First passage times for Slepian process with linear and piecewise linear barriers

In this paper, we derive explicit formulas for the first-passage probabilities of the process S(t) = W(t) − W(t + 1), where W(t) is the Brownian motion, for linear and piece-wise linear barriers on arbitrary intervals [0,T]. Previously, explicit formulas for the first-passage probabilities of this process were known only for the cases of a constant barrier or T ≤ 1. The first-passage probabilities results are used to derive explicit formulas for the power of a familiar test for change-point detection in the Wiener process.

This process is often called Slepian process and can be expressed in terms of the standard Brownian motion W (t) by (1.1) the authors show applications in queuing theory. Another important application of the Karlin-McGregor identity deals with finding boundary crossing probabilities for various scan statistics, see Naus (1982), Glaz et al. (2009), and Noonan and Zhigljavsky (2020). The structure of the paper is as follows. In Section 2.2, we provide an expression for F a,b (T | x) for integer T and in Section 2.4 we extend the results for non-integer T . In Sections 3 and 4, we extend the results to the case of piecewise-linear barriers. In Section 5, we outline an application to a change-point detection problem; this application was our main motivation for this research. In the Appendix, we provide detailed proofs of all theorems.

Linear barrier a + bt
The key result of this section is Theorem 1, where an explicit formula is derived for the first-passage probability F a,b (T | x) defined in Eq. 1.2 under the assumption that T is a positive integer, T = n. First, we formulate a lemma that is key to the advances of this paper and can be obtained from Katori (2011, p. 5) or Katori (2012, p.40). In this lemma, we use the notation ϕ s (z) := 1 √ 2πs e −z 2 /(2s) (2.1) for the normal density with variance s. For the standard Brownian motion process W (t), ϕ s (a − c)dc = Pr(W (s) ∈ dc | W (0) = a) is the transition probability. We shall also use W n+1 = {x = (x 0 , . . . , x n ) ∈ R n+1 : x 0 < x 1 < . . . < x n } for the so-called Weyl chamber of type A n , see Fulton and Harris (2013) for details.
Lemma 1 is an extension of the Karlin-McGregor identity of (1959), when applied specifically to the Brownian motion, and accommodates for different drift parameters μ i of W i (t).

Corollary 1 Under the same assumptions as Lemma 1, we have
Proof Denote the transition density for the process W i (t) by ϕ s,μ i (a − c); that is, Using the relation ϕ s,μ i (a − c) = ϕ s (a − c + μ i s) and dividing both sides of Eq. 2.2 by Pr(W μ (s) ∈ dc | W μ (0) = a), we obtain the result.

Linear barrier a + bt with integer T
Let ϕ(t) = ϕ 1 (t) and Φ(t) = t −∞ ϕ(u)du be the density and the c.d.f. of the standard normal distribution. Assume that T = n is a positive integer. Define and let μ i , a i and c i be i-th components of vectors μ, a and c respectively (i = 0, 1, . . . , n). Note that we start the indexation of vector components at 0.
Theorem 1 For any integer n ≥ 1 and x < a, where μ, a and c are given in Eq. 2.4.
Theorem 1 is a special case of Theorem 3 with (using the notation of Theorem 3) n = T and T = 0. Theorem 1 is formulated as a separate theorem as it is the first natural extension of Shepp's results of (1971). Indeed, if b = 0 then μ = 0 and Eq. 2.5 coincides with Shepp's formula (2.15) in (1971) expressed in the variables y i = x i + ia (i = 0, 1, . . . , n).

Linear barrier a + bt with non-integer T
In this section, we shall provide an explicit formula for the first-passage probability F a,b (T | x) defined in Eq. 1.2 assuming T > 0 is not an integer. Represent T as T = m + θ, where m = T ≥ 0 is the integer part of T and 0 < θ < 1. Set n = m + 1 = T . Let ϕ θ (t) and ϕ 1−θ (t) be as defined in Eq. 2.1. Define the (n + 1)-and n-dimensional vectors as follows: μ 1 = μ is as defined in Eq. 2.4, and let a 1i and c 1i be i-th components of vectors a 1 and c 1 respectively (i = 0, 1, . . . , n). Similarly, let a 2i and c 2i be i-th components of vectors a 2 and c 2 respectively (i = 0, 1, . . . , m). Recall that we start the indexation of vector components at 0.

