E l e c r o n i c J o u r n a l o f P r o b a b i l i y Vol. 4 999 Paper no. 7, pages 25. Journal URL hp://www.mah.washingon.edu/~ejpecp Paper URL hp://www.mah.washingon.edu/~ejpecp/ejpvol4/paper7. abs.hml MODERATE DEVIATIONS TYPE EVALUATION FOR INTEGRAL FUNCTIONALS OF DIFFUSION PROCESSES R. Lipser Deparmen of Elecrical Engineering-Sysems Tel Aviv Universiy, 69978 Tel Aviv, Israel lipser@eng.au.ac.il V. Spokoiny Weiersrass Insiue for Applied Analysis and Sochasics Mohrensr. 39, 7 Berlin, Germany spokoiny@wias-berlin.de Absrac We esablish a large deviaions ype evaluaion for he family of inegral funcionals κ ΨXsgξ sds,, where Ψ and g are smooh funcions, ξ is a fas ergodic diffusion while X is a slow diffusion ype process, κ, /2. Under he assumpion ha g has zero barycener wih respec o he invarian disribuion of he fas diffusion, we derive he main resul from he moderae deviaion principle for he family κ gξ sds, which has an independen ineres as well. In addiion, we give a preview for a vecor case. Keywords Large deviaions, moderae deviaions, diffusion AMS 99 subjec classificaion: 6F Submied o EJP on April 3, 999. Final version acceped on Sepember 28, 999.
Inroducion In his paper, we consider a wo scaled diffusion model wih independen Wiener processes V and W : dξ = bξ d + σξ dv dx = F X, ξ d + GX, ξ dw. 2 The fas componen ξ is assumed o be an ergodic Markov process while he slow componen X is a diffusion ype process governed by he fas process ξ = ξ and independen of i Wiener process W = W. Under appropriae condiions, a sochasic version of he Bogolubov averaging principle holds see [2], ha is, he slow process is averaged wih respec o he invarian densiy of he fas one, say pz. In oher words, he X process is approximaed by a Markov diffusion process X wih respec o some Wiener process W : wih he averaged drif and diffusion parameers F x = F x, zpzdz, Gx = R dx = F X d + GX dw 3 R G 2 x, zpzdz /2. Le us assume funcions F and G are unknown and indicae here a saisical procedure for esimaion of he averaged funcion F from he observaion of he slow process X. The averaging principle suggess he following recipe: o proceed wih he pah of X as if i is he pah of X. For insance, i is well known he kernel esimae wih kernel K and bandwidh h F x = K X x h K X x h dx d for F x via X, T so ha F x = K X x h dx T K X x h d is aken as esimae of F x via X, T. An asympoic analysis, as, for F x leads o sudy of properies for inegral funcionals X F x, ξ K x ds h and X F i x, ξ K x ds, i =, 2, h where F i x, y is he i -h derivaive of F x, y in x. Namely, assuming ha boh T and h depend on : T = T, h = h wih T, h ɛ, we need o show ha for any of funcions F x, y, F x, y, F 2 x, y specified as Hx, y he inegral [ Hx, ξ ] Hx, y py dy K X x h ds 2
goes o zero faser han κ wih some κ > see [8]. For fixed x, denoe by gξ = Hx, ξ Hx, y py dy and by ΨX = K X x h. Then he desired propery holds, if for insance wih T we have P T κ ΨXsgξ sds z { exp z 2 } cons. T 2κ. We avoid a sraighforward verificaion of his asympoic and prefer o consider firs he special case, Ψ, which is of an independen ineres and allows o clarify he main idea and o simplify he exposiion. So, le S,κ = κ gξs ds, 4 where g = gz is an arbirary funcion wih zero barycener wih respec o he invarian densiy p. Our approach o he asympoic analysis employs, so called, Poisson decomposiion in he form used in [4] for proving he cenral limi heorem CLT: S,κ = e,κ + /2 κ S, 5 where e,κ is a negligible process and S is a coninuous maringale wih he predicable quadraic variaion S,κ is close o a linear increasing funcion γ, so ha he S,κ process is approximaed in he disribuion sense by Wiener process wih he diffusion parameer γ. The Poisson decomposiion allows o analyze he asympoic behavior, in, of S,κ under κ /2. We exclude wo exreme poins κ = /2 and κ = and emphasize only ha for κ = /2 he family S, obeys he CLT see e.g [4] while for κ = a family of occupaion measures of ξ,, obeys he large deviaion principle LDP see [7] or e.g. [6] and