FAST CONVERGENCE OF QUASI-MONTE CARLO FOR A CLASS OF ISOTROPIC INTEGRALS

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1 MATHEMATICS OF COMPUTATION Volume 7, Numbe 233, Pages S ()1231-X Aticle electoically published o Febuay 23, 2 FAST CONVERGENCE OF QUASI-MONTE CARLO FOR A CLASS OF ISOTROPIC INTEGRALS A. PAPAGEORGIOU Abstact. We coside the appoximatio of d-dimesioal weighted itegals of cetai isotopic fuctios. We ae maily iteested i cases whee d is lage. We show that the covegece ate of quasi-mote Calo fo the appoximatio of these itegals is O( log /). Sice this is a wost case esult, compaed to the expected covegece ate O( 1/2 )ofmotecalo, it shows the supeioity of quasi-mote Calo fo this type of itegal. This is much faste tha the wost case covegece, O(log d /), of quasi-mote Calo. 1. Itoductio The Mote Calo method (MC) is fequetly used fo multidimesioal itegatio. The expected eo of MC, usig itegad evaluatios, is of ode 1/2 idepedet of the dimesio. Howeve, this covegece is ot fast, ad a lage umbe of evaluatios may be ecessay. Quasi-Mote Calo (QMC) methods evaluate the itegad at detemiistic poits i cotast to MC methods, which use adom poits. The detemiistic poits ae, oughly speakig, uifomly spead because they belog to low discepacy sequeces. The Koksma-Hlawka iequality states that the wost case QMC eo fo multivaiate itegatio is of ode log d /, whee is the umbe of itegad evaluatios ad d is the dimesio. A simila boud fo the aveage eo of multidimesioal itegatio is show by Woźiakowski [18]. Niedeeite [6], ad Dmota ad Tichy [3] ae authoitative efeeces o low discepacy sequeces, thei popeties, ad thei applicatios to umeical itegatio. ThecoceabouttheQMCeoisthatlog d / becomes huge whe is fixed ad d is lage as sometimes happes i pactice. This has cotibuted to the belief that QMC methods should ot be used fo high-dimesioal poblems [1]. Howeve, tests by Paskov ad Taub [12] ad Paskov [13] showed that QMC methods ca be vey effective fo high-dimesioal itegals aisig i computatioal fiace. They used QMC methods to appoximate 36-dimesioal itegals equied fo picig a collatealized motgage obligatio. Othe papes epotig the success of QMC methods fo poblems i fiace iclude [4, 7, 1]. A suvey of the state of the at may be foud i Chapte 4 of the moogaph by Taub ad Weschulz [17]. Received by the edito Mach 2, Mathematics Subject Classificatio. Pimay 65D3, 65D32. Key wods ad phases. Multidimesioal itegatio, quadatue, Mote Calo methods, low discepacy sequeces, quasi-mote Calo methods. This eseach has bee suppoted i pat by the NSF. 297 c 2 Ameica Mathematical Society

2 298 A. PAPAGEORGIOU Oe of the hypotheses advaced to explai the success of QMC methods is that the fiacial poblems ae oisotopic sice some dimesios ca be fa moe impotat tha othes. I a ecet pape, Sloa ad Woźiakowski [14] used this fact to obtai a possible theoetical explaatio fo the supisigly good pefomace of QMC methods fo poblems i fiace. Papageogiou ad Taub [11] used QMC-GF, a QMC method usig poits fom the geealized Faue 1 sequece [15], fo a model isotopic poblem suggested by a physicist B. Keiste [5]. Thei tests o high dimesioal istaces of this poblem (d agig fom 25 to 1) showed the supeioity of QMC-GF ove MC ad ove Keiste s poposed quadatue ules. Tests o the same poblem by Novak et al. [9] showed that QMC-GF pefoms extemely well compaed to NEW [8], a itepolatoy algoithm fo multidimesioal itegatio of smooth fuctios. I this pape we pove that the wost case speed of covegece of QMC fo a class of isotopic fuctios (which icludes the oes tested i [11]) is of ode log /. Thus, QMC has two advatages ove MC fo this class of itegals: QMC coveges as log / while MC coveges as 1/2. The wost case eo of QMC is O( log /) while oly the expected eo of MC is O( 1/2 ). We summaize the emaide of this pape. Fo the eade s beefit we biefly list cetai popeties of low discepacy sequeces i the secod sectio. The poblem is fomulated i the thid sectio, ad fast covegece is pove i the last sectio. 