Georgian Mathematical Journal Volume 15 (2008), Number 4, 713–730
SEQUENTIAL DETECTION OF DRIFT CHANGE FOR BROWNIAN MOTION WITH UNKNOWN SIGN HANS RUDOLF LERCHE AND ILSE MAAHS
Abstract. We study tests of power one for the following change-point problem. Suppose one observes a process W which is either a Brownian motion without drift or a Brownian motion that has zero drift up to a random time τ after which with equal probability the drift becomes either θ or −θ, where the value of θ > 0 is known. The distribution of τ is also assumed to be known. We search for a stopping time T ∗ that minimizes an appropriate Bayes risk and give a solution that is asymptotically optimal, when the cost of observation tends to zero. 2000 Mathematics Subject Classification: 62L15, 62C10, 60G40. Key words and phrases: Bayes problems, tests of power one, Brownian motion, change point, sequential detection.
1. Introduction Let W be a Brownian motion which has either zero drift or a drift change at an unknown random time τ . We examine the testing problem H0 : there is never a drift change, H1 : there is a drift change to either θ or −θ with equal probability at a random time τ , where θ > 0 is assumed to be known. We consider the Bayesian setting, combining the probability of type I error 2 with the expected cost of stopping delay. The cost is taken as c θ2 (c > 0) per observation unit. This leads to the expected loss ¤ θ2 £ L(c, T ) := P∞ (T < ∞) + c E (T − τ )+ , (1) 2 where T is a stopping time of W , P∞ is the measure of the Brownian motion without drift and E means taking expectation with respect to P , the measure which describes the drift change. The goal is to minimize L(c, T ) over all suitable stopping times T . We will provide an asymptotic expansion of the minimal risk L∗c as well as an asymptotically optimal stopping time, when the cost c tends to zero. Our approach has its origin in Lerche [8] and is similar to that of Beibel, [1]–[2], and Beibel–Lerche, [3]–[4]. 2. General Setup and Main Result We are now more specific with our definitions. Let (Ω, A, Q) denote a probability space on which a standard Brownian motion B = (Bt , t ≥ 0) is defined. c Heldermann Verlag www.heldermann.de ISSN 1072-947X / $8.00 / °
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Let Y denote a random variable with two outcomes ±θ with equal probability. Let τ be a random variable with a given distribution ρ on R+ . We assume that B, Y and τ are independent under the measure Q. We define Wt = Bt +Y (t−τ )+ and F W = (FtW , t ≥ 0) as filtration generated by W . Let P denote the image measure of Q under W . The process W = (Wt , FtW , 0 ≤ t < ∞, P ) is a Brownian motion which has drift zero up to a random time τ and drift Y after it. By Girsanov’s theorem a further measure P∞ can be constructed on F W under which W is a standard Brownian motion (see [2, p. 460]). The statistical problem described in the introduction can be stated as follows: The process W is observed and one seeks for a stopping time T of W which detects the change at the unobservable time τ quickly and reliable. As criterium we take the Bayes risk L(c, T ) given in (1). Introduce the minimal risk over M, the set of all F W -stopping times T . Let L∗c := inf L(c, T ). Let us define the T ∈M
stopping time S 1 of W as c
½ S 1 := inf t ∈ c
R+ 0
dP ¯¯ 1 : ≥ dP∞ FtW c
¾ .
The main result (Theorem 6.3) states that ³ ´ L∗c = L c, S 1 + o(c) . c
3. Rewriting the Bayes Risk In this section we derive an alternative representation of the Bayes risk L(c, ·). We show for all F W -stopping times T with L(c, T ) < ∞ there holds " à # ! dP ¯¯ c L(c, T ) = E gc (2) + VT , dP∞ FTW 2 where gc (x) := x1 + c log x and (Vt ; 0 ≤ t < ∞) is a non-negative increasing process. Then (2) transforms the initial optimal stopping problem to a “perturbed” generalized parking problem (see [3]). For t ∈ R+ let µ 2 ¶ Z∞ θ + ψt := exp − (t − s) cosh (θ (Wt − Wt∧s )) ρ(ds) . 2 0
Using Girsanov’s formula one can show in a way similar to [2, p. 460] that ψt =
dP ¯¯ dP∞ FtW
P∞ -a.s. holds.
