Stopping Brownian Motion without Anticipation as Close as Possible to its Ultimate Maximum

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1 Theory Probab. Al. Vol. 45, No.,, (5-36) Research Reor No. 45, 999, De. Theore. Sais. Aarhus Soing Brownian Moion wihou Aniciaion as Close as Possible o is Ulimae Maximum S. E. GRAVERSEN 3, G. PESKIR 3, A. N. SHIRYAEV 3 Le B = (B ) be sandard Brownian moion sared a zero, and le S = max r B r for. Consider he oimal soing roblem V3 = inf E B S ) where he infimum is aken over all soing imes of B saisfying. We show ha he infimum is aained a he soing ime 3 = inf 8 9 j S B z3 where z3 = :... is he unique roo of he equaion wih '(x) = (= )e x = 48(z3) z3'(z3) 3 = and 8(x) = R x '(y) dy. The value V 3 equals 8(z3). The mehod of roof relies uon a sochasic inegral reresenaion of S, ime-change argumens, and he soluion of a free-boundary (Sefan) roblem.. Formulaion of he roblem Imagine he real-line movemen of a Brownian aricle sared a during he ime inerval [; ]. Le S denoe he maximal osiive heigh ha he aricle ever reaches during his ime inerval. As S is a random quaniy whose values deend on he enire Brownian ah over he ime inerval, is ulimae value is a any given ime [; ) unknown. Following he Brownian aricle from he iniial ime onward, he quesion arises naurally as o deermine a ime when he movemen should be erminaed so ha he osiion of he aricle a ha ime is as close as ossible o he ulimae maximum S. In his aer we resen he soluion o his roblem if closeness is measured by a mean-square disance. To formulae he roblem more recisely, le B = (B ) be a sandard Brownian moion (B = ; E(B ) = ; E(B ) = ) defined on a robabiliy sace (; F ; P ), and le IF B = (F B ) denoe he naural filraion generaed by B. Leing M denoe he family of all soing (Markov) imes wih resec o IF B saisfying, he roblem is o comue (.) V3 = inf M E B max B and o find an oimal soing ime (he one a which he infimum in (.) is aained). * Cenre for Mahemaical Physics and Sochasics, suored by he Danish Naional Research Foundaion. MR 99 Mahemaics Subjec Classificaion. Primary 6G4, 6J65, 6L5. Secondary 6J5, 6J6, 34B5. Key words and hrases: Brownian moion, oimal soing, aniciaion, ulimae maximum, free-boundary (Sefan) roblem, he Iô-Clark reresenaion heorem, Markov rocess, diffusion. goran@imf.au.dk

2 The soluion of his roblem is resened in Theorem. below. I urns ou ha he maximum rocess S = (S ) given by (.) S = su s B s and he CUSUM-ye refleced rocess S B = (S B ) lays a key role in he soluion. The oimal soing roblem (.) is of ineres, for examle, in financial mahemaics and financial engineering where an oimal decision (i.e. oimal soing ime) should be based on a redicion of he fuure behaviour of he observable rocess (asse rice, index, ec.). The argumen also carries over o many oher alied roblems where such redicions lay a role.. The resul and roof The main resul of he aer is conained in he nex heorem. Below we le (.) '(x) = e x = and 8(x) = x '(y) dy denoe he densiy and disribuion funcion of a sandard normal variable. (x IR) Theorem. Consider he oimal soing roblem (.) where (B ) Then he value V3 is given by he formula is a sandard Brownian moion. (.) V3 = 8(z3) = :73... where z3 = :... is he unique roo of he equaion (.3) 48(z3) z3'(z3) 3 = and he following soing ime is oimal (see Figures -5): 8 (.4) 3 = inf j S B 9 z3 where S is given by (.) above. Proof. Since S = su s B s is a square-inegrable funcional of he Brownian ah on [; ], by he Iˆo-Clark reresenaion heorem (see e.g. [].9) here exiss a unique IF B -adaed rocess H = (H ) saisfying E(R H d) < such ha (.5) S = a + H db where a = E(S ). Moreover, he following exlici formula is known o be valid: (.6) H = 8 S B for (see e.g. [3].93 and [].365, or Secion 3 below for a direc argumen).

