Probabilistic Choice over a Continuous Range: An Econometric Model Based on Extreme-Value Stochastic Processes

Transcription

1 Probablst Choe over a Cotuous Rage: A Eoometr Model Based o Extreme-Value Stohast Proesses Stephe R Cosslett Departmet of Eooms Oho State Uversty (Revsed September 988) Abstrat The behavour of a dvdual whose hoe set ossts of a real terval [ 0, T ] (for example, tme or loato) a be modelled terms of radom utlty maxmzato, lke probablst models of dsrete hoe The hose pot t maxmzes a utlty futo defed over the hoe set The utlty futo has a determst part whh s a futo of observed varables ad ukow parameters to be estmated, ad a stohast part whh s kow to the deso-maker but ot to the eoometra Spefyg a probablty dstrbuto for the stohast part the dues a probablty dstrbuto of the hoe t over the terval [ 0, T ] The resultg hoe probabltes wll deped o observed varables ad o ukow parameters, whh a the be estmated from data o observed hoes I the lass of models proposed here, the stohast utlty s represeted by a otuous-tme radom proess whose ftedmesoal dstrbutos are based o the geeralzed extreme-value dstrbuto Frst some geeral results are preseted, showg how the dstrbuto futos of the stohast proess determe the dstrbuto futo of maxmum utlty, whh tur determes the probablty that utlty s maxmzed a gve terval of the hoe set The we vestgate the propertes of a partular lass of extreme-value stohast proesses, whh are show to have otuous sample paths wth probablty oe Ths makes them plausble addates for the radom part of a utlty futo ( otrast wth, for example, a whte-ose proess) Results are establshed for the dstrbuto of the utlty maxmum Uder sutable regularty odtos o the systemat part of the utlty futo, a algorthm leads to I would lke to thak D MFadde for suggestg the problem of maxmzg a otuous-tme stohast utlty futo Ths researh was supported part by the Ceter for Eoometrs ad Deso Sees at the Uversty of Florda

2 omputatoally tratable, but ot smple, expressos for the hoe probabltes These hoe probabltes are show to geerate a probablty measure o [ 0, T ] Some speal ases have to be addressed: pots wth dsrete hoe probabltes, tervals wth zero hoe probablty, ad addtoal regularty odtos that esure the exstee of a hoe probablty desty futo Followg the formal results, we gve explt expressos for the hoe probabltes some smple ases, to llustrate the effets of peaks, valleys, ad steps the systemat part of the utlty futo Fally, we dsuss the questo of parameter estmato, ad gve some odtos for osstet maxmum lkelhood estmato the ase of dsretzed data o the hoe varable Keywords: hoe probablty model, otuous hoe, geeralzed extreme value, radom utlty Address: Stephe R Cosslett, Departmet of Eooms, Oho State Uversty, Columbus, OH 430

3 Itroduto Ths paper presets a model of probablst hoe over a otuous rage of alteratves, based o maxmzato of a uderlyg utlty futo The dvdual who makes the hoe kows the utlty futo, ad so hs hoe s etrely determst The eoometra, o the other had, aot observe the utlty futo, ad so models t as a stohast proess over the hoe set, terms of observed varables ad ukow parameters A model of the stohast utlty futo the dues a probablty dstrbuto for the loato of ts maxmum Ths probablty dstrbuto s the hoe probablty model If t s ot too omplated, ts parameters a the be estmated from data o observed hoes As a llustrato, suppose that a dvdual has to shedule a atvty to start at tme t a terval [ 0, T ] There s a observable vetor futo x ( that expresses the attrbutes of the atvty as futos of t For example, f the atvty s a automoble trp ad t s departure tme, the oe ompoet of x ( mght be the tme take to reah the destato There s also a observed vetor z of haratersts of the dvdual, whh prple help to determe the struture of hs utlty futo Se there wll always be other relevat varables that are ot observed, the hoe t aot be uquely determed from x ( ) ad z Istead, the best we a do s to model the hoe probabltes p ( = p( t x, z), where p ( dt s the probablty that t wll be the small terval [ t, t + dt] Ths s obvously very muh lke dsrete hoe modellg, the dfferee beg the otuous hoe set [ 0, T ] stead of hoe from a fte set of alteratves A large lass of dsrete hoe probablty models has bee developed based o radom utlty maxmzato (see, for example, MFadde, 984) Eah dvdual hooses amog the alteratves as f he were maxmzg some objetve futo, whh we may all utlty Idvdual values of ths futo are modelled as radom vetors draw aordg to some probablty dstrbuto, odtoal o observed varables ad o a ukow set of parameters That leads to a hoe probablty model: the probablty that a alteratve s hose s just the probablty that ts ompoet of the radom utlty vetor s greater tha that of ay other alteratve Several osderatos go to the spefato of the uderlyg probablty dstrbuto Amog others, we would lke ompatblty wth the requremets of eoom theory; heret plausblty of the struture of the model; eough flexblty that we are ot led to a otradto of observed data; ad omputatoal tratablty of

4 the hoe probabltes ad of the parameter estmates Computatoal tratablty s obvously mportat, ad wll be oe of the oers of ths paper Nevertheless, t s geerally held that a hoe probablty model ought to be ompatble wth some uderlyg utlty maxmzato model, as opposed to a purely ad ho set of probabltes, ad t s from that pot of vew that we propose a model of hoe over a otuous rage Of ourse there s always some stohast utlty futo orrespodg to ay gve hoe probablty model p ( : for example, U ( = { t = t } where t s a radom varable wth probablty desty futo p We shall therefore have to dsuss what propertes a stohast proess ought to have order to be a reasoable model for a stohast utlty futo A dsrete hoe model s based o the dstrbuto of a radom utlty vetor, wth dmeso equal to the umber of alteratves the hoe set For otuous hoe, we have to spefy stead the dstrbuto of a stohast proess o [ 0, T ] If oe starts from some stadard model suh as a Gaussa Markov proess for the stohast utlty, oe soo rus to severe dffultes beause there s o aalyt expresso for the probablty dstrbuto of the maxmum, exept some very speal ases A dfferet le of attak s to study the behavor of dsrete hoe models as the umber of alteratves beomes large Be-Akva ad Wataatada (98), amog others, proposed the otuous logt model for spatal hoe: t s obtaed from the ovetoal multomal logt model by allowg the umber of alteratves to ted to fty, sums beg replaed by the orrespodg tegrals Aother approah s that take by Small (987): the rage of hoes s parttoed to a umber of tervals, eah of whh a the be osdered as a dsrete alteratve The resultg dsrete hoe model s the represeted by Small s ordered geeralzed extreme value (OGEV) model, whh s a partular ase of the geeralzed extreme value model trodued by MFadde (978, 98) Ths allows for a realst patter of seral orrelato betwee the radom utltes orrespodg to the sequee of the tervals, otrast to the multomal logt model These prevous studes, however, dd ot address the questo of whether the hoe probabltes are ompatble wth maxmzato of some reasoable uderlyg stohast proess The model preseted here s related to both of these last two approahes It s based o the GEV model, exteded to a otuous rage of alteratves, ad t leads to