2. Low discepacy sequeces Discepacy is a measue of deviatio fom uifomity of a sequece of poits. I paticula, the discepacy of poits x 1,...,x [, 1] d, d 1, is defied by D (d) = D (d) (x 1,...,x )=sup A(E; ) λ(e) E, whee the supemum is take ove all the subsets of [, 1] d of the fom E = [,t 1 ) [,t d ), t j 1, 1 j d, λ deotes the Lebesgue measue, ad A(E; ) deotes the umbe of the x j that ae cotaied i E. A detailed aalysis of low discepacy sequeces ca be foud i [3, 6, 15] ad i the efeeces theei. A sequece x 1,x 2,... of poits i [, 1] d is a low discepacy sequece if D (d) )d c(d)(log, >1, whee the costat c(d) depeds oly o the dimesio d. The Koksma-Hlawka iequality establishes a elatio betwee low discepacy sequeces ad multivaiate itegatio (see [6]). If f is a eal fuctio defied o [, 1] d of bouded vaiatio V (f) i the sese of Hady ad Kause, the fo ay sequece x 1,...,x [, 1) d we have f(x) dx 1 f(x i ) [,1] V (f)d(d). d 1 The geealized Faue ad the Sobol low discepacy sequeces ae icluded i FINDER, a Columbia Uivesity softwae system, ad ae available to eseaches upo equest by witig the autho.

3 FAST CONVERGENCE OF QUASI-MONTE CARLO 299 So fa, we have discussed the discepacy of a sequece of poits with espect to the Lebesgue measue. We biefly discuss the case whee the uifomity of a sequece is assessed with egad to a pobability measue µ ad itoduce some otatio that we will use late. We assume that the suppot of µ is R + ad that µ is absolutely cotiuous with espect to the Lebesgue measue. Fo >1, let x i R +, i =1,...,, be ay give poits. Defie the diffeece betwee the empiical distibutio (appoximatig µ usig the poits x i )adthe measue µ by R µ (E) = A(E; ) µ(e), E R +, whee A(E; ) deotes the umbe of x i cotaied i E ad does ot deped o µ but depeds oly o the poits x i. The discepacy of the poits x i, i =1,...,, with espect to the pobability measue µ is defied by D µ, = D µ, (x 1,...,x )=sup R µ (E), E whee the supemum is take ove all sets of the fom E =[,x), x R +. Fo x we use the followig otatio: µ(x) = µ([,x)), R µ (x) = 1 [,x)(x i ) µ(x) = 1 [,µ(x))(µ(x i )) µ(x), whee 1 A deotes the chaacteistic fuctio of a set A. Thus, give a low discepacy sequece (with espect to the Lebesgue measue) t i [, 1], i =1, 2,...,the sequece x i = µ 1 (t i ) R +, i =1, 2,..., has discepacy D µ,, with espect to the measue µ, ad satisfies D µ, (x 1,...,x )=D (1) (t 1,...,t ), >1. Fo bevity, whe d = 1 we will wite D istead of D (1). 3. Poblem fomulatio We coside the appoximatio of a weighted high-dimesioal itegal of the fom I d (f) = f( x )e x 2 (1) dx, R d whee d is the dimesio, f : R R, ad deotes the Euclidea om i R d. We also assume that f is such that the itegal (1) is well defied ad f exists a.e. The itegal (1) ca be educed, via a chage of vaiable, to a oe-dimesioal itegal, which ca ofte be solved aalytically, e.g., f = cos. We do ot do this because we wat to assess the pefomace of QMC methods fo d-dimesioal itegatio. I [11], the empiical covegece ate of QMC is popotioal to 1, as if it sees that this is eally a oe-dimesioal poblem. I cotast, the empiical covegece ate of MC emais popotioal to 1/2 ;itdoesotseethatthe poblem is eally oe dimesioal.

