hal , version 5-4 Oct 2012

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1 Auhor mauscrip, published i "Advaces i Applied Probabiliy 13 3" ERROR BOUNDS FOR SMALL JUMPS OF LÉVY PROCESSES EL HADJ ALY DIA Absrac The pricig of opios i expoeial Lévy models amous o he compuaio of expecaios of fucioals of Lévy processes I may siuaios, Moe-Carlo mehods are used However, he simulaio of a Lévy process wih ifiie Lévy measure geerally requires eiher o rucae small jumps or o replace hem by a Browia moio wih he same variace We will derive bouds for he errors geeraed by hese wo ypes of approximaio Key words Approximaio of small jumps, Lévy processes, Sorohod embeddig, Spizer ideiy AMS subjec classificaios 6G51, 65N15 JEL classificaio C, C15 hal-517, versio 5-4 Oc 1 1 Iroducio I he rece years, he use of geeral Lévy processes i fiacial models has grow exesively see, 5, 11 A variey of umerical mehods have bee subsequely developed, i paricular mehods based o Fourier aalysis see 4, 1, 13, 15 Noeheless, i may siuaios, Moe-Carlo mehods have o be used The simulaio of a Lévy process wih ifiie Lévy measure is o sraighforward, excep i some special cases lie he Gamma or Iverse Gaussia models I pracice, he small jumps of he Lévy process are eiher jus rucaed or replaced by a Browia moio wih he same variace see 1, 7, 8, 16, 18 The laer approach was iroduced by Asmusse ad Rosisi 1, who showed ha, uder suiable codiios, he ormalized cumulaed small jumps asympoically behave lie a Browia moio The purpose of his aricle is o derive bouds for he errors geeraed by hese wo mehods of approximaio i he compuaio of fucios of Lévy processes a a fixed ime or fucioals of he whole pah of Lévy processes We also derive bouds for he cumulaive disribuio fucios These bouds ca be used o deermie which ype of approximaios o use, sice replacig small jumps by Browia is more ime-cosumig if we use Moe Carlo mehods Our bouds ca be applied o derive approximaio errors for loobac, barrier, America or Asia opios Bu his laer poi will o be developed, ad is lef o aoher paper The characerisic fucio of a real Lévy process X wih geeraig riple γ, b, ν is give by Ee iux = exp { iγu b u + + e iux 1 iux1 x 1 νdx }, where γ R, b, ad ν is a Lévy measure The process X is he idepede sum of a drif erm γ, a Browia compoe bb, ad a compesaed jump par wih Lévy measure ν The process X has fiie resp ifiie aciviy if νr < resp νr = + For < 1, he process X is defied by X = γ + bb + X s 1 { Xs >} xνdx < x 1 Uiversié Paris-Es, Laboraoire d Aalyse e de Mahémaiques Appliquées, UMR CNRS 85, 5 bd Descares, Champs-sur-Mare, Mare-la-Vallée, Frace diaeha@gmailcom 1

2 E H A DIA The process X is obaied from X by subracig he compesaed sum of jumps o exceedig i absolue value Le R = X X 11 The process R is a Lévy process wih characerisic fucio Ee iur = exp { e iux 1 iux } νdx x I holds E R = ad Var R = σ, where σ = x νdx x hal-517, versio 5-4 Oc 1 Noe ha lim σ = The behavior of σ whe goes o is ow for classical models VG, NIG, CGMY As oed i Example 3 of 1, if νdx = x 1 α Lxdx, where α, ad L is slowly varyig a, he i holds σ L + L / α 1/ 1 α/ ; cosequely, lim σ/ = + We also defie he process ˆX by ˆX = X + σŵ,, where Ŵ is a sadard Browia moio idepede of X We aim o sudy he behavior of he errors made by replacig X by X or ˆX, wih respec o he level These errors are sudied for he process X a a fixed dae ad for is ruig remum Se, for ay, M = X s, M = Xs, ˆM = ˆX s Uless saed oherwise, X is a Lévy process wih geeraig riple γ, b, ν The paper is orgaized as follows I he ex secio, we will sudy he errors resulig from he rucaio of he compesaed sum of small jumps The resuls of ha secio are based o esimaes for he momes of R We also derive a esimae for he expecaio E M M, by usig Spizer s ideiy I Secio 3 we sudy he errors resulig from Browia approximaio The process X will be approximaed by he process ˆX A major resul of Secio 3 is Theorem, which saes a error boud for he expecaio of a fucio of he remum This resul is he cosequece of Theorem 37, which relies o he Sorohod embeddig heorem Trucaio of he compesaed sum of small jumps I his secio, we will sudy he errors resulig from he approximaio of X by X These errors are relaed o he momes of R Defie σ = max σ, 1 The ex resul will be useful for may proofs i his paper Proposiio 1 Le X be a Lévy process ad R defied i 11 The E R 4 = x 4 νdx + 3 σ, x

