On the sequential testing and quickest change-pointdetection problems for Gaussian processes

Transcription

1 Pavel V. Gapeev and Yavor I. Stoev On the sequential testing and quickest change-pointdetection problems for Gaussian processes Article Accepted version Refereed Original citation: Gapeev, Pavel V. and Stoev, Yavor I. 17. On the sequential testing and quickest changepointdetection problems for Gaussian processes. Stochastics: an International Journal of Probability and Stochastic Processes. ISSN DOI: 1.18/ Informa UK Limited This version available at: Available in LSE Research Online: November 17 LSE has developed LSE Research Online so that users may access research output of the School. Copyright and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. Users may download and/or print one copy of any articles in LSE Research Online to facilitate their private study or for non-commercial research. You may not engage in further distribution of the material or use it for any profit-making activities or any commercial gain. You may freely distribute the URL of the LSE Research Online website. This document is the author s final accepted version of the journal article. There may be differences between this version and the published version. You are advised to consult the publisher s version if you wish to cite from it.

2 To appear in Stochastics: An International Journal of Probability and Stochastic Processes pp. On the sequential testing and quickest change-point detection problems for Gaussian processes Pavel V. Gapeev Yavor I. Stoev We study the sequential hypothesis testing and quickest change-point disorder detection problems with linear delay penalty costs for certain observable time-inhomogeneous Gaussian diffusions and fractional Brownian motions. The method of proof consists of the reduction of the initial problems into the associated optimal stopping problems for onedimensional time-inhomogeneous diffusion processes and the analysis of the associated free boundary problems for partial differential operators. We derive explicit estimates for the Bayesian risk functions and optimal stopping boundaries for the associated weighted likelihood ratios and obtain their exact asymptotic growth rates under large time values. 1 Introduction The problems of statistical sequential analysis seek to determine the distributional properties of continuously observable stochastic processes with minimal costs. The problem of sequential testing for two simple hypotheses about the drift rate of an observable Gaussian process is to detect the form of its drift rate from one of the two given alternatives. In the Bayesian formulation of this problem, it is assumed that these alternatives have an a priori given distribution. The problem of quickest change-point or disorder detection for an observable Gaussian process is to find a stopping time of alarm τ which is as close as possible to the unknown time of change-point θ at which the local drift rate of the process changes from one form to another. In the classical Bayesian formulation, it is assumed that the random time θ takes the value with probability π and is exponentially distributed given that θ >. Such problems found London School of Economics, Department of Mathematics, Houghton Street, London WCA AE, United Kingdom; p.v.gapeev@lse.ac.uk University of Michigan, Department of Mathematics, 53 Church Street, Ann Arbor, MI 4819, United States; ystoev@umich.edu Mathematics Subject Classification 1: Primary 6G4, 6J6, 34K1. Secondary 6M, 6C1, 6L15. Key words and phrases: Sequential testing, quickest change-point disorder detection, likelihood ratio, time-inhomogeneous diffusion process, optimal stopping problem, curved boundary, free boundary problem, a change-of-variable formula with local time on curves, fractional Brownian motion, fundamental martingale. Date: 16 Jan 17 1

3 applications in many real-world systems in which the amount of observation data is increasing over time see, e.g. Carlstein, Müller, and Siegmund [5] for an overview. The sequential testing and quickest change-point detection problems were originally formulated and solved for sequences of observable independent identically distributed random variables see, e.g. Shiryaev [4, Chapter IV, Sections 1 and 3]. The first solutions of these problems in the continuous-time setting were obtained in the case of observable Wiener processes with constant drift rates see Shiryaev [4, Chapter IV, Sections and 4]. The standard disorder problem for observable Poisson processes with unknown intensities was introduced and solved in Davis [9], under certain restrictions on the model parameters. Peskir and Shiryaev [34, 35] solved both problems of sequential analysis for Poisson processes in full generality see also [36, Chapter VI, Sections 3 and 4]. The method of solution of these problems was based on the reduction of the associated optimal stopping problems to the equivalent free boundary problems for ordinary integro-differential operators and a unique characterization of the Bayesian risks by means of the smooth and continuous fit conditions for the value functions at the optimal stopping boundaries. Further investigations of both problems for observable Wiener processes were studied in [18, 19] in the finite horizon setting, and for certain more general time-homogeneous diffusions in [, 1] on infinite time intervals. These two classical problems of sequential analysis for the case of observable compound Poisson processes, in which the unknown characteristics were the intensities and distributions of jumps, were investigated in Dayanik and Sezer [11, 1]. Some multidimensional extensions of the problems with several observable independent compound Poisson and Wiener processes were considered in Dayanik, Poor, and Sezer [1] and in Dayanik and Sezer [13]. Other formulations of the change-point detection problem for Poisson processes for various types of probabilities of false alarms and delay penalty costs were studied in Bayraktar, Dayanik, and Karatzas [1]. More general versions of the standard Poisson disorder problem were solved by Bayraktar, Dayanik, and Karatzas [], where the intensities of the observable processes changed to certain unknown values. These problems for observable jump processes were solved by successive approximations of the value functions of the corresponding optimal stopping problems. The same method was also applied for the solution of the disorder problem for observable Wiener process in Sezer [38], in which disorder occurs at one of the arrival times of an observable Poisson process. The aim of this paper is to address these problems of statistical sequential analysis in their Bayesian formulations for certain Gaussian processes with non-stationary increments. We formulate a unifying optimal stopping problem for the appropriate time-inhomogeneous diffusion likelihood ratio processes and show that the optimal stopping times are the first times at which these processes exit from certain regions restricted by time-dependent curved boundaries. It is verified that the value functions and the stopping boundaries provide a unique solution of the equivalent free boundary problem for a partial differential operator of parabolic type. Since the latter problem does not admit an explicit solution, we consider an associated auxiliary ordinary differential free boundary problem in which the time variable plays the role of a parameter. We derive analytic expressions for the value functions and optimal boundaries in the auxiliary problem and specify their exact asymptotic behavior under large time values. The resulting solutions determine the asymptotic growth rates of the value functions and optimal stopping boundaries in the original time-inhomogeneous problems. The paper is organized as follows. In Section, we formulate a unifying optimal stopping problem for the time-inhomogeneous diffusion likelihood ratio processes and show how this

