J Al Probab Vol 34, No 1, 1997, (66-73) Prerint Ser No 1, 1994, Math Inst Aarhus On Wald-Tye Otimal Stoing for Brownian Motion S RAVRSN and PSKIR The solution is resented to all otimal stoing roblems of the form: where B (B t ) t su jb j c is standard Brownian motion and the suremum is taken over all stoing times for B with finite exectation, while the ma : R +! R satisfies jxj 1 cjxj + d for some d R with c > being given and fixed The otimal stoing time is shown to be the hitting time by the reflecting Brownian motion jbj (jb t j) t of the set of all ( aroximate ) imum oints of the ma x 7! (jxj) cx The method of roof relies uon Wald s identity for Brownian motion and simle real analysis arguments A simle roof of the Dubins-Jacka-Schwarz-She-Shiryaev ( square root of two ) imal inequality for randomly stoed Brownian motion is given as an alication 1 Introduction The aim of this aer is to resent the solution to a class of Wald s tye otimal stoing roblems for Brownian motion, and from this deduce some shar inequalities which give bounds for the exectation of functionals of randomly stoed Brownian motion in terms of the exectation of the stoing time More recisely, let B (B t ) t be standard Brownian motion defined on the robability sace (; F ; P ) Then in this aer we find the solution to all otimal stoing roblems of the following form Maximize the exectation: (11) jb j over all stoing times for B with finite exectation, where the measurable ma : R +! R satisfies jxj 1 cjxj + d for some d R with c > being given and fixed It is shown that the otimal stoing time is the hitting time by the reflecting Brownian motion jbj (jb t j) t of the set of all ( aroximate ) imum oints of the ma x 7! (jxj) cx The result just indicated will be resented in more detail in Section, as well as extended to all continuous local martingales by using the standard time change method To conclude the introduction we should like to say that our main emhasis in this aer is on simlicity of our solution to the roblem under consideration Nevertheless, we will see in Section 3 that our c AMS 198 subject classifications Primary 64, 6J65 Secondary 64, 644, 6H5 Key words and hrases: Brownian motion (Wiener rocess), otimal stoing (time), Wald s identity for Brownian motion, Burkholder-undy s inequality, Doob s otional samling theorem, the concave conjugate, continuous (local) martingale, time change, the square root of two imal inequality goran@imfaudk 1
method is flexible enough to rovide a simle roof of the Dubins-Jacka-Schwarz-She-Shiryaev inequality for Brownian motion which was firstly found in [] (and indeendently in [4]), and then roved by an entirely different method in [3] Wald s otimal stoing for Brownian motion 1 In this section we resent the solution to the otimal stoing roblem (11) For simlicity, we shall only consider the case where (jxj) jxj for <, and it will be clear from our roof below that the case of general ma ( satisfying the boundedness condition ) could be treated by exactly the same method Thus, if B (Bt)t is standard Brownian motion, then the roblem under consideration in this section is the following Maximize the exectation: B (1) c over all stoing times for B with finite exectation, where < and c > are given and fixed First, it should be noted that in the case, we find by Wald s identity (see [5]) for Brownian motion ( jb j ( ) ) that the exression in (1) equals (1c)( ) Thus taking n or for n 1, deending on whether < c < 1 or 1 < c < 1, we see that the suremum equals +1 or resectively If c 1, then the suremum equals, and any stoing time for B is otimal These facts solve the roblem (1) in the case The solution in the general case < < is formulated in the following theorem Theorem 1 ( Wald s otimal stoing for Brownian motion ) Let B (Bt)t be standard Brownian motion, and let < < and c > be given and fixed Consider the otimal stoing roblem: () su B c where the suremum is taken over all stoing times for B with finite exectation Then the otimal stoing time (at which the suremum is attained) in () may be defined as follows: n (3) ;c 3 1 o 1() inf t > : jbtj c Moreover, for all stoing times for B with finite exectation we have: () (4) jb j c The uer bound in (4) is the best ossible c Proof iven < < and c >, denote: (5) V(; c) jb j c whenever is a stoing time for B Then by Wald s identity for Brownian motion we find
