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beekman1975asymptotic Asymptotic Distributions for the Ornstein-Uhlenbeck Process Author(s): John A. Beekman Source: Journal of Applied Probability, Vol. 12, No. 1 (Mar., 1975), pp. 107-114 Published by: Applied Probability Trust Stable URL: http://www.jstor.org/stable/3212412 . A...

beekman1975asymptotic
Asymptotic Distributions for the Ornstein-Uhlenbeck Process Author(s): John A. Beekman Source: Journal of Applied Probability, Vol. 12, No. 1 (Mar., 1975), pp. 107-114 Published by: Applied Probability Trust Stable URL: http://www.jstor.org/stable/3212412 . Accessed: 16/11/2013 06:49 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. . Applied Probability Trust is collaborating with JSTOR to digitize, preserve and extend access to Journal of Applied Probability. http://www.jstor.org This content downloaded from 210.34.5.240 on Sat, 16 Nov 2013 06:49:14 AM All use subject to JSTOR Terms and Conditions J. Appl. Prob. 12, 107-114 (1975) Printed in Israel C Applied Probability Trust 1975 ASYMPTOTIC DISTRIBUTIONS FOR THE ORNSTEIN-UHLENBECK PROCESS JOHN A. BEEKMAN, Ball State University, Muncie, Indiana Abstract This paper gives the asymptotic distributions, as the time period grows infinite, of the first exit times above a fixed constant and from upper and lower constant boundaries for the Ornstein-Uhlenbeck stochastic process. The results of a large amount of numerical analysis illustrate the asymptotic forms. ORNSTEIN-UHLENBECK PROCESS; RANDOM WALK; FIRST EXIT TIMES 1. Introduction The purpose of this paper is to give the asymptotic distributions, as t -+ 00, of the first exit times above a fixed constant and from upper and lower constant boundaries for the Ornstein-Uhlenbeck stochastic process. The Laplace trans- forms for these distributions were given by the author in [2]. These problems were also studied by Wang and Uhlenbeck (1945), Siegert (1951), Darling and Siegert (1953), Bellman and Harris (1951), Mehr and McFadden (1965), and Breiman (1966). Other papers treating similar problems for Gaussian processes in general (some of which refer to the Ornstein-Uhlenbeck process) are by Fortet (1943), Newell (1962), Pickands (1967), Mandl (1968), Orey (1970), (1972), Marcus (1972), Qualls and Watanabe (1972), Sweet and Hardin (1970), and Durbin (1971). Let {X(t),0 ? t < oo} be the Ornstein-Uhlenbeck process with transition density function p(x, s; y, t) = P[X(t) y IX(s) = x] ={2rcA(st)}exp{- [y - xexp[ - k(t - s)]]2 2A(s, t) where A(s, t) = a2{1 - exp [ - 2k(t - s)]}, and a2 > 0, k > 0. In [4] Breiman obtained an asymptotic distribution for Ta = sup{t: sup IX(s)I < a}, as t -+ + 0o. Ost Received in revised form 13 February 1974. Research partially supported by N.S.F. Grant GP-29005. 107 This content downloaded from 210.34.5.240 on Sat, 16 Nov 2013 06:49:14 AM All use subject to JSTOR Terms and Conditions 108 JOHN A. BEEKMAN Theorem 1 of [4] shows that, as t -+ + co, P[Fa > t jX(O) = 01] ,' e-'P(a) where fl(a) is determined by three conditions. In particular, f(1) = 1, and (J(3 - VJ6)) = 2. This paper will give two companion theorems. 2. First exit time distribution for an upper boundary Let E.= sup{t: sup X(s) < a}, a > 0. Theorem 1. With the above definition, as t -, + oo, P[Ea > t jX(0) = 0] = aet' + o(e(P-6)), > 0, where 0(a) is the zero in - 1 < A < 0 of f(a,).) = f (t - a)t eat-~tdt, and e = 0/( - f), where 0 = lim (A; - Pf )f tx+ 1e-' dt/f(a, ). Proof. From (5.8) of [2], P[ sup X(s) < a] = 1 - e-~" a P2 1{D-(0)/D_.