4 problem arises from the Bayesian sequential testing and quickest change-point detection settings. We formulate an equivalent free boundary problem and derive explicit solutions of the auxiliary ordinary free boundary problems which have the time variable as a parameter. In Section 3, we study the asymptotic behavior of the resulting estimates for stopping boundaries under large time values, by means of deriving their Taylor expansions with respect to the local drift rate of the observable processes. These estimates determine the upper rates of decrease and the lower rates of increase for their true counterparts given certain assumptions on the parameters of the model. In Section 4, we apply these results to the models with observable fractional Brownian motions, by proving that the optimal stopping times have finite expectations. The latter fact is needed for the verification that the solution of the free boundary problem provides the solution of the original optimal stopping problem, which is done in Section 5. Preliminaries In this section, we give a formulation of the unifying optimal stopping problem for a onedimensional time-inhomogeneous regular diffusion process and consider the associated partial and ordinary differential free boundary problems..1 For a precise formulation of the problem, let us consider a probability space Ω, G, P with a standard Brownian motion B = B t t. Let Φ = Φ t t be a one-dimensional timeinhomogeneous diffusion process with the state space [,, which is a pathwise strong solution of the stochastic differential equation dφ t = ηt, Φ t dt + ζt, Φ t db t Φ = φ,.1 where ηt, φ and ζt, φ > are some continuously differentiable functions of at most linear growth in φ on [,. Let us consider an optimal stopping problem with the value function τ ] V t, φ = inf E t,φ [GΦ t+τ + F Φ t+s ds,. τ where E t,φ denotes the expectation under the assumption that Φ t = φ, for some φ [,. Here, the gain function Gφ and the cost function F φ are assumed to be non-negative, continuous and bounded, Gφ is concave and continuously differentiable on, c c, for some c [, ], and the infimum in. is taken over all stopping times τ such that the integral above has a finite expectation, so that E t,φ τ < holds. Such time-inhomogeneous optimal stopping problems for diffusion processes within a finite horizon setting have been considered in McKean [7], van Moerbeke [4], Jacka [3], Broadie and Detemple [4], Myneni [8], Peskir [33, 3], and [18, 19] among others see also Peskir and Shiryaev [36, Chapter VII] and Detemple [14] for an overview and further references. Other time-inhomogeneous optimal stopping problems with infinite time horizon were recently considered in [17]. Example.1 Sequential testing problem. Suppose that we observe a continuous process X = X t t of the form X t = θµt + B t, where µt > is increasing and two times continuously differentiable function for t >, µ =, and B = B t t is a standard 3

5 Brownian motion which is independent of the random variable θ. We assume that P θ = 1 = π and P θ = = 1 π holds for some π, 1 fixed. The problem of sequential testing of two simple hypotheses about the values of the parameter θ can be embedded into the optimal stopping problem of. with Gφ = aφ b/1 + φ and F φ = 1, where a, b > are some given constants see, e.g. [4, Chapter IV, Section ] and [36, Chapter VI, Section 1]. In this case, the likelihood ratio process Φ takes the form Φ t = π t 1 π L t with L t = exp µ s dx s t µ s ds,.3 and thus solves the stochastic differential equation in.1 with the coefficients ηt, φ = µ tφ /1 + φ and ζt, φ = µ tφ, where the process B = B t t defined by B t = X t t µ sφ s 1 + Φ s ds.4 is the innovation standard Brownian motion generating the same filtration F t t as the process X. Example. Quickest change-point detection problem. Suppose that we observe a continuous process X = X t t of the form X t = µt µθ + +B t, where µt > is increasing and two times continuously differentiable function for t >, µ =, and B = B t t is a standard Brownian motion which is independent of the random variable θ. We assume that P θ = = π and P θ > t θ > = e λt holds for all t, and some π, 1 and λ > fixed. The problem of quickest detection of the change-point parameter θ can be embedded into the optimal stopping problem of. with Gφ = 1/1 + φ and F φ = cφ/1 + φ, where c > is a given constant see, e.g. [4, Chapter IV, Section 4] and [36, Chapter VI, Section ]. In this case, the likelihood ratio process Φ takes the form Φ t = L t π t e λt 1 π + λe λs ds L s with t t L t = exp µ µ s s dx s ds,.5 and thus solves the stochastic differential equation in.1 with the coefficients ηt, φ = λ1 + φ + µ tφ /1 + φ and ζt, φ = µ tφ, where the innovation standard Brownian motion B = B t t is given by.4. Observe also that the problem setting in both Examples.1 and. above includes the canonical case of the observable Brownian motion with the linear drift function µt = ρt, for some constant ρ > see, e.g. [4, Chapter IV] and [36, Chapter VI]. Note that the timedependent function µt µθ + can be considered in the models in which certain known seasonal effects may change the local drift rate of the observable process. Another possible natural extension µt θ + of the original linear drift function µt = ρt is not studied in the paper. The problem setting in the case of observable diffusion processes in which the local drift rate depends on the running value of the observations was considered in [] and [1]. 4