so, due o he conracion principle of Varadhan [25], he family S,κ,, a leas for bounded g, obeys he LDP as well. In conras o boh, he case < κ < /2 preserves he large deviaion ype propery, which is he same as for a family of Wiener processes paramerized by diffusion parameer 2κ γ. In oher words, he case < κ < /2 guaranees, so called, moderae deviaion evaluaion for he family S,κ,. For esablishing he moderae deviaion principle MDP, we use he condiions on he drif and diffusion parameers b and σ see A- in Secion 2 proposed by Khasminskii, [3] and modified by Vereennikov, [26]. These condiions allow o verify ha boh e,κ and S,κ γ are exponenially negligible processes wih he rae of speed 2κ. Formally we could apply a recen Pukhalskii s resul [23], which being adaped o he case considered is reformulaed as: if S and γ are exponenially indisinguishable wih he rae of speed 2κ, hen he family 2κ S, obeys he same ype of LDP as /2 κ γ W does, ha is wih he rae of speed 2κ and he rae funcion of Freidlin-Wenzell s ype see [2]: for absoluely coninuous funcion ϕ Jϕ = 2γ ϕ 2 d. Neverheless, we give here anoher proof of he same implicaion, in which he role of he fas convergence of S o γ is discovered wih more deails and migh be ineresed by iself. For he inegral funcional κ T κ ΨX gξ d = ΨX gξ d we also apply he Poisson decomposiion 5 ΨX de,κ 3 + /2 κ ΨX d S
in which he firs erm is he Iô inegral wih respec o he semimaringale e,κ while he second one is he Iô inegral wih respec o he maringale S. As for Ψ, he main conribuion comes from he second erm and many deails of proof are borrowed from Ψ. Resuls on he MDP for processes wih independen incremens are well known from Borovkov, Mogulski [2], [3] and Chen [5], Ledoux [5]. For he depended case, he MDP esimaions have araced some aenion as well. Some perinen MDP resuls can be found in: Bayer and Freidlin [] for models wih averaging, Wu [27] for Markov processes, Dembo [8] for maringales wih bounded jumps, Dembo and Zajic [9] for funcional empirical processes, Dembo and Zeiouni [] for ieraes of expanding maps. The paper is organized as follows. In Secion 2, we fix assumpions and formulae main resuls. Proofs of he main resuls are given in Secion 3. Taking ino accoun an ineres o he vecor case seing, in Secion 4 we give a preview for he MDP wih a vecor fas process. Acknowledgemen The auhors hank S. Pergamenshchikov and A. Vereennikov for helpful remarks and suggesions leading o significan improvemen of he paper. 2 Assumpions. Formulaion of main resuls For a generic posiive consan, noaion l will be used hereafer. We fix he following assumpions. The iniial condiions ξ and X for he Iô equaions and 2 respecively are deerminisic and independen of. A- The funcion gx is coninuously differeniable and gx l + x. A-. Funcions b, σ are coninuously differeniable b once, σ wice; 2. σ 2 x is uniformly posiive and bounded; is derivaives are bounded as well; 3. here exis consans C > and c > such ha for x > C xbx c x 2 b 2 x + b xσ 2 x /cb 2 x. A-2 Funcions F = F x, z, G = Gx, z are bounded, coninuous, and Lipschiz coninuous in x uniformly in z. A-3 Funcion Ψ = Ψx is wice coninuously differeniable and bounded joinly wih is derivaives he value Ψ = Ψx is involved in he formulaion of he main resul. x A-4 For some d > and every >, T d and for a chosen κ, 2 lim T 2κ =. 4
I is well known from [4], [24] ha under A- he process ξ is ergodic wih he uniquely defined invarian densiy pz = cons. exp in addiion o A- assume Then, he funcion { vz = 2 z by σ 2 z is well defined and bounded. Se γ = σ 2 y dy } Our main resul is formulaed in he heorem below. R. Under A-, we have R gz pzdz < and gzpzdz =. 6 2 z σ 2 gypydy 7 zpz R v 2 zσ 2 zpzdz. 8 Theorem Assume A--A-4, 6. Then for every z > and κ, /2 lim T 2κ log P Ψ T κ ΨXsgξ sds z z2 2γ. For Ψ, we give more refined evaluaion. Le us recall he definiion of he LDP in he space C of coninuous funcions on [, plied by he local remum opology: rx, X = n 2 n X X, X, X C. Following Varadhan, [25], he family S,κ = n, for fixed κ, /2 is said o obey he LDP in he meric space C, r wih S,κ he rae of speed 2κ and he rae funcion J κ = J κ ϕ, ϕ C, if. level ses of J κ are compacs in C, r;. for any open se G from C, r lim 2κ log P S,κ G inf J κϕ; ϕ G 2. for any closed se F from C, r lim 2κ log P S,κ F inf J κϕ. ϕ F In he case J κ J, < κ < 2, he family S,κ is said o obey he MDP in C, r wih he rae funcion J. Theorem 2 Assume A-., A-, 6, and < κ < /2. Then, he family S,κ obeys he MDP in he meric space C, r wih he rae of speed 2κ and he rae funcion Jϕ = 2γ ϕ 2 d, dϕ = ϕ d, ϕ =, oherwise. 9 5