-( - a)}du, where D,(z) represents a parabolic cylinder function (Weber function). (See [1], for example.) Now 1 - P[ sup X(s) < a] = P[ sup X(s) _ a] = P[Ea _ t]. Therefore P[Ea. t] = e-2 .- 1 {D-)x(O)IDx(- a)}du. By the complex inversion formula, we have Equation (E): -PE. t]f = - -_*a2 71 {D_(O)/ID_x( - a)} =e e di, > 0. 2ni ,-to D x(- a) This content downloaded from 210.34.5.240 on Sat, 16 Nov 2013 06:49:14 AM All use subject to JSTOR Terms and Conditions Asymptotic distributions for the Ornstein-Uhlenbeck process 109 Let f(A) = e-*a2D_x(O)ID_( - a). From page 343 of [19], we see that D_x(b) is an analytic function of A for Real A > - 1. So f(A) is analytic for Real A > - 1 except for zeros of Dx( - a) for Real A < 0. Since D_x(x) satisfies the self- adjoint Sturm-Liouville equation 'D"(x) + (I - x2)() = (x), and the boundary conditions lim lI- l(D(x) = 0, it follows that zeros of D_( - a) are also characteristic values of the Sturm-Liouville system. Therefore, the poles of f(A) are simple and real. Let P3(a) be the position of the largest pole. We will show that - 1 < #(a) < 0. We will now transform (E) by integrating along the contour of the rectangle with corner points y+? iT, P - 6 ? iT, y > 0, 6 positive and appropriately small, and then allow T - c oo. By Cauchy's theorem we obtain, taking account of the residue at the pole A = P of the integrand, d1 -6+ioo * P[E. < t] = e9 + 2-a i- ef(A)dA where 0 = limx,.,+ (A - )f(A). Since we will show that D_x(O)/D_( - a) is bounded on the new line of integration, the last integral is absolutely convergent, and we have the asymptotic relation dP[E, e t] Be'+ O(e• ). In general, if d P[X < x] = ce-bx, then P[X ? x] = 1 - e-bxc/b and P[X > x] = e-bxcb. Therefore, P[Ea > t] = ae"' + O(ef-j'6t) for a = 0/(- p). On page 119 of [1] will be found the following integral representation for D-x( - a): -+a2 1* - a) e eat-t2 t•-ldt, Reall > 0. Therefore, for Real A > 0, This content downloaded from 210.34.5.240 on Sat, 16 Nov 2013 06:49:14 AM All use subject to JSTOR Terms and Conditions 110 JOHN A. BEEKMAN 0 000 e -D '-(O)ID_(- a) =a j e-+t2tx-ldt/ eat-t2t•-dt. We now wish to develop an expression to continue the above into Real A < 0. For 1 > 0, use of integration by parts yields e-fa2D-(0)/D _x( - a) = J e- •t d dt ee"t-+t2t(t - a)dt. By transforming variables, one easily obtains fo0 e-t2t2+1dt = 2FT(1 + 'A) which is positive for A > - 2. Let g(A) = ea- t2 t(t )d- a)dt et2 tz+ dt for all A. For Real A > - 2, g(A) is analytic. Moreover, g(0) = 1, and g(o) monotonically approaches - coo as A - - 1. Therefore, g(A) has a unique zero in A. for - 1 < A < 0. Furthermore, 1/g(A) is the analytic continuation of f(A) from 1 > 0 to Real A > - 1. For 6 > 0 and sufficiently small, 1/g(P - 6) 1 < I l/g(P - 16) 1 < oo. 3. Numerical work for #(a) and o The zeros of f(a, A) were approximately calculated for a = 0.5, 1,2, 3. For each fixed ao, numerical integrations (described below) were performed until a A, and A2 were found such that f(ao, *) A- 0.0000k and f(ao,A2) - 0.00j with k < 3, j < 3, 1k -j I ? 1. The zero P(ao) was then approximated as R(A1 + A2). Since f(a, A) converges for A > - 1, the approach of t to 0 was handled as follows: f (a,) = lim g(t; a)dt where g(t; a) = (t - a)t"eat-+2. Moreover, there exists T(a) such that t > T(a) Sg(t;a) < 10-5. Thus T(0.5)= 6.2, T(1) = 6.75, T(2) = 8, T(3) = 9.1. It is easy to verify that g(T + k + 1;a)/g(T + k; a) < [1 + 1/(T - a)]-r-aT-0'5s for k = 0,1,2, ... This content downloaded from 210.34.5.240 on Sat, 16 Nov 2013 06:49:14 AM All use subject to JSTOR Terms and Conditions Asymptotic distributions for the