4 3 A. PAPAGEORGIOU We obtai a equivalet itegal ove the cube [, 1] d.wehave (2) I d (f) = f( x )e x 2 dx =2 d/2 f( y / 2)e y 2 /2 dy R d R d = π d/2 f( y / 2) e y 2 /2 dy = R (2π) d/2 πd/2 f d (φ 1 ) 2 (t j )/2 dt, d [,1] d j=1 whee φ is the cumulative omal distibutio fuctio with mea ad vaiace 1, φ(u) = 1 u e s2 /2 ds, u [, ]. 2π Let t i =(t i1,...,t id ) [, 1] d, i =1,...,, be ay detemiistically chose sample poits. Let x i =(x i1,...,x id ) R d, be such that x ij = φ 1 (t ij ), j = 1,...,d, i =1,...,. We appoximate the itegal (1) by the QMC method (3) I d, (f) = πd/2 f( x i / 2). We deive the eo equatio ad the covegece ate of the method I d, fo the followig class of fuctios. Defiitio 1. F is the class of fuctios f : R R, such that I d (f) <, f is absolutely cotiuous, f exists a.e., ad ess sup { f () : R} M, whee M is a costat. The examples oigially cosideed by Keiste [2, 5] ad late by Papageogiou ad Taub [11], ad Novak et al. [9] belog to F sice f = cos. The example f() =(1+ 2 ) 1/2 i[2,9]alsobelogstof. 4. Speed of covegece I this sectio we deive the eo ad the covegece ate of the method (3) fo the itegal (1) i the class F.Wehave (4) I d (f) = 2 d/2 f( x / 2)e x 2 /2 dx R d = c d 2 d/2 f(/ 2) d 1 e 2 /2 d = π d/2 f(/ 2)µ ()d, whee c d =2π d/2 /Γ(d/2) ad µ ( ) is the desity fuctio of the distibutio µ of = x, x R d.notethat 2 = x 2 follows the chi-squae distibutio (see [16] fo the elatioship betwee µ ad the chi-squae distibutio). Fist we coside itegals with Gaussia weights ad deive the eo of a method that uses the aveage of fuctio evaluatios, at abitay poits, to appoximate them. The we show the eo of the method I d, i the class F,ad deive cetai auxiliay iequalities fo the measue µ. We coclude the sectio by

5 FAST CONVERGENCE OF QUASI-MONTE CARLO 31 showig that fo a paticula choice of the sample poits the covegece of the method I d, is O( log /). Lemma 1. Let h : R R, d 1, be a fuctio such that R d h( x )e x 2 dx <, h is absolutely cotiuous, ad h exists a.e. Let x i R d, i =1..., be ay poits, 1. The Rd h( x ) e x 2 /2 (2π) d/2 dx 1 h( x i )= R µ ()h () d, whee µ is defied i (4) ad R µ () = 1 1 [,)( x i ) µ(), R +. Poof. Niedeeite [6] exhibits the eo of a quasi-mote Calo method appoximatig the itegal of a diffeetiable fuctio. We apply a simila techique. Fo, d 1cosidex i R d, i =1,...,.The Rd h( x ) e x 2 /2 (2π) dx 1 h( x d/2 i ) = h()µ () d 1 h( x i ) 1 = h(µ 1 (t)) dt 1 h( x i ) 1 = g(t) dt 1 g(s i ),s i = µ( x i ), g= h µ 1 1 = R(t)g (t) dt, R(t) = 1 1 [,t) (s i ) t 1 = R(t) dh(z) dz (µ 1 ) (t) dt z=µ 1 (t) 1 = R(t) dh(z) dµ 1 (t) dz = = which completes the poof. z=µ 1 (t) R(µ())h () d R µ ()h () d, I the poof of the Lemma 1 we have used the quatity R µ () =R(µ()), R +, which is bouded fom above by the discepacy D of the poits µ( x i ), i = 1,...,. This suggests that good sample poits ca be obtaied by appopiately tasfomig oe-dimesioal low discepacy sequeces as we will see below. We ow tu ou attetio to the itegal I d. Fo each f F, we defie the eo of the method I d, (f) by e(i d,,f)= I d (f) I d, (f).