5 omputatoally tratable (although ot always smple) hoe probabltes Explt expressos for the hoe probabltes are worked out some speal ases as a llustrato Somewhat surprsgly, the otuous logt model a emerge as a speal ase f the attrbute vetor x ( vares suffetly slowly over tme Frst we vestgate the geeral problem of defg the hoe probablty futo for a lass of stohast utlty proesses, subjet to some otuty odtos We propose the followg restrto o the proess, f we wat to plausbly terpret ts realsatos as utlty futos: wth probablty oe, sample paths should be otuous, exept for a (fxed) fte set of pots where there may be jumps More geeral models are of ourse possble: oe ould, for example, allow the set of dsotuty pots to be radom also, although t s ot kow whether a tratable model of ths kd a be foud Further regularty odtos ould also be proposed, whh mght ot hold for the model as presetly formulated: for example, although the fte-dmesoal dstrbuto futos of the stohast proess are otuous, they do ot orrespod to absolutely otuous probablty measures Further researh should show what addtoal regularty odtos a be mposed o models of ths type Geeral results are preseted Seto, for the ase where the utlty futo s the sum of a uobserved stohast proess ad a parametr futo of observed varables The dstrbuto futo of maxmum utlty, G, s related to the fte-dmesoal dstrbutos of the stohast proess, ad the probablty of the evet that utlty s maxmzed some terval J of the hoe set, P ( ), s expressed terms of G I Seto 3, we trodue a spef stohast proess based o the GEV model Some bas propertes of the proess are derved, ludg otuty probablty Usg a result of de Haa (984) o a spetral represetato of max-stable proesses, we the fd the almost sure otuty of sample paths referred to above Next we apply the geeral results of Seto to the GEV model of Seto 3 I Seto 4 we derve a expresso for the hoe probabltes, defed o tervals of the hoe set, terms of the parametr part of the utlty futo We also show that these hoe probabltes a be exteded to a probablty measure I Seto 5 we osder some speal ases of the hoe probablty futo: steps or sharp peaks the parametr utlty futo a lead to dsrete pots whh have postve hoe probablty; there s the possblty of shadows, e, tervals for whh the hoe probablty s zero; ad fally, uder 0 J A reet paper by Dagsvk (988) also exteds the geeralzed extreme value model to a otuous hoe set 3

6 sutable assumptos, there s the regular ase where a hoe probablty desty futo a be defed To llustrate the ostruto of the hoe probabltes, some smple examples are worked out expltly Seto 6 Maxmum lkelhood estmato of the parameters s dsussed brefly Seto 7, for the ase where observatos have bee ategorzed to dsrete tervals The Appedx otas proofs of the formal results, together wth some prelmary lemmas Fally, we should emphasze that the model preseted here s ot meat to be deftve; rather, t demostrates a approah that may lead to a varety of more geeral models of stohast utlty maxmzato Choe probabltes ad stohast utlty otuous tme Let U ( t, z) be the odtoal dret utlty of someoe wth haratersts z, odtoed o t [ 0, T ], where t s the startg tme of the atvty That s, U ( t, z) s the utlty attaed by optmzg over all other desos whle keepg t fxed I ths paper we osder oly the addtvely separable ase U ( t, z) = V ( x(, z, θ) + W ( () where V s a kow parametr futo of x ( (the observed haratersts of the atvty f started at tme ad of z, wth ukow parameter vetor θ The other term, W ( ), s a uobserved stohast proess The dstrbuto of W s supposed to be kow up to a parameter vetor φ; t may also deped o z but we assume t does ot deped o x ( ) The o-stohast part V ( t, θ) = V ( x(, z, θ) s assumed to satsfy the followg regularty odtos (V) V ( t, θ) s otuous t, exept possbly at a fte set of pots that does ot deped o x, z or θ (V) At pots of dsotuty t, the left ad rght lmts V ( t, θ) ad V ( t+, θ) exst, ad V ( t, θ) = max{ V ( t, θ), V ( t+, θ) } (V3) V ( t, θ) / t s otuous t, exept possbly at a fte set of pots that does ot deped o x, z or θ (V4) V ( t, θ) s otuous θ uformly t Codtos V ad V3 allow, for example, a peewse lear parametrzato of V There a also be jumps V at fxed tmes, whh ould represet, for example, tme- 4

7 varyg user fees Codto V mples that V s upper semotuous t Wthout ths last property, the utlty futo mght ot have a maxmum ad the hoe problem would be ll-defed Codto V4 s ot eeded for formulatg the hoe probablty model, but wll be used later whe we ome to estmate θ The fte-dmesoal dstrbutos of W are F t,, t k ( w,, w k ) = Pr{ W ( t ) w, =,, k } () for t [ 0, T], =,, k, ad k {,,} Wthout loss we a arrage that t < t < < t k For otatoal oveee we have ot expltly show the depedee o φ ad z The fte-dmesoal dstrbuto futos must satsfy the osstey odto lm Ft t ( w,, wk ) = Ft,, t, t,, t ( w,, w, w,, w ) (3) w,, k + k + k for eah {,, k } ad k It the follows that there s a separable stohast proess W o [ 0, T ] wth those fte-dmesoal dstrbutos (see, for example, Bllgsley (986), Seto 38) Separable meas that there s a outable dese set D [ 0, T ], suh that for ay t [ 0, T ] ad for almost ay sample path (e, exept for a set of sample paths that has probablty zero) there s a sequee { t, t, } of pots D wth t t ad w( t ) w( (4) (Ths s dfferet from otuty, whh requres (4) to hold for ay sequee that overges to The odto that W should be separable stll may ot defe the proess uquely However, the results that we obta wll apply to ay separable stohast proess W that has the fte-dmesoal dstrbutos of the model By tself the exstee of a separable stohast proess s ot very useful, beause t holds for ay osstet set of fte-dmesoal dstrbutos (ludg, for example, those orrespodg to the depedet logt model), ad beause t says lttle about the propertes of the stohast proess W If W s to be a plausble addate for the uobserved part of the utlty futo U (, z), t should satsfy some regularty propertes For example, a udesrable property of models wth whte-ose type proesses, where W ( t ) ad W ( t ) are depedet for ay t t, s that the utlty at the hose pot s fte I the extreme-value model for W proposed the ext seto, we fd that the sample paths are otuous wth probablty oe I partular, ths mples that maxt { W ( } s fte wth probablty oe 5

8 To avod repetto, we shall stop sayg wth probablty oe : ay evet E s to be terpreted as E N, where N s some fxed evet wth probablty oe I those results that deped o otuty, N wll be the evet that the sample path W ( s otuous Proposto (a) If W s separable, ad f V satsfes odto V, the U s separable ( If sample paths of W are otuous, ad f V satsfes odtos V V, the sample paths of U are upper semotuous The proof s omtted: these results follow dretly from the deftos Now osder the maxmzato of U( Let M = { t U ( U ( s) for all s [0, T ]} (5) If U ( s upper semotuous the M s o-empty, beause the hoe set s ompat For ay terval J [ 0, T ], let P ( J ) = Pr{ M } (6) 0 J If M s a sgle pot (e, f there are o multple maxma) ths s the same as the hoe probablty P( J ) = Pr{ t J }, (7) but we should ot assume ths advae To see that the probablty (6) s well defed, we eed the followg result Lemma Suppose U s separable, ad ts sample paths are upper semotuous Let [ 0, T ] J be a terval of [ 0, T ] The { M J } s a elemet of R, the σ-feld geerated [0, ] by ylders R T [ 0, T ] Beause the fte-dmesoal dstrbutos defe a probablty measure o R, (6) defes P 0( J ) for eah terval J [ 0, T ] Obvously P 0 ([0, T]) = ad P 0 ( ) = 0, but we aot yet olude that P 0 exteds to a probablty measure For example, t does ot follow that P 0 ( J J ) = P0 ( J) + P0 ( J ) for dsjot tervals J ad J, beause there mght be pots of M both tervals We shall vestgate the propertes of P 0 whe W s defed by the extreme-value model of Seto 3 I Seto 4, P 0( J ) s evaluated for ay terval J [ 0, T ] The we shall fd that P 0 ( J ) + P0 ( J ) =, whh meas P 0 ( J ) s uambguously detfed wth the hoe probablty for that terval, e, P ( J ) = P( ) We shall also verfy that 0 J 6