6 . It follows from the general theory of optimal stopping for Markov processes see, e.g. [36, Chapter I, Section.] that the optimal stopping time in the problem of. is given by τ = inf{s V t + s, Φ t+s = GΦ t+s }.6 whenever it exists. We further search for an optimal stopping time of the form τ = inf{s Φ t+s / g t + s, h t + s}.7 for some functions g t < h t to be determined see, e.g. Section 14] for a time-inhomogeneous finite-horizon setting. [36, Chapter IV,.3 By means of standard arguments see, e.g. [4, Chapter V, Section 5.1], it can be shown that the infinitesimal generator L of the process t, Φ = t, Φ t t is given by the expression L = t + ηt, φ φ + ζ t, φ φφ.8 for all t, φ,. In order to find analytic expressions for the unknown value function V t, φ from. and the unknown boundaries g t and h t from.7, we use the results of general theory of optimal stopping problems for continuous time Markov processes see, e.g. [4, Chapter III, Section 8] and [36, Chapter IV, Section 8]. We formulate the associated free boundary problem LV t, φ = F φ for gt < φ < ht.9 V t, gt+ = Ggt and V t, ht = Ght instantaneous stopping.1 V t, φ = Gφ for φ < gt and φ > ht.11 V t, φ < Gφ for gt < φ < ht.1 LGφ > F φ for φ < gt and φ > ht.13 for some gt < c < ht and all t. Note that the superharmonic characterization of the value function see, e.g. [4, Chapter III, Section 8] and [36, Chapter IV, Section 9] implies that V t, φ from. is the largest function satisfying with the boundaries g t and h t. Moreover, since the system in may admit multiple solutions, we need to use some additional conditions which would uniquely determine the value function and the optimal stopping boundaries for the initial problem of.. For this reason, we will need to assume that the smooth-fit conditions φ V t, gt+ = φ Ggt and φ V t, ht = φ Ght smooth fit.14 hold for all t >. We further provide an analysis of the parabolic free boundary problem of.9-.13, satisfying the conditions of.14, and such that the resulting boundaries are continuous and of bounded variation. Since such free boundary problems cannot normally be solved explicitly, the existence and uniqueness of classical as well as viscosity solutions of the variational in- 5

7 equalities, arising in the context of optimal stopping problems, have been extensively studied in the literature see, e.g. Friedman [16], Bensoussan and Lions [3], Krylov [5], or Øksendal [3]. Although the necessary conditions for existence and uniqueness of such solutions in [16, Chapter XVI, Theorem 11.1], [5, Chapter V, Section 3, Theorem 14] with [5, Chapter VI, Section 4, Theorem 1], and [3, Chapter X, Theorem 1.4.1] can be verified by virtue of the regularity of the coefficients of the diffusion process in.1, the application of these classical results would still have rather inexplicit character. We therefore continue with the following verification assertion related to the free boundary problem formulated above, which is proved in the Appendix. Theorem.3 Let the process Φ be a pathwise unique solution of the stochastic differential equation in.1. Suppose that the functions Gφ and F φ are bounded and continuous, and G is concave and continuously differentiable on, c c, for some c [, ]. Assume that the couple g t and h t, such that g t < c < h t, together with V t, φ; g t, h t form a solution of the free boundary problem of.9-.14, while the boundaries g t and h t are continuous and of bounded variation. Define the stopping time τ as the first exit time of the process Φ from the interval g t, h t as in.7, and assume that E t,φ τ < holds. Then, the value function V t, φ takes the form { V t, φ; g t, h t, if g t < φ < h t V t, φ =.15 Gφ, if φ g t or φ h t with V t, φ; g t, h t = E t,φ [GΦ t+τ + τ ] F Φ t+s ds,.16 and the boundaries g t and h t are uniquely determined by the smooth-fit conditions of Note that the solution of the free boundary problem in cannot be found in an explicit form for the sequential testing and quickest change-point detection problems formulated in Examples.1 and. above, due to the presence of a time derivative in the infinitesimal generator L. This is in contrast with the canonical time-homogeneous setting where the solution is explicit see, e.g. Theorems 1.1 and.1 in [36, Chapter VI]. In this respect, let us introduce the function V t, φ and the boundaries ĝt and ĥt which satisfy the second-order ordinary differential equation LV t, φ = F φ + t V t, φ for gt < φ < ht,.17 and the conditions of.1-.14, where the variable t plays the role of a parameter. We further provide a connection of the original and the auxiliary free boundary problems associated with the differential equations in.9 and.17, respectively. In particular, we will show that, under certain conditions, the lower and upper optimal stopping boundaries ĝt and ĥt of the auxiliary problem provide lower and upper estimates of the optimal stopping boundaries g t and h t of the original problem. Moreover, we can use the same techniques for finding ĝt and ĥt as in the time-homogeneous setting. 6

8 Let us first state the corresponding verification assertion for the modified free boundary problem which directly follows from Theorem.3. Corollary.4 Let the process Φ be a pathwise unique solution of the stochastic differential equation in.1. Suppose that the functions Gφ and F φ are bounded and continuous, and G is concave and continuously differentiable on, c c, for some c [, ]. Assume that the couple ĝt and ĥt, such that ĝt < c < ĥt, together with V t, φ; ĝt, ĥt form a unique solution of the ordinary differential free boundary problem of , the derivative t V t, φ; ĝt, ĥt exists and is continuous, and the boundaries ĝt and ĥt are continuous and of bounded variation. Then, the function V t, φ defined by { V t, φ; ĝt, V t, φ = ĥt, if ĝt < φ < ĥt Gφ, if φ ĝt or φ ĥt.18 is the value function for the optimal stopping problem V t, φ = inf τ E t,φ [GΦ t+τ.19 τ + F Φ t+s t V t + s, Φt+s I ] Φ t+s ĝt + s, ĥt + s ds where I denotes the indicator function and the stopping time τ of the form τ = inf{s Φ t+s / ĝt + s, ĥt + s}. is optimal in.19, whenever the integral above is of finite expectation, and τ = otherwise. Remark.5 Let us fix some t and assume that t V t + s, φ holds for all s and φ ĝt + s, ĥt + s. Then, the value function V t + s, φ of the auxiliary optimal stopping problem in.19 represents a lower estimate for the value function V t + s, φ of., i.e. V t+s, φ V t+s, φ for all s and φ >. Indeed, it follows from the fact that t V t + s, φ for all s and φ ĝt + s, ĥt + s that the stopping times τ over which the infimum is taken in.19 include those for which E t,φ τ < holds. Hence, comparing the right-hand sides of. and.19, and using again the property t V t + s, φ, we obtain V t + s, φ V t + s, φ for all s and φ >. It thus follows from the structure of the optimal stopping times τ and τ in.7 and. that the inequality τ τ should hold P t,φ -a.s.. In this case, the optimal stopping boundaries ĝt + s and ĥt + s from. are lower and upper estimates for the original optimal stopping boundaries g t + s and h t + s in.7, that is ĝt + s g t + s and h t + s ĥt + s for all s. We also note that, by means of the arguments based on the comparison results for stochastic differential equations similar to the ones used in [, Lemma.1] and [1, Lemma 3.1], it can be shown that that g t + s is increasing decreasing and h t + s is decreasing increasing whenever µ t + s is increasing decreasing. Example.6 Sequential testing problem. Let us first solve the free-boundary problem 7

