6, 6, 3, 0, 1, 1, 0, 1, 3, 5, 6, 4, 3, 1, 9, 8, 7, 4, 9, 9, 3, 2, 4, 3, 8, 6, 4, 6, 0, 5, 6, 7, 4, 7, 0, 3, 2, 3, 9, 5, 7, 0, 4, 1, 2, 4, 7, 9, 3, 3, 0, 2, 6, 2, 5, 1, 8, 3, 7, 9, 1, 4, 0, 5, 7, 7, 2, 7, 9, 8, 7, 0, 4, 5, 1, 6
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OFFSET
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2,3
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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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E(T_{a,0}) = sqrt(Pi/2)*integrate_{0..a} (1 + erf(t/sqrt(2)))*exp(t^2/2) dt.
E(T_{a,0}) = (1/2)*sum_{k >= 1} (sqrt(2)*a)^k/k!*Gamma(k/2).
E(T_{a,0}) = (1/2)*(Pi*erfi(a/sqrt(2)) + a^2 * 2F2(1,1; 3/2,2; a^2/2)), where erfi is the imaginary error function, and 2F2 the hypergeometric function.
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EXAMPLE
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10.42840939799594900415536630110135643198749932438646...
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MATHEMATICA
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Ex[T[a_, 0]] := (1/2)*(Pi*Erfi[a/Sqrt[2]] + a^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, a^2/2]); RealDigits[Ex[T[2, 0]], 10, 100] // 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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