6 32 A. PAPAGEORGIOU We also defie the wost case eo of this method i the class F by e(i d, )=supe(i d,,f). f F Accodigly, the quatity e(i d,1,f) is the eo we obtai usig a sample of size 1. This quatity ca also be viewed as the iitial eo of the method I d, (f) whe >1. Lemma 2. Fo f F,themethodI d, appoximates the itegal I d with eo e(i d,,f)= πd/2 R µ ()f 2 R (/ 2) d +, whee µ is defied i (4), R µ () = 1 1 [,)( x i ) µ(), adx i R d, i = 1,..., ae abitay but fixed sample poits. Poof. The poof follows fom Lemma 1 by settig h( ) =π d/2 f( / 2). Theoem 1. Fo the class of fuctios F, the eo of the method (3) satisfies e(i d, )= e(i d,1) R µ () d, ρ( x 1 ) R + whee ρ( x 1 )= R + 1 [,) ( x 1 ) µ() d, adx i R d, i =1,..., ae abitay but fixed sample poits. Poof. Usig Lemma 2 ad by cosideig a fuctio f F such that f = M ad R µ ()f (/ 2) wedeivee(i d, ). Similaly, we deive e(i d,1 )adthepoof follows. We poceed to obtai bouds fo 1 µ(), >, o which the value of the quatity R + R µ () d depeds. We have 1 µ() = µ (y) dy = µ ( + y) dy = c d (2π) d/2 ( + y) d 1 e (+y)2 /2 dy ( = γ d d 1 e 2 /2 y ) d 1 +1 e y e y2 /2 dy, γ d = c d (2π) d/2 γ d d 1 e 2 /2 γ d d 1 e 2 /2 w 1, e wy e y2 /2 dy, w = d 1, ad 2 d 1 whee the last iequality holds by vitue of the fact that e yz dy = z 1, z>, ad e y2 /2 1. Sice 2 d 1, we coclude that w 1 d/ (fo d =1the above expessio holds fo w = 1) ad that (5) 1 µ() dγ d d 2 e 2 /2 = d µ (), d 1.

