Deviation probability bounds for fractional martingales and related remarks

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1 Deviaion pobabiliy bounds fo facional maingales and elaed emaks Buno Sausseeau Laboaoie de Mahémaiques de Besançon CNRS, UMR Roue de Gay 253 Besançon cedex, Fance Absac In his pape we pove exponenial inequaliies also called Bensein s inequaliy fo facional maingales. As an immediae coollay, we will discuss weak law of lage numbes fo facional maingales unde divegence assumpion on he β vaiaion of he facional maingale. A non ivial example of applicaion of his convegence esul is poposed. Key wods: Facional maingales, exponenial inequaliy, law of lage numbes 2 MSC: 6G22, 6G48, 6B12 1. Inoducion The noion of facional maingales has been inoduced in Hu e al. 29 whee he auhos poved an exension of Lévy s chaaceizaion heoem o he facional Bownian moion. The pupose of his sho communicaion is o invesigae exponenial inequaliies of Bensein s ype and hei applicaions o laws of lage numbes fo facional maingales. Moe pecisely when we fix α 1 2, 1 2, if M = M is a coninuous local maingale, he defined by M α = sα dm s will be called a facional maingale pocess M α = M α povided ha he above inegal exiss. Fo a fixed ime, we can conside he ue maingale Zu u defined by Zu = u sα ξ s dw s whee W = W is a F Bownian moion and ξ = ξ is a pogessively measuable and squae inegable pocess. Hee is conside as a fixed paamee fo he maingale Z. As a consequence of he classical exponenial inequaliy, one can easily obain some deviaion pobabiliy bounds. Fo example if α <, i is clea ha P M α u, ξ τ 2 dτ ν 2 exp u2 4 2α ν Hence i is easy o pove some exponenial inequaliies when is fixed see also Remak 2. Ou aim is o invesigae some deviaion bounds fo sup s M s α. This will be no moe a saighfowad applicaion of he esul in he maingale case when α =. We ecall ha fo he maingale M, if i vanishes a ime =, hen P sup M s a 2 exp a2, 1 s 2c if c is a consan such ha M c fo all see Revuz and Yo, 1999, Execice 3.16, Chape 4. And so ou wok will concen he exension of exponenial inequaliies simila o 1 fo facional maingales. Since on any ime ineval, he pocess M α has finie nonzeo vaiaion of. addess: buno.sausseeau@univ-fcome.f Buno Sausseeau Pepin submied o Elsevie May 3, 212

2 ode β = 2/1 + 2α we shall y o use some quaniies elaed o he β vaiaion in he saemen of ou Bensein s ype inequaliy. This will epesen he main esul of his pape and i is he pupose of Theoem 1. As an applicaion of hese exponenial inequaliies, we will have a discussion aound weak law of lage numbes fo facional maingales. We hink ha using ou esul on he deviaion pobabiliy bounds fo facional maingale is a fis sep o he sudy of law of lage numbes fo facional maingales. So in Poposiion 2 we esablish ha sup s M s α / M α β, ends o in pobabiliy povided ha he β vaiaion M α β, ends o infiniy fase han a fo some a >. Of couse his is no a classical condiion bu we pesen a non ivial applicaion of his weak law of lage numbes in Poposiion 3. To end his discussion, a elaed convegence esul is given in Poposiion 4 unde he moe convenional assumpion on he divegence of he quadaic vaiaion of he undelying maingale. Moe pecisely, we shall pove ha sα ξ s dw s / s2α ξ 2 sds ends o almos suely when α > and unde he assumpion ha + ξ 2 sds = + almossuely. In a sense, he law of lage numbes fo he maingale ξ sdw s is ansfeed o is facional maingale M α. The pape is oganized as follows. In he following secion we pecise ou noaions and we sae ou esuls. Two poofs will be given in Secion 3 and Secion Noaions and main esuls We follow he eminology of Hu e al. 29. Le Ω,F,P be a complee pobabiliy space equipped wih a coninuous filaion F such ha F conains he P negligeable evens. Le β 1 and le X = X 1 be a coninuous adaped pocess. The β vaiaion of X on he ime ineval [,] is denoed by X β, and is defined as he limi in pobabiliy if i does exis of S [,] n β,n := X n i X n i 1 β i=1 whee fo i =,..., n, n i = i n. If he convegence holds in L1, we say ha he β vaiaion exiss in L 1. A paamee α 1 2, 1 2 is fixed and we denoe β = α. We noice ha β 1,+ and β > 2 when α <. Le M α = M α a facional maingale of ode α. This means ha M α is a coninuous F adaped pocess such ha hee exiss a coninuous local maingale M = M wih s2α d M s < a.s. fo all, and M α = s α dm s. 2 If α, 1 2, he above inegal always exiss as a Riemann-Sieljes inegal. In ode o ensue he exisence M α when α 1 2,, we assume he following hypohesis in all he sequel. Hypohesis I. The coninuous local maingale M is of he fom M = ξ s dw s whee W = W is a F Bownian moion and ξ = ξ is a pogessively measuable pocess such ha fo all E ξ s β ds < fo some β > β, if α < ; E ξ 2 s ds <, if α >. 2