3 . Associae wih H he square-inegrable maringale M = (M ) given by (.7) M = By he maringale roery of M H s db s. and he oional samling heorem, we obain (.8) E(B S ) = EjB j E(B M ) + EjS j = E( ) E(B M ) + = E H d + for all M (recall ha S jb j). Insering (.6) ino (.8) we see ha (.) can be rewrien as (.9) V3 = inf M E! B FS d + where we denoe F (x) = 48(x) 3. Since S B = (S B ) is a Markov rocess for which he naural filraion IF SB coincides wih he naural filraion IF B, i follows from general heory of oimal soing (see [4]) ha in (.9) we need only consider soing imes which are hiing imes for SB. Recalling moreover ha S B jbj by Lévy s disribuional heorem (see e.g. [].3) and once more aealing o general heory, we see ha (.9) is equivalen o he oimal soing roblem (.) V3 = inf M E F jb j d! +. In our reamen of his roblem, we firs make use of a deerminisic change of ime.. Moivaed by he form of (.), consider he rocess = ( ) given by (.) = e B e. By Iˆo s formula we find ha (.) d = d + where he rocess = ( ) is given by (.3) = is a (srong) soluion of he linear sochasic differenial equaion d e s db e s = e db s. s As is a coninuous Gaussian maringale wih mean zero and variance equal o, i follows by Lévy s characerisaion heorem (see e.g. [].4) ha is a sandard Brownian moion. We hus may conclude ha is a diffusion rocess wih he infiniesimal generaor given by (.4) IL = z d dz + d dz. Subsiuing = e s in (.) and using (.), we obain 3

4 (.5) V 3 = inf M E! e s F j s j ds + uon seing = log(= ). I is clear from (.) ha is a soing ime wih resec o IF B if and only if is a soing ime wih resec o IF. This shows ha our iniial roblem (.) reduces o solve (.6) W 3 = inf E! e s F j s j ds where he infimum is aken over all IF -soing imes wih values in [; ]. This roblem belongs o he general heory of oimal soing for ime-homogeneous Markov rocesses (see [4]). 3. To calculae (.6) define (.7) W 3 (z) = inf E z! e s F j s j ds for z IR, where = z under P z, and he infimum is aken as above. General heory combined wih basic roeries of he ma z 7! F (jzj) roms ha he soing ime (.8) 3 = inf f > : j j z 3 g should be oimal in (.7), where z 3 > is a consan o be found. To deermine z 3 and comue he value funcion z 7! W 3 (z) in (.7), i is a maer of rouine o formulae he following free-boundary (Sefan) roblem: (.9) IL W (z) = F (jzj) for z (z3 ; z 3 ) (.) W (6z 3 ) = (insananeous soing) (.) W (6z 3 ) = (smooh fi) where IL is given by (.4) above. We shall exend he soluion of (.9)-(.) by seing is value equal o for z = (z 3 ; z 3 ), and hus he ma so obained will be C everywhere on IR bu a z 3 and z 3 where i is C. Insering IL from (.4) ino (.9) leads o he following equaion: (.) W (z) + zw (z) W (z) = F (jzj) for z (z 3 ; z 3 ). The form of he equaion (.) and he value (.6) indicaes ha z 7! W 3 (z) should be even; hus we shall addiionally imose (.3) W () = and consider (.) only for z [; z 3 ). The general soluion of he equaion (.) for z is given by (.4) W (z) = C (+z ) + C z'(z) + (+z )8(z) + 8(z) 3=. 4