Theorem 2 For x < a and non-integer
A proof of Theorem 2 is provided in Appendix A.1. If b = 0 then the above formula for F a,b (T | x) coincides with Shepp's formula (2.25) in Shepp (1971) expressed in variables x i = u i +ia and y i = v i +ia (i = 0, 1, . . . , n). For m = 0 and hence T = θ, Theorem 2 agrees with results in Zhigljavsky and Kraskovsky (1988), Bischoff and Gegg (2016), and Deng (2017).

Boundary crossing probability
In this section, we provide an explicit formula for the first-passage probability for S(t) with a continuous piecewise linear barrier, where not more than one change of slope is allowed. For any non-negative T , T and real a, b, b we define the piecewise- for an illustration of this barrier, see Fig. 1. We are interested in finding an expression for the first-passage probability We only consider the case when both T and T are integers. The case of general T , T can be treated similarly but the resulting expressions are much more complicated.
Define the (T + T +1)-dimensional vectors as follows: and let a 3i and c 3i be i-th components of vectors a 3 and c 3 respectively (i = 0, 1, . . . , T + T ).

Theorem 3 For x < a and any positive integers T and T , we have
The proof of Theorem 3 is included in the appendix, see Appendix A.2. Note that if b = b then Eq. 3.4 reduces to Eq. 2.5 with n = T + T .

Two particular cases of Theorem 3
Below we consider two particular cases of Theorem 3; first, the barrier is See Figs. 2 and 3 for a depiction of both barriers. As we demonstrate in Section 5, these cases are important for problems of change-point detection.

Boundary crossing probability
Theorem 3 can be generalized to the case when we have more than one change in slope. In the general case, the formulas for the first-passage probability become very complicated; they are already rather heavy in the case of one change in slope.
In this section, we consider just one particular barrier with two changes in slope.
As will be explained in Section 5, the corresponding first-passage probability is important for some change-point detection problems.
Define the four-dimensional vectors as follows: and let a 4i and c 4i be i-th components of vectors a 4 and c 4 respectively (i = 0, 1, 2, 3).

Theorem 4 For any real
For the proof of Theorem 4, see Appendix A.3.

A particular case of Theorem 4
In this section, we consider a special barrier B(t; h, 0, −μ, μ) (depicted in Fig. 4), which will be used in Section 5. In the notation of Theorem 4, a = h, b = 0, b = −μ, b = μ and we obtain

Another linear barrier with two changes in slope
For real h and μ, define the barrier B(t; h, 0, 0, −μ, μ) by A. Zhigljavsky, J. Noonan The barrier B(t; h, 0, 0, −μ, μ) looks similar to the barrier depicted in Fig. 4, except the constant part is two units long. The corresponding first-passage probability will be important in Section 5.

Theorem 5 For any real h, μ and x < h
The proof of Theorem 5 is very similar to the proof of Theorem 4.

