Statistics of the first passage area functional for an Ornstein-Uhlenbeck process

We consider the area functional defined by the integral of an Ornstein-Uhlenbeck process which starts from a given value and ends at the time it first reaches zero (its equilibrium level). Exact results are presented for the mean, variance, skewness and kurtosis of the underlying area probability distribution, together with the covariance and correlation between the area and the first passage time. Amongst other things, the analysis demonstrates that the area distribution is asymptotically normal in the weak noise limit, which stands in contrast to the first passage time distribution. Various applications are indicated.


Introduction
First passage problems play a special role in the theory of stochastic processes [1].
The study of first passage functionals of Brownian motion [2] is an important subdiscipline and one which finds many applications. This includes the analysis of lattice polygons and queueing systems [3,4], snowmelt dynamics [5], DNA breathing [6], polymer translocation [7] and barrierless reaction kinetics [8]. Conditioning on a fixed first passage time provides insight into interface fluctuations [9], laser cooling of atoms [10] and colloidal suspensions [11]; the associated techniques being developed explicitly in [12,13], along with studies of functional correlations in [14,15]. A recent overview of key results, which also provides some important new insights into asymptotic behaviour, may be found in [16].
In this paper, we are interested in the first passage area functional associated with the relaxation of an Ornstein-Uhlenbeck process to zero (its equilibrium level, also known as the mean reversion level). The area functional A is defined by the time integral of the process starting from a given initial value up to the first passage time T to reach zero. Both A and T are well-defined random variables. The former arises, either directly or through natural extension, in the modelling of two-dimensional diffusion [17], random acceleration [18][19][20], inelastic collapse [21], discrete-time persistence [22] and neuron dynamics [23,24]. An understanding of the statistics of this area variable is therefore relevant to a variety of different problems.
A specific example which is easy to visualise is that of a queueing system which features balking. This term refers to when 'customers' arriving at a queue decide not to join because the queue is too long. The context can be quite general; for example, an everyday example is the road traffic jam, where balking refers to when forewarned drivers find an alternative route instead of joining the jam. At large scale the Ornstein-Uhlenbeck process is a good approximation to the evolution of the queue length of a critically balanced queue when the rate of balking is proportional to the queue length [25][26][27]. One then identifies T as the duration (or lifetime) of the queue and A as the total waiting time of all the customers involved in the queue during its lifetime, an important operational measure of overall inefficiency or 'cost'.
The first passage time probability density is known exactly for the specific problem in hand [28][29][30]; see also below. It is unlikely, however, that the area probability density has a simple, closed-form solution. To make progress, here we derive exact expressions for the mean, variance, skewness and kurtosis of A . This goes significantly beyond the account given in [17] where only the mean was derived.
The behaviour in the weak noise limit is of special interest and we shall establish that the law of A is asymptotically normal in this limit. This contrasts with the law of T which is not asymptotically normal.
The paper is structured as follows. In Section 2, the basic theory is developed, culminating in an explicit expression for the double Laplace transform of the joint probability density of A and T . In Section 3, we summarise the key results relating to the first passage time T . In Section 4, the main new results relating to the area A are presented, together with analysis of the correlation between A and T . Finally, in Section 5, conclusions are drawn.

General theory
The process we wish to study can be modelled by the Langevin equation denoting the Wiener process. The first passage time to reach zero is defined by ; ; This quantity satisfies a backward Fokker-Planck equation subject to the boundary conditions ( , , 0  (3). Inspection of (2) shows that ( 0, , ) The formal solution of (3) is Here, ( ) D z ν is a parabolic cylinder (Weber) function which satisfies [31] ( ) , which is given in Section 3.
Unfortunately, it seems quite impractical to invert (4) when 0 at least in any simple form. One can, however, make progress in evaluating various statistical quantities without carrying out the inversion. For example, the generating functions for the basic moments are given by where the connection to the Laplace transform follows from the structure of (2). In pursuing this approach, the following representation for ( ) where, after integrating the lower integral by parts, one has Written in this way, the singular behaviour as , 0 s p → has been regularised, making subsequent calculations somewhat easier. Much of what is to come is based on manipulating (6, 7) to derive formal power series whose coefficients are the statistical quantities of interest. In this context, setting 0 p = in (6, 7) is justified.