Ornstein- Uhlenbeck process 111 From this relation, and Maclaurin's inequality for series, g(t; a)dt < I g(t;a) < g(T;a)[1 + 1/(T - a)]eaT-r-0.5 e- Tgk=T k=T = g(T; a)[1 + 1/(T - a)]ea-2T-051/(1 - e-). For a < 3, if T 6, [1 + 1/(T - a)]ea-2T-0'5(1 -e-) < 0.0002 and thus fg(t; a)dt < 10-. The integrals Jf1,,g(t; a)dt were approximated by the generalized Simpson's rule. The 0 limits were approximated by approaching P in steps A = - fl > 0. The terminal values 0A are listed. Even though much computer time was expended, different final values of A were needed to reach the same rough measure of stability of the 8A's. a Pf(a) OA A a 0.5 - 0.665 252 5 0.268 829 0.000 007 5 0.404 101 1.0 - 0.389 946 0 0.220 038 0.000 005 0 0.564 278 2.0 -0.097 328 0 0.099 662 0.000 003 0 1.023 981 3.0 -0.011 609 7 0.014 416 0.000 000 3 1.241 720 Mandl has approached this asymptotic distribution in a different manner in Chapter V of [8]. On page 145, he lists the characteristic roots for a = 2, 3, 4. The above values for P(2) and P(3) agree to three decimals for a = 2 and five decimals for a = 3. Mandl's numerical work does not consider the two boundary cases treated in the next two sections. 4. First exit time distribution for non-symmetric boundaries For a > 0, b > 0, a # b, let Ea.-b = sup {t: - b < inf X(s) ? sup X(s) < a}. O t X(0) = 0]. Un- fortunately, it seems very difficult to prescribe conditions on a and b such that a zero P(a, b) comparable to #(a) will exist. For example, preliminary calculations suggest P(a, b) exists for a > 2, b > 1, but not for a ? 1.5, b = 1. It therefore seems necessary to assume the existence of P(a, b) as stated below. Theorem 2. Assume that a zero (w.l.o.g. the largest) Pf(a, b) exists in - 1 < A < 0 of the function h(a, b, t) = f( - a, A)f( - b, 2) - f(a, 2)f(b, 2) for f(x, 2) defined in Theorem 1. Then, as t -+ co, P[Ea -b > t X(O) = 0] = aeP' + O(e(a-'t), 3 > 0, This content downloaded from 210.34.5.240 on Sat, 16 Nov 2013 06:49:14 AM All use subject to JSTOR Terms and Conditions 112 JOHN A. BEEKMAN for or = 0/[ - fl(a, b)] where 0 = lim (A - f)k(a, b, A)/h(a, b, A) and k(a, b, A) = f(0, ){f( - a,) - f(a, ) + f( - b, A) - f(b, A)}. Proof. Using (5.7) from [2], and techniques used in the proof of Theorem 1, d 1 fY+ioo -d=P[Ea.- b t]=27_ efY(A)dA, y > 0, dt 2if-too and f(A) = {e-jb2[D,_(a) - D_-(- a)]D_x(0) + e- +a2[D_(b) - D_x(- b)]D_x(0)}/g(a, b), where g(x, y) = [D_x(x)D_x(y) - D_x(- x)D_ x(- y)]. Again, fG() is analytic for Real A > - 1 except for zeros of the denominator for Real) A 0. It is ad- vantageous to rewrite g(x, y) as ?[D_x(y) + Dx(- y)][D_x(x)- D-x(- x)] + [D_-x(y) - Dx( - y)] [D_x(x) + D_x(- x)]. For fixed y, g(x, y) satisfies a self-adjoint Sturm-Liouville equation and boundary conditions; for fixed x, g(x, y) performs similarly. Hence, it follows that the zeros of g(x, y) are also characteristic values of either Sturm-Liouville system. Therefore, the poles of f(A) are simple and real. Let fl(a, b) be the position of the largest pole. We have assumed that - 1 < p < 0. For A > 0, f(A) can be expressed through integrals. Let g(x) equal the reciprocal of that expression. In terms of the f(a, A) from Theorem 1, f( - a, A)f( - b, A) - f(a, A)f(b, A) f(0, )[f( - a, A) -f(a,)A) +f( - b, A) -f(b, A)] Let us examine the denominator of g(A). f(- a, A) -f(a, A) = f (t + a)t e-at-a dt - f (t - a)teat-2 dt + - a (t - a)t'xea*t-it2dtJ. Each of the quantities in brackets is positive for - 1 < ) < O. Since f(0,A ) > 0 and f( - b, 2) - f(b, A) > 0, the denominator of g(2) is positive for - 1 < ) < 0. For Real) > - 1, A # 0, g(A) is analytic. We have assumed that g(A) has at least one zero for - 1 < A < 0. Furthermore, 1/g(A) is the analytic, continuation This content downloaded from 210.34.5.240 on Sat, 16 Nov 2013 06:49:14 AM All use subject to JSTOR Terms and Conditions Asymptotic distributions for the Ornstein-Uhlenbeck process 113 of f(A) from ) > 0 to Real A > - 1,) A 0. If fl, is the largest zero in - 1 < A < 0, 1 1 for 6 > 0, and sufficiently small. 