login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A250720 Decimal expansion of Var(T_{2,0}), the variance of the "first-passage" time required for an Ornstein-Uhlenbeck process to cross the level 2, given that it started at level 0. 1

%I #11 Jun 03 2017 15:33:16

%S 1,0,5,2,7,5,2,0,3,5,4,8,8,1,8,3,9,1,5,1,2,2,7,2,5,3,2,7,7,2,1,2,3,2,

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

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

%N Decimal expansion of Var(T_{2,0}), the variance of the "first-passage" time required for an Ornstein-Uhlenbeck process to cross the level 2, given that it started at level 0.

%C 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.

%H Steven R. Finch, <a href="/A249417/a249417.pdf">Ornstein-Uhlenbeck Process</a>, May 15, 2004. [Cached copy, with permission of the author]

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/DigammaFunction.html">Digamma Function</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Ornstein%E2%80%93Uhlenbeck_process">Ornstein-Uhlenbeck process</a>

%F 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,

%e 105.27520354881839151227253277212321654867002519525248129...

%t digits = 104; 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 -> 2*digits]; RealDigits[Var[T[2, 0]], 10, digits] // First

%Y Cf. A249417, A249445, A249449, A250721.

%K nonn,cons

%O 3,3

%A _Jean-François Alcover_, Nov 27 2014

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 09:04 EDT 2024. Contains 371240 sequences. (Running on oeis4.)