7 FAST CONVERGENCE OF QUASI-MONTE CARLO 33 I a simila way we deive a lowe boud fo 1 µ(). We have ( 1 µ() = γ d d 1 e 2 /2 y ) d 1 +1 e y e y2 /2 dy γ d d 1 e 2 /2 (6) e y e y2 /2 dy ( γ d d 1 e 2 /2 1 1 ) 3,>, whee the last iequality ca be foud i [16, p. 174]. Theoem 2. Thee exist detemiistic poits x i R d, i =1,...,, fo which the eo of the method I d, fo the itegal I d is bouded as follows: [ e(i d, ) e(i ( ) ] d,1) c d 2log ρ(ζ) Γ(d/2) + d (1 + o(1)), whee c is a costat, ρ(ζ) = R + 1 [,) (ζ) µ() d, ad ζ = µ 1 (1/2), i.e., e(i d, )=O( log /). Poof. Let t i [, 1), i =1,...,,beumbes with discepacy D = c/, whee c 1/2 is a costat. Fo istace, these umbes ca be tems of a low discepacy sequece o a (t, m, 1)-et. It is show i [6] that the discepacy of these poits is give by D = 1 2 +max 1 i t (i) 2i 1 2, whee t (1) t (i) t () deotes the odeed sequece of the poits. This implies that the discepacy of the sequece { t τ i = i if t i < 1 (4) 1 t i (4) 1,,...,, othewise caot exceed c/ ad its maximum tem satisfies τ () < 1 (4) 1. Hece, without loss of geeality, we assume that t () < 1 (4) 1. Coside x i R d such that µ( x i ) = t i, i = 1,...,. Let = () = max 1 i { x i }, assume that is sufficietly lage so that 2 > d 1ad coside the eo equatio of Theoem 1. We have e(i d, ) e(i d,1 ) = 1 ρ( x 1 ) R + R µ () d 1 ρ(ζ) R µ () d, R + because ρ(s) ρ(ζ), s R +,foζ = µ 1 (1/2). Thus, ρ(ζ) e(i d,) e(i d,1 ) R µ () d + [1 µ()] d. Sice R µ () c/, R +,weestimatethefisttemoftheaboveequatioby c /. We estimate the secod tem usig (5) ad R µ (z) dz ad 1 µ( ) c/. Thus, [1 µ(z)] dz d[1 µ()], d 1, ρ(ζ) e(i d,) e(i d,1 ) c ( + d).

8 34 A. PAPAGEORGIOU Coside the fuctio q = g() =d µ () We estimate its ivese, g 1,by h(q) = = dγ d d 2 e 2 /2, > d 1. { [ 2 log(a d q 1 )+ d 2 1/2 log log(a d q )]} 1 = 2 log(a d q 2 1 )(1 + o(1)), whee a d = d/γ(d/2). Ideed, g(h(q)) = dγ d2 (d 2)/2 { log(a d q 1 )+ d 2 } (d 2)/2 log log(a d q 1 ) a d 2 q [log(a d q 1 )] (d 2)/2 = q(1 + o(1)), as q. Fom (5) ad (6) we have tight bouds fo µ(), ( 1 d g() 1 1 ) 1d [1 d g() 1 ] 2 1 µ() g(), > d 1. The fuctio g is deceasig fo 2 >d 2, which implies that g 1 [1 µ( )]. We substitute by the value g 1 [1 µ( )] i the eo estimate to obtai ρ(ζ) e(i d,) e(i d,1 ) c (g 1 [1 µ( )] + d). Sice we have assumed that t () < 1 (4) 1 ad µ( )=t (),wehave1 µ( ) > (4) 1. Thisimpliesthat = g 1 [1 µ( )] = 2 log(a d )(1 + o(1)), which completes the poof. We do ot kow if the boud of Theoem 2 is shap. The poof of Theoem 2 ot oly shows how good sample poits ca be obtaied but also how the quality of ay sample ca be assessed by calculatig the discepacy of its poits. If the discepacy D µ, of the sample poits x i, i =1,...,, is small, the the eo of the method I d, will be small. The quatity 2 log(a d ), a d = d/γ(d/2), i the eo boud of Theoem 2 is small i pactice. Fo d as small as 6 a sample of size >1 8 is equied fo 2log(ad ) >d. Coollay 1. Fo 2 log(a d ) d, a d = d/γ(d/2), we have e(i d, ) e(i d,1) ρ(ζ) 2d c. Coollay 2. If is small so that = () d, the we have e(i d, ) e(i d,1) ρ(ζ) [ d + d] c.