5. Numerical work for #(a, b) and a It is apparent that fl(a, b) = fl(b, a). This may seem surprising for a process subject to drift. But it is explained once the drift pattern is studied more closely. Thus E{X(t) X(s)= x} = xe-k(t-s) (for the positive k of Section 1) which implies a drift downward (upward) for x > 0 (x < 0). The techniques employed earlier were used to compute the following table. a b fl(a,b) OA A aA 2.0 1.0 - 0.733 638 0.757 306 0.000 238 1.032 261 3.0 1.5 - 0.235 225 0.217 168 0.000 025 0.923 235 Because of the much greater computer time involved in computing the f(a, b)'s, A1 and A2 were found such that h(ao, bo, A!) - 0.00 k and h(ao, bo, -2) - 0.00j with k < 7, j j 6, 6k - jl - 8. Linear interpolation was used to determine the fl's. For all values of a, T's were computed such that f g(t; a)dt < 10-8. Combining Sections 3 and 5, we deduce that as t -+ co P[E2 > t] _ 1.024e-0-097t , P[E3 > t] - 1.242e-?'012t P[E2 _. > t] - 1.032e-0.734, P[E3,-1.s > t] _- 0.923e-0',235t These are in agreement with the fact that P[Ea,-b > t] < P[Ea > t] for a >0, b > 0. Acknowledgements The author gratefully acknowledges several helpful suggestions from Professor Leo Breiman, and the large volume of difficult numerical work done by Mr. Steven Jost and the Ball State University Computer Center. References [1] BATEMAN MANUSCRIPT PROJECT (1953) Higher Transcendental Functions. Vol. 2, McGraw- Hill, New York. [2] BEEKMAN, J. A. (1967) Gaussian Markov processes and a boundary value problem. Trans. Amer. Math. Soc. 126, 29-42. [3] BELLMAN, R. AND HARRIS, T. (1951) Recurrence times for the Ehrenfest model. Pacific J. Math. 1, 179-193. [4] BREIMAN, L. (1966) First exit times from a square root boundary. Proc. Fifth Berkeley Symp. Math. Statist. Prob. 2, Part 2, 9-16, University of California Press, Berkeley. This content downloaded from 210.34.5.240 on Sat, 16 Nov 2013 06:49:14 AM All use subject to JSTOR Terms and Conditions 114 JOHN A. BEEKMAN [5] DARLING, D. A. AND SIEGERT, A. J. F. (1953) The first passage problem for a continuous Markov process. Ann. Math. Statist. 24, 624-639. [6] DURBIN, J. (1971) Boundary-crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test. J. Appl. Prob. 8, 431-453. [7] FORTET, R. (1943) Les fonctions al6atoires du type de Markoff associ6es A certaines equations lineaires aux deriv6es partielles du type parabolique. J. Math. Pures Appl. 22, 177-243. [8] MANDL, P. (1968) Analytic Treatment of One-Dimensional Markov Processes. Springer- Verlag, Prague. [9] MARCUS, M. B. (1972) Upper bounds for the asymptotic maxima of continuous Gaus- sian processes. Ann. Math. Statist. 43, 522-533. [10] MEHR, C. B. AND McFADDEN, J. A. (1965) Certain properties of Gaussian processes and their first-passage times. J. R. Statist. Soc. B 27, 505-522. [11] NEWELL, G. F. (1962) Asymptotic extreme value distributions for one dimensional diffusion processes. J. Math. Mech. 11, 481-496. [12] OREY, S. (1970) Growth rate of Gaussian processes with stationary increments. Bull. Amer. Math. Soc. 76, 609-611. [13] OREY, S. (1972) Growth rate of certain Gaussian processes. Proc. Sixth Berkeley Symp. Math. Statist. Prob. University of California Press, Berkeley. [14] PICKANDS, J., III (1967) Maxima of stationary Gaussian processes. Z. Wahrschein- lichkeitsth. 