9 in with Gφ = aφ b/1 + φ and F φ = 1 as in Example.1 above. For this, we follow the arguments of [4, Chapter IV, Section ] and [36, Chapter VI, Section 1] and integrate the second-order ordinary differential equation in.17 twice with respect to the variable φ/1 + φ as well as use the conditions of.1 and.14 at the upper boundary ĥt to obtain V t, φ; ĝt, ĥt = b 1 + φ ĥt µ t 1 + ĥt φ Υĥt 1 + φ Ψĥt + Ψφ,.1 where we denote Ψφ = 1 φ 1 + φ ln φ and Υφ = φ 1 + ln φ,. φ for all φ >. Then, applying the conditions of.1 and.14 at the lower boundary ĝt, we obtain that the functions ĝt and ĥt solve the system of arithmetic equations aµ t gt 1 + gt = bµ t ht 1 + gt Υht 1 + ht gt + Ψht Ψgt, gt b + aµ t = Υht Υgt,.4 which is equivalent to the system b aµ t = ht + 1 ht gt 1 gt,.5 bµ t = ht + ln ht gt ln gt,.6 for all t >. It is shown in [4, Chapter IV, Section ] and [36, Chapter VI, Section 1] that the system in.5-.6 admits the unique solution < ĝt < b/a < ĥt <, for any µ t and t fixed. Moreover, by using the implicit function theorem, we can differentiate.5-.6 to get from which we deduce that b a µ tµ t = h t h t h t g t + g t g t,.7 b µ tµ t = h t + h t ht g t g t gt,.8 g t = µ tµ tb ahtg t gt + 1ht gt and h t = µ tµ tb agth t ht + 1ht gt.9 holds for all t >. In particular, we also obtain that the partial derivative t V t, φ exists and is continuous. It can also be shown directly from the analysis of the system in.5-.6 that ĝt is increasing decreasing and ĥt is decreasing increasing whenever µ t is increasing 8

10 decreasing. Example.7 Quickest change-point detection problem. Let us now solve the freeboundary problem in with Gφ = 1/1 + φ and F φ = cφ/1 + φ as in Example. above, where we set ĝt = for all t. For this, we follow the arguments of [4, Chapter IV, Section 4] or [36, Chapter VI, Section ] and integrate the second-order ordinary differential equation in.17 twice with respect to the variable φ/1 + φ as well as use the conditions of.1 and.14 at the upper boundary ĥt to obtain V t, φ; ĥt = 1 ĥt 1 + ĥt + Ct y 1 + x exp Λt Hy Hx dx dy,.3 φ 1 + y x where we denote Ct = c λ 1 + x, Λt =, and Hx = ln x µ t µ t x,.31 for all t and φ >. It thus follows from the condition of.14 that the boundary ĥt solves the arithmetic equation Ct ht 1 + x exp Λt Hht Hx dx = 1,.3 x for all t. It is shown in [4, Chapter IV, Section 4] and [36, Chapter VI, Section ] that the equation in.3 admits the unique solution λ/c ĥt, for any µ t and t fixed. Moreover, by using the implicit function theorem, we can also obtain that ĥt is continuosly differentiable, as well as the partial derivative t V t, φ exists and is continuous. It can also be shown directly from the analysis of the equation in.3 with the notation of.31 that ĥt is decreasing increasing whenever µ t is increasing decreasing. 3 Asymptotic behaviour of the stopping boundaries In this section, we are interested in how the optimal stopping boundaries ĝt and ĥt in the modified problem behave asymptotically with respect to the derivative µ t of the drift function µt in Example.1 and Example., as t. More precisely, we will obtain the limits and the asymptotic expansions of ĝt and ĥt with respect to µ t in some particular cases, when either µ t or µ t holds as t. Observe that in the canonical case of the observable Brownian motion with linear drift µt = ρt we have µ t = ρ, for some constant ρ >, so that the results of this section automatically provide an asymptotic analysis of the time-constant optimal stopping boundaries a = a ρ and b = b ρ with respect to the signal-to-noise ratio coefficient ρ. Example 3.1 Sequential testing problem. Let us introduce the function W x which is 9

11 the inverse of e x x, and thus, solves the equation e W x W x = x for x 3.1 see, e.g. [8, Formula 1.5]. Note that W x is strictly increasing and satisfy the properties W =, and W x as x, and it has the asymptotic series expansion W x lnx lnlnx as x 3. see, e.g. [8, Formula 4.19]. Then, by solving the quadratic equation in.5 for ht, we obtain that ĝt and ĥt satisfy gt h ± t = gt + 1 gt + b aµ t ± gt + b aµ t 1, where ĥt = ĥ t or ĥt = ĥ+t, for all t. Hence, by substituting the expression of 3.3 into the formula of.6 and taking exponentials on both sides, we have that ĝt satisfies the following equation gt + 1 gt + b aµ t ± 4 = W e gt+bµ t / gt, gt + 1 gt + b aµ t which contains both the positive and negative branch of the function on the left-hand side, depending on the root which we have chosen for ĥt in 3.3. If we rearrange the terms and square both sides of the expression in 3.4, we get that ĝt should satisfy 1 + W e gt+bµ t / gt = gt + 1 gt + b aµ t W e gt+bµ t / gt, 3.5 for all t. Let us first consider the case in which b > a and µ t holds as t. If we assume that ĥt = ĥ t, by using the assumption that b > a and < ĝt < b/a, we obtain that ĥ t, which contradicts the fact that b/a < ĥt < holds for all t. It follows that ĥt = ĥ+t and ĝt should solve the equation in 3.4 with the positive branch of the function taken on the left-hand side. Hence, the left-hand side of the expression in 3.4 converges to t as t, so that /ĝt eĝt+bµ holds by virtue of the properties of the function W x defined in 3.1. In particular, the functions on both sides of 3.5 converge to with the same speed, and thus, the following expression holds W eĝt+bµ t /ĝt b aµ t + ĝt + 1 ĝt as t

