Large Deviations for Perturbed Reflected Diffusion Processes

Transcription

1 Large Deviations for Perturbed Reflected Diffusion Processes Lijun Bo & Tusheng Zhang First version: 31 January 28 Research Report No. 4, 28, Probability and Statistics Group School of Mathematics, The University of Manchester

2 Large deviations for perturbed reflected diffusion processes Lijun Bo 1,2 and Tusheng Zhang 3, 1. Department of Mathematics, Xidian University, Xi an 7171, China 2. School of Mathematical Sciences, Nankai University, Tianjin 371, China 3. Department of Mathematics, University of Manchester Oxford Road, Manchester M13 9PL, England, U.K. January 31, 28 Abstract In this article, we establish a large deviation principle for the solutions of perturbed reflected diffusion processes. The key is to prove a uniform Freidlin-Ventzell estimates of perturbed diffusion processes. Keywords: Large deviations; perturbed diffusion processes; uniform Freidlin-Ventzell estimates. AMS Subject Classification: 6F1, 6J27 1 Introduction In [6], Doney and Zhang obtained the existence and uniqueness of the solutions for the following perturbed diffusion and perturbed reflected diffusion equations: X t = x + σx s db s + bx s ds + α X s, t [, 1], 1.1 st and Y t = y + ay s db s + α Y s + L t, t [, 1], 1.2 st Corresponding author. address: tzhang@maths.man.ac.uk 1

3 where α, 1, x R, y R + are deterministic, b, σ : R R and a : R + R are bounded Lipschitz continuous function, L t, t [, 1]} is non-decreasing with L = and χ Ys =}dl s = L t, t [, 1]. We may think of L t, t [, 1]} the local time of the semimartingale Y t, t [, 1]} at point zero. B t, t [, 1]} is a 1-dimensional standard Brownian motion on a completed probability space Ω, F, F t t, P. The perturbed reflected Brownian motion was first introduced by Le Gall and M. Yor, in [8, 9], and subsequently studied by Carmona et al. in [2, 3], Perman and Werner in [12] and Chaumont and Doney in [4]. Consider the small noise perturbations of 1.1 and 1.2: and Xt = x + σxsdb s + bxsds + α Xs, t [, 1], 1.3 st Y t = y + ays db s + α Ys + L t, t [, 1]. 1.4 st The aim of this paper is to establish a large deviation principle LDP for the laws of X and Y on the space of continuous functions equipped with uniform topology, respectively. Our approach will be based on a classical result of Azencott [1], which can be stated as follows: Proposition 1.1 Let E i, d i i = 1, 2 be two Polish spaces and Φ i : Ω E i, > i = 1, 2 be two families of random variables. Assume that 1 Φ 1, > } satisfies a large deviation principle with the rate function I 1 : E 1 [, ]. 2 There exists a map K : I 1 < } E 2 such that for every a <, K : I 1 a} E 2 is continuous. 3 For any R, δ, a >, there exist ρ > and > such that, for any h E 1 satisfying I 1 h a and, P d 2 Φ 2, Kh δ, d 1 Φ 1, h ρ exp R. 1.5 Then Φ 2, > } satisfies a large deviation principle with the rate function Ig = infi 1 h; Kh = g}. 2

4 The estimate 1.5 is also known as the uniform Freidlin-Ventzell estimates. We would like to mention that some works in the literature motivated our present study. In [7], Doss and Priouret considered the LDP for the small noise perturbations of reflected diffusions, through checking uniform Freidlin-Ventzell estimates. Further, Millet et al. used the Freidlin-Ventzell estimates to obtain the LDP of a class of anticipating stochastic differential equations in [1]. Recently, in [11], Mohammed and Zhang studied a LDP for small noise perturbed family of stochastic systems with memory. In this article, the appearance of the terms α st Xs and α st Ys make 1.3 and 1.4 different from the equations studied in the literature. To deduce the uniform Freidlin-Ventzell estimates for 1.3 and 1.4, the estimates concerning the perturbed terms α st Xs and α st Ys are the key in the paper. The rest of the paper is organized as follows. Section 2 is for the large deviation principle of perturbed diffusion processes. The large deviation estimates for the perturbed