3 ad for ay real q > ERROR BOUNDS FOR SMALL JUMPS OF LEVY PROCESSES 3 E R q K q, σ q, where K q, is a posiive cosa which depeds oly o q ad Proof Le c R deoe he h cumula of R The c 1 R = E R =, ad, for ay, c R = x x νdx oe ha c R = Var R = σ See Proposiio 1 of Subsiuig io he geeral formula µ 4 = c 4 + 4c 3 c 1 + 3c + 6c c 1 + c 4 1 cf 3 below, where, here ad below, µ ad c deoe he h mome ad h cumula of a disribuio, respecively, gives he firs par of he proposiio We ow prove he secod par Le = q/ Sice < q/ 1, E R q E R q by Jese s iequaliy for cocave fucios I hus suffices o prove he resul for he case q =, N; i fac, for ay N, i holds hal-517, versio 5-4 Oc 1 E R K, σ The las iequaliy ca be proved by iducio as follows I is rivial for =, 1, Suppose ha holds for all < m The, by he well-ow resul see eg Theorem of 14 µ m = m 1 = m 1 µ c m, m 1, 3 for all m we have recall ha c 1 R = m m 1 E R m E R c m R = Hece, i view of he iducio hypohesis, i suffices o show ha c m R σ m Sice m, we have c m R = x xm νdx, ad hece c m R x m νdx x m σ m x x νdx The proposiio is hus esablished 1 Esimaes for smooh fucios Le X be a Lévy process ad f a C-Lipschiz fucio where C > The, E f X f X CE R C E R C σ

4 4 E H A DIA Noe ha we do o as ha f X be iegrable If f is more regular, sharper esimaes ca be derived, as show i he followig proposiio Proposiio Le X be a ifiie aciviy Lévy process 1 If f C 1 R ad saisfies E f X <, ad if here exiss β > 1 such ha,1 E f X + θr f X β 1 β is fiie ad iegrable wih respec o θ o, 1, he E f X f X = o σ If f C R ad saisfies E f X + E f X <, ad if here exiss β > 1 such ha,1 E f X + θr f X β 1 β is fiie ad iegrable wih respec o θ o, 1, he E f X f X = σ Ef X + o σ hal-517, versio 5-4 Oc 1 Noe ha, if f has bouded derivaives or f is he expoeial fucio ad e βx is iegrable, where β > 1, he codiios i he above proposiio are saisfied Recall ha he rucaio of small jumps is used whe νr = I ypical applicaios, we have lim if σ/ >, so ha o σ is i fac o σ Proof To prove par 1, we firs wrie f X f X as f X f X = 1 f X + θr f X R dθ + f X R 4 bytheorem74of17, R as Sice R ad X areidepede,e f X R = For ay 1 < α < β, by Hölder s iequaliy, E f X + θr f X By Lyapuov s iequaliy, R E f X + θr f X α 1 α E R 1 E f X + θr f X α α E f X + θr f X β 1 α α 1 β α 1 α Furher, he assumpio,1 E f X + θr f X β < implies ha he { f collecio X + θr f X α is uiformly iegrable; hece, sice },1 f X + θr f X α as as, E f X + θr f X α poiwise for θ, 1 Therefore, by domiaed covergece, 1 lim E f X + θr f X α 1 α dθ = Combied wih Proposiio 1, i hus follows ha 1 E f X + θr f X R dθ = o σ