9 P 0 satsfes the outable addtvty property o tervals of [ 0, T ], from whh t follows that P 0 exteds to a probablty measure, e, the hoe probablty P (A) s defed for ay Borel set A [ 0, T ] The ostruto used to prove lemma does ot lead to a pratal method of evaluatg P 0 ( J ) The followg results show how, at least prple, P 0( J ) a be ostruted from lmts of fte-dmesoal dstrbuto futos Of ourse they are useful oly f the lmts are tratable ad satsfy the dfferetablty odto of Lemma 3 Frst we eed the dstrbuto of the maxmum utlty, G( u ; V ) = Pr{ sup U ( u} t [0, T ] (8) = Pr{ W ( u V ( for all t [0, T]} Lemma Suppose W s separable, ad odto V holds Let J be a terval of [ 0, T ] Let J = { t,, t m ( ) }, =,,, deote a olleto of fte subsets of J The there s a outable dese set D suh that f J D the Pr{ supu ( u} = lm F{ J }[ u V (,, u V ( tm( ) )] t J where F s the fte-dmesoal dstrbuto futo of W J (9) If fat W s otuous ad V also satsfes odto V, the we a use ay outable dese D ostrutg the sets J, for example R 0 (the ratoal umbers) Next, let J be a ope terval wth respet to [0, T] ad defe U = supu (, U = sup U ( ), (0) t t J t J where J s the omplemet of J wth respet to [ 0, T ] Oe we have foud G, the dstrbuto of maxmum utlty, the we also have the jot dstrbuto of U ad U, beause Pr{ U u, U u } = Pr{ W ( u V ( for all t J, W ( u V ( for all t J = Pr{ W ( u V ( + ( u u ) { J} for all t} = G u ; V ( u u ) { }) () ( J where { J } s the dator futo of J But P J ) = Pr{ U > } () 0 ( U } 7

10 so P 0( J ) a be foud from the jot dstrbuto () just as the stohast utlty model of bary hoe The result s the ext lemma Lemma 3 Let G( ; V ) be the dstrbuto futo of maxmum utlty, defed by (8), ad let J be a ope terval wth respet to [ 0, T ] Suppose there s a otuous futo G suh that, gve ay pot u ad ay δ > 0, ( G ( u ; V ε{ J}) G( u ; V )) G ( u ) δ (3) ε for all suffetly small ε > 0 ad all u some eghborhood of u The 0( J ) du G ( u P = ) (4) To smplfy the otato, the explt depedee of G o V ad J has bee dropped To get ay further, we eed a spef model for W 3 The GEV stohast proess ad ts propertes We ow trodue a spef model for W (, the stohast part of the utlty futo () Defe the futos H k ( y,, yk ; t,, tk ) terms of the ftedmesoal dstrbuto futos () by F w w t,, t k k k k ( w,, w ) = exp{ H ( e,, e k ; t,, t )} (3) for k =,, Suppose t A [ 0, T ], where A s ay fte set of pots The the followg odtos o the futos H k are suffet for the futos F, wth { t,, tk} A, to be a set of probablty dstrbuto futos (MFadde, 978, 98) (G) H k ( y,, yk ; t,, tk ) s oegatve ad learly homogeeous y,, yk for + y R, =,, k (G) lm Hk ( y,, yk ; t,, t y k ) = for ay {,, k } (G3) If { (),, ( r)} {,, k }, the the mxed partal dervatve H k / y( ) y( r) exsts, ad s oegatve for odd r ad opostve for eve r (G4) H y,, y,0, y,, y ; t,, t = k ( + k k ) H k + ( y,, y, y+,, yk ; t,, t, t,, tk ) for k For dsrete hoe, these odtos defe the lass of geeralzed extreme value (GEV) hoe probablty futos (MFadde, 978) The lear homogeety odto G r 8

11 s ot eessary for the futos F to be dstrbuto futos, exept that t mples H ( 0; = 0, but t allows oe to derve a explt expresso for the hoe probabltes Codto G3 esures that there s a oegatve desty futo orrespodg to the dstrbuto futo F Codto G4 restates the osstey odto (3) I the otuous ase t [ 0, T ], osder a extreme value stohast proess (EVSP) wth fte dmesoal dstrbutos gve by (3) wth ths H k : H k ( y,, yk ; t,, tk ) = dt max{ y exp[ λ ( t ]} (3) where the maxmum s over odtos: =,, k The futo λ (τ) satsfes the followg (L) λ (τ) s symmetr, wth λ ( 0) = 0 (L) λ (τ) s otuous ad strtly ovex for all τ the terval τ 0, τ ) ( 0 ( τ 0, τ0 0 < τ 0 If τ 0 <, the λ ( τ) = for all τ outsde the terval ) (L3) λ(τ) s dfferetable for all τ τ 0, τ ) exept possbly at τ = 0 ( 0, where (L4) λ ( τ) ad λ( τ) / τ as τ τ0 The symmetry odto L s ot essetal, but smplfes some of the proofs To gve a smple example of a futo λ satsfyg these odtos, let λ τ; σ) = log[( σ τ ) ] (33) + 0 ( where x max{ x,0}, ad σ > 0 s a parameter The τ = / σ + 0 The futos H k defed by (3) learly satsfy odtos G, G, ad G4, but they do ot satsfy odto G3 for k 3 beause the maxmum futo does ot have the eeded hgher-order dervatves Ths meas that there s ot a desty futo orrespodg to the futo F these ases To show that F s fat a dstrbuto futo, osder the GEV model defed by k α H k ( y,, yk ; t,, tk ) = dt ( y exp[ λ ( t ] ) (34) = wth α > These futos H k satsfy odtos G G4, so that the orrespodg futos F defed by (3) are dstrbuto futos Now / α 9

12 / α lm α = max{ } (35) α for 0, =,, k, ad overgee s uform (,, k ) provded the are bouded Therefore the tegrad (35) overges uformly t to the tegrad of (3), ad so H k H k ad F F potwse as α The futo F defed by (3) ad (3) s the the lmt of a sequee of dstrbuto futos, ad so s tself a (possbly defetve) dstrbuto futo But H ( 0,,0; t,, ) = 0 mples F t,, t k (,, ) = ad therefore F s a proper dstrbuto futo k t k Now let W, the radom part of the odtoal dret utlty futo (), be a separable stohast proess o [ 0, T ] wth fte-dmesoal dstrbuto futos defed by (3) ad (3) Propertes of W are gve the ext four lemmas Some results follow dretly from the defto The margal dstrbuto of W ( s type I extreme value, where F ( w) = exp( e t 0 λ ( τ) w ) = d τ e, (36) whh s fte by lemma A ( the appedx) If t t τ0, the W ( t ) ad W ( t ) are depedet The orrelato oeffet betwee W ( t ) ad W ( t ) reases from 0 to as t t dereases from τ 0 to 0; ts values a be omputed umerally for ay partular hoe of λ Lemma 3 Let W be a stohast proess o [ 0, T ] wth fte-dmesoal dstrbuto futos defed by (3) ad (3), ad suppose the futo λ (3) satsfes odtos L L4 The W s otuous probablty Cotuty probablty meas that, for ay gve t, plm[ W ( t + δ) W ( ] 0 as δ 0 Lemma 3 Let W be a stohast proess satsfyg the odtos of Lemma 3 The W has the same fte dmesoal dstrbutos as the proess W defed by W ( = max{ X λ ( T } k k k (37) 0