To rewrite the Bayes risk L(c, T ) we derive a stochastic differential equation for log ψt in terms of observable quantities. For t ∈ R+ let Rt = Y 1{τ ≤t} denote
SEQUENTIAL DETECTION OF DRIFT CHANGE FOR BROWNIAN MOTION
the momentary drift of the process W at time t, ¶ µ 2 Zt ¯ W θ −1 c ¯ Rt = E[Rt Ft ] = ψt θ exp − (t − s) sinh(θ(Wt − Ws )) ρ(ds) 2
715
(3)
0
and ¯ c2 = E[R2 ¯F W ] = ψ −1 θ2 R t t t t
Zt
µ
¶ θ2 exp − (t − s) cosh(θ(Wt − Ws )) ρ(ds) . (4) 2
0
Further let W be the innovation process Zt cs ds . R
W t := Wt −
(5)
0
By a well-known result (see Liptser and Shiryaev [10, p. 297]) this is a standard Brownian motion under the probability measure P relative to the filtration F W . Proposition 3.1. For all t ∈ R+ 0 there holds P -a.s. 1 ³ c ´2 c d log(ψt ) = Rt dW t + Rt dt . 2 Proof. Let Zt Z t ³ ´2 cs dW s − 1 cs ds . R Zet∞ := exp− R 2 0
(6)
(7)
0
We have Zt Z t ³ ´2 ¡ ¢2 c Rs ds = E Y 1{τ ≤s} | FsW ds 0
0
Zt ≤
¡ ¢ E Y 2 1{τ ≤s} | FsW ds ≤ θ2 t < ∞ .
0
According to the Novikov condition (see Karatzas and Shreve [7]) the process (Zet∞ , Gt , 0 ≤ t < ∞, P ), where G = (Gt ; t ≥ 0) is the right continuous version of the usual augmentation F W , is a martingale with mean 1. By Girsanov’s W with theorem there exists a distinct measure Pe∞ on F∞ dPe∞ ¯¯ = Zet∞ P -a.s. ¯ dP FtW W for all t ∈ R+ 0 , where F∞ = σ(Ws , 0 ≤ s < ∞). Since Zt ³ ´ ct ds , Wt = W t + R 0
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H. R. LERCHE AND I. MAAHS
it follows by Girsanov’s theorem, that (Wt , FtW, 0 ≤ t < ∞, Pe∞ ) is a standard Brownian motion. According to Theorem 4.25 in Jacod, Shiryaev [6, p. 110] W the measures Pe∞ and P∞ coincide on F∞ . ³ ´−1 In particular P -a.s. ψt = Ze∞ for all t ∈ R+ . ¤ t
We will now use Proposition 3.1 to rewrite L(c, T ). Theorem 3.2. It holds for all F W -stopping times T with ET < ∞ we have ¶ ZT µ ³ ´ 2 2 £ ¤ θ 1 c2 − R cs E (T − τ )+ = E log ψT + R ds . s 2 2 0
Proof. According to Proposition 3.1 ZT ZT ³ ´2 1 cs dW s + E cs ds . E log ψT = E R R 2 0
(8)
0
Further by the Jensen inequality the relations ZT ZT ³ ´2 ¯ £ ¤ cs ds ≤ E E Y 2 1{τ ≤s} ¯F W ds ≤ θ2 ET < ∞ R E s 0
0
are valid whence ZT cs dW s = 0 . R
E
(9)
0
Finally, we have ¤ 1 θ2 £ E (T − τ )+ = E 2 2 =
1 E 2
Z∞ Y 2 1{τ ≤s} 1{s≤T } ds 0 Z∞
¯ £ ¤ 1{s≤T } E Y 2 1{τ ≤s} ¯FsW ds
0
1 = E 2
ZT c2 ds . R s 0
Combined with (8) and (9) this yields the result.
¤
Corollary 3.3. Let c > 0. For all F W -stopping times T with L(c, T ) < ∞ there holds ¶ ZT µ ³ ´ 2 c c2 − R cs ds L(c, T ) = E gc (ψT ) + R s 2 0
where gc (x) =
1 x
+ c log x.
SEQUENTIAL DETECTION OF DRIFT CHANGE FOR BROWNIAN MOTION
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Proof. From L(c, T ) < ∞ it follows immediately that ET < ∞ as well as P (T < ∞) = 1. This yields ¤ £ £ £ ¤ ¤ P∞ (T < ∞) = E∞ 1{T S o , c c e e e β(c) β(c) c β(c) where the last inequality follows from Step 1 and the fact that 1c minimizes gc . 1 Noting that β(c) = c(1 + log 1c ) we obtain e ! Ã ¶ ³ ³ ´ µ ³ ´ ´ 1 1 e e e e + c log β(c) P Sc ≤ Sβ(c) + c + c log L c, Sc ≥ P Sc > Sβ(c) e e e c β(c) µ ¶ ³ ´ 1 e e = c P Sc ≤ Sβ(c) log β(c) + c + c log . e c ³ ´ Step 3: There holds that lim P Sec ≤ Sβ(c) = 0. e c→0
According to Proposition 6.1 as well as in the definition of Sec , we have for c→0 ³ ´ 1 L c, Sec − c2 ≤ L∗c ≤ c log + O(c) . c For c → 0, this result combined with the inequality proved in Step 2, leads to ³ ´ ³ ´ L c, Sec − c − c log 1c 0 ≤ P Sec ≤ Sβ(c) ≤ e e c log β(c) c2 + c log 1c + O(c) − c − c log 1c c O(1) ≤ = + . e e e c log β(c) log β(c) log β(c)
SEQUENTIAL DETECTION OF DRIFT CHANGE FOR BROWNIAN MOTION
Since
µ
1 lim c + c log c→0 c
¶
µ = lim
x→∞
we have
µ e = lim log lim log β(c)
c→0
c→0
1 log x + x x
1 c + c log 1c
729
¶ = 0,
¶ = ∞.
and the result follows.