5 ERROR BOUNDS FOR SMALL JUMPS OF LEVY PROCESSES 5 Par 1 of he proposiio he follows from 4 usig Fubii s heorem We ow prove he secod par of he proposiio Usig Taylor s formula we ge E f X f X = E f X X X + = E f X R + 1 = E 1 = E 1 +E 1 X X f x X xdx f X + θr 1 θ R dθ f X + θr 1 θ R dθ f X 1 θ R dθ f X + θr f X 1 θ R dθ hal-517, versio 5-4 Oc 1 The firs expecaio afer he las equaliy sig is equal o σ Ef X while he secod oe ca be show o be o σ by followig he proof of par 1 The proposiio is proved Remar 3 Assume ha X is a iegrable ifiie aciviy Lévy process ad ha f C 1 R wih f beig C-Lipschiz The E f X f X Cσ Ideed, E f X R = by he assumpios o X ad f, E f X <, ad so he resul follows direcly from 4 usig 1 E f X f X E f X + θr f X R dθ We will cosider ow he case of he remum process Proposiio 4 Le X be a Lévy process ad f a K-Lipschiz fucio The Proof We have E f X s f E f M f M K σ Xs KE KE X s R s K E Rs X s Noe ha R is a càdlàg marigale So, usig Doob s iequaliy, we ge E f X s f Xs K E R = K σ

6 6 E H A DIA Remar 5 Suppose ha X is a iegrable Lévy process ad f a fucio from R + R o R, K-Lipschiz wih respec o is secod variable The Ef τ, X τ Ef τ, Xτ τ T, τ T, K σ, where T, deoes he se of soppig imes wih values i, For a proof, he reader is referred o 9, pp The boud i Proposiio 4 migh o be opimal This is wha suggess he followig resul Theorem 6 Le X be a iegrable ifiie aciviy Lévy process The E M M = o σ Proof Usig Spizer s ideiy see Proposiio 1 i Secio 3 of 1 for deails, we have hal-517, versio 5-4 Oc 1 I holds E M M = X + s X s + = X s + R s + X s + = EX s + = X s + R s1 X s +R s > X s1 X s > E ds E Xs + ds s s X s + Xs + ds s = X s + R s 1 X s > +1 X s +R s >,X s 1 X s +R s,x s > X s 1 X s > = X s + R s 1 X s +R s >,X s 1 X s +R s,x s > + R s 1 X s > = R s X s + 1 X s +R s >,X s +1 X s +R s,x s > + R s 1 X s > Se Is = E X s + Xs + Thus, sice E Rs1 X s > = by idepedece, I s E R s X s + By he lef iequaliy, E M M We ow prove ha E M M = o σ Sice R s X s + R s 1 X s < R s, we ge I s E R s 1 X s < R s Hece, by Cauchy-Scwarz iequaliy, Thus, I s E R s 1 E 1 X s < R s 1 = σ sp X s < R s 1 E M M σ P Xs < R s 1 ds s

7 ERROR BOUNDS FOR SMALL JUMPS OF LEVY PROCESSES 7 Sice νr =, Rs as ad X s X s as wih X s HeceP Xs < R s 1 as Therefore, by domiaed covergece, lim P X s < R s 1 ds s =, ad so E M M = o σ I fiacial applicaios, he fucio f i Proposiio 4 is o always Lipschiz, as for call loobac opio where he fucio is expoeial Hece he followig proposiio Proposiio 7 Le X be a Lévy process ad p > 1 If Ee pm <, he E e M e M C p, σ, hal-517, versio 5-4 Oc 1 where C p, is a posiive cosa idepede of Lemma 8 Le p > If Ee pm <, he < 1 Ee pm < Remar 9 For ay p >, Ee pm < if ad oly if x>1 epx νdx < The oly if par follows from Theorem 53 of 17, oig ha e px e pm For he if par, decompose X as he idepede sum X = Y + Z + Z of Lévy processes, where Y has Lévy measure ν { x 1}, ad Z ad Z are pure jump wih Lévy measures ν {x>1} ad ν {x< 1}, respecively Here ν E deoes he resricio of ν o E Noeha M Y s +Z ; huse e pm E e p Y s E e pz I ca be deduced from Theorems 53 ad 518 of 17 ha E e p Ys is fiie; so is E e pz by he former heorem, uder he assumpio ha x>1 epx νdx < Hece E e pm < Proof Proof of Lemma 8 For, 1, defie R = X X 1 The process R is he compesaed sum of jumps belogig o, 1 i absolue value So Ee pm Ee p X 1 s +p R s Ee p X1 s Ee p R s ByhypohesisadRemar9,oighaRemar9holdsalsofor M 1,Eep X1 s < We eed o boud Ee p R s idepedely of We have Ee p R s + p R s = E! = = 1 + pe R s + + = p! E R s

