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A249417
Decimal expansion of E(T_{1,0}), the expected "first-passage" time required for an Ornstein-Uhlenbeck process to cross the level 1, given that it started at level 0.
10
2, 0, 9, 3, 4, 0, 6, 6, 4, 9, 6, 7, 8, 3, 2, 1, 8, 0, 6, 9, 2, 0, 1, 6, 1, 8, 1, 1, 2, 5, 0, 0, 8, 1, 8, 2, 8, 6, 0, 0, 5, 4, 6, 9, 0, 5, 2, 0, 7, 9, 5, 8, 5, 2, 0, 5, 3, 0, 2, 3, 7, 8, 0, 6, 6, 8, 9, 4, 7, 2, 6, 9, 5, 7, 8, 0, 3, 9, 2, 8, 1, 0, 3, 7, 5, 5, 7, 5, 9, 5, 8, 6, 6, 0, 4, 3, 1, 2, 2, 0, 5, 6, 5
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
Steven R. Finch, Ornstein-Uhlenbeck Process, May 15, 2004. [Cached copy, with permission of the author]
Michael Kopp, Elma Nassar, Etienne Pardoux, Phenotypic lag and population extinction in the moving-optimum model: insights from a small-jumps limit, Journal of Mathematical Biology (2018), Vol. 77, Issue 5, 1431-1458.
FORMULA
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.
EXAMPLE
2.09340664967832180692016181125008182860054690520795852...
MATHEMATICA
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[1, 0]], 10, 103] // First
CROSSREFS
Cf. A249418.
Sequence in context: A199287 A198735 A071120 * A189963 A156649 A197330
KEYWORD
nonn,cons
AUTHOR
STATUS
approved