¤
Below we show that Sβ(c) is asymptotically optimal, that is, the risk connected with Sβ(c) differs from the minimal risk L∗c only in an error of the order o(c). Theorem 6.3. For c → 0 there holds ¡ ¢ L∗c = L c, Sβ(c) + o(c) 1 θ2 + cK(θ, ρ) + c Eτ − cE log N∞ + o(c) . c 2 Proof. According to Proposition 5.2 the stopping time Sb grows for large b as 2 log b. By l’Hˆopital’s rule we have θ2 = c + c log
¡ ¢ e log c + c log 1c log β(c) lim = lim = lim c→0 log β(c) c→0 c→0 log c
log 1c c+c log 1 c
1 c
c log 1c = 1. c→0 c + c log 1 c
= lim
2 θ2
log 1c for c → 0. Since, by Proposition 6.2 ³ ´ and the remarks at the beginning of this section, lim P Sec ≤ Sβ(c) = 0 and e c→0 ³ ´ P Sec ≤ Sβ(c) = 1 holds for 0 < c ≤ 1, Sec converges in probability to ∞. Hence by Corollary 4.10 Z∞ µ ³ ´2 ¶ c2 − R cs R lim E ds = 0 . s
So, both Sβ(c) and Sβ(c) grow as e
c→0
ec S
¡ ¢ ¡ ¢ Since 1c minimizes the function gc , we have gc 1c ≤ gc ψSec . Further EVt is increasing in t, so EVSβ(c) ≤ EV∞ . Thus for c → 0 ³ ´ ¡ ¢ ¡ ¢ ∗ e 0 ≤ L c, Sβ(c) − Lc ≤ L c, Sβ(c) − L c, Sc + c2 µ ¶ ¡ ¢ c 1 c ≤ gc + EV∞ − gc ψSec − EVSec + c2 c 2 2 ∞ Z µ ³ ´2 ¶ c 2 c cs ds + c2 = o(c) . ≤ E Rs − R 2 ec S
As desired we obtain
¡ ¢ L∗c = L c, Sβ(c) + o(c) .
The second equality follows directly from Corollary 5.3
¤
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H. R. LERCHE AND I. MAAHS
Acknowledgement We thank the referee for his valuable criticism and suggestions. References 1. M. Beibel, Bayes-Optimalit¨at in Trend¨anderungsmodellen mit kontinuierlicher Zeit. Dissertation, Univ. Freiburg, 1994. 2. M. Beibel, Sequential change-point detection in continuous time when the post-change drift is unknown. Bernoulli 3(1997), No. 4, 457–478. 3. M. Beibel and H. R. Lerche, A new look at optimal stopping problems related to mathematical finance. Empirical Bayes, sequential analysis and related topics in statistics and probability (New Brunswick, NJ, 1995). Statist. Sinica 7(1997), No. 1, 93–108. 4. M. Beibel and H. R. Lerche, Sequential Bayes detection of trend changes. Foundations of statistical inference (Shoresh, 2000), 117–130, Contrib. Statist., Physica, Heidelberg, 2003. 5. C. Dellacherie and P. A. Meyer, Probabilities and potential. B. Theory of martingales. (Translated from the French) North-Holland Mathematics Studies, 72. NorthHolland Publishing Co., Amsterdam, 1982. 6. J. Jacod and A. N. Shiryaev, Limit theorems for stochastic processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 288. Springer-Verlag, Berlin, 1987. 7. I. Karatzas and E. Shreve, Brownian motion and stochastic calculus. Second edition. Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1991. 8. H. R. Lerche, An optimal property of the repeated significance test. Proc. Nat. Acad. Sci. U.S.A. 83(1986), No. 6, 1546–1548. 9. H. R. Lerche, The shape of Bayes tests of power one. Ann. Statist. 14 (1986), no. 3, 1030–1048. 10. R. S. Liptser and A. N. Shiryaev, Statistics of random processes. I. General theory. (Translated from the Russian) Applications of Mathematics, Vol. 5. Springer-Verlag, New York–Heidelberg, 1977.
(Received 4.09.2007; revised 21.10.2008) Authors’ address: Department for Mathematical Stochastics University of Freiburg Eckerstr. 1, 79104 Freiburg, Germany E-mail:
[email protected] [email protected]