13 where ( X k, Tk ) s a eumerato of the pots the Posso proess o R [0, T] wth testy measure ( e x dx) ds (38) Ths result s proved by de Haa (984, theorem 3) for max-stable proesses, a lass of proesses whh ludes the preset model, uder the assumpto of otuty probablty The testy measure (38) meas that the umber of pots (or evets) ourrg the tme terval ( t, t + d ad wth levels the terval ( x, x + dx) s a Posso radom varable wth rate e x dx dt The total umber of evets s fte, beause the total rate for the Posso proess s fte, but the umber of evets wth levels exeedg ay gve x s fte, wth probablty oe De Haa alls (37) the spetral represetato for max-stable proesses It has a obvous heurst terpretato: the radom ompoet of utlty s the upper evelope of a sequee of bumps, wth shape gve by, ourrg at radom tmes ad wth radom heghts However, a separable stohast proess s ot uquely determed by ts fte-dmesoal dstrbutos, so W does ot eessarly have the represetato W Lemma 33 Let W ( ad ( X k, Tk ) be defed as lemma 3 The, wth probablty oe, W has the represetato W ( = max{ X λ ( T } > k K k k (39) where K s a (radom) fte set of pots of the Posso proess Lemma 34 Let W be a separable stohast proess satsfyg the odtos of lemma 3 The sample paths of W are otuous, wth probablty oe The represetato (39) atually mples stroger regularty propertes for W : wth probablty oe, sample paths are dfferetable exept at a (radom) fte set of pots Now we a derve the dstrbuto of the maxmum value of U (, usg lemma For ay terval J [ 0, T ], defe the evelope futo q( ; J ) by q ( t ; J ) sup{ V ( s) λ ( t s)} (30) s J For the speal ase J = [ 0, T ], let qt () = qt (;[ 0, T]) (3)

14 Lemma 35 Let U ( = V ( + W (, where V s a futo satsfyg odtos V ad V, ad W s a separable stohast proess satsfyg the odtos of lemma 3 Let G ( u ; V ) be the dstrbuto futo of maxmum utlty, defed by (8) The u q( ( e dt e ) G ( u ; V ) = exp (3) wth q ( defed by (30) (3) 4 Choe probabltes: geeral formulato A geeral expresso for P ( ), where J s a ope terval wth respet to [ 0, T ], 0 J was foud Seto (lemma 3) Now we evaluate ths expresso whe the stohast proess W s defed by the extreme-value model of Seto 3 Usg the expresso (3) for G ( u ; V ), the dstrbuto futo of maxmum utlty, we fd where u ( e K( J ; ) ) G( u ; V v { J}) = exp v (4) ( max{ q( t ; J ) v, q( t ; J }) K ( J ; = dt exp ) (4) J deotes the omplemet of J wth respet to [ 0, T ], ad q ( t ; J ) s defed by (30) Beause the expoet (4) fatorzes to a term depedg oly o u ad a term depedg oly o v, t s lear that the uform dfferetablty odto of lemma 3 wll be satsfed provded oly that K ( J ; s rght-dfferetable wth respet to v at v = 0 Suppose that K ( J ) = [ K( J ; ] + v = 0 (43) v does fat exst (as wll be show) The plus sg dates the rght dervatve The G ( u) = v + [ G( u ; V v { J}) ] = G( u ; V ) e u K ( J ) v = 0 (44) The depedee o u s ow explt, so the tegrato (4) a be doe dretly The result s K( J ) P 0( J ) =, (45) q( dt e

15 where we have used the detty q( K ( J ;0) = dt e To vestgate the dfferetablty of the tegral K ( J ;, we defe R( J ; = { t q( t ; J ) > q( t ; J ) + v ad q( > } (46) for v 0, ad also R ( J ) = R( J ;0) The R s a terval, as gve by the ext lemma Agle brakets wll be used to deote tervals that may or may ot lude ther ed pots For example, ( b meas that the terval s ether ( a, or ( a, b] We adopt the oveto that ( a,, ( a, b] ad [ a, are ull f a b, ad [ a, b] s ull f a > b Lemma 4 Suppose J s the ope terval ( a,, wth 0 a < b T, ad R ( J ; s defed as (46) Assume that odtos V V ad L L4 hold The there are futos η ( t, ad ζ ( t,, rght-otuous v at v = 0, suh that whe v 0 (( ; = η(, ζ( b, R (47) If J s a ope terval wth respet to [ 0, T ], there are some speal ases where J s ot of the form ( a,, but the method used to prove lemma 4 a stll be appled For oveee of otato, we rewrte the tervals [ 0, b ), ( a, T ], ad [ 0, T ] as ( 0,, ( a, T + ), ad ( 0, T + ), where 0 < b T ad 0 a < T The (47) holds for these ases also, f we defe η( 0, v ) = τ0 ad ζ( +, = T + τ0 T (48) The remag ase s trval: f J = the R ( J ; = Usg (47), we a rewrte (4) as K( J ; = η( τ 0 dt exp[ q( t ; J )] + e v ζ( b, η( dt exp[ q( t ; J )] (49) + T +τ 0 ζ( b, dt exp[ q( t ; J the ase where ζ ( b, η(, ad otherwse )] K ( J ; = dt exp[ q( t ; J )] (40) 3

16 Lemma 4 Suppose J = ( [0, T ], ad K ( J ; s defed by (4) The, uder the odtos of lemma 4, K ( J ; s rght-dfferetable wth respet to v at v = 0, wth dervatve where ( J ) = ζ η ( ( a) q( K dt e (4) η ( a) = η( 0) ad ζ ( = ζ( b,0) terms of the futos η ( t, ad ζ ( t, defed lemma 4 I the proof of lemma 4 we show that η ( a) ζ(, so the tegral (4) s properly defed From the proof of lemma 4, we see that η (a) s the ross-over pot betwee q ( t ; [0, a] ) ad q ( t ; ( T ]), e, t = η(a) s the soluto of q ( t ; [0, a]) = q( t ; ( T]) Smlarly, ζ ( s the ross-over pot betwee q ( t ; [0, ) ad q ( t ; [ b, T ]) As before, the ases where J has the form [ 0, b ), ( a, T ], or [ 0, T ] a be hadled the same way, usg (48) for the approprate lmts of tegrato If J = the K 0 From (45) ad (4), we fally get ( q( (( b ) dt e q( P = ζ 0 ) dt e (4) η( a) Ths s smlar to the otuous logt model, but wth the evelope futo q replag the systemat utlty V ad wth R ( ( ) replag the rage of tegrato ( a, The ext step s to fd P 0( J ) for ay terval J [ 0, T ], ad show that t s equal to the hoe probablty P (J ) We start by fdg P ( ) for a losed terval, usg P0 ([ b]) = lm P0 ( ( a /, b + / ) ) (43) Lemma 43 Suppose the stohast proess W s defed by the extreme-value model of seto 3, ad odtos V ad V hold The ( ( a) q( q( P = η 0 ([ b] ) dt e dt e (44) ζ 0 J = for 0 < a b < T, where η ad ζ are defed lemma 4 The method used to prove lemma 43 a also be used for tervals ope at oe ed ad losed at the other If J = ( b] wth 0 a < b < T the the lmts of tegrato are 4