Formulation of the problem
In this section, we illustrate the natural appearance of the first-passage probabilities for the Slepian process S(t) for piece-wise linear barriers and in particular the barriers considered in Sections 3.2 and 4.2. Suppose one can observe the stochastic process X(t) (t ≥ 0) governed by the stochastic differential equation (5.1) where ν > 0 is the unknown (non-random) change-point and μ = 0 is the drift magnitude during the 'epidemic' period of duration l with 0 < l < ∞; μ and l may be known or unknown. The classical change-point detection problem of finding a change in drift of a Wiener process is the problem Eq. 5.1 with l = ∞; that is, when the change (if occurred) is permanent, see for example (Pollak and Siegmund 1985;Moustakides 2004;Polunchenko 2018;Polunchenko and Tartakovsky 2010).
In Eq. 5.1, under the null hypothesis H 0 , we assume ν = ∞ meaning that the process dX(t) has zero mean for all t ≥ 0. On the other hand, under the alternative hypothesis H 1 , ν < ∞. In the definition of the test power, we will assume that ν is large. However, for the tests discussed below to be well-defined and approximations to be accurate, we only need ν ≥ 1 (under H 1 ).
In this section, we only consider the case of known l, in which case we can assume l = 1 (otherwise we change the time-scale by t → t/ l and the barrier by B → B/ √ l). When testing for an epidemic change on a fixed interval [0, T ] with l unknown, one possible approach is to construct the test statistic on the base of max 0<s<t<T [W (t) − W (s)], the maximum over all possible choices of l and locations. This idea was discussed in Siegmund (1986), where asymptotic approximations are offered. The case when l is unknown is more complicated and the first-passage probabilities that have to be used are more involved.
We define the test statistic used to monitor the epidemic alternative as The stopping rule for S 1 (t) is defined as follows where the threshold h is chosen to satisfy the average run length (ARL) constraint E 0 (τ (h)) = C for some (usually large) fixed C (here E 0 denote the expectation under the null hypothesis). Since l is known, for any μ > 0 the test with the stopping rule Eq. 5.2 is optimal in the sense of the Abstract Neyman-Pearson lemma, see Theorem 2, Grenander (1981, p. 110).
is stochastically equivalent to the Slepian process S(t) of Eq. 1.1. Under H 0 , ES 1 (t) = 0 for all t ≥ 0 and under H 1 we have

Approximation for E 0 (τ (h ))
The problem of construction of accurate approximations for E 0 (τ (h)) was addressed in Noonan and Zhigljavsky (2019). For completeness, we briefly review the approach.
Consider the unconditional probability (taken with respect to the standard normal distribution): Under H 0 , the distribution of τ (h) has the form (1 − Φ(h))δ 0 (ds) + q h (s)ds , s ≥ 0, where δ 0 (ds) is the delta-measure concentrated at 0 and is the first-passage density. This yields There is no easy computationally convenient formula for q h (t) as expressions for F h,0 (s) are very complex. One of the simplest (yet very accurate) approximation for F h,0 (s) takes the form: (1). Using Eq. 5.4, we approximate the density q h (s) by Subsequent evaluation of the integral in Eq. 5.3 yields the approximation . (5.5) Numerical study shows that the approximation Eq. 5.5 is very accurate for all h ≥ 3. Setting h = 3.63 in Eq. 5.5 results in C 500.

Approximating the power of the test
In this section we formulate several approximations for the power of the test Eq. 5.2 which can be defined as (5.6) where P 1 denotes the probability measure under the alternative hypothesis. Define the piecewise linear barrier Q ν (t; h, μ) as follows The barrier Q ν (t; h, μ) is visually depicted in Fig. 5. The power of the test with the stopping rule Eq. 5.2 is then First passage times for Slepian process...