9 FAST CONVERGENCE OF QUASI-MONTE CARLO 35 Poof. Whe d we caot diectly use (5) as we did i the poof of Theoem 2. Usig R µ () c/, R +,wehave R µ () d d R µ () d + d [1 µ()] d c d + d[1 µ( d)] c d + d[1 µ( )] c [ d + d], which completes the poof. Coollaies 1 ad 2 show coditios that elate the size of the dimesio d ad thesamplesize ad how these coditios affect the eo of the method I d,. These coditios ca be iteestig i pactice. I all cases, the size of D µ, ad the coditios of the above two coollaies ca be easily checked to yield pactical umeical eo estimates. Fo istace, the discepacy of the poits used by QMC-GF i [11] is small fo 1 6 ad d 1. This implies that they ca be used to efficietly evaluate itegals of fuctios i the class F. Thus, simulatio esults epotig fast covegece eve whe d is lage as i [9, 11] ca be explaied. Ackowledgmets I wish to thak F. Cubea, S. Tezuka, J. Taub ad G. Wasilkowski fo thei commets which geatly impoved this pape. Refeeces [1] Batley, P., Fox, B.L., ad Niedeeite, H. (1992), Implemetatio ad Tests of Low- Discepacy Sequeces, ACM Tas. o Modelig ad Compute Simulatio, 2:3, [2] Capstick, S., ad Keiste, B.D. (1996), Multidimesioal quadatue algoithms at highe degee ad/o dimesio, Joual of Computatioal Physics, 123, CMP 96:7 [3] Dmota, M., Tichy, R.F. (1997), Sequeces, discepacies ad applicatios, Lectue Notes i Mathematics, 1651, Spige, New Yok. MR 98j:1157 [4] Joy, C., Boyle, P.P., ad Ta, K.S. (1996), Quasi-Mote Calo Methods i Numeical Fiace, Maagemet Sciece, 42, No. 6, [5] Keiste, B.D. (1996), Multidimesioal Quadatue Algoithms, Computes i Physics, 1:2, [6] Niedeeite, H. (1992), Radom Numbe Geeatio ad Quasi-Mote Calo Methods, CBMS-NSF Regioal Cofeece Seies i Applied Math. No. 63, SIAM. MR 93h:658 [7] Niomiya, S., ad Tezuka, S. (1996), Towad eal-time picig of complex fiacial deivatives, Applied Mathematical Fiace, 3, 1 2. [8] Novak, E., Ritte, K. (1996), High dimesioal itegatio of smooth fuctios ove cubes, Nume. Math., 75, MR 97k:6557 [9] Novak, E., Ritte, K., Schmitt, R., Steibaue, A. (1997), O a ecet itepolatoy method fo high dimesioal itegatio, Pepit Uivesity of Elage. [1] Papageogiou, A., ad Taub, J.F. (1996), Beatig Mote Calo, Risk, 9:6, [11] Papageogiou, A., ad Taub, J.F. (1997), Faste evaluatio of multi-dimesioal itegals, Computes i Physics, Nov./Dec., [12] Paskov, S.H. ad Taub, J.F., Faste Valuatio of Fiacial Deivatives, Joual of Potfolio Maagemet, Fall, 1995, [13] Paskov, S.H. (1997), New Methodologies fo Valuig Deivatives, i MathematicsofDeivative Secuities, S. Pliska ad M. Dempste eds., Isaac Newto Istitute, Cambidge Uivesity Pess, Cambidge, UK, CMP 98:6

10 36 A. PAPAGEORGIOU [14] Sloa, I.H., ad Woźiakowski, H. (1998), Whe Ae Quasi-Mote Calo Algoithms Efficiet fo High Dimesioal Itegals?, J. Complexity, 14(1), [15] Tezuka, S. (1995), Uifom Radom Numbes: Theoy ad Pactice, Kluwe Academic Publishes, Bosto. [16] Tog, Y.L. (199), The Multivaiate Nomal Distibutio, Spige Velag, New Yok. MR 91g:621 [17] Taub, J.F. ad Weschulz, A.G. (1998), Complexity ad Ifomatio, Cambidge Uivesity Pess, Cambidge, UK. CMP 99:13 [18] Woźiakowski, H. (1991), Aveage case complexity of multivaiate itegatio, Bulleti of the Ameica Mathematical Society, 24, MR 91i:65224 Depatmet of Compute Sciece, Columbia Uivesity, New Yok, NY 127 addess: ap@cs.columbia.edu