17 η (a) ad η (, whle f J = [ wth 0 < a < b T they are ζ (a) ad ζ ( To keep the otato osstet, we defe ζ ( 0) = η(0 ) = τ0 ad η( ) = ζ( T + ) = T + τ0 T (45) For ay terval J [ 0, T ], we have J = J J, where J ad J are dsjot tervals (or ull) Usg the represetato of P 0 that we have just foud, t s lear that 0 ( 0 0 = P J ) + P ( J ) + P ( J ) (46) But from the deftos (6) ad (7), P ( J ) P( ) Ths s ompatble wth (46) oly f 0 J P ( J ) = P0 ( J ), (47) whh also mples that a maxmum utlty te betwee ay two tervals has zero probablty For all pratal purposes, P (J ) eeds to be defed oly whe the hoe set J s a terval However, t may be of some theoretal terest to show that P (J ) exteds to a probablty measure over the real terval [ 0, T ] Ths requres the followg outable addtvty property Lemma 44 Suppose J, s a sequee of dsjot tervals of [ 0, T ] suh that, J U J = s also a terval of = [ 0, T ] The, uder the odtos of lemma 43, J 0 ( J ) = P0 ( J ) = P (48) From (48), ad the results P 0 ( ) = 0 ad P 0 ([0, T]) =, t follows that P 0 defes a probablty measure o the lass of fte dsjot uos of tervals of [ 0, T ] The by stadard results (see, for example, Bllgsley, 986, seto 3) P 0 has a uque exteso to a probablty measure o the lass of Borel sets [ 0, T ] 5 Choe probabltes: speal ases I ths seto we dsuss three speal ases that a arse the hoe probablty model gve by (4) ad (44): () tervals that are ever hose; () dsrete pots that are hose wth ozero probablty; ad (3) tervals o whh a hoe probablty desty a be defed () If R ( ( ) s empty or ossts of a sgle pot, or equvaletly f η ( a) = ζ(, the P ( ( ) = 0 Ths typally ours f at some earby pot s the parametr utlty 5

18 V (s) s muh larger tha ts value the terval ( a, If some pot t ( s to be hose, the the path of the stohast proess W must rse steeply from s to t, so that U ( > U ( s), but must ot be too steep at t, so that U ( a be a maxmum Aordg to lemma 33, the sample paths of W are made up of segmets of the futo λ, so the requred urvature may ot be attaable We a say that suh a terval, whh s ever hose, s the shadow of some earby peak of the futo V Examples are gve the ext seto () If R ([ t, t] ) s a o-degeerate terval, or equvaletly f η ( > ζ(, the the probablty dstrbuto of t has a dsrete ompoet P ({ t}) > 0 Ths a happe oly at pots where V s rregular, as follows Lemma 5 Uder the odtos of lemma 43, a eessary odto for η ( > ζ( s that V s ot dfferetable at t If V has (uequal) left ad rght dervatves at t, the further eessary odtos are that V ( t+ ) < V ( t ) ad max{ V ( t+ ), V ( t )} λ (0+) Although these odtos are eessary, they are ot suffet beause t may be the shadow of a hgher value of V elsewhere Typally there wll be dsrete hoe probabltes at pots where V has a step or a sharp peak, as the examples the ext seto Restrtg the hoe set to [ 0, T ] s equvalet to settg V ( = outsde that terval Cosequetly there are also dsrete probabltes assoated wth the ed-pots t = 0 ad t = T, exept the speal ase where V ( as t approahes the edpots Heurstally, these dsrete probabltes orrespod to orer solutos of the utlty maxmzato problem, where utlty s dereasg at t = 0 or reasg at t = T I some applatos ths mght ot be the approprate way to model the ed-pot effets, whh ase oe would use oly the odtoal probabltes for a teror soluto, Pr{ t J t (0, T )} The assumpto about how V behaves at the ed-pots s obvously less mportat f the utlty a be defed over a larger rego tha the terval o whh (odtoal) hoe probabltes are eeded (3) Uder some addtoal assumptos, we a fd a probablty desty futo for t, e, p ( = dp([0, t]) / dt (D) The dervatve V ( exsts at t (D) If V ( λ (0+), the the seod dervatve V ( exsts at t (D3) The seod dervatve λ ( exsts for all t 0, τ ) ( 0 6

19 (D4) There s a eghborhood N t of t suh that P ( J ) > 0 for all proper tervals J N t (e, there s o shadowg N t ) Lemma 5 Suppose odtos D D4 hold at some pot t The, uder the odtos of lemma 43, dη( V ( = f V ( > λ (0+), (5) dt λ ( η( dη( = dt f V ( < λ (0+) (5) Dfferetatg P ([0, t] ), as gve by (44), we fd the probablty desty futo dη p ( = exp[ q( η)] ds exp[ q( s)] (53) dt where the dervatve of η = η( s gve by (5) or (5) If V ( = λ (0+), the p ( may be dsotuous at t From the proof of lemma 5, t s lear that we get (5) o the sde where V ( s larger, ad (5) o the sde where V ( s smaller I the ase V ( < λ (0+), we have η ( t ) = t Ths mples q ( η ) = V (, so the probablty desty redues to V ( q( s) p ( = e ds e (54) If (54) holds for all t some terval J, the the hoe probablty futo odtoal o J redues to the otuous logt model, V ( p( t J ) = e ds e (55) J V ( s) for t J Ths shows that, uder sutable restrtos, the otuous logt model s after all ompatble wth maxmzato of a stohast proess wth (almost surely) otuous sample paths The smplfed model (55) s, however, ulkely to be useful for estmato beause V wll be a futo of ukow parameters θ, ad the odto that V ( λ (0+) for all t J wll hold oly for restrted values of θ 6 Costruto ad examples of hoe probabltes I prate the haratersts x ( are ot observed as futos of t over the terval [ 0, T ] Istead, they are usually terpolated from observatos at some fte set of pots We shall assume lear terpolato, for smplty quadrat or other forms of 7

20 terpolato would make some of the followg expressos algebraally more omplated, but the hoe probabltes would stll be omputatoally tratable For the same reaso we assume that the o-stohast ompoet of the utlty futo s lear x (, ( x(, z, θ) = β ( z, θ V ( = V ) x ( (6) The V ( s peewse lear t We should allow for dsotutes V ( : tmevaryg pres, for example, are typally step futos Let I = ( t, t+ ), =,, m, (wth t 0 = 0 ad t m + = T ) be a set of tervals o eah of whh V ( s lear, V ( = a + b ( t t ), t < t < t+ (6) If V s dsotuous at t = the, aordg to odto V, V t ) s the larger of t a + b ( t t ) ad a Aordg to odto V3, the oeffets a ad b are otuous θ The examples whh follow are based o the partular hoe (33) for the futo λ the GEV model of seto 3: λ ( τ) = log( σ τ) for 0 τ < τ0 (63) wth σ > 0 ad τ 0 = / σ Computato of the hoe probabltes hges o the evelope futo q defed by (3), so frst we eed a systemat proedure for ostrutg q We have q s) = max q( s; I ) (64) ( ( where q( s; I ) = max t t t + { a + b ( t t ) s } (65) Let ψ (s) be the maxmzg value of t, so that ( ) ( ) qs ( ; I) = a + b ψ ( s) t λ s ψ ( s) (66) Whe λ s gve by (63), the soluto s t f s t d ψ ( s) = s + d f t d s t + d (67) t + f s t + d where 8