7, 190-223. [15] QUALLS, C. AND WATANABE, H. (1972) Asymptotic properties of Gaussian processes. Ann. Math. Statist. 43, 580-596. [16] SIEGERT, A. J. F. (1951) On the first passage time probability problem. Phys. Rev. 81, 617-623. [17] SWEET, A. L. AND HARDIN, J. C. (1970) Solutions for some diffusion processes with two barriers. J. Appl. Prob. 7, 423-431. [18] WANG, M. C. AND UHLENBECK, G. E. (1945) On the theory of Brownian motion, II. Rev. Mod. Phys. 17, 323-342. [19] WHITTAKER, E. T. AND WATSON, G. N. (1935) A Course of Modern Analysis. Cambridge University Press, Cambridge. This content downloaded from 210.34.5.240 on Sat, 16 Nov 2013 06:49:14 AM All use subject to JSTOR Terms and Conditions Article Contents p. 107 p. 108 p. 109 p. 110 p. 111 p. 112 p. 113 p. 114 Issue Table of Contents Journal of Applied Probability, Vol. 12, No. 1 (Mar., 1975), pp. 1-217 Front Matter A Linear Birth and Death Process under the Influence of Another Process [pp. 1-17] Shock Models with Underlying Birth Process [pp. 18-28] General Stochastic Epidemic with Recovery [pp. 29-38] On the Extinction Times of Varying and Random Environment Branching Processes [pp. 39-46] A Branching Process with Disasters [pp. 47-59] On Quasi-Stationary Distributions for Multi-Type Galton-Watson Processes [pp. 60-68] The Markovian Self-Exciting Process [pp. 69-77] Some Counting and Interval Properties of the Mutually-Exciting Processes [pp. 78-86] On a Quickest Detection Problem with Costly Information [pp. 87-97] On a Class of Parabolic Differential Equations Driven by Stochastic Point Processes [pp. 98-106] Asymptotic Distributions for the Ornstein-Uhlenbeck Process [pp. 107-114] On a Continuous/Discrete Time Queueing System with Arrivals in Batches of Variable Size and Correlated Departures [pp. 115-129] Short Communications A Note on the Age-Dependent Branching Process with Immigration [pp. 130-134] Branching Processes with Varying and Random Geometric Offspring Distributions [pp. 135-141] Multivariate Exponential Distributions Based on Hierarchical Successive Damage [pp. 142-147] On the Distribution of the Inter-Record Times in an Increasing Population [pp. 148-154] Some Remarks on Probability Inequalities for Sums of Bounded Convex Random Variables [pp. 155-158] The Behavior near the Origin of the Supremum Functional in a Process with Stationary Independent Increments [pp. 159-160] Truncation Approximation of the Limit Probabilities for Denumerable Semi-Markov Processes [pp. 161-163] Distribution of Minimum Path [pp. 164-166] Some Non-Stationary Point Processes with Stationary Forward Recurrence Time Distribution [pp. 167-169] Two Dependent Poisson Processes Whose Sum Is Still a Poisson Process [pp. 170-172] A Supplement to the Strong Law of Large Numbers [pp. 173-175] On Gordin's Central Limit Theorem for Stationary Processes [pp. 176-179] Further Results on Kendall's Autoregressive Series [pp. 180-182] The Signal-Noise Problem: A Solution for the Case That Signal and Noise Are Gaussian and Independent [pp. 183-187] Weak Convergence in an Appointment System [pp. 188-194] On a Certain Type of Network of Queues [pp. 195-200] Busy Period of a Finite Queue with Phase Type Service [pp. 201-204] A Finite Dam with Variable Release Rate [pp. 205-211] Emptiness Times of a Dam with Stable Input and General Release Function [pp. 212-217] Back Matter
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