3 Proofs 3. Exponenial ermaringale Hereafer, random processes are assumed o be defined on some sochasic basis Ω, F, F = F, P wih he general condiions see, e.g. [9], Ch.,. Le M be a coninuous local maringale wih he predicable quadraic variaion M. I is well known ha M is a coninuous process and λ R Z λ = exp λm λ2 2 M is a coninuous local maringale. Being posiive, he Z process is a ermaringale see e.g. Problem.4.4 in [8] and herefore for every Markov ime τ on he se {τ = }, Z τ = lim Z We apply his propery for he following useful Lemma Le τ be a sopping ime and A be an even from F. EZ τ λ.. If here exiss a posiive consan α so ha M τ 2 M τ α on he se A, hen P A e α. 2. Le η and B be posiive consans so ha M τ η, M τ B on he se A. Then P A exp η2. 2B 3. If for fixed T >, B > i holds M T B, hen P M η, M T B 2 exp η2. 2B 4. If for fixed T >, B > i holds M T B on he se A, hen P M η, M T B, A 2 exp η2. 2B Proof:. By virue of, EI A Z τ P Ae α and he resul holds. 2. Analogously, EI A Zτλ P A exp λη λ2 B 2 P A exp η 2 2B and he asserion follows. 3. Inroduce Markov imes τ ± = inf{ : ±M η}, where inf{ } =, and wo ses A ± = {τ ± T, M T B}. Since by 2. P A ± exp, i remains o noe only ha { M η} A + A. η2 2B 4. The proof is he same as for 3. wih A ± = {τ ± T, M T B} replaced by A ± = {τ ± T, M T B, A}. 6
3.2 Proof of Theorem 2 3.2. Poisson decomposiion for S,κ Le S,κ and vy be defined in 4 and 7 respecivelly. Se uz = z vydy, z R. I is well known see, e.g. [4] ha he invarian densiy pz of he fas process saisfies he Fokker-Plank-Kolmogorov equaion = bzpz + 2 σ2 zpz or, in he equivalen form, 2 σ2 zpz = bzpz. Lemma 2 Under A-, A-, and 6, he Poisson decomposiion holds S,κ wih S = e,κ = κ [uξ uξ ]. = /2 κ S +e,κ vξ sσξ sdv s 2 Moreover, for every > and a suiable consan l, e,κ l κ + ξ and so s e,κ s l κ + ξs, >. Proof: Le us consider he conjugae o equaion wih g from 4: s 2 σ2 zv z + bzvz = gz. I is clear he funcion v, defined in 7, is one of soluions of his equaion. The Iô formula, applied o uξ, gives he required decomposiion uξ = uξ + gξsds + /2 vξsσξ sdv s. Under 6 vz is bounded funcion and so, he las saemen of he lemma holds. 3.2.2 Exponenial negligibiliy of e,κ We esablish here he exponenial negligibiliy lim 2κ log P 3 is derived from Lemma 2 and he following lemma below. Lemma 3 Under A-, for every η > and sufficienly large L, lim 2κ log P κ ξ > η =, lim 2κ log P T e,κ > η =. 3 ξs 2 ds > L =. 7
Proof: We sar wih useful remarks:. he second saemen of he lemma is valid provided ha for some posiive consan C and L large enough lim 2κ log P T hereafer we will use he consan C from he assumpion A-.3; 2. he firs saemen of he lemma is valid, if for all L I ξs > C ξs 2 ds > L = 4 lim 2κ log P 2 κ ξ 2 > η 2, ξs 2 ds LT =. 5 So, only 4 and 5 will be verified below. By he Iô formula we have 2 κ ξ 2 = 2 κ ξ 2 + 2κ 2ξsbξ s + σ 2 ξs ds + 3/2 2κ M, 6 where M = 2ξ sσξsdv s. Le us noe ha 2ξ bξ + σ 2 ξ is bounded above. In fac, for ξ > C he value 2ξ bξ is negaive see A- and, since σ 2 is bounded, for ξ > C we have 2ξ bξ + σ 2 ξ σ 2 ξ cons. For ξ C he value 2 ξ bξ + σ 2 ξ is bounded as well. For noaion convenience, a posiive consan r is chosen such ha 2ξ bξ + σ 2 ξ r. Then, 6 implies 2 κ ξ 2 2 κ ξ 2 + 2κ rt + 3/2 2κ M. Se so small ha η 2 > 2κ rt + 2 κ ξ 2 and noe ha for any { { 2 κ } ξ 2 > η 2, ξs 2 ds LT M η2 rt 3/2 2κ /2 /2 ξ 2, } ξs 2 ds LT. { } T Denoe by A = ξ s 2 ds LT and apply Lemma. Since on he se A we have M T = 4 ξ s 2 σ 2 ξsds llt, by Lemma he inequaliy holds i.e. 5 is valid. P 2 κ ξ 2 > η 2, ξs 2 ds LT { η 2 rt /2 ξ 2 2 exp 3/2 2κ /2 2lLT Nex, he funcion b obeys he following propery see A-: 2 }, 8