21 ( σ b ) + d = sg ( b ) (68) (wth d = 0 f b = 0 ) From (66) ad (67), we see that q ( s; I ) s lear for s [ t d, t + d ] ad s log lear outsde that terval, wth a maxmum at t + f b > 0 or at t f b < 0 For example, f b σ the a + b ( t+ t ) + log[ σ s t+ ] f t+ d s < t+ + σ q ( s; I ) = a + b ( s t + d ) log( b / σ) f t d s t+ d (69) a + log[ σ( t s)] f t σ < s t d For eah ad j, defe M, j = { s q( s; I ) q( s; I j ) ad q( s; I) > } (60) By the argumet used to prove lemma 4, M, j s a terval I fat t s always a odegeerate terval beause there s a terval where q ( s; I ) > but q ( s; I j ) = The M, j,, j = ( t σ m f < j, j =, j + σ M m, t ) f > j The pot m, j s foud by omparg eah of the segmets of q ( s; I ) (as equato 69) wth eah of the segmets of q ( s; I j ) to determe f ad where they terset or, possbly, ode The we a determe the (possbly degeerate) tervals S I = s q ( s; I ) = q ( s)} = M, j j { (6) The evelope futo a be omputed as (wth m q( s) = q( s; I ) [ s { S S (6) S m+ = + }] = ), whh allows for the possblty that adjaet tervals may overlap Havg foud q, we a determe the dsrete probablty pots, f ay The oly addates are the pots t, = 0,,, m + If V s otuous at t, the R ({ t }) = t{ S S } (63) If V s dsotuous at t, the 9

22 ( t σ, t d ) t S f V ( t ) = V ( t + ) R ({ t}) = (64) ( t d, t + σ ) t S f V ( t ) = V ( t ) If R ({ t }) s o-degeerate, e, ζ ( t ) < η( t ) P [ t, t ]) gve by (44) (, the t has a dsrete hoe probablty Elsewhere ( 0, T ) there s a probablty desty gve by (53) If t t t+, ad f t s ot a dsrete probablty pot, the exp [ q ( t d )] f t d S p ( = ds exp[ q( s)] (65) 0 otherwse Example ( step ) 0 f 0 t < T / V ( = (66) f T / t T For ths example, osder the ase where 0 ad σ T / The R ({0}) = σ (, 0), R ({ T /}) = ( t, T /), ad R ({ T}) = ( T, T + σ ), orrespodg to dsrete probabltes, wth t = T / σ ( e ) The resultg probabltes are ad P({0}) P({ T /}) = ( e P({ T}) = /(σ D ), = e e /(σ D ), ) /(σ D ), p( = / D 0 e / D f f f 0 < t t t < t < T / T / < t < T where σ D = ( T /)( + e ) + (e + e ) /( ) The terval (t, T/) s the shadow of the step at t = T/ 0

23 Example ( peak ) t / T f 0 t T / V ( = (67) ( T / T f T / t T where > 0 For ths example, osder the ase where /T > σ, so that there s ozero dsrete probablty at the peak The P({ 0}) = P({ T}) = σt /(8 D ), P({ T / }) = e D σt σ 4, ad p( = σt D σt D t exp T ( T exp T f f 0 < t < T / T / < t < T where e σt D = + ( ) 4 e σ Apart from a ostat fator, ths desty futo s the same as the otuous logt model, but the dsrete probablty ompoets are of ourse dfferet If the peak at t = T / were rouded, so that V ( were otuous there, the the dsrete ompoet at t = T / would dsappear For a rouded peak, however, the Jaoba (5) would o loger be ostat, so the desty p ( would dffer from that of the otuous logt model Example 3 ( valley ) t / T f 0 t T / V ( = (68) ( T / T f T / t T where > 0 For ths example osder the ase where / T > σ > / T (whh mples > ) I ths ase there s shadowg of some, but ot all, of the valley The resultg probabltes are

24 P ({0}) = P({ T}) = D σt σ 3 8 ad p( = σt D 0 σt D 3 3 t exp T ( T exp T f f f T 0 < t < d T d T + d T < t < + d < t < T 3 where d 3 = σ T σt σt D 3 = + + exp + σ 4 σt I ths example the shadowg s due to the steepess of the sdes of the valley, ad would geerally happe eve f the peewse lear V were replaed by a smoother futo 7 Parameter estmato: dsrete observatos Gog bak to the geeral ase, we brefly osder the problem of estmatg the parameter vetor θ (6) from a radom sample of observatos o x, z ad t I prate the hoe t s ot usually observed as a otuous varable Istead, the hoe set [ 0, T ] s parttoed to a fte set of tervals J, =,,, ad we observe the dex of the hose terval (e, t J ) The tervals J eed ot be the same as the tervals I used the terpolato of V ( ; fat there s some smplfato f they are ot the same beause the ay dsrete probablty pot (other tha 0 ad T) s the teror of a hoe terval J I that ase the problem redues to estmato of a dsrete hoe model, wth P x, z, θ ) = P( J ) (7) ( Although we are bak to the dsrete ase, t s a dsrete hoe model whh the stohast utlty has a ovarae struture ompatble wth a uderlyg otuous-

25 tme stohast proess that has almost surely otuous sample paths Estmato from data where t tself s observed wll be dsussed elsewhere A dsrete hoe model lke (7) s usually estmated by maxmum lkelhood estmato Let Θ deote the parameter spae, N the sample sze, ad L m, N ( θ) the log lkelhood futo for m dsrete alteratves The θ $ N s determed (ot eessarly uquely) by L ˆ m, N ( θn ) = max Lm, N ( θ) θ Θ f a maxmum exsts or, f ot, θ $ N s a lmt pot of a sequee θ } satsfyg lm L, ( θ, ) = sup L, ( θ) m N N θ Θ m N { N, Suh a estmator exsts almost surely, provded θ $ N s allowed to be o the boudary of Θ, ludg f eessary a pot at fty Suffet sets of odtos for strog osstey of θ $ N are gve by several authors I partular, see odtos () (4) MFadde (984, seto 3) These odtos do ot requre the probabltes to be bouded away from zero Although the osstey odtos are qute mld, the requremet that P ( x, z, θ) should be otuous θ (exept possbly o a set of ( x, z) wth zero probablty) has to be heked ay partular model From assumpto V4, ad from the defto of q as a evelope futo, t a readly be show that q (s) s otuous θ uformly s, ad that ds exp[ q( s)] s otuous θ However, we also eed to show that the lmts of tegrato η ad ζ are otuous θ The followg addtoal assumptos wll be used (C) The seod dervatve λ (τ ) exsts for all τ ( 0, τ0 ) ad has a strtly postve lower boud (C) The futo t, t, ) defed by ( θ V t, θ) V ( t, θ) = ( t t ) ( t, t, ) (7) ( θ s otuous θ uformly t ad t for all t [ t 0, T ] ad all t [ 0, t0 ], where t 0 s some spefed pot [ 0, T ] Lemma 7 Suppose that odtos V V4, L L4, ad C C hold The η ( t 0 ) ad ζ t ) are otuous θ ( 0 3