3 Unde Hypohesis I, he inegal appeaing in 2 always exiss as a Riemann-Sieljes inegal. This is a consequence of Hu e al. 29, Lemma 2.2 and he fac ha he ajecoies of M ae α Hölde coninuous on finie ineval. Moeove, by Theoem 2.6 and Remak 2.7 of Hu e al. 29, he β vaiaion of M α exiss in L 1 and M α β, = c α ξ s β ds whee c α depends only on α. The explici fom of he consan c α is given in Hu e al. 29 bu his is no impoan in ou wok. Neveheless we sess he poin ha unde Hypohesis I, he expession of M α is given by M α = s α ξ s dw s. 3 Moeove, using Hölde s inequaliy one deduces he following elaions beween he β vaiaion of M α and he quadaic vaiaion of he undelying maingale M: M c 2/β α β 2 β M α 2/β β, when α < ; M α β, c α 2 β 2 M β/2 when α >. Ou main esul which is a genealizaion of Bensein s inequaliy o facional maingales is saed in he nex heoem. Theoem 1. We assume Hypohesis I. We denoe C = /2 2. Fo any posiive funcion ν and any L 1 he following exponenial inequaliies hold. i When α < we have P sup s M α s L c 1 β β 2ββ ν 1/2, ξ τ β dτ 2/β ν C exp wih c 1 defined in 24 and κ 2 = 4πββ /β β 3. { κ2 L 2 β β ββ ii When α >, fo any ε,α i holds ha P sup M s α L 2 6 κ α ε ν 1/2, ξ τ 2 dτ ν C exp { κ2 L 2 } s 2α ε wih κ = π/2 1/2 ε 3/2. iii If we assume ha he pocess ξ is bounded by c almos-suely, hen fo any α 1 2, 1 2 and any ε, α P sup M s α L 2 6 κ c 1/2+α ε C exp { κ2 L 2 } s 1+2α 2ε, 6 wih κ = π/2 1/2 ε 3/2. Fomally, he above inequaliies ae consisen asympoically when gows o infiniy wih he classic ones ecalled in 1 when α = o equivalenly β = β = 2. Fo example, one can pu L 1/2+ε wih ε = 1/4 in 5. Remak. The above esul have a saighfowad poof if we ae ineesed by exponenial inequaliies wihou he supemum wih espec o s [,]. Fo example o show ha P M α u, ξ τ β dτ 2/β u 2 ν 2 exp 4 C β,β 2β β/ββ ν } 4 5 3

4 when α <, i suffices o emak ha, by Hölde s inequaliy, ξ τ β dτ 2/β ν implies ha s 2α ξ 2 sds C β,β 2β β /ββ ν. Thus he inequaliy is a consequence of he classical exponenial inequaliy when one consides he maingale Z u u defined by Z u = u sα ξ s dw s is conside as a fixed paamee. The unifom deviaions saed in heoem 1 ae a lile bi moe complicaed han he one we descibed in he above emak. Thei poofs ae posponed in Secion 3. As a coollay of he above heoem, we obain a weak law of lage numbes fo facional maingales. Poposiion 2. Unde Hypohesis I, le M α be a facional maingale wih α 1 2, 1 2 having he expession M α = sα ξ s dw s. We assume ha he pocess ξ is bounded by a consan c. Then we have he following weak law of lage numbes: suppose ha hee exiss a > such ha M α β, lim a = + almos-suely hen he following convegence holds in pobabiliy sup s M α s M α β,. Poof. Wih η > we use 6 o wie ha sup s M s α sup s M s α P η P η, M α M α β, M α β, a β, + P M α β, a P sup M s α η a + P M α β, a s η 2 2a C exp + P M α 128 c α 2ε β, a. 7 I suffices o choose ε closed o 1/2 + α such ha a > 1 + 2α 2ε and he fis em in he igh hand side of 7 ends o as goes o infiniy. The second em in 7 ends o because lim a M α β, = + almos-suely. The following poposiion povides a non ivial example of applicaion of he above esul. One efes o Nuala 26 fo deails abou facional Bownian moion. Poposiion 3. Fo H,1, le B H = B H be a facional Bownian moion adaped wih espec o he filaion F and le Φ be a bounded coninuous funcion fom R o R. Wih α 1 2, 1 2, he facional maingale N α defined by N α weak law of lage numbes Poof. As egad o 7, one have o find a > such ha P N α β, a. = sα ΦB H s dw s saisfies he sup s N s α P N α. 8 β, 4