The first passage time statistics
We begin by highlighting certain key details of the statistics of the first passage time variable T . The inversion of (4) when 0 s = is known exactly, although the derivation is usually by other means [28][29][30]: An important feature of this result is the essential singularity as 0 → T . One can write the moment generating function based on (5, 6) as From this, one can expand to derive the following integral expressions for the first two moments: where 0.577... γ = is Euler's constant. As mentioned in the Introduction, an important point of focus is the behaviour in the weak noise limit, which corresponds to α → ∞ .
Based on (A4, A5) from the Appendix one has Importantly, (12,13) imply that the variance of the first passage time is bounded above by 2 T  T  T T , these asymptotes may also be derived directly using (8).
The structure of (12, 13) foreshadows the limiting form of the higher order moments, which may be obtained as follows. As α → ∞ , one has from (9) in conjunction with (A6) from the Appendix: where the constants ( ) These may be evaluated with the help of the identity [31]  ....
A little thought convinces one that these constants ( )

The area statistics
We now turn to the main topic of the paper, that of deriving the statistics of the area variable A . Noting from (5) that It follows at once that the first area moment is given by This surprisingly simple result was first identified in [17]. It is interesting to note that this is what one would obtain for the corresponding deterministic problem, i.e.  For the second central moment 2 ( ) C α A , which is simply the variance, one can quickly derive the following expression based on (7,19): The last step draws on (10). This is plotted in figure 1 and the behaviour as α → ∞ follows from (12) (18), the second area moment itself is given by One can readily show that 2 ( ) α A satisfies (17).
To characterize the distribution more fully, it is useful to study the third and fourth central moments. For the former we have from (7, 19): The asymptotic behaviour as α → ∞ is easily extracted: For the fourth central moment the analysis is slightly more demanding. One has based on (7, 19): which can be written using (11) as Based on (13) the α → ∞ behaviour is then given by One can then show using (7) and (A6) from the Appendix that 2 2 The first two terms of (25) are trivial, since 1 The important observation, as determined from (24), is that all cumulants ( ) n K α A for 3 n ≥ tend to constant values as α → ∞ , e.g., ....
Based on the following well-known relationship between cumulants and central moments: this means that for the limiting central moments one has These results confirm that the area distribution is asymptotically normal, in the sense that there is a suitably rescaled area variable Â such that which is based on the reduction of (25,26) to the Gaussian forms Having established what we set out to achieve, we conclude by considering the matter of correlations. One naturally expects a positive correlation between A and T on physical grounds; typically, the larger the first passage time is, the larger the corresponding area will be. One can study this based on (2) and using the Laplace transform (6) In turn, one can define the correlation coefficient .  [14], with the observation that the strong noise limit 0 α → is not the same in Correlation coefficient α both cases. The decay to zero as α → ∞ is a signal that, unlike the first passage time variable, the area variable is not so tightly clustered around its mean value.

Discussion
We have derived exact results for the key statistical measures relating to the distribution of the area integral of the Ornstein-Uhlenbeck process up to the first passage time to reach its equilibrium level. In addition, the correlation function between the area and the first passage time has been computed. The results should find application as they stand. One could extend the analysis relatively easily to discuss the area statistics when the first passage time relates to a level which differs from the equilibrium level. This has been done in some detail for the statistics of the first passage time itself; for a recent overview see [33,34].
The precise form of the area probability density remains an open question. By appealing to the governing differential equation which may be deduced from (3): one can study the deviations from normality as α → ∞ as well as postulate the behaviour in the extreme tails as 0 → A (essential singularity, guided by the Brownian motion result [3,16]; see also [33]) and → ∞ A (exponential decay, guided by the dominant singularity of the Laplace transform (4)): where 0.57279... φ = is the smallest positive root of 2 ( 2 ) 0 D φ φ − = [21]. In the absence of a full solution, more formal confirmation and extension of these asymptotes would be a welcome next step. One approach might be to adapt the methods developed in [16], but we do not pursue this any further here.