by cy, y > C and cy, y < C which provides he boundedness for he posiive par of he funcion z bydy. Therefore for every fixed posiive ν one can choose a posiive consan c ν such ha he funcion ψc ν, z = c ν ν z bydy is nonnegaive. Since he funcion b is smooh, he funcion ψc ν, z is wice coninuously differeniable in z and he Iô formula, applied o ψc ν, ξ, gives ψc ν, ξ = ψc ν, ξ ν[b 2 ξ s + 2 b ξ sσ 2 ξ s]ds M, where M = νbξ sσξsdv s is he coninuous maringale wih he predicable quadraic variaion M = νbξ sσξs 2 ds. Le us esimae below he value MT 2 M T. Observe ha M T 2 M T = [ψc ν, ξt ψc ν, ξ ] + ν[b 2 ξ s + 2 b ξsσ 2 ξs] 2 νbξ sσξs 2 ds. 7 An appropriae lower bound for he righ side of 7 is consruced as follows. The nonnegaive value ψc ν, ξt is excluded from he righ side of 7. Then, wih C from assumpion A-, we find + [ ν{b 2 ξ s + 2 b ξ sσ 2 ξ s} 2 νbξ sσξ s 2] ds ν{b 2 ξ s + 2 b ξ sσ 2 ξ s} 2 νbξ sσξ s 2 I ξ s Cds [ ν{b 2 ξ s + 2 b ξ sσ 2 ξ s} 2 νbξ sσξ s 2] I ξ s > Cds. 8 Due o assumpion A-, here exiss a posiive consan Hν, C, depending on ν and C, such ha ν{b 2 ξ s + 2 b ξ sσ 2 ξ s} 2 νbξ sσξ s 2 I ξ s C Hν, C and hus, he firs erm in he righ side of 7 is larger han Hν, CT. The second erm in he righ side of 7 is evaluaed below by using A-: for x > C b 2 x + 2 b xσ 2 x /cb 2 x b 2 xσ 2 x lb 2 x b 2 x c 2 x 2. 9
Hence, wih ν = ν = /cl, i holds Therefore, on he se A = [ ν{b 2 ξ s + 2 b ξ sσ 2 ξ s} 2 νbξ sσξ s 2] I ξ s > Cds [ ν b 2 ξs c 2 ] lν I ξ 2 s > Cds b 2 ξ s 2c 2 l I ξ s > Cds ξ s 2 2l I ξ s > Cds { } T I ξ s > C ξs 2 ds > LT we have M T 2 M T ψc ν, ξ Hν, CT + LT 2l. Since Hν, C is independen of L and ξ is fixed, he value L is chosen so large o provide ψc ν, ξ + Hν,CT < LT 2l. Then, wih chosen L, by Lemma we have and 4 follows. P A exp LT 2l + Hν, CT + ψc ν, ξ Remark We emphasize one esimae which is useful for verifying he saemen of Theorem. For small and L large enough and any T 2κ log P κ { η ξ 2 T 2κ 2 T } > η cons. 2 κ 2κ 9 2 3.2.3 Maringale S The process S = vξ sσξsdv s, defined in Lemma 2, is he coninuous maringale wih he predicable quadraic variaion S = v 2 ξ sσ 2 ξ sds. 2 By assumpion A- he random process ξ is ergodic in he following sense see e.g. [4]: for every coninuous and bounded funcion h and fixed P lim hξ sds = R hzpzdz. Hence, wih γ defined in 8, P lim S = γ, >. The proof of Theorem 2 requires a sronger ergodic propery.
Lemma 4 Assume A-. Then for every T > and η > lim 2κ log P S γ η =. Proof: Se gx = v 2 xσ 2 x γ and noe ha S γ = gξ sds := S. 22 Since he funcion g is bounded, coninuously differeniable, and R gzpzdz =, he funcion 2 z vz = σ 2 zpz gypydy compare 7 is coninuously differeniable and bounded as well. Define also he funcion uz = z vydy. The same argumens, which have been applied for he proof of Lemma 2, provide he Poisson decomposiion wih S = /2 S + e 23 e = [uξ s uξ ] S = vξ sσξ sdv s. 24 Wih < we have < κ and so similarly o he proof of 3 we obain lim 2κ log P e > η =. Thus, i suffices o check ha lim 2κ log P S > η =. The process S is he coninuous maringale and is predicable quadraic variaion fulfills S T lt. Then by Lemma P S > η 2 exp η2 2lT and he required asserion follows. 3.3 The MDP We are now ready o complee he proof of Theorem 2. Due o Lemma 3 he families /2 κ S,κ and /2 κ S are exponenially indisinguishable wih he rae of speed 2κ, ha is if one of family obey he MDP wih he rae of speed 2κ, hen he anoher family possesses he same propery. We will examine he MDP for he family of maringales /2 κ S. To his end, we apply he Dawson-Gärner heorem see e.g. [] and [22] which saes ha i suffices o check ha he family /2 κ S obeys he MDP for every T > in he meric space C [,T ], r T r T is he uniform meric on [, T ] wih he rae of speed 2κ and he rae funcion T J T ϕ = 2γ ϕ 2 d, dϕ = ϕ d, ϕ = 25, oherwise.