8 8 E H A DIA By Doob s iequaliy R is a càdlàg marigale Ee p R s 1 + p E R s + + = 1 + p E R + p +! E R = p Var + p R +E! R = p x νdx +Ee p R < x 1 p σ1 +Ee p R +Ee p R p E R! 1 hal-517, versio 5-4 Oc 1 I hus suffices o show ha < 1 Ee β R < for ay β R Ideed, we have Ee β R = exp { e βx 1 βx } νdx < x 1 a mome-geeraig fucio of a compesaed compoud Poisso process By Taylor s heorem, e βx 1 βx = β x e βξ / for ay x 1, where ξ is some umber bewee ad x This complees he proof, as i implies ha { Ee β R β } exp e β x νdx x 1 Proof Proof of Proposiio 7 By he mea value heorem, we have e M e M = M M e M, where M is bewee M ad M Le q be defied such ha 1 p + 1 q = 1 E e M e M E M M e M E Rs e M E q 1 Rs q Ee p M Hece, usig Doob s iequaliy ad he Proposiio 1, we ge E e M e M q q 1 E R q 1 q Ee p M 1 p C p, σ 1 p 1 E e pm + e pm p, where C p, deoes a cosa depedig o p ad We coclude he proof by Lemma 8

9 ERROR BOUNDS FOR SMALL JUMPS OF LEVY PROCESSES 9 Esimaes for cumulaive disribuio fucios For cumulaive disribuio fucios, bouds are expeced o be bigger However, i some cases we ca ge similar resuls as i Lipschiz case I he firs resul below, we assume local boudedess of he probabiliy desiy fucio of he Lévy process X ad is remum process M a a fixed ime The regulariy of he probabiliy desiy fucio of a Lévy process is sudied i 17, 3 For he remum process see 6, 9 Proposiio 1 Le X be a Lévy process 1 If b >, he P X x P X x 1 σ x R πb If X has a locally bouded probabiliy desiy fucio ad x R, he for ay q, 1, P X x P X x C x,,qσ 1 q, hal-517, versio 5-4 Oc 1 where, here ad below, C x,,q deoes a posiive cosa depedig o x, ad q 3 If M has a locally bouded probabiliy desiy fucio o, + ad x >, he for ay q, 1/, P M x P M x C x,,q σ 1 q Lemma 11 Le X ad Y be wo rv s We assume ha X has a bouded desiy i a eighbourhood of x R, ad here exiss p > such ha E X Y p is fiie The here exiss a cosa K x >, such ha for ay > Proof We have P X x P Y x K x + E X Y p p P X x P Y x = P X x, Y < x P X < x, Y x We will sudy he above erms o he righ of he equaliy P X x, Y < x = P x X < x + X Y = P x X < x + X Y, X Y +P x X < x + X Y, X Y > P x X < x + +P X Y > Suppose ha X has a bouded desiy f i he ierval x, x +, > fixed, ad le { K x = max f, 1 } x x+ By cosiderig he cases < ad separaely, i is readily checed ha P x X < x + K x,