5 The hree condiions W (z 3 ) = W (z 3 ) = W () = deermine consans C ; C and z 3 uniquely; i is easily verified ha C = 8(z 3 ) ; C =, and z 3 is he unique roo of he equaion (.3). Insering his back ino (.), we obain he following candidae for he value (.7): (.5) W (z) = 8(z 3 )(+z ) z'(z) + (z )8(z) 3= when z [; z 3 ], uon exending i o an even funcion on IR as indicaed above (see Figure ). To verify ha his soluion z 7! W (z) coincides wih he value funcion (.7), and ha 3 from (.8) is an oimal soing ime, we shall noe ha z 7! W (z) is C everywhere bu a 6z 3 where i is C. Thus by he Iˆo-Tanaka formula we find: (.6) e W ( ) = W ( ) + e sil W ( s ) W ( s ) ds + e s W ( s ) d s. Hence by (.) and he fac ha IL W (z) W (z) = > F (jzj) for z = [z 3 ; z 3 ], uon exending W o 6z 3 as we lease and using ha he Lebesgue measure of hose > for which = 6z 3 is zero, we ge: (.7) e W ( ) W ( ) e s F (j s j) ds + M R where M = (M ) is a coninuous local maringale given by M = es W ( s ) d s. Using furher ha W (z) for all z, a simle alicaion of he oional samling heorem in he soed version of (.7) under P z shows ha W 3 (z)w (z) for all z. To rove equaliy one may noe ha he assage from (.6) o (.7) also yields: (.8) = W ( ) 3 e s F (j s j) ds + M 3 uon using (.9) and (.). Since clearly E z ( 3 ) < and hus E z ( 3 ) < as well, and z 7! W (z) is bounded on [z 3 ; z 3 ], we can again aly he oional samling heorem and conclude ha E z (M 3 ) =. Taking he execaion under P z on boh sides in (.8) enables one herefore o conclude W 3 (z) = W (z) for all z, and he roof of he claim is comlee. From (.5)-(.7) and (.5) we find ha V 3 = W 3 () + = (8(z 3 )) + = 8(z 3 ). This esablishes (.). Transforming 3 from (.8) back o he iniial roblem via he equivalence of (.9), (.) and (.5), we see ha 3 from (.4) is oimal. The roof is comlee. Remarks:. Recalling ha S B jbj we see ha 3 is idenically disribued as he soing ime e = inf f > : jb j = z 3 g. This imlies E( 3 ) = E(e ) = EjB e j = (z 3 ) E( e ) = (z 3 ) (E( 3 )), and hence we obain (.9) E( 3 ) = (z 3) + (z 3 ) = :55... Moreover, using ha (B 4 6B +3 ) is a maringale, similar argumens show ha 5

6 (.3) E(3) = From (.9) and (.3) we find (.3) Var(3) = (z3) 6 +5(z3) 4 (+(z3) )(3+6(z3) +(z3) 4 ) = :36... (z3) 4 (+(z3) ) (3+6(z3) +(z3) 4 ) = :5... (.3) V = inf EB max B s = s. For he sake of comarison wih (.) and (.9) i is ineresing o noe ha + = :8... wih he infimum being aained a = =. For his, recall from (.8) and (.6) ha (.33) E(B S ) = E! s B s FS ds + s where F (x) = 48(x)3. Using furher ha S B jbj, elemenary calculaions show! (.34) E(B S ) jbs j = 4 E 8 ds 3+ s r = 4 arcan s ds3+ s r r = 4 arcan + arcan! () ++. Hence (.3) is easily verified by sandard means. 3. In view of he fac ha 3 from (.8) wih z 3 = :... from (.3) is oimal in he roblem (.7), i is ineresing o observe ha he unique soluion of he equaion F (^z) = is given by ^z = :67... Noing moreover ha he ma z 7! F (z) is increasing on [; ) and saisfies F () =, we see ha F (z) < for all z [; ^z) and F (z) > for all z > ^z. The size of he ga beween ^z and z3 quanifies he endency of he rocess jj o reurn back o he favourable region [; ^z) where clearly i is never oimal o so. 4. The case of a general ime inerval [; T ] easily reduces o he case of a uni ime inerval reaed above by using he scaling roery of Brownian moion imlying (.35) inf E T B max B = T inf E T B max B which furher equals o T (8(z3) ) by (.). Moreover, he same argumen shows ha he oimal soing ime in (.35) is given by (.36) 3 = inf 8 T j S B z3 T where z3 is he same as in Theorem.. 5. The maximum funcional in he argumen above can be relaced by oher funcionals. The 9 6