For fixed T, for he verificaion of he above-menioned MDP we use well known implicaion see e.g. in [7] Theorem.3 formally reformulaed here for he MDP case: Exponenial ighness Local MDP Le us recall he definiions of exponenial ighness and local MDP. } = MDP. 26 Following Deushel and Sroock [6] see also Lynch and Sehuraman [2], he family /2 κ S, is exponenially igh in C [,T ], r T wih he rae of speed 2κ, if here exiss a sequence of compacs K j j : K j C [,T ] such ha lim j lim 2κ log P /2 κ S C [,T ] \ K j = 27 Effecive sufficien condiions for 27 are known from Pukhalskii [22] lim L lim δ lim 2κ log P lim 2κ log P /2 κ /2 κ δ where is aken over all sopping imes τ T. S > L = 28 S τ+ S τ > η =, η >, 29 Following Freidlin and Wenzell [2], he family /2 κ S, obeys he local MDP in C [,T ], r T wih he rae of speed 2κ and he local rae funcion J T, if for every ϕ C [,T ] lim δ lim inf δ lim 2κ log P lim inf 2κ log P /2 κ S ϕ δ J T ϕ 3 /2 κ S ϕ δ J T ϕ. 3 3.3. Verificaion of 28 Inroduce Markov imes wih inf{ } = σ ± L, = inf { > : /2 κ S { > +L < L and noe ha 28 holds, if lim lim 2κ log P σ ± L L, T =. Taking ino accoun he saemen of Lemma 4, is suffices o verify lim L lim 2κ log P σ ± L, } T, S γ η =, η >. 32 2
We consider separaely cases ±. Since S is he coninuous maringale, he posiive process z λ = exp λ /2 κ S λ2 2 2κ S is he local maringale and he ermaringale as well, ha is Ez σ +. Now, wrie L, Ez σ + λi σ + L, L, T, S γ η. Under λ >, he random variable z σ + λ L, is evaluaed below on he se {σ ± L, T, S γ η} as: z σ + λ exp λl L, λ 2 2 2κ S T exp λl λ2 2 2κ L γt + η while he choice of λ = 2κ γt +η implies 2κ log P σ + L, T, S γ η L2 γt + η, L. The proof for he case is similar. 3.3.2 Verificaion of 29 Due o he saemen of Lemma 4, i suffices o show ha, as long as η, lim lim 2κ log P /2 κ S τ+ S τ > η, S γ η. δ δ Le us firs ake τ =. Se σ ± η, = inf { /2 κ δ { > : /2 κ S { > +η < η } and use he obvious inclusion S > η, S γ η } {σ η, ± δ, S γ η }. δ The inequaliy Ez σ ± η, λ implies Ez σ ± η, λi σ η, ± δ, S γ η. δ λ2 λη 2κ The lower bound z σ + η, λ e 2 η +γt δ is valid for any posiive λ on he se {σ η, + δ, S γ η } while he choice of λ = δ 2κ log P σ η, + δ, δ The same upper bound holds wih σ η, and so ha 2κ log P σ η, ± δ, δ η 2κ η +γt δ implies η 2 S γ η 2γη + T δ. S γ η log 2 2γη + T δ. η 2 3
Furhermore, for < τ T, we have he same esimae: 2κ log P /2 κ S τ+ S τ > η, S γ η δ η 2 log 2 2γη + T + δ δ. 33, In fac, one can consider a new maringale S τ+ = S τ+ S τ wih respec o he filraion F τ+ wih S, = S τ+ S τ, and apply he same argumens. The righ side of 33 ends o, as δ, η, and 29 holds. 3.3.3 Verificaion of 3 By virue of Lemma 4, we check only lim η J T ϕ. lim δ lim 2κ log P /2 κ S ϕ δ, S γ η For ϕ he lef side of 34 is equal o. Hence, for ϕ he desired upper bound holds. Le ϕ =. Inroduce he coninuous local maringale M = /2 κ 2κ 34 λsd S s wih M = λ 2 sd S s, where λ is piece wise consan righ coninuous funcion. The process z λ = e M 2 M is he posiive local maringale wih Ez T λ. This inequaliy implies Ez T λi /2 κ S ϕ δ, S γ η. Now, we find a lower bound for he random value z T λ on he se U,δ,η = { /2 κ S ϕ δ, S γ η }. Since λ is righ coninuous having limi o he lef funcion of bounded variaion on [, T ], he Iô formula for λ S is valid: Applying i, we find M T = = λt S T = λsd S s + [ 2κ /2 κ λt S T [ 2κ λt ϕ T ] ϕ s dλs S sdλs. ] S sdλs + 2κ λt {/2 κ S T ϕ T } 4
M T = = 2κ { /2 κ S s ϕ s }dλs 2κ 2κ S s dλ 2 s λ 2 T S T T λ 2 sγds + λ 2 T { 2 S T γt } { S s γs}dλ 2 s. Hence, here is a consan l, depending on T and λ, so ha on he se U,δ,η, ha is on U,δ,η and hereby M T M T [ 2κ λt ϕ T 2κ [ 2 ] ϕ s dλs λ 2 sγds + lη ], he following nonrandom lower bound akes place lδ 2κ log z T λ 2κ λt ϕ T ϕ s dλs λ 2 sγds lδ lη 2 2κ log P λt ϕ T λt ϕ T /2 κ S ϕ δ, S γ η ϕ s dλs 2 ϕ s dλs 2 λ 2 sγds lδ lη λ 2 sγds, δ, η. 35 If ϕ is no absoluely coninuous funcion, one can choose a sequence of piece wise consan and righ coninuous funcions λ n s so ha he righ side of 35 ends o along wih n. If ϕ is absoluely coninuous funcion he righ side of 35 is ransformed ino Uλ, ϕ = λs ϕ s 2 λ2 sγds and hus, J T ϕ = Uλ, ϕ. 3.3.4 Verificaion of 3 We use he obvious lower bound P /2 κ S ϕ δ P λ /2 κ S ϕ δ, S γ η and prove lim inf η J T φ. lim inf lim inf 2κ log P δ /2 κ S ϕ δ, S γ η 36 5
I is clear ha he verificaion of 36 is required only for absoluely coninuous funcions ϕ wih ϕ = and J T ϕ <. Moreover, his class of funcions can be reduced o wice coninuously differeniable funcions ϕ wih ϕ =. In fac, if ϕ = and J T ϕ < bu ϕ is absoluely coninuous only and even ϕ is unbounded, hen one can choose a sequence ϕ n, n of wice coninuously differeniable funcions wih ϕ n lim J T ϕ n = J T ϕ. If for every n we have n lim inf η J T ϕ n, lim inf lim inf 2κ log P δ such ha lim n r T ϕ, ϕ n =, /2 κ S ϕ n δ, S γ η hen, choosing n so ha for n n i holds r T ϕ, ϕ n δ 2, by virue of he riangular inequaliy r T /2 κ S, ϕ rt /2 κ S, ϕ n + r T ϕ, ϕ n we ge P /2 κ S ϕ δ, S γ η P /2 κ S ϕ n δ 2, S γ η. 37 Hence lim inf η lim inf lim inf 2κ log P δ J T ϕ n J T ϕ, n. /2 κ S ϕ δ, S γ η Thus, le ϕ be wice coninuously differeniable funcion wih ϕ =. Se λ = ϕ he maringale M = /2 κ M T = 2κ λ sd S s wih ϕ 2 γ 2 d S = 2κ ϕ 2 γ 2 v2 ξ σ 2 ξ d cons. γ and define and so, z λ = e M 2 M is he maringale, Ez T λ =. We use his equaliy o inroduce new probabiliy measure P : dp = z T λ dp. Since z T λ >, P -a.s., no only P P bu also P P wih dp = z T λ dp. Wrie P /2 κ S ϕ δ, S γ η = { /2 κ S ϕ δ} { S γ η } z T λ dp. On he se { /2 κ S ϕ δ} { S γ η } he random variable z T λ possesses nonrandom lower bound l is a generic consan: z T λ exp { 2κ J T ϕ + lδ + lη }. 6
Therefore 2κ log P /2 κ S ϕ δ, S γ η J T ϕ lδ + η + 2κ log P /2 κ S ϕ δ, S γ η. To finish he proof, i remains o check ha for every δ >, η > lim P /2 κ S ϕ > δ =. 38 lim P S γ > η = 39 I is well known see e.g. Theorem 4.5.2 in [9] ha he random process S, being P - coninuous maringale, is ransformed o P -coninuous semimaringale wih he decomposiion S = A + N, where N is he coninuous local maringale having S as he quadraic variaion and he drif A is defined via he muual variaion zλ, S of maringales zλ and S as: Hence A = /2 κ /2 κ S ϕ = A = ϕ s γ d S s and we find ha = γ z s λ d zλ, S s. ϕ s γ d S s ϕ + /2 κ N ϕ s d S s γs + /2 κ N = γ ϕ T S γ Therefore, wih a suiable consan l, we obain /2 κ S ϕ l ϕ s S s γsds + /2 κ N. S γ + /2 κ N. As was menioned above N T coincides wih S T P and P -a.s. and so, i is bounded P -a.s. Now, by he Doob inequaliy E is he expecaion wih respec o P P /2 κ N > η 2κ η 2 E N T,, η >. The laer propery allows o conclude ha 38 holds provided ha 39 is valid. 7
Thus, only 39 remains o prove. Le us recall ha S γ obeys he Poisson decomposiion see 23 and 24 S γ = /2 S + e 4 wih e = [uξ s uξ ] and S = vξ sσξ sdv s. The maringale S obeys P - decomposiion S = A + N wih A = z s λ d zλ, S and coninuous local maringale N wih N S P - and P -a.s.. As was menioned in Subsecion 3.2.3, S T is bounded. Then, by he Doob inequaliy P /2 N > η η 2 E N T,, η >. 4 Show now ha lim P /2 A > η =, η >. 42 To his end, we find an upper bound for A. Since dz λ = z λ λ d S, we ge A = ϕ s γ d S, S s, where he muual variaion S, S of P -maringales S and S is given, due o 2 and 24, by he formula S, S = vξ svξ sσ 2 ξ sds. The boundedness of v, v, and σ 2 implies A cons. and so 42 holds. To finish he proof of 39, i remains o check lim P e > η =. 43 From he definiion of e i follows he exisence of posiive consans L, L 2 so ha e L + L 2 ξ. Therefore, we prove below lim P ξ > η =, η >. 44 The verificaion of 44 uses he semimaringale decomposiion of ξ wih respec o P. Se Φ = σξsdv s. The P -maringale Φ is he semimaringale wih respec o P wih he 8
decomposiion: Φ = L + M, where M is coninuous local maringale wih M = Φ P - and P -a.s. and L = where V, Hence is P -Wiener process. P : By he Iô formula we find z s λ d zλ, U s. Consequenly, L = κ M = ϕ s γ vξ sσ 2 ξ sds σξ sdv, s, dξ = bξ + +κ ϕ γ vξ σ 2 ξ d + σξ dv, ξ is deerminisic. 2 ξ 2 = 2 ξ 2 + 2 +2 3/2 ξ sbξ s + +κ ϕ γ vξ σ 2 ξ + 2 σ2 ξ ds ξ sσξ sdv, s. Due o A-, ξsbξ s + +κ ϕ γ vξ σ 2 ξ + 2 σ2 ξ ds is bounded by a posiive consan independen of. The Iô inegral τ n n be is localizing sequence. Then ξ sσξ sdv, s E 2 ξ τ n 2 2 ξ 2 + cons., n is coninuous local P -maringale. Le and by he Faou lemma E 2 ξ 2 2 ξ 2 + cons. Consequenly, wih σ 2 l, 9/4 E ξs 2 σ 2 ξsds lt /4 ξ 2 + 5/4 cons.. Then, by he Doob inequaliy P 3/2 ξsσξ sdv s > η 4lT /4 ξ 2 + 5/4 cons. η 2,. Consequenly 44 holds. 3.4 Proof of Theorem As for Ψ, he proof uses he Poisson decomposiion from Lemma 2: κ gξ s ds = e,κ + /2 κ S, which for U = κ ΨX sgξsds implies Poisson ype decomposiion U = ΨX s de,κ s + /2 κ ΨXs d S s, 45 9
where he firs erm in his decomposiion is he Iô inegral wih respec o he semimaringale e,κ. Since he funcion Ψ is wice coninuously differeniable, we decompose also ΨX s de,κ s applying he Iô formula o ΨX e,κ : ΨX sde,κ s = ΨX e,κ e,κ s [ Ψ XsF Xs, ξs + 2 Ψ XsG 2 Xs, ξs] ds e,κ s := U + U 2 + U 3. Ψ X sgx s, ξ s dw s Now, wih U 4 = /2 κ ΨX s d S s, we arrive a he final decomposiion U = U + U 2 + U 3 +U 4 and show ha U 4 delivers he main conribuion in he required esimae announced in Theorem : lim T 2κ log P U 4 while he ohers U i, i =, 2, 3 are exponenially negligible: Ψ T κ z z2, z > 46 2γ lim T 2κ log P U i > δψ T κ =, δ >, i =, 2, 3. 47 Lemma 5 Under he assumpions of Theorem, 47 holds. Proof: i=: Since by A-4 T d >, i suffices o verify only ha lim T 2κ log P > δ =, δ >. U Recall ha e,κ l κ + ξ see Lemma 2. Then, by virue of he upper bound U Ψ e,κ, i suffices o check only ha lim T 2κ log P κ ξ > δ =. By A-4 lim T 2κ =, so ha applying 2 we find, wih a suiable consan r, as. T 2κ log P κ ξ > δ rd 2κ{ δ 2 d 2κ 2 d } 2 κ 2κ 2
i=2: Wrie U 2 e,κ Ψ XsF Xs, ξs + 2 Ψ XsG 2 Xs, ξs ds. Due o A-2 and A-4 here exiss a posiive consan l so ha U 2 he proof is compleed as for i=. i=3: The U 3 variaion U 3 = U 3 T l l e,κ and process is he coninuous local maringale wih he predicable quadraic 2ds. e,κ s Ψ XsGX s, ξs By virue of assumpions A-2 and A-4, e,κ 2. The same argumens, as were used for he proof for i=, yield T 2κ log P U 3 /2 T > δ cons.d 2κ{ δ 2 d 2κ 2 d } 2 κ 2κ. Taking now δ = 2 κ, we arrive a he upper bound: wih a suiable consan l T 2κ log P U 3 /2 T > κ 2 ld 2κ{ κ d 2κ 2 d } 2 κ 2κ which ends o as. Hence, i remains o prove only ha for every δ > T 2κ log P U 3 > δ, U 3 /2 T κ 2,. To his end, we apply Lemma : and so ha as. P U 3 > δ, U 3 /2 T κ 2 2 exp δ2 2 κ { T 2κ log 2 exp δ2 } 2 κ T 2κ 2κ δ2 log 2 d 2 κ, The proof of 46 uses he following auxiliary saemen which slighly exends he resul of Lemma 4. Lemma 6 Under assumpions of Theorem lim T 2κ log P S γ > δ =, δ >. Proof: We apply he Poisson decomposiion S γ = /2 S +e used in he proof of Lemma 4, and ake ino accoun ha e l+ ξ and S is he coninuous maringale wih S T lt. Since κ > for small, he argumens, used in he proof of he saemen i= 2
from Lemma 5, provide limt 2κ log P ξ > δ =. Nex, since S T lt, P -a.s. by Lemma we have P /2 S > δ 2 exp δ2 2lT, so ha T 2κ log P /2 S > δ δ 2 2lT 2κ, 2ld 2κ δ2 and he required asserion holds. To complee he proof of Theorem, i remains o check he validiy of 46. The U 4 is he coninuous maringale wih he predicable quadraic variaion process which is evaluaed above as: U 4 = 2κ ΨXs 2 d S s U 4 T 2κ Ψ 2 S T 2κ Ψ 2 γt + 2κ Ψ 2 S γ. Le us noe U 4 T Lemma 2κ Ψ 2 γt + δ on he se A = { S γ δ}. Then, by P U 4 Ψ T κ z, A exp Therefore T 2κ log P U 4 Ψ T κ z, A P U 4 { Ψ T κ z 2 P U 4 z 2 T 2κ 2 2κ [γt, z >. 48 + δ] z2 2[γ+δ/T ]. Since also } Ψ T κ z, A P Ω \ A and by Lemma 6 lim T 2κ log P Ω \ A =, i remains o recall only ha T d see A-4, so ha z2 2[γ+δ/T ] z2 2[γ+δ/d] z2 2γ, δ. 4 Appendix. Vecor case I his Secion, we formulae wihou proof he resul in MDP evaluaion for he vecor diffusion process ξ wih respec o vecor Wiener process V wih he uni diffusion marix boh ξ and V are valued in R d : dξ = bξ d + σξ dv The assumpions, under which he MDP for he family S,κ = gξ s ds holds are more resricive. All elemens of vecor- and marix- funcions b and σ are Lipschiz coninuous funcions. We use wo condiions from Pardoux and Vereennikov, [2]. 22 κ
A σ : a = σσ is nonsingular marix and here exis posiive consans λ, λ + such ha for any x R d \ {} x = x x < λ σσ x x x, x λ + x A b : recurrence condiion here exis posiive consans C, r and α > such ha for x > C bx, x r x α. x I is shown in [2] ha under A σ and A b ξ is an ergodic process wih he unique invarian measure, µ. We assume he funcion g is coninuous, bounded, and R d gxµdx =. As in he scalar case, we exploi he Poisson decomposiion applying he resul from [2] on he Poisson equaion Lu = g, where L = 2 aij x xi xj + b i x xi is he diffusion operaor. I is shown in [2] ha he Poisson equaion obeys a bounded soluion he gradien of which u = ux x,..., u iz x d has bounded componens. Inroduce γ = uxσxσ x xµdx 49 R d and noe ha since he marix σσ is nonsingular γ > for any u wih u. Theorem 3 Under he seing of his Secion and /2 < κ <, he family S,κ, obeys he MDP in he meric space C, r wih he rae of speed 2κ and he rae funcion 9 wih γ from 49. References [] Bayer, U., Freidlin, M.I. 977 Theorems on large deviaions and sabiliy under random perurbaions DAN USSR. 235, N 2, pp. 253 256. [2] Borovkov, A.A., Mogulski, A.A. 978 Probabiliies of large deviaions in opological vecor space I., Siberian Mah. J. 9, pp. 697 79. [3] Borovkov, A.A., Mogulski, A.A. 98 Probabiliies of large deviaions in opological vecor space II.. Siberian Mah. J. 9, pp. 2 26. 23
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