10 1 E H A DIA for ay > Thus, usig Marov s iequaliy, we ge P X x, Y < x K x + Similarly, usig P x X < x K x, i holds ha P X < x, Y x K x + Lemma 11 is hus esablished Proof Proof of Proposiio 1 We have E X Y p p E X Y p p P X x P X x = P X x, X < x P X < x, X x 5 I holds ha P X x, X < x = P x X X X < x = P x R bb + X bb < x hal-517, versio 5-4 Oc 1 Noe ha bb is idepede of X bb ad R, ad 1 πb is a upper boud of he probabiliydesiy fucio of bb The, by codiioig o he pair R, X bb, i ca be cocluded ha P x R bb + X bb < x 1 πb E R Therefore, usig ha E R σ, Similarly P X x, X < x 1 πb σ P X < x, X x = P x X < x X X = P x σb + X σb < x R 1 πσ σ Hece par 1 of he proposiio follows from 5 We ow prove par of he proposiio Le p > By Lemma 11 followed by Proposiio 1, here exis posiive cosas K x, ad K p, such ha P X x P X x K x, + E X X p p = K x, + E R p p σ p K x, + K p, for ay > Choosig = σ p p+1 yields P X x P X x max K x,, K p, σ p p+1, p

11 ERROR BOUNDS FOR SMALL JUMPS OF LEVY PROCESSES 11 ad so he resul follows sice p/p + 1 ca be chose arbirarily i, 1 We ow prove par 3 of he proposiio Le p > 1 By Lemma 11, here exiss a cosa K x, > such ha P M x P M x K x, + E M M p for ay > O he oher had E M M p E p p X s Xs p = E Rs So by Doob s iequaliy, we have, usig he cosa K p, from par, hal-517, versio 5-4 Oc 1 p E M M p p E R p 1 p p p K p, σ p p 1 Par 3 of he proposiio he follows by choosig = σ p p+1 3 Approximaio of he compesaed sum of small jumps by a Browia moio I his secio we will replace R by a Browia moio This mehod gives beer resuls, subjec o a covergece assumpio I fac, Asmusse ad Rosisi proved 1, Theorem 1 ha, if X is a Lévy process, he he process σ 1 R coverges i disribuio o a sadard Browia moio, whe, if ad oly if for ay > Codiio 31 is implied by he codiio σ σ lim = 1 31 σ σ lim = + 3 The codiios 31 ad 3 are equivale if ν does o have aoms i some eighbourhood of zero 1, Proposiio 1 31 Esimaes for smooh fucios The errors resulig from Browia approximaio have o bee much sudied i he lieraure, a leas heoreically There are some resuls which we ca fid i 7, 8 Proposiio 31 Le X be a ifiie aciviy Lévy process ad > 1 If f C 1 R ad saisfies E f X <, ad if here exiss β > 1 such ha,1 E f X + θσŵ f X β 1 β ad,1 E f X + θr f X β 1 β are fiie ad iegrable wih respec o θ o, 1, he E f X f ˆX = o σ

12 1 E H A DIA If f C R ad saisfies E f X +E f X <, ad if here exiss β > 1 such ha,1 E f X + θσŵ f X β 1 β ad 1,1 E f X + θr f X β β are fiie ad iegrable wih respec o θ o, 1, he E f X f ˆX = o σ hal-517, versio 5-4 Oc 1 Examples of fucios saisfyig he above codiios are oed afer Proposiio Proof Wecosiderolypar Theproofforpar1issimilar ByProposiio, we have E f X f X = σ Ef X + o σ O he oher had, usig he same reasoig as i he proof of Proposiio we will replace R by σŵ we ge E fx + σŵ f X = σ Ef X + o σ Hece E fx f ˆX = o σ The combiaio of Proposiio 6 of 7 ad he Spizer s ideiy for Lévy processes Proposiio 1 of 1 leads o he followig resul Proposiio 3 Le X be a iegrable ifiie aciviy Lévy process The EM E ˆM 33σρ 1 + log ρ, where ρ = σ 3 x x 3 νdx Remar 33 Uder codiio 3, we have lim ρ = ad, i ur, σρ 1 + log = o σ ρ Proof Le, Usig Spizer s ideiy for Lévy processes, we have EM E ˆM EX + s E = ds ˆX s + ds s s EX s + E ˆX s ds + s + EX s + E ˆX s ds + s O he oe had, EX s + E ˆX s + X E s + Rs + Xs + σŵs + E Rs σŵs sσ 1 + π