12 Furthermore, taking into account the asymptotic series expansion of 3., we see that W eĝt+bµ t /ĝt bµ t + ĝt + lnĝt as t. 3.7 Since ĝt is bounded from above by b/a for all t and using the equation of 3.3 for ĥt, we therefore conclude that ĝt and aµ t ĥt bµ t as t. 3.8 Let us now consider the case in which b < a and µ t holds as t. Since the function on the left-hand side of.5 converges to as t, taking into account the fact that ĝt < b/a < ĥt holds for t, we obtain that ĝt as t. Assuming that t W /ĝt eĝt+bµ does not converge to implies that there exists a sequence t n n N, such that t n and ĝt n = Oe bµ t n / as n. Now, when ĥt = ĥ+t, we obtain that ĥt n as n, while the assumption that the right-hand side of 3.4 does not converge to leads to contradiction. On the other hand, when ĥt = ĥ t, we obtain that ĥt, which contradicts the assumption that b/a < ĥt < holds for all t. We therefore obtain t that W /ĝt eĝt+bµ, and by the same considerations as in the case b > a above, regarding the asymptotic behaviour of both the left and right sides of 3.5, we obtain 3.8. Let us finally consider the case in which µ t holds as t. Since the left-hand side of.6 converges to in this case, by using the fact that the function x + ln x is strictly increasing for x >, and < ĝt < b/a < ĥt < holds for all t, we may conclude that ĝt b/a and ĥt b/a holds as t. Example 3. Quickest change-point detection problem. Integrating by parts and using the notations of.31, we obtain y 1 + x Ct exp Λt Hy Hx dx = cy Q Λt 1, Λt/y 1, 3.9 x λ Λt + 1 where we denote Qz, y = zy z e y Γz, y with Γz, y = y e u u z 1 du, 3.1 for all z and y. In this case, the expression in.3 takes the form Q Λt 1, Λt/ht ht 1 = λ Λt + 1 c, 3.11 for all t. We also recall the properties of the function Qz, y in [41, Section 9] see also [, Section.5] and note that Qz, y 1 as well as Qz, = 1 holds for all z. Let us first consider the case in which µ t, and thus Λt as t. Since λ/c ĥt holds, we have Λt/ĥt, so that Q Λt 1, Λt/ĥt 1 as t. Therefore, by using the fact that ĥt satisfies the equation in 3.11, we get that ĥt 11

13 holds as t. Suppose that µ t, so that Λt holds as t. Then, using the property Qz, y 1, it follows from 3.11 that ĥt λ c as t. 3.1 Let us now determine the exact rate of increase for ĥt in the case in which µ t as t. In this case, the expression in.3 can be written as Λt ht 1 + x exp Λt Hx dx = λ Λt x c exp Hht, 3.13 for t. Then, using the definition of the function Hx in.31, we obtain the expansion on the right-hand side of 3.13 in the form λ c exp Λt Hĥt under µ t. Note that the assumption of λ ĥtλt, 3.14 c lim sup ĥt Λt = 3.15 t implies that there exists a sequence t n n N, such that t n and expλt n Hĥt n as n. Since we have ĥt, there exists t such that λ/c < ĥt holds for all t t. Moreover, since the function Hx is strictly increasing for x >, by evaluating the left-hand side of 3.13 at ĥt, we obtain that ĥt ĥt > λ c 1 + x Λt exp Λt Hx x x d exp Λt Hx > λ c dx = ĥt x d exp ΛtHx exp Λt Hĥt λ exp ΛtH c 3.16 holds for all t t. This fact means that the leading term of the left-hand side of 3.13 is larger than the leading term on the right-hand side of 3.13 along the sequence t n as n, and thus, the assumption of 3.15 cannot be satisfied. Therefore, we conclude that ĥtλt is finite under ĥt and Λt, so that ln ĥt = Oµ t as t. 4 The fractional Brownian motion setting In this section, we apply the asymptotic results obtained above to demonstrate the existence of solutions in the problems of sequential analysis for an observable fractional Brownian motion with linear drift. In particular, we will prove that the optimal stopping time τ has a finite expectation. Observe that we consider the application of the results obtained above on an 1

14 optimal stopping problem for the time-inhomogeneous diffusion process which is driven by a fundamental martingale associated with the observable fractional Brownian motion. Note that several optimal stopping and sequential analysis problems for fractional Brownian motions were considered by Elliot and Chan [15], Prakasa Rao [37], Çetin, Novikov and Shiryaev [6], and Chronopoulou and Fellouris [7]. Example 4.1 Sequential testing problem. Suppose that in the setting of Example.1 the observable continuous process X Y H = Yt H t is given by Yt H = θρt + Bt H, where B H = Bt H t is a fractional Brownian motion with parameter H 1/, 1 independent of θ, and ρ > is a constant. Introduce the process M H = M H t t by M H t = Z H t c 1 t where the process Z H = Z H t t is defined by Z H t = t ρ s1 H Φ s 1 + Φ s ds with M H t = Z H t = c 1t H H, 4.1 s 1/ H t s 1/ H HΓ3/ HΓH + 1/ dy H s and c 1 = Γ3/ H HΓH + 1/Γ H, 4. and the likelihood ratio process Φ is given by Φ t = π 1 π L t with L t = exp ρzt H ρ Z H t, 4.3 for all t. It follows from the result of [9, Theorem 3.1] that the process M H is a fundamental martingale with respect to the filtration F t t and thus admits the representation with respect to the innovation standard Brownian motion B of the form M H t = t c 1 s 1/ H db s and B t = 1 t s H 1/ dm H s, 4.4 c1 for all t see, e.g. [9, Section 5.]. In this case, the process Φ from 4.3 satisfies the stochastic differential equation in.1 with ηt, φ and ζt, φ as in Example.1, where µ t = ρ c 1 t 1/ H holds for all t. Hence, the analysis from the previous section can be applied for the drift rate µ t when 1/ < H < 1 as t. Let us fix a starting time t and introduce the deterministic time change βt, s with the rate µ s defined as βt, s = t+s t µ u du c 1ρ t + s H t H, 4.5 H and its inverse γt, s shifted by t, so that βt, γt, s t = s holds for all s. Since the process Φ satisfies the stochastic differential equation of.1, by applying the time-change 13