out that the exression in (5) may be equivalently written in the following form: jxj cx (6) V (; c) Z 1 1 dp B (x) whenever is a stoing time for B with () < 1 Our next ste is to imize the ma x 7! D(x) jxj cx over R For this, note that D(x) D(x) for all x R, and therefore it is enough to consider D(x) for x > We have D (x) x 1 cx for x >, and hence we see that D attains its imal value at the oint 6(c) 1() Thus it is clear from (6) that the otimal stoing time in () might be defined by (3) This comletes the first art of the roof Finally, inserting 3 3 ;c from (3) into (6), we easily find: V 3(; c) D (c) 1() () c This establishes (4) with the last statement of the theorem, and the roof is comlete Remark The receding roof shows that the solution to the roblem (11) in the case of general ma ( satisfying the boundedness condition ) could be found by using exactly the same method: The otimal stoing time is the hitting time by the reflecting Brownian motion jbj (jb t j) t of the set of all (aroximate) imum oints of the ma x 7! D(x) (jxj) cx ( Here aroximate stands to cover the case ( in an obvious manner ) when D does not attain its least uer bound on the real line) In the remaining art of this section we will exlore some consequences of the inequality (4) in more detail For this, let a stoing time for B with ( ) < 1 and < < be given and fixed Then from (4) we get: (7) jb j inf c> c( ) + c () It is elementary to comute that this infimum equals ( ) In this way we obtain: (8) jb j 1 ( < ) with the constant 1 being the best ossible in all of the inequalities ( Observe that this also follows by Wald s identity and Jensen s inequality in a straightforward way) Next consider the case < < 1 Thus we shall look at V (; c) instead of V (; c) in (5) and (6) By the same argument as for (6) we obtain: c jb j V (; c) Z 1 1 cx jxj dp B (x) where < < 1 The same calculation as in the roof of Theorem 1 shows that the ma x 7! D(x) cx jxj attains its imal value over R at the oint 6(c) 1() Thus as in the roof of Theorem 1 we find: 3
From this inequality we get: su c> c jb j c( ) + () c () c jb j The same calculation as for the roof of (8) shows that this suremum equals ( ) Thus as above for (8) we obtain: (9) 1 jb j ( < 1 ) with the constant 1 being the best ossible in all of the inequalities ( Observe that this also follows by Wald s identity and Jensen s inequality in a straightforward way) 3 The revious calculations together with conclusions (8) and (9) indicate that the inequality (4)+(7) rovide shar estimates which are otherwise obtainable by a different method that relies uon convexity and Jensen s inequality ( see Remark 4 below ) This leads recisely to the main oint of our observation: The revious rocedure can be reeated for any measurable ma satisfying the boundedness condition In this way we obtain a shar estimate of the form: jb j 1 ( ) where is a ma to be found ( by imizing and minimizing certain real valued functions of real variable ) We formulate this more recisely in the next corollary Corollary 3 Let B (B t ) t Then for any stoing time for B the inequality holds: (1) be standard Brownian motion, and let : R!R be a measurable ma jb j 1 inf c> c( ) + su xr jxj 1 cx and is shar whenever the right-hand side is finite Similarly, if H : R!R is a measurable ma, then for any stoing time for B with finite exectation the inequality holds: (11) su c> c( ) + inf xr H and is shar whenever the left-hand side is finite jxj 1 cx H jb j 1 Proof It follows from the roof of Theorem 1 as indicated in Remark and the lines above following it (or just straightforward by using Wald s identity) It should be noted that the boundedness condition on the mas and H is contained in the non-triviality of the conclusions Remark 4 As noted by a referee, if we set H(x) ( x) 1 cx su xr jxj inf x for x, then we have: cx H(x) ~ H(c) 4
where ~ H denotes the concave conjugate of H Similarly, we have: inf c> Thus (1) reads as follows: (1) H c( ) + su xr jb j 1 jxj 1 cx H ( ~ ) inf c( ) ~ H(c) H ~ ( )1 Moreover, since ~H is the (smallest) concave function which dominates H, it is clear from a simle comarison that (1) also follows by Jensen s inequality This rovides an alternative way of looking at (1) and (11) and clarifies (7)+(8) ( A similar remark might be directed to (11) with (9)) Note that (1) gets the following form: jb j 1 ( ) whenever x 7! ( x) is concave on R + Remark 5 By using the standard time change method, we can generalize and extend the inequalities (1) and (11) to cover the case of all continuous local martingales Let M (M t ) t be a continuous local martingale with the quadratic variation rocess [M ] ([M ] t ) t such that M, and let, H : R +! R be measurable functions Then for any t > for which ([M ] t ) < 1 the inequalities hold: (13) (14) su c> 1 jmt j inf c> c ([M ] t ) + inf xr c ([M ] t ) + su xr H jxj 1 cx jxj 1 cx H 1 jm t j and are shar whenever the right-hand side in (13) and the left-hand side in (14) are finite To rove the sharness of (13) and (14) for every given and fixed t >, consider M t B t+ with > and being the hitting time of some > by the reflecting Brownian motion jbj (jb t j) t Letting! 