13 ERROR BOUNDS FOR SMALL JUMPS OF LEVY PROCESSES 13 O he oher had, i follows from Proposiio 6 of 7 ha EX s + E ˆX s + Aσρ, hal-517, versio 5-4 Oc 1 wih A < 165 cosider he fucio fx = x + Therefore, EM E ˆM 1 + σ + Aσρ log π 165σ + ρ log The las expressio is miimal for = 4ρ, ad so he desired resul follows by subsiuio 3 Esimaes by Sorohod embeddig We will use a powerful ool o prove he resuls of his secio This is he Sorohod embeddig heorem We will begi by defiig some useful oaios Defiiio 34 We defie x x4 νdx β = σ 4, βp,θ β1 = β 1 6 log β 1 3 = β pθ p+4θ log β θ p+4θ , β = β p + 1 log β 1 4, Remar 35 Noe ha uder codiio 3, we have lim β = The proof of Proposiio 3 cao be exeded o he Lipschiz fucios, because he reformulaio of he Spizer ideiy for Lévy processes cao be applied i ha case We have o use aoher mehod Defie V j, = R j R j 1, j = 1,,, so ha R / = j=1 V j,, = 1,, The V j, are iid wih he same disribuio as R /, hece E V j, = ad Var V j, = σ / Thus, by Sorohod s embeddig heorem Theorem 1 of 19, see p 163, here exis posiive iid rv s τ j, j = 1,,, ad a sadard Browia moio, ˆB, such ha he parial sums R / ad he ˆB τ1+ +τ, = 1,,, have he same joi disribuios; moreover, E τ 1 = Var V 1, ad Eτ 1 4EV 4 1, 33 Furher, oe ha he σŵ/ ad ˆB σ /, = 1,,, have he same joi disribuios Se T = τ τ, T = σ This seig will be used i all of he subseque resuls Theorem 36 Le X be a iegrable ifiie aciviy Lévy process, ad f a Lipschiz fucio The Ef M Ef ˆM C σ β1,

14 14 E H A DIA where C is a posiive cosa idepede of Proof Se If = E f If = f E X X s f f Because f is, say, K-Lipschiz, we ca show ha f X f X + σŵ X s + σŵs X K + σŵ Rs + σ Ŵs hal-517, versio 5-4 Oc 1 As he righ had side expressio is iegrable, by domiaed covergece we ca deduce ha lim + If = I f I holds ha If = f E KE KE 1 X + ˆB T f X + ˆB T ˆB T ˆB T X + ˆB T X + ˆB T Par 1 of he followig heorem cocludes he proof Theorem 37 Le X be a ifiie aciviy Lévy process The: I holds ha I holds ha lim E + 1 lim E + 1 ˆB T ˆB T ˆB T ˆB T C σ β1 C σ β For ay reals p 1 ad θ, 1, i holds ha lim E + 1 ˆB T ˆB T p C p,θ, σ p βp,θ I he above, C ad C p,θ, are cosas idepede of This heorem is he mai resul of his secio Lemma 38 Le X be a ifiie aciviy Lévy process The, for ay >, lim P T T > 4σ 4 β + 1

15 ERROR BOUNDS FOR SMALL JUMPS OF LEVY PROCESSES 15 hal-517, versio 5-4 Oc 1 Proof As T T = i=1 τ i E τ i, by Kolmogorov s iequaliy P T T > Var T T 1 Var τ 1 Eτ 1 4 4E R, where he las iequaliy follows from 33 The proof he follows from Proposiio 1 Proof Proof of Theorem 37 For >, we have E ˆB T ˆB T = I 1 + I, wih 1 I 1 = E ˆB T ˆB T 1 {1 T T 1 } I = E ˆB T ˆB T 1 {1 T T >} 1 O { 1 T T }, se, for fixed, s 1 = T T s = T T We have s 1 s s 1 + Le j be such ha j s 1 < j + 1 We have s 1 s j + If j s 1 s j + 1, we have ˆB s1 ˆB s ˆBs1 ˆB j + ˆBj ˆB s j σ +1 j u j+1 ˆB u ˆB j If j s 1 j + 1 s j +, we have ˆB s1 ˆB s ˆBs1 ˆB j + ˆBj ˆB j+1 + ˆBj+1 ˆB s 3 ˆB u ˆB j j σ + j u j+1 Hece I 1 3E j σ = 3E 1 j σ + +3 j u j+1 j 1 u j ˆB u ˆB j ˆB u ˆB j 1