15 formula for Itô integrals from [3, Theorems and 8.5.7], we obtain Φ γt,s = Φ t exp B s s s + Φ γt,u du 1 + Φ γt,u with Bs = γt,s t µ u db u, 4.6 where B = B s s is a standard Brownian motion with respect to the filtration F γt,s s. Therefore, by using the definition of τ in. and taking into consideration the time change βt, s from 4.5, we conclude that the stopping time βt, τ with respect to the filtration F γt,s s can be represented as { βt, τ = inf s B s s s + Φ γt,u du + ln Φ t / } ln ĝγt, s, ln ĥγt, s, Φ γt,u for all t. Assume that b a in Example.1. In this case, noticing from 4.5 that γt, s and using the fact that ĝt b/a and ĥt b/a as t, it follows that for any ε > there exists t > large enough such that the inequalities b a ε < ĝγt, s < b a < ĥγt, s < b a + ε 4.8 hold for all t > t and s. Let us now fix an arbitrary ε > such that ε < b/a and further assume that t > t. Then, introducing the sets of sample paths A = {ω Ω ĝt < Φ t < ĥt}, A s = { ω A ĝγt, s < Φγt,s < ĥγt, s}, and C s = { ω Ω Φγt,s b/a < ε }, 4.9 and using the inequalities in 4.8, we get the inclusion A s C s for any s. Therefore, by the definition of the event C s, for the upper bounds c 1 ε and c ε defined below, we have c 1 ε b aε a + b aε < Φ γt,s 1 + Φ γt,s < b + aε a + b + aε c ε, for ω A s, 4.1 for any ε >. It follows from the notations in 4.6 and the structure of the event A that A s D s holds, where we set { D s = ω Ω B s s ĝγt, s ĥγt, s ln c ε s, ln c 1 ε s }, 4.11 ĥt ĝt for all s. Define the stopping time τ as { τ = inf s B s s ĝγt, s ĥγt, s / ln c ε s, ln c 1 ε s }, 4.1 ĥt ĝt 14

16 and notice that the stopping times βt, τ = βt, τω and τ = τω admit the representations { βt, τω = sup s ω } { A u and τω = sup s ω D u }, 4.13 u s u s for any ω Ω. Then, it follows from the inclusion A s D s for s that βt, τ τ holds. Because of the assumption b a, we can choose ε < b/a such that either 1 ε > b/a holds when b < a or 1 + ε < b/a holds when b > a. Hence, assuming that b < a, we have 1/ c ε >. Thus, it follows from the expressions in 4.8 and 4.1 that τ τ holds, where we set τ = inf { s B s lnb aε lnaĥt + 1/ c εs }, 4.14 which is a stopping time with polynomial moments of all orders see, e.g. [39, Chapter IV]. Therefore, it follows from the fact that βt, τ τ τ holds and the structure of the time change in 4.5 that E t,φ τ E t,φ γt, τ t < is satisfied, and we get the same inequalities in the case of b > a, similarly. Let us now prove that t V t, φ; ĝt, ĥt > holds for all φ ĝt, ĥt and t > large enough. For this purpose, by differentiating the expression in.1 and using the expressions in. and.9, we get t V t, φ; ĝt, ĥt = H 1Ψĥt Ψφ/tµ t 4.15 ĥt µ t 1 + ĥt φ H 1ξ ĥt ĥ tĥt + 1 H 1Ξt, φ + =, 1 + φ t ĥt tµ t where we denote Ξt, φ = φ + ln φ ĥt ln ĥt + φ Υ ĥt Υφ ĥt φb aĝt φ ĥt ĝt1 + φ, for all t > and φ >. It is clear that Ξt, ĥt = holds, and thus, we obtain from the expressions in.4 and.6 that Ξt, ĝt = µ t b + ĝta + b 1 + ĝt + b aĝt 1 + ĝt = b aĝt 1 + ĝt 1 µ t, 4.17 holds for t >. Since b/a > ĝt > is satisfied, and there exists t > such that µ t < holds for all t t, we have Ξt, ĝt > for t t. Then, by differentiating the expression in 4.16, we get φ Ξt, φ = φ b aĝt1 + ĥt Υĥt Υφ ĥt ĝt, 4.18 for all t > and φ >. Observe that, since Υφ is an increasing function, it follows that φ Ξt, φ changes its sign at most once in the region φ ĝt, ĥt for all t t. It is easily 15

17 seen that the inequality φ Ξt, ĥt < holds, which means that either Ξt, φ is decreasing for φ ĝt, ĥt or there exists some φ ĝt, ĥt such that Ξt, φ is increasing for φ ĝt, φ ] and decreasing for φ φ, ĥt. Hence, since Ξt, ĝt > and Ξt, ĥt = holds, we get that Ξt, φ > is satisfied in both cases for φ ĝt, ĥt and t t. For 1/ < H < 1, it follows from the expressions in 4.15 that the inequality t V t, φ; ĝt, ĥt > holds for all φ ĝt, ĥt and t t. We can therefore apply the assertions of Remark.5 and use the fact that E t,φ τ < to obtain that E t,φ τ E t,φ τ < holds when the starting time t satisfies t > t t. Example 4. Quickest disorder detection problem. Suppose that in the setting of Example. the observable continuous process X Y H = Yt H t is given by Yt H = ρt θ + + Bt H, where B H = Bt H t is a fractional Brownian motion with parameter H 1/, 1 independent of θ, and ρ > is a constant. In this case, the likelihood ratio process Φ is given by Φ t = L t π t e λt 1 π + λe λs ds with L t = exp ρzt H ρ Z H t, 4.19 L s for all t. Therefore, by using the same reasoning as in Example 4.1, we obtain that the process Φ from 4.19 satisfies the stochastic differential equation in.1 with ηt, φ and ζt, φ as in Example., where µ t = ρ c 1 t 1/ H holds for all t. Hence, the analysis from the previous section can be applied for the drift rate µ t when 1/ < H < 1 as t. Let us fix a starting time t and define the deterministic time change βt, s and its inverse γt, s as in 4.5 for all s. By using the expression in 4.19, we get that Φ s Φ e λs L s holds for all s. Therefore, if we define the stopping time τ as τ = inf{s Φ e λt+s L t+s ĥt + s}, 4. we have that τ τ holds, where τ is defined in.. In order to simplify further notations, we define the process Φ = Φ s s by Φ s = Φ e λγt,s L γt,s for s. Since the process L has the form of 4.19, by applying the time-change formula for Itô integrals from [3, Theorems and 8.5.7], we obtain Φ s = Φ exp B s s s + λγt, s t + Φ u 1 + Φ du, 4.1 u where the process B = B s s defined in 4.6 is a standard Brownian motion. Therefore, by using the definition of τ in 4. and taking into consideration the time change from 4.5, the stopping time βt, τ can be represented as { βt, τ = inf s B s s s } + λγt, s t + Φ u 1 + Φ du + ln Φ ln ĥγt, s. 4. u Since γt, s as t, it follows from 3.1 that for any ε > there exists t > large 16

18 enough such that the inequalities λ c < ĥγt, s < λ c + ε 4.3 hold for all t > t and s. Let us now fix an arbitrary ε > and further assume that t > t. By using the fact that Φ is a nonnegative process, we obtain from 4.5 that the inequalities λγt, s t + s Φ u s H 1 + Φ du λγt, s t λ u c 1 ρ 1/ H 4.4 hold for all s. Define the random variable t as λ + cε t = sup s + ln s c Φ λ s H c 1 ρ and notice that it follows from the inequalities in 4.3 and 4.4 that 1/ H s +, 4.5 ĥγt, s s Φ u ln λγt, s t Φ 1 + Φ du + s u < t s 4.6 holds for all s and any t > t fixed. Subsequently, we obtain from 4. that βt, τ τ, where we set τ = inf{s B s t s}, 4.7 for any t > t. Moreover, by introducing the event A = {ω Ω Φ < ĥt}, we also obtain that βt, τ = on Ω \ A, and hence, we conclude that βt, τ τ IA holds. Since we have that t > on the event A and t < P t,φ -a.s., for 1/ < H < 1, we get that τ IA has polynomial moments of all orders see, e.g. [39, Chapter IV]. Therefore, it follows from the fact that βt, τ βt, τ τ IA holds and the structure of the time change in 4.5 that E t,φ τ E t,φ γt, τ t < is satisfied. Let us finally show that t V t, φ; ĥt > holds for all φ, ĥt and t >. For this purpose, differentiating the expression in.3 and using the expressions in.3 and 3.9, we get t V t, φ; ĥt = ĥt = φ ĥt φ Ct t y + 1 cy λy + 1 t y x x for all φ < ĥt and t >. Note that we also have exp ΛtHy Hx dx dy 4.8 Q Λt 1, Λt/y Λt + 1 x a e x Γa, x = x a e x e u u a 1 du = x dy e xu u + 1 a 1 du,

19 for a < and x >. It is shown by differentiation of the expressions in 4.9 that the function x a e x Γa, x is decreasing in x and increasing in a see, e.g. [, Section.5] for similar results. Hence, the function y x + 1 Q x 1, x x x+1 = y e Γ x/y x 1, x y y y 4.3 is decreasing in x for x, y >, where the functions Qz, y and Γz, y are defined in 3.1. Recall that, for 1/ < H < 1, the function µ t is decreasing, so that Λt is increasing in t. Hence, by using the formulas from 4.3, we obtain from the expressions in 4.8 that V t, φ; ĥt is increasing in t that leads to tv t, φ; ĥt > for all φ, ĥt and t >. We can therefore apply the assertions of Remark.5 and use the fact that E t,φ τ < to conclude that E t,φ τ E t,φ τ <, when the starting time t satisfies t > t. 5 Appendix Let us now prove the verification assertion stated in Theorem.3 above. Proof: In order to verify the assertions stated above, let us denote by V t, φ the right-hand side of the expression in.15. Then, using the fact that the function V t, φ satisfies the conditions of by construction, we can apply the local time-space formula from Peskir [31] see also [36, Chapter II, Section 3.5] for a summary of the related results and further references to obtain V t + u, Φ t+u + + u u F Φ t+s ds = V t, φ + M u + K u 5.1 LV + F t + s, Φ t+s I Φ t+s g t + s, Φ t+s h t + s ds for all t, where the process M = M u u defined by M u = u V φ t + s, Φ t+s ζt + s, Φ t+s I Φ t+s g t + s, Φ t+s h t + s db s 5. is a continuous local martingale with respect to the probability measure P t,φ. Here, the process K = K u u is given by K u = 1 u + 1 φ V t + s, g t + s I Φ t+s = g t + s dl g s 5.3 u φ V t + s, h t + s I Φ t+s = h t + s dl h s where φ V t+s, g t+s = V φ t+s, g t+s+ V φ t+s, g t+s, φ V t+s, h t+s = V φ t + s, h t + s+ V φ t + s, h t + s, and the processes l g = l g u u and l h = l h u u 18