Fig. 5 Graphical depiction of the boundary Q ν (t; h, μ)
Consider the barrier B(t; h, 0, −μ, μ) of Section 4 with t ∈ [0, 3]. Define the conditional first-passage probability The denominator in Eq. 5.7 is very simple to compute, see Eq. 2.6 with b = 0 and a = h. The numerator in Eq. 5.7 can be computed by Eq. 4.4. Computation of γ 3 (x, h, μ) requires numerical evaluation of a two-dimensional integral, which is not difficult.
To assess the impact of the final line-segment in the barrier B(t; h, 0, −μ, μ) on the power (the line-segment with gradient μ in Fig 5, t ∈ Then we make the approximation P(h, μ) γ 1 (0, h, μ), where the quantity F h,0,−μ (1, 1| 0) can be computed using Eq. 3.6 with b = μ. The denominator can be computed using Eq. 2.6 with b = 0 and a = h.
In Table 1, we provide values of P(h, μ), γ 2 (0, h, μ) and γ 1 (0, h, μ) for different μ, where the values of h have been chosen to satisfy E 0 (τ (h)) = C for C = 100, 500, 1000; see Eq. 5.5 regarding computation of the ARL E 0 (τ (h)). Since the values in Table 1 are given to three decimal places, these values of P(h, μ) can be obtained from either γ 3 (0, h, μ) or γ 4 (0, h, μ); both of these two approximations provide a better accuracy than 3 decimal places. Comparing the entries of Table 1 we can observe that the quality of the approximation P(h, μ) γ 2 (0, h, μ) is rather good, especially for large μ. By looking at the columns corresponding to γ 1 (0, h, μ), one can also see the expected diminishing impact which the final line-segment in B(t; h, 0, −μ, μ) has on power, as μ increases. However, for small μ the contribution of this part of the barrier to power is significant suggesting it is not be sensible to approximate the power of our test with γ 1 (0, h, μ).
To summarize the results of this section, for approximating the power function P(h, μ), we propose one the following two approximations: a very accurate approximation γ 3 (0, h, μ) requiring numerical evaluation of a two-dimensional integral and γ 2 (0, h, μ), a less accurate but simpler approximation requiring evaluation of a one-dimensional integral only. The approximation P(h, μ) γ 4 (0, h, μ) is extremely accurate but too costly whereas the approximation γ 1 (0, h, μ) is less accurate than γ 2 (0, h, μ) but slightly cheaper, requiring the numerical evaluation of a two-dimensional integral. The approximation P(h, μ) γ 1 (0, h, μ) has been studied mainly for assessing the impact which the final line-segment in B(t; h, 0, −μ, μ) has on the power. Then the event Ω can be equivalently expressed as Ω = Ω 1 ∩ Ω 2 with Under the conditioning introduced in Eq. A.1 we have for i = 0, 1, . . . , m + 1 and j = 0, 1, . . . , m: Now under the above conditioning, the processes are independent and so the conditional probability of Ω in Eq. A.1 becomes a product of the conditional probabilities of Ω 1 and Ω 2 . Therefore, Eq. A.1 becomes ×Pr {W (0) ∈ du 0 , . . . , W (m + 1) ∈ du m+1 , W (θ) ∈ dv 0 , . . . , The region of integration for the variables u i in Eq. A.2 is determined from the following chain of inequalities: Whence, the upper limit of integration with respect to u i+1 is infinity and the lower limit for the integral with respect to u i+1 , i = 1, . . . , m is given by the formula u i −a−ib. For the variables v j in Eq. A.2, we have the following chain of inequalities v 0 < v 1 +a +bθ < . . .< v m +m(a + bθ)+ (m − 1)m 2 b < v m+1 +(m + 1)(a + bθ)+ (m + 1)m 2 b .
Once again, the upper limit of integration with respect to v i+1 is infinity and the lower limit for the integral with respect to v i+1 (i = 0, . . . , m) is For v 0 , the upper and lower limits of integration are infinite. Now using Eq. 2.3 with n = m + 1 we obtain where ϕ θ (·) is given in Eq. 2.1, a 1 and c 1 are given in Eq. 2.7. Similarly, using Eq. 2.3 with n = m we have where ϕ 1−θ (·) is given in Eq. 2.1, a 2 and c 2 are given in Eq. 2.8. The third probability in the right-hand side of Eq. A.2 is simply and collating all terms, we obtain the result.
The conditioning introduced in Eq. A.6 results in: From this, we can express Eq. A.6 as (A.7) The region of integration for Eq. A.7 is determined from the following inequalities (see proof of Eq. 2.5 for similar discussion): x 1 < x 2 + a + b < x 3 + 2a + 2b + b < x 4 + 3a + 3b + 2b + b .
Thus, the upper limit of integration is infinity for all x i . For integration with respect to x 4 , the lower limit is x 3 − a − b − b − b . For integration with respect x 3 , the lower limit is x 2 − a − b − b . Finally, for x 2 , the lower limit is x 1 − a − b = −x − a − b. Now using Eq. 2.3 with n = 3 we obtain Pr Ω|W 0 (0) = 0, . . . , W 3 (0) = 3a + 2b + b + x 3 , W 0 (1) μ 4 , a 4 and c 4 are given in Eq. 4.2. The second probability in the right-hand side of Eq. A.7 is 3 i=1 ϕ(x i − x i+1 )dx i+1 . Using the fact and collecting all results we complete the proof.