16 16 E H A DIA The rv s j 1 u j ˆB u ˆB j 1 1 j σ are iid wih he same dis- +3 ribuio as u ˆBu ad, i ur, u 1 ˆBu The I 1 3 E 1 j σ V j, +3 hal-517, versio 5-4 Oc 1 where V j σ 1 j are iid rv s wih he same disribuio as +3 u 1 ˆBu O he oher had, we ow ha if Vj 1 j m are iid rv s saisfyig Ee αv 1 < where α is a posiive real, he where g : x 1, + E V j = E 1 j m E V j g 1 j m mee αv 1, 1 α logx Ideed, sice g is cocave, we have = Eg 1 j m g e αv j e αv j 1 j m g E 1 j m m g E j=1 = g mee αv 1 I our case V 1 = u 1 ˆBu So e αv j, because g is o-decreasig e αv j, by Jese s iequaliy, because g is o-decreasig V 1 ˆB u + ˆB u u 1 u 1 For α, 1/8, we have Ee αv α 1 Ee u 1 ˆBu + u 1 ˆB u Ee 4α ˆB u 1 u 1 Ee 4α u 1 ˆB u 1 = Ee 4α u 1 ˆB u = 1 8α 1 The las equaliy follows from u 1 ˆBu χ 1 upousighemome-geeraig fucio of he χ 1 disribuio, give by 1 β 1 for β < 1 I follows sraighforwardly from he above ha, for α, 1 8, σ I 1 C α log + 3,

17 1 where C α = 3 α I E E ERROR BOUNDS FOR SMALL JUMPS OF LEVY PROCESSES log1 8α log3 ˆBT ˆB 1 T P ˆBT E Rs + E E R 1 + E ˆBσ 4 σ P Cosider ow I We have s σ T T > ˆBT P T T > 1 1 ˆBs P 1 P 1 T T >, 1 T T > 1 1 T T > 1 where he fourh iequaliy is obaied usig Doob s iequaliy So, by Lemma 38, we have 1 1 hal-517, versio 5-4 Oc 1 Hece lim E + 1 lim I 4 4σ 4 1 β σ + ˆBT ˆB T Cα log σ σσ β Par 1 ow follows by leig C = max C α, 8 ad choosig = σ β 1 3 For he proof of pars ad 3 of he heorem, we refer he reader o 9, pp However, some small correcios are eeded i he proof of par 3 i order o comply wih he defiiio of βp,θ Remar 39 Leig θ = 1/ ad p = 1, i he defiiio of βp,θ, we see ha par 3 of Theorem 3 parially geeralizes pars 1 ad I may be releva o oe here ha for par 3 he proof used he fucio gx = α 1 logx p, whereas for pars 1 ad i used he fucio gx = α 1 logx p/, p = 1,, respecively The followig resul follows direcly from par 1 of Theorem 3 Proposiio 31 Le X be a iegrable ifiie aciviy Lévy process, ad f a Lipschiz fucio The Ef X Ef ˆX C β1σ, where C is a posiive cosa Proof We have R = d ˆBT, σŵ = d ˆBT So, if f is K-Lipschiz, we have Ef Ef X Ef ˆX = X + ˆB T Ef KE ˆB T ˆB T We coclude wih Theorem 37 For o-lipschiz fucios, we have he followig resul X + ˆB T