20 defined by and l g u l h u 1 = P t,φ lim ε ε 1 = P t,φ lim ε ε u u I g t + s ε < Φ t+s < g t + s + ε ζ t + s, Φ t+s ds 5.4 I h t + s ε < Φ t+v < h t + s + ε ζ t + s, Φ t+s ds 5.5 are the local times of Φ at the curves g t and h t, at which V φ t, φ may not exist. It follows from the concavity and continuous differentiability of the gain function Gφ in., and the stopping time τ in.7, that the inequalities φ V t, g t and φ V t, h t should hold for all t, so that the continuous process K defined in 5.3 is non-increasing. We may therefore conclude that K u = can hold for all u if and only if the smooth-fit conditions of.14 are satisfied. Using the assumption that the inequality in.13 holds for the function Gφ with the boundaries g t and h t, we conclude that LV + F t, φ holds for any φ g t and φ h t. Moreover, it follows from the conditions in.1-.1 that the inequality V t, φ Gφ holds for all t, φ [,. Thus, for any stopping time τ such that E t,φ τ <, the expression in 5.1 yields the inequalities GΦ t+τ + τ F Φ t+s ds K τ V t + τ, Φ t+τ + τ F Φ t+s ds K τ V t, φ + M τ. 5.6 Let τ n n N be the localizing sequence of stopping times for the process M such that τ n = inf{u M u n}. Then, taking the expectations with respect to the probability measure P t,φ in 5.6, by means of the optional sampling theorem see, e.g. [6, Chapter III, Theorem 3.6] or [4, Chapter I, Theorem 3.], we get the inequalities τ τn ] E t,φ [GΦ t+τ τn + F Φ t+s ds K τ τn 5.7 E t,φ [V t + τ τ n, Φ t+τ τn + τ τn F Φ t+s ds K τ τn ] V t, φ + E t,φ M τ τn = V t, φ. Hence, letting n go to infinity and using Fatou s lemma, we obtain τ ] E t,φ [GΦ t+τ + F Φ t+s ds K τ E t,φ [V t + τ, Φ t+τ + τ F Φ t+s ds K τ ] V t, φ 5.8 for any stopping time τ such that E t,φ τ < and E t,φ K τ >, and all t, φ [,, where K τ = holds whenever the conditions of.14 are satisfied. By virtue of the structure of the stopping time in.7 and the conditions of.11, it is readily seen that the equalities in 5.6 hold with τ instead of τ when either φ g t or φ h t, respectively. Let us now show that the equalities are attained in 5.8 when τ replaces τ and the smooth- 19

21 fit conditions of.14 hold for g t < φ < h t. By virtue of the fact that the function V t, φ and the boundaries g t and h t solve the partial differential equation in.9 and satisfy the conditions in.1 and.14, it follows from the expression in 5.1 and the structure of the stopping time in.7 that GΦ t+τ τ n + τ τ n V t + τ τ n, Φ t+τ τ n + F Φ t+s ds 5.9 τ τ n F Φ t+s ds = V t, φ + M τ τ n holds for g t < φ < h t. Hence, taking expectations and letting n go to infinity in 5.9, using the assumptions that Gφ is bounded and the integral in.16 is of finite expectation, we apply the Lebesgue dominated convergence theorem to obtain the equality τ ] E t,φ [GΦ t+τ + F Φ t+s ds = V t, φ 5.1 for all t, φ [,. We may therefore conclude that the function V t, φ coincides with the value function V t, φ of the optimal stopping problem in. whenever the smooth-fit conditions of.14 hold. In order to prove the uniqueness of the value function V t, φ and the boundaries g t and h t as solutions to the free-boundary problem in with the smooth-fit conditions of.14, let us assume that there exist other continuous boundaries of bounded variation gt and ht such that gt < c < ht holds. Then, define the function Ṽ t, φ as in.15 with Ṽ t, φ; gt, ht satisfying and the stopping time τ as in.7 with gt and ht instead of g t and h t, respectively, such that E t,φ τ <. Following the arguments from the previous part of the proof and using the fact that the function Ṽ t, φ solves the partial differential equation in.9 and satisfies the conditions of.1 and.14 with gt and ht instead of gt and ht by construction, we apply the change-of-variable formula from [31] to get Ṽ t + u, Φ t+u + + u u F Φ t+s ds = Ṽ t, φ + M u 5.11 LṼ + F t + s, Φ t+s I Φ t+s / gt + s, ht + s ds where the process M = M u u defined as in 5. with Ṽφt, φ instead of V φ t, φ is a continuous local martingale with respect to the probability measure P t,φ. Thus, taking into account the structure of the stopping time τ, we obtain from 5.11 that GΦ t+ τ τn + τ τn Ṽ t + τ τ n, Φ t+ τ τn + F Φ t+s ds 5.1 τ τn F Φ t+s ds = Ṽ t, φ + M τ τn

22 holds for gt < φ < ht and the localizing sequence τ n n N of M such that τ n = inf{u M u n}. Hence, taking expectations and letting n go to infinity in 5.1, using the assumptions that Gφ and F φ are bounded and the integral in.16 taken up to τ is of finite expectation, by means of the Lebesgue dominated convergence theorem, we have that the equality τ ] E t,φ [GΦ t+ τ + F Φ t+s ds = Ṽ t, φ 5.13 is satisfied. Therefore, recalling the fact that τ is the optimal stopping time in. and comparing the expressions in 5.1 and 5.13, we see that the inequality Ṽ t, φ V t, φ should hold for all t, φ [,. We finally show that gt and ht should coincide with g t and h t. By using the fact that Ṽ t, φ and V t, φ satisfy.1-.1, and Ṽ t, φ V t, φ holds for all t, φ [,, we get that g t gt and ht h t. Then, inserting τ τ n into 5.11 in place of u and using the assumptions that Gφ is bounded and the appropriate integrals are of finite expectation, by means of the arguments similar to the ones above, we obtain τ ] E t,φ [Ṽ t + τ, Φ t+τ + F Φ t+s ds = Ṽ t, φ 5.14 τ + E t,φ LṼ + F t + s, Φ t+s I Φ t+s / gt + s, ht + s ds for all t, φ [,. Thus, since we have Ṽ t, φ = V t, φ = Gφ for φ = g t and φ = h t, and Ṽ t, φ V t, φ, we see from the expressions in 5.1 and 5.14 that the inequality τ E t,φ LṼ + F t + s, Φ t+s I Φ t+s / gt + s, ht + s ds, 5.15 should hold. Due to the assumption of continuity of gt and ht, we may therefore conclude that g t = gt and h t = ht, so that Ṽ t, φ coincides with V t, φ for all t, φ [,. Acknowledgments. The authors are grateful to the Editor for his encouragement to prepare the revised version of the paper and two anonymous Referees for their valuable suggestions which helped to improve the presentation of the paper. The paper was essentially written when the first author was visiting the Department of Mathematical Stochastics of the University of Freiburg in Germany. The hospitality and financial support from the Alexander von Humboldt Foundation through the fellowship for experienced researchers is gratefully acknowledged. This research was also supported by a Small Grant from the Suntory and Toyota International Centres for Economics and Related Disciplines STICERD at the London School of Economics and Political Science. 1

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