26 The uform otuty odto (7) wll be satsfed by most parametr models for V ( t, θ), provded of ourse that t = t0 s ot oe of the pots where V a be dsotuous t The model dsussed seto 6, where V s gve by (6) ad λ by (63), learly satsfes the odtos of lemma 7 provded that the steps V (f ay) are ot at the boudares of hoe tervals J I fat, ths provso s ot eessary for the model of seto 6: eve f V ( t, θ) s dsotuous t at t = t0 for some values of θ, the ross-over pots η ( t 0 ) ad ζ ( t 0 ) a be determed (by eumeratg all the possble ases) ad are foud to be otuous θ I that model, therefore, the P ( x, z, θ) are always otuous θ I prple t s also mportat to hek the detfato odto (odto (4) MFadde, 984, Seto 3), but t s geerally qute dffult to fd verfable odtos that make θ detfable Eve the model defed by (6) ad (63), oe a ostrut pathologal examples whh, say, P ( x, z, θ) = for a rage of θ The usual approah s to suppose a dsrete model to be detfed exept whe there s a obvous trasformato of θ that leaves the expeted log lkelhood varat The hoe probabltes P ( x, z, θ) are geerally ot dfferetable wth respet to θ a eghborhood of θ 0, eve after exludg a set of ( x, z) wth probablty zero I Example of seto 6, for example, P ([0, T /]) s otuous but ot dfferetable wth respet to = ( x, z, θ) at = σt / As θ vares over a eghborhood of θ 0, the values of ( x, z) at whh = σt / wll geeral rage over a set wth ozero probablty Ths lak of dfferetablty meas that the usual theorems o asymptot ormalty of θ $ N are ot applable, ad further vestgato of the asymptot propertes of θ $ N s eeded It may also be eessary to use gradet-free methods to maxmze L N (θ) 4

27 Appedx The appedx otas proofs of the lemmas the text (exept where a spef referee s gve), together wth some prelmary results that are eeded the proofs Maxmum of a stohast utlty futo Proof of Lemma Frst, suppose that J s a ope terval wth respet to [ 0, T ] (e, J s a ope terval wth respet to R, exept that f a ed-pot s 0 or T the that edpot may be luded) The omplemet of J wth respet to [ 0, T ], deoted J, s the uo of at most two losed tervals Beause U ( s upper semotuous ad ompat, sup U ( s) = U ( s) for some s J, s J J s ad therefore { M J } = { U ( > sup U ( s) for some t J } = U u R 0 { U ( > u s J for some t J } { sup U ( s) u} s J (A) where R 0 s the set of ratoal umbers Usg the separablty of U wth respet to D, we a rewrte ths as { } M J = U U { ( ) } I U { ( ) } U t > u U s u, (A) u R 0 t J D s J D whh s a elemet of [ 0, T ] R Next suppose that J s a losed terval, say [ a, b] wth a b We have I = { M J } = { M } (A3) J where J = ( a /, b + / ) [0, T ] (A4) Eah of the sets M J } s of the form (A), ad therefore (A3) s also a elemet of R [ 0, T ] { A smlar argumet ompletes the proof for the ase where the terval J s ope at oe ed ad losed at the other 5

28 Proof of Lemma Let W be separable wth respet to some outable dese set D Wthout loss we a hoose D to ota the dsotuty pots of V, so U s also separable wth respet to D Let J be a sequee of fte sets suh that J J D The ad therefore sup U ( u t J sup t J D U ( u Pr sup U ( u Pr sup U ( u = Pr supu ( u, (A5) t J t J D t J where the last equalty follows from separablty Rewrtg the left-had sde of (A5) as { W ( u V ( for all t J } = F [ u V ( t ),, u V ( t )] Pr { J } m( ) the gves the result (9) Proof of Lemma 3 For ay B > 0, dvde ( B, B) to tervals of wdth ( N =,,) ad set v N = ε N evet, ( 0, ±,, ± mn ) ε = N N = where mn = B/ εn Defe the A = U ( v, v + ] ad U < v } (A6), N {, N, N, N where U = supu (, U = sup U ( ) t t J t J The mn > U ad B < U B} = lm Pr{ A, N } N = m Pr{ U (A7) As (), we have Pr { A, N } = G( v, N ; V εn { J}) G( v, N ; V ) (A8) Gve a pot u ( B, B], defe j = j( u, N) so that u s the terval v, v + ] N ( j, N j, N Applyg the uform dfferetablty odto (3) to the rght had sde of (A8), lm Pr{ Aj u, N ), N } = G( ε N N ( u at eah pot u, ad therefore, by the domated overgee theorem, ) (A9) 6

29 lm N m N = m N B, N } = du G ( u) B Pr{ A (A0) Lettg B (A7) ad (A0) gves U } = du G ( u Pr{ U > ) (A) Aordg to the defto (6) ths s just equal to P ( ), provded that J s a ope terval Propertes of the otuous-tme GEV model The followg prelmary lemmas establsh useful results volvg the futo λ, whh satsfes assumptos L L4 Lemma A The tegral d τ exp[ λ( τ)] s fte Proof Choose ay pot τ 0, ) The 0 < λ ( τ ) < Strt ovexty of λ ( τ0 mples λ ( τ) > λ( τ) + ( τ τ) λ ( τ) ad therefore 0 J τ 0 0 d τ e λ( τ) < [ λ ( τ )] exp[ λ( τ ) + τ λ ( τ )] whh s fte as requred Lemma A If y > 0, y > 0, ad 0 < t t < τ0, the the equato y exp[ t ] = y exp[ λ( t t)] (A) has exatly oe otrval soluto for t (Trval solutos are those wth λ t t ) = λ t t ) = ) ( ( Proof Suppose wthout loss that t < t Set s = t t, ad τ = t t If τ s outsde the terval ( s τ 0, τ0 ) the oe or both of λ (τ) ad λ ( τ s) are fte, so ay soluto would be a trval oe If τ s τ 0, τ ), we are lookg for solutos of ( 0 f s τ ) = log ( y / y ) (A3) ( where we have defed f s ( τ ) = λ( τ) τ s) 7

30 If τ 0 s fte, the λ ( τ 0 s) s fte, ad so f s (τ) as τ τ0 from below If τ 0 =, the λ ( τ s ) as τ (from odto L4); the equalty f s ( τ ) > s λ ( τ s) (from strt ovexty of λ) the shows that f s (τ) as τ τ0 ths ase also By a smlar argumet, f s (τ) as τ s τ0 from above Thus the otuous futo f s (τ) s ubouded o the terval ( s τ 0, τ0 ), ad therefore (A3) has at least oe soluto that terval If s < τ0 ad τ s oe of the tervals ( s τ0, 0) or ( s, τ0 ), the f ( τ) > 0 beause λ (τ) s strtly reasg (from odtos L ad L3) If τ ( 0, s) the odtos L L3 mply λ ( τ) > 0 ad λ ( τ s ) < 0, so aga f s ( τ) > 0 The futo f s (τ) s otuous o ( s τ 0, τ0 ), so the postve dervatve mples that t s strtly reasg Therefore (A3) has at most oe soluto that terval Combg these results, t follows that there s just oe otrval soluto Lemma A3 Let τ = ρ( ε, δ) be the soluto of λ ( τ) + ε = λ( τ + δ), τ < τ (A4) 0 for ay gve ε > 0 ad δ ( 0, τ0 ) (By Lemma A, ths soluto exsts ad s uque) The ρ( ε, δ) as δ 0 + λ( τ) d τ e = o( δ) (A5) Proof Frst, ote that the upper lmt of tegrato a be haged to τ 0 Se λ (τ) s strtly reasg (from odtos L ad L3), λ ( ρ) < ( ε / δ) < λ ( ρ + δ), where ρ s the soluto of (A4), ad therefore λ ( ρ + δ) as δ 0 + Beause λ ( τ) as τ τ0 (odto L4) ad s fte for τ < τ0, t follows that ρ + δ τ0 as δ 0 + From strt ovexty of λ, ad therefore λ( τ) > λ( ρ + δ) + ( τ ρ δ) λ ( ρ + δ) > λ( ρ + δ) + ( τ ρ δ) ε / δ s 8