5 Fo ha sake, we will make use of he local ime L H, y of B H a y R defined heuisically fo as L H, y = δ y B H s ds. I is known see Beman, 1973/1974; Geman and Hoowiz, 198 ha, y L H, y exiss and is joinly coninuous in, y. By he self-similaiy popey of he facional Bownian moion, he disibuions of L H, y and 1 H L H 1, y H ae equal. Using he occupaion imes fomula, we may wie ha ΦB H s β ds = d 1 1 ΦB H u β du By he bi-coninuiy of he local ime, we finally obain 1 1 H = Φ H Bu H β du in disibuion = Φ H y β L H 1, ydy R = 1 H Φz β L H 1, z H dz. R ΦB H s β ds in disibuion. Consequenly we have P N α β, a N α β, = P 1 H a 1 H R Φz β dz L H 1, P R Φz β dz L H 1, = as soon as a < 1 H. I emains o emak ha such a choice of a is always possible and he convegence 8 is hus a consequence of Poposiion 2. To end his discussion abou he law of lage numbes fo facional maingales, one has o menion he following esul. I has been used in Sausseeau 211 o invesigae asympoic popeies of a nonpaameic esimaion of he dif coefficien in facional diffusion. Poposiion 4. Le ξ = ξ s s is one dimensional, adaped pocess wih espec o he filaion geneaed by a sandad Bownian moion W = W, such ha fo any T >, T ξ2 sds <. When α > and + ξ 2 sds = + almos-suely, we have lim sα ξ s dw s s2α ξ 2 s ds = almos-suely. 9 we noice ha he assumpion on he divegence of he quadaic vaiaion of he maingale ξ sdw s is moe common. Neveheless his esul is no a saighfowad applicaion of he echniques used in he maingale case when α =. The poof of 9 is based on a facional vesion of he Toepliz lemma and is posponed in Secion Poof of Theoem 1 Exponenial inequaliies fo coninuous maingales have aaced a lo of aenion: see fo example Caballeo e al. 1998; Lipse and Spokoiny 2 and de la Peña Due o ou facional famewok, he echnics used in he afoemenioned woks ae useless. Ou mehodology is closed o he one used in he poof of Theoem 2 fom Nuala and Rovia 2 see also Rovia and Sanz-Solé 1996; Sowes Tha being said we need he following lemma. 5

6 Lemma 5. Le ε > saisfying α < ε < 1. Then hee exiss a consan C = C α,ε such ha u + h α u α C h ε u α ε, u >, h >. 1 Poof. Wih h = xu, Inequaliy 1 is equivalen o x α C x ε, x >. Accoding o he cases we need o pove ha { FC x := x α C x ε, x > when α > and G C x := x α C x ε, x > when α <. 11 If we choose C such ha C α ε { } x 1 ε sup x 1 + x 1 α, hen F C and G C ae negaive and 11 is hen ue. I is easy o check ha we may find a consan C saisfying he above inequaliy and ha is independen of ε. Now we pove Theoem 1. Poof. We follow he agumens developed in Nuala and Rovia 2. The inequaliy 4 will be a consequence of he Chebyshev exponenial inequaliy involving he andom vaiable sup s M s α. So he fis sep is o apply he Gasia-Rodemich-Rumsey inequaliy in ode o have bounds on his andom vaiable. Wih Ψx = exp x 2 /4 and p a coninuous, non-negaive funcion on, such ha p =, Lemma 1.1 in Gasia e al. 197/1971 eads as follows: fo all s we have s 4B M s α M α 8 Ψ 1 y 2 dpy 12 povided ha B := Ψ M α s M α p s dsd <. The funcion p will be chosen lae. We noice ha he funcion Ψ 1 is defined fo u Ψ as Ψ 1 u = sup{v; Ψv u}. Wih ln + he funcion defined by ln + z = maxlnz, fo z, he inequaliy B/y 2 expln + B/y 2 implies ha 1/2 B B Ψ 1 y 2 2 ln + y 2. Fuhe calculaions show ha ln + B y 2 1/2 2 1/2 { ln + B 1/2 + ln + y 2 1/2 }. Since M α =, we deduce fom 12 ha { ln sup M s α 2 9/2 + B 1/2 + ln + y 2 } 1/2 dpy. 13 s In he following we will need an esimae of he expecaion of he andom vaiable B. This will be possible because a maingale wih bounded quadaic vaiaions will appea by means of he incemens of M α. We fix and fo any < s < we wie M α s M α = 6 g s, τ dw τ

7 wih g s, τ = ξ τ s τ α 1 {<τ s} + ξ τ s τ α τ α 1 {τ }. We fis noice ha g s, τ 2 dτ = s τ 2α ξ τ 2 dτ + s τ α τ α 2 ξ τ 2 dτ 14 In ode o have some esimaes of he quaniy g s,τ 2 dτ, we ea diffeen cases accoding o he sign of α and accoding o he assumpion we made on he pocess ξ. Case i: we assume Hypohesis I and α < Wih β > β fom Hypohesis I, we denoe p = β /2 > 1 and q = β /β 2 is conjugae. Le ε > o be fixed lae. Saing fom 14, we use Lemma 5 o wie [ ] g s, τ 2 dτ s 2ε s τ 2α 2ε ξτ 2 dτ + τ 2α 2ε ξτ 2 dτ 15 s 2ε[ ] I 1 + I 2 16 wih obvious noaions. Now we choose ε such ha 1 + 2αq 2εq >. We emak ha such a choice is always possible. Indeed αq 2εq = [β ββ β εββ ] 2 and hen choosing ε of he fom ε = aβ β/ββ wih a,1, we emak ha We chose a = 1/2, hencefoh ε is fixed as By Hölde s inequaliy we obain wih I 1 < 1 + 2αq 2εq < 2β β ββ 2. ε = 1 β β 2 ββ. 1/q s τ 2α 2εq dτ 1/p ξτ 2p dτ C β,β s 2α 2ε+1/q ξ 2 L 2p, 17 C β,β = [ ββ ] β 2 β 2 β β. Similaly we obain he following esimaion fo I 2 : I 2 C β,β 2α 2ε+1/q ξ 2 L 2p,. 18 Repoing 17 and 18 in 16 yields On he even A = { g s, τ 2 dτ 2 C β,β s 2ε 2α 2ε+1/q ξ 2 L 2p,. 19 ξ2p τ dτ 1/p ν } i holds ha g s, τ 2 dτ 2 C β,β s 2ε 2α 2ε+1/q ν. 7

8 Now i is clea ha he funcion p mus be defined as py = 2C β,β 1/2 α ε+1/2q ν 1/2 y ε 2 and fo fixed < s, we conside he maingale M = M u u defined by M u = u Is quadaic vaiaion saisfies fo any u M u g s, τ ps dw τ. 21 g s, s 2 ps ds 1 almos-suely on A. Le W be he Dambis, Dubins-Schwaz Bownian moion associaed o he maingale M such ha M u = W M u. We have E [ Ψ M α s M α p s Consequenly EB1 A 2 1/2 2 and 1 A ] [ M 2 = E exp 4 [ E exp 2 1/2. 1 A ] ] 1 4 sup W 2 1 A 1 Eexp{1 A ln + B} 1 + E1 A exp{ln + B} /2 2 := C. We use he inequaliy ln + y 2 + 1/2 y ε 1 dy 2 1/2 z 1/2 e εz dz = 1 π := κ, 22 ε 2ε in ode o ewie 13 wih p defined in 2 as sup s [ M s α 2 9/2 ln + B 1/2 ε + κ ] 2C β,β 1/2 α ε+1/2q ν 1/2 [ ln + B 1/2 ε + κ] c 1 23 wih c 1 = 32 C 1/2 β,β α ε+1/2q ν 1/2. Now we end he poof wih Chebishev s exponenial inequaliy. Fo L 1 we have { P sup M s α 2Lκc 1, A P ln + B 1 } 2 2Lκc1 s 2ε κ A c 1 E [ exp 1 A ln + B ] { } 2 κ 2L 1 exp C exp { κ2 L 2 }. We ecall ha 4ε = 2α + 1/q = 2β β/ββ and he expession 4 is a consequence of he above inequaliy wih he noaion 2ε ε c 1 = 2 11/2 π 1/2 ββ β β 3/2 [ ββ ] β 2 2β 2 β β. 24 8