18 18 E H A DIA Proposiio 311 Le X be a ifiie aciviy Lévy process ad p > 1 If Ee pm <, he for ay x R ad for ay θ, 1 E e M x + E e ˆM x C p,θ, σ β p,θ p, p 1 where C p,θ, is a posiive cosa idepede of Proof Defie M = X + R,, ˆM = X We ow ha lim + M, = M as ad lim + ˆM U = X + ˆB T, Û, = X + σŵ = ˆM + ˆB T as Se hal-517, versio 5-4 Oc 1 So M = d U ad ˆM, = d Û, By he mea value heorem, we have e U e Û, = U Û, eū,, where Ū, is bewee U ad Û, Se + I = E e U x E eû, + x Thus I E e U e Û, E U Û, eū, E ˆB T ˆB T E E E ˆB T ˆB T eū, ˆB T ˆB T ˆB T ˆB T p 1 1 p 1 p p 1 p p 1 p Ee pū, 1 p 1 1 p E e pm + e p 1, ˆM p 1 1 p E e pm + e p ˆM 1 p Bu E e pm + e p ˆM E e pm + e pσ Ŵs e pm Ee pm + e p σ Ee pm e p σ E e pm + e pm

19 ERROR BOUNDS FOR SMALL JUMPS OF LEVY PROCESSES 19 So usig domiaed covergece, Theorem 37 ad Lemma 8, we ge E e M x + E e ˆM + + x = lim + e E M, x E ˆM e + = lim e eû, E U x E + C p,θ, σ 1 1 β p,θ p p 1 + x + x hal-517, versio 5-4 Oc 1 33 Esimaes for cumulaive disribuio fucios The bouds obaied i his secio are beer ha hose obaied by rucaio, provided ha codiio 3 is saisfied Proposiio 31 Le X be a ifiie aciviy Lévy process Below, he cosas C ad C x,,q,θ are idepede of 1 If b >, he P X x P ˆX x C σ β1 x R If X has a locally bouded probabiliy desiy fucio ad x R, he for ay pair of reals θ, 1, q, 1/, P X x P ˆX x q Cx,,q,θ σ 1 q β 1 1,θ q 3 If M has a locally bouded probabiliy desiy fucio o, + ad x >, he for ay pair of reals θ, 1, q, 1/, P M x P ˆM x q Cx,,q,θ σ 1 q β 1 1,θ q Proof Recall ha R =d ˆBT ad σŵ = d ˆBT Se Y = X + ˆB T, Ŷ = X + ˆB T Thus P P X x P ˆX x = Y x P = P Ŷ x Y x, Ŷ < x P Y < x, Ŷ x I holds ha P Y x, Ŷ < x = P x Y Ŷ Ŷ < x = P x ˆBT ˆB T bb + Ŷ bb < x By cosrucio, bb is idepede of Ŷ bb ad of ˆBT ˆB T Furher, 1 b is a upper boud of he probabiliy desiy fucio of bb π By codiioig o he pair ˆBT ˆB T, Ŷ bb, i ca hus be cocluded ha P Y x, Ŷ < x 1 b π E ˆBT ˆB T

20 E H A DIA hal-517, versio 5-4 Oc 1 Aalogously, i also holds ha P Y < x, Ŷ x 1 b π E ˆBT ˆB T We ge he firs par of he proposiio by usig Theorem 37 We ow prove he secod par of he proposiio Le p 1 By Lemma 11, here exiss K x, > such ha, for ay >, P Y x P Ŷ x Kx, + E Y Ŷ p p E ˆB T ˆB p T = K x, + p Hece, give θ, 1, by Theorem 37 here exiss a cosa C p,θ, > such ha σ P Y x P Ŷ p βp,θ x Kx, + C p,θ, p Choosig = σ p p+1 β p,θ 1 p+1 yields P Y x P Ŷ x max Kx,, C p,θ, σ p p+1 β p,θ 1 p+1 The resul he follows by subsiuig p = 1/q 1 For he hird par of he proposiio, we use he oaio of Proposiio 311 Noe ha P M x P ˆM x = lim = lim P M P U x P ˆM, x, x P Û x Le p 1 ad pu I x, = x, x + Usig he proof of Lemma 11, we have P U, E U x P Û x Û, p P U I x, + p E 1 ˆBT ˆB T p P M I x, + p, for ay > By he assumpio o M, here exiss a cosa K x, > such ha P M I x, < K x, for ay > Combied wih Theorem 37, leig yields lim P U, x P Û x K x, + C σ p βp,θ p,θ, p, for some cosa C p,θ, > So as i par, he resul follows by choosig = σ p p+1 β p,θ 1 p+1 REFERENCES

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