1 and using ( integrability ) roerties of ( in the context of Corollary 3 ), by Burkholder-undy s inequalities (see [1]) and uniform integrability arguments we (eventually) finish with the inequalities (1) and (11) for otimal, at least in the case when allows the limiting rocedures which are required ( the case of general could then follow by aroximation ) Thus the sharness of (13)+(14) follows from the sharness of (1)+(11) 3 Alications In this section, as an alication of our method and results obtained, we shall resent a simle roof of the Dubins-Jacka-Schwarz-She-Shiryaev (square root of two) imal inequality for randomly stoed Brownian motion which was firstly found in [] ( and indeendently in [4] ), and then roved by an entirely different method in [3] The method of attack in the roof of (31) and (3) is based uon the trick of icking u the two martingales (33) and (35) with the roerties 5
desired We shall begin by stating the two inequalities to be roved Let B (B t ) t be standard Brownian motion, and let be a stoing time for B with finite exectation Then the following inequalities are shar: (31) (3) t jb t j t B t 1 We shall first deduce these inequalities by our method, and then show their sharness by icking u the otimal stoing times ( for which the equalities are attained ) Our aroach to the roblem of establishing (31) is motivated by the fact that the rocess ( st B s B t ) t is equally distributed as the reflecting Brownian motion rocess (jb t j) t for which we have found otimal bound (4) ( from where by (7) we get (8) with 1 ), while (B ) whenever ( ) < 1 These observations clearly lead us to (31), at least for some stoing times To extend this to all stoing times, we shall use a simle martingale argument Proof of (31): Set S t st B s for t Since (B t) t t is a martingale, and (S t B t ) t is equally distributed as (jb t j) t, we see that: St 1 (33) Z t c B t t + 14c is a martingale ( with resect to the natural filtration which is known to be the same as the natural filtration of B ) Using (B ), by Doob s otional samling theorem and the elementary inequality xct c(x t) + 14c, we find: for any bounded stoing time S c 1 S B c 1 Z ) (Z ) 14c Hence we get: S 1 inf c> c( )+14c for any bounded stoing time Passing to the limit, we obtain (31) for all stoing times with finite exectation This comletes the roof of (31) Next we extend (31) to any continuous local martingale M (M t ) t with M For this, note that by the time change and (31) we obtain: (34) M s B [M ] s B s q 1 [M ] t st st s[m ] t for all t > 3 In the next ste we will aly (34) to the continuous martingale M defined by: (35) M t jb j jb j 1 F t^ for t In this way we get: 6
(36) t<1 jb j 1 jb j Ft^ We now ass to the roof of the square root of two inequality Proof of (3): Since A x + x A for < x < t r r jb t j t<1 jb j t<1 jb t^ j jb j 1 Ft^ 1 1 jb j jb j + jb j ( ) r jb j jb j 1 A, by (36) we find: t<1 1 + jb j 1 jb j + 1 jb j ( ) This establishes (3) and comletes the first art of the roof 4 To rove the sharness of (31) one may take the stoing time: 9 3 1 inf 8 t > : jb t j a jb j Ft^ for any a > Then the equality in (31) is attained It follows by Wald s identity Note that for any a > the stoing time 3 1 could be equivalently (in distribution) defined by: 3 1 inf t > : B s B t a st 5 To rove the sharness of (3) one may take the stoing time: 3 inf t > : jb s jjb t j a st for any a > Then it is easily verified that t 3 jb t j 1 a and ( 3 ) a ( see [3] ) Thus the equality in (3) is attained, and the roof of the sharness is comlete RFRNCS [1] BURKHOLDR, D: L: and UNDY, R: F: (197): xtraolation and interolation of quasilinear oerators on martingales: Acta Math: 14 (49-34) [] DUBINS, L: : and SCHWARZ, : (1988): A shar inequality for sub-martingales and stoing times: Astérisque 157-158 (19-145) [3] DUBINS, L: B: SHPP, L: A: and SHIRYAV, A: N: (1993): Otimal stoing rules and imal inequalities for Bessel rocesses: Theory Probab: Al: 38 (6-61) 7
[4] JACKA, S: D: (1991): Otimal stoing and best constants for Doob-like inequalities I: The case 1 : Ann: Probab: 19 (1798-181) [5] WALD, A: (1945): Sequential tests of statistical hyotheses: Ann: Math: Statist: 16 (117-186) Svend rik raversen Deartment of Mathematical Sciences University of Aarhus, Denmark Ny Munkegade, DK-8 Aarhus matseg@imfaudk oran Peskir Deartment of Mathematical Sciences University of Aarhus, Denmark Ny Munkegade, DK-8 Aarhus homeimfaudk/goran goran@imfaudk 8