9 Case ii: we assume Hypohesis I and α > By Hypohesis I, ξ2 τ dτ exiss almos-suely. Using 14, we eplace 15 by g s, τ 2 dτ s 2α ξτ 2 dτ + s 2ε τ 2α 2ε ξτ 2 dτ sup s s 2α ξτ 2 dτ + s 2ε 2α 2ε ξτ 2 dτ 2s 2ε 2α 2ε ξτ 2 dτ 25 wih < ε < α. The es of he poof is simila wih he following modificaions. We use he maingale M defined by 21 wih he new funcion p defined by py = 2 1/2 α ε ν 1/2 y ε. On he even { ξ2 τ dτ ν }, M has also a quadaic vaiaion bounded by 1. The inequaliy 23 is eplaced by [ ln M s α + B 1/2 ε + κ] 32 α ε ν 1/2 whee κ is defined in 22. The es of he poof is idenical. Case iii: we assume ha ξ is bounded When α 1 2,, hee exiss ε > such ha 1 + 2α 2ε >. Then fom 15, i is easy o see ha 19 may be eplaced by g s, τ 2 dτ 2 c 2 2ε 1+2α 2ε. 26 When α >, 25 may also be eplaced by 26. The es of he poof is simila o he pevious case wih he help of he funcion p defined by py = 2c 1/2 1/2+α ε y ε. 4. Poof of Poposiion 4 The esul is based of he following facional vesion of he Toepliz lemma. Lemma 6. Le α >. Le x be a coninuous eal funcion such ha lim x = x and le γ be a measuable and posiive. Then i holds ha povided ha lim γ sds = +. α 1 s γ d x s ds α 1 s γ d ds x, Poof. Le ε > and A be such ha x s x < ε fo s > A. We denoe C A = sup s A x s x. By Fubini s heoem s α γ s ds = α s α 1 γ d ds, and we wie fo > A α 1 s γ d x s ds α 1 s γ d ds x sα 1 γ d x s x ds sα 1 γ d ds ε + C A A sα 1 γ d ds sα 1 γ d ds. 27 9

10 Anohe applicaion of Fubini s heoem implies ha A A α 1 ds γ d A = γ [ α A α ] d sα 1 ds γ d α γ d A α γ d α γ d α A γ d /2 α γ d α A γ d /2 α /2 γ d and he las em ends o as ends o. We epo his convegence in 27 and we obain he esul. Now we pove 9. Poof. By he sochasic Fubini heoem and consequenly s α ξ s dw s = α s α 1 ξ dw ds sα ξ s dw s s2α ξsds = α 1 s ξ2 d ξ2 2 α 1 s ξ2 d ds Since i is assumed ha ξ 2 sds = + almos-suely, ξ dw ξ2 d and he genealized Toepliz lemma 6 implies 9. a.s. ξdw ds d. Refeences Beman, S. M. 1973/1974. Local nondeeminism and local imes of Gaussian pocesses. Indiana Univ. Mah. J. 23, Caballeo, M. E., Fenández, B., and Nuala, D Esimaion of densiies and applicaions. J. Theoe. Pobab. 11, 3, de la Peña, V. H A geneal class of exponenial inequaliies fo maingales and aios. Ann. Pobab. 27, 1, Gasia, A. M., Rodemich, E., and Rumsey, J., H. 197/1971. A eal vaiable lemma and he coninuiy of pahs of some Gaussian pocesses. Indiana Univ. Mah. J. 2, Geman, D. and Hoowiz, J Occupaion densiies. Ann. Pobab. 8, 1, Hu, Y., Nuala, D., and Song, J. 29. Facional maingales and chaaceizaion of he facional Bownian moion. Ann. Pobab. 37, 6,

11 Lipse, R. and Spokoiny, V. 2. Deviaion pobabiliy bound fo maingales wih applicaions o saisical esimaion. Sais. Pobab. Le. 46, 4, Nuala, D. 26. The Malliavin calculus and elaed opics, Second ed. Pobabiliy and is Applicaions New Yok. Spinge-Velag, Belin. Nuala, D. and Rovia, C. 2. Benoulli 6, 2, Lage deviaions fo sochasic Volea equaions. Revuz, D. and Yo, M Coninuous maingales and Bownian moion, Thid ed. Gundlehen de Mahemaischen Wissenschafen [Fundamenal Pinciples of Mahemaical Sciences], Vol Spinge-Velag, Belin. Rovia, C. and Sanz-Solé, M The law of he soluion o a nonlinea hypebolic SPDE. J. Theoe. Pobab. 9, 4, Sausseeau, B Nonpaameic infeence fo facional diffusion. AXiv e- pins Sowes, R. B Lage deviaions fo a eacion-diffusion equaion wih non-gaussian peubaions. Ann. Pobab. 2, 1,