7 inegral funcional is an examle which urns ou o have a rivial soluion. Seing I R = B d we find by Iô s formula ha he following analogue of (.5) is valid: (.37) I = Denoing () db. M = R (s) db s i follows as in (.8) ha (.38) E(B I ) = EjB j E(B M ) + EjI j = E( ) + =3 for all M. Hence we see ha (cf. (.4) below): (.39) inf M E(B I ) = = = :8... and ha he infimum is aained a 3 =. 6. From he oin of view of mahemaical saisics, he "esimaor" B of S is biased, since E(B ) = for all bu E(S ) 6=. I is hus desirable o consider he values (.4) V3 e = inf a+b air ; M E S and V e = inf E a+b S air ; and comare hem wih he values from (.) and (.3). However, by using ha E(B ) = we also find a once ha a 3 = E(S ) is oimal in (.4) wih e V3 = V 3 = = :9... and ev = V = = : Sochasic inegral reresenaion of he maximum rocess In his secion we resen a direc derivaion of he sochasic inegral reresenaion (.5) and (.6) (cf. [3] and [] ). For he sake of comarison we shall deal wih a sandard Brownian moion wih drif given by (3.) B = B + where is a real number. The maximum rocess S associaed wih B is given by (3.) S = su B s. s. To derive he analogue of (.5) and (.6) in his case, we shall firs noe ha saionary indeenden incremens of B imly (3.3) E S j F B = S + E = S + E = S + E su B s s S + F B + su (B s B ) (S B ) F B s S (z x) + z=s ; x=b. 7

8 R Using furher he formula E(X c) + = P fx > zg dz, we see ha (3.3) reads as c (3.4) E S j F B = S + S B where we use he following noaion: (3.5) F (z) = P8 S z9 and he ma f = f (; x; s) is defined accordingly. F (z) dz := f (; B ; S ). Alying Iˆo s formula o he righ-hand side of (3.4), and using ha he lef-hand side defines a coninuous maringale, we find uon seing a = E(S ) ha (3.6) E S j F B = a + = (s; B s ; S s ) db s F s (S s B s ) dbs as a non-rivial coninuous maringale canno have ahs of bounded variaion. This reduces he iniial roblem o he roblem of calculaing (3.5). 3. The following exlici formula is well-known (see e.g. [].368 or [5] ): z (3.7) F () (z) = 8 Insering his ino (3.6) we obain he reresenaion e 8 z z (). (3.8) S = a + where he rocess H is exlicily given by (3.9) H = 8 (S B ) () H db + e (S B ) 8 (S B ) (). Seing = in his exression, we recover (.5) and (.6). 4. Noe ha he argumen above exends o a large class of rocesses wih saionary indeenden incremens (including Lévy rocesses) by reducing he iniial roblem o calculaing he analogue of (3.5). In aricular, he following "redicion" resul deserves a secial noe. I is derived in exacly he same way as (3.4) above. Le X = (X ) T be a rocess wih saionary indeenden incremens sared a zero, and le us denoe S = max s X s for T. If E(S T ) < hen he redicor E(S T jf X ) of S T based on he observaions f X s j s g is given by he following formula: (3.) ES T j F X where F T(z) = P fs T zg. = S + S X F T(z) dz 8

9 REFERENCES [] KARATAS, I. and SHREVE, S. (998). Mehods of Mahemaical Finance. Sringer-Verlag. [] REVU, D. and YOR, M. (994). Coninuous Maringales and Brownian Moion. (Second Ediion) Sringer-Verlag. [3] ROGERS, L. C. G. and WILLIAMS, D. (987). Diffusions, Markov Processes, and Maringales; Volume : Iô s Calculus. John Wiley & Sons. [4] SHIRYAEV, A. N. (978). Oimal Soing Rules. Sringer-Verlag. [5] SHIRYAEV, A. N. (999). Essenials of Sochasic Finance (Facs, Models, Theory). World Scienific. Svend Erik Graversen Dearmen of Mahemaical Sciences Universiy of Aarhus, Denmark Ny Munkegade, DK-8 Aarhus maseg@imf.au.dk Alber Shiryaev Seklov Mahemaical Insiue Gubkina sr Moscow Russia shiryaev@mi.ras.ru Goran Peskir Dearmen of Mahemaical Sciences Universiy of Aarhus, Denmark Ny Munkegade, DK-8 Aarhus home.imf.au.dk/goran goran@imf.au.dk 9

10 z z3 z3 z 7! W3(z) Figure. A comuer drawing of he ma (.7). The smooh fi (.) holds a z3 and z3. Figure. A comuer simulaion of a Brownian ah (B (!)) wih he maximum being aained a = :5.

11 Figure 3. A comuer drawing of he maximum rocess (S (!)) ah from Figure. associaed wih he Brownian Figure 4. A comuer drawing of he difference rocess (S (!)B (!)) from Figures -3.

12 z3 7! z3 3 Figure 5. A comuer drawing of of he oimal soing sraegy (.4) for he Brownian ah from Figures -4. I urns ou ha 3 = :6 in his case (cf. Figure ).