31 τ ρ 0 ( ) d τ e λ τ < < exp[ λ( ρ + δ) + ε] exp[ λ( ρ + δ) + ε] ( δ / ε) τ ρ 0 d τexp[ ( τ ρ) ε / δ] (A6) As show above, ρ + δ τ0 as δ 0 +, ad therefore exp[ ρ + δ)] 0 Therefore the left-had tegral (A6) s o (δ), as requred Remark I lemma A3 we had ε > 0 ad δ > 0 By symmetry, ρ ( ε, δ) = ρ( ε, δ) = ρ( ε, δ) δ = ρ( ε, δ) δ, ad so (A5) holds also whe ε < 0 ad δ 0 If ε > 0 ad δ 0, or f ε < 0 ad δ 0 +, the orrespodg result s ρ( ε, δ) λ( τ) τ e d = o( δ) (A7) Proof of Lemma 3 Fx t, ad let δ 0 Frst, osder the subsequee of strtly postve terms, δ > 0 Defe F( w, w ; δ) = Ft, t+ δ( w, w ) (A8) = exp( d τ max{exp[ λ( τ + δ) w ], exp[ λ( τ) w ]} ) As before, let τ = ρ( ε, δ) be the soluto of (A4) Whe τ < ρ( w w, δ) the frst expoet the tegrad of (A8) s larger, ad whe τ > ρ( w w, ) the seod expoet s larger, so (A8) beomes Now F( w, w ; δ) = exp e w ρ( w w, δ) e w d τ exp[ λ( τ + δ)] ρ( w w, δ) δ d τ exp[ λ( τ)] (A9) Pr{ W ( W ( t + δ) < ε} = du [ F ( u, u + ε; δ) F ( u, u ε; δ)] (A0) where F ( w, w F( w, w ; δ) ; δ ) = (A) w Evaluatg the tegral (A0) gves 9

32 Pr{ W ( W ( t + δ) < ε} = ρ( ε, δ) ρ( ε, δ) d τ exp[ λ( τ + δ)] + e d τ exp[ λ( τ + δ)] ε ρ( ε, δ) d τ exp[ λ( τ)] (A) ρ( ε, δ) ρ( ε, δ) d τ exp[ λ( τ + δ)] + e d τ exp[ λ( τ + δ)] ε ρ( ε, δ) d τ exp[ λ( τ)] Usg lemmas A ad A3, we see that Pr{ W ( W ( t + δ) < ε} as δ 0 + The same result s obtaed whe δ 0, ad the ase δ = 0 s trval Thus for ay sequee δ 0, as requred plm [ ( t + δ ) W ( ] = 0 W Proof of Lemma 33 Suppose frst that t s the terval J = [ a + h] [0, T ], where 0 < h < τ 0 Let N ( x) be the umber of pots of the Posso proess wth X k > x ad T k J, ad let N ( x) be the umber of pots wth X k > x h) The the radom x varables N ad N have Posso dstrbutos wth rates he ad T exp[ λ ( h) x] respetvely Defe the sequee of evets A( x, m ) = {0 < N( x) m } { N( x) m } (A3) for =,, Choose x suffetly small (e, large egatve) suh that Pr{ N = 0} < (3), ad the m suffetly large that Pr{ N > m } ad Pr{ N > m } are both less tha (3), so that Pr{ A( x, m )} > The Pr { A ( x, m )} = U so that, wth probablty oe, at least oe of the evets A x, m ) must our ( Thus for some x there s at least oe evet wth X k > x ad T k J Frst, ths mples W ( fte for t J, beause the wdth of the terval s less tha τ 0 Seodly t mples that, whe t J, oly evets wth X k > x h) a otrbute to the maxmum (37) But there are a fte umber (at most m ) of suh evets It 30

33 follows that, wth probablty oe, (39) holds for t J Se [ 0, T ] a be overed wth a fte umber of tervals lke J, (39) holds wth probablty oe for all t [ 0, T ] Proof of Lemma 34 Cosder the stohast proess proesses W ad almost every sample path w ( W defed lemma 3 The W have the same fte-sample dstrbutos Aordg to lemma 33, s the maxmum of a fte umber of otuous futos, e, { xk tk } wth k K ad wth t k t < τ0, ad therefore w ( s otuous wth probablty oe Now f a stohast proess has otuous sample paths wth probablty oe, the so do all other separable stohast proess wth the same fte-dmesoal dstrbutos (see, for example, Bllgsley, 986, theorem 38), partular W The ext results volve the evelope futos q ( t ; J ) ad q (, defed by (30) ad (3) Lemma A4 Suppose odtos V V ad L hold Let J [ 0, T ] be a losed terval If q ( t ; J ) >, the there s a pot s J suh that q( = V ( s) t s) Proof For some gve t wth q ( t ; J ) >, let s, s, be a sequee of pots J suh that V ( s ) t s ) q( t ; J ) The we a hoose a subsequee suh that s s J From odtos V V we have V ( s ) V ( s± ) V ( s) <, ad from odto L we have λ ( t s ) λ( t s), so that q( t ; J ) V ( s) t s) But from the defto (30) we must have q( t ; J ) V ( s) t s), ad therefore q ( t ; J ) = V ( s) t s) as requred Proof of Lemma 35 I lemma, let J be the terval [ 0, T ] ad let the dstrbuto futo F be gve by (3) ad (3) The u ( e dt exp[ q( t ; J ) ( u ; V ) lm exp G = )] (A4) where the futo q ( t ; J ) s defed by (30), ad the fte sets J ( =,,) satsfy [ 0, T ] D where D s some outable dese set Now osder J lm d t exp[ q( t ; J )] (A5) Suppose q ( > From the defto (30), A B mples q( t ; A) q( t ; B), so the tegrad s mootoally reasg at eah t By lemma A4, there s some s = s( [0, T ] suh that q( = V ( s) t s) 3

34 Gve ε > 0, we a the fd some δ > 0 suh that [ V ( s) t s) ] [ V ( s ) t s )] < ε for all s at least oe of the tervals [ s δ, s] ad [ s, s + δ], aordg to odtos V ad L For suffetly large, there wll be pots of J these tervals, ad therefore q q( t ; J ) < ε ( Thus q( t ; J ) q( for eah t at whh q ( > If q ( =, the so s q ( t ; J ) for all The lmt of (A5) s the just d t exp[ q( ], by mootoe overgee Substtutg ths (A4), we get the result (3) Choe probabltes Next, we defe { t V ( s = q( s), t [0, ]} ψ ( s) = T, (A6) where s s suh that q (s) >, ad q s the evelope futo (3) Aordg to lemma A4, ψ (s) s o-empty We shall eed the set R ( J ; defed by (46), ad also the omplemetary set { t q( t ; J ) q( t ; J ) + v ad q( ) > } R ( J ; = t, (A7) where J [ 0, T ] s a terval I the speal ase v = 0 we wrte R ( J ) = R( J ;0) ad R ( J ) = R ( J ;0) Lemma A5 Suppose t ψ( s ), where q ( s ) > ( =,), ad assume that the futo λ satsfes odtos L L4 The t > t mples s s Proof Frst, defe { s V ( t ) s t ) V ( t ) s t ) ad λ( s < } S( t, t) = If 0 < t t < τ0, the from lemma A there s just oe soluto of V t ) s t ) = V ( t ) s ) ( t (A8) for whh λ ( s < Call the soluto s 0 It s lear that S ( t, t ) les to the rght of s 0 f t > t, ad to the left f t > t If t t τ0, the S ( t, t ) s just the rego where V t ) s ) s fte Thus S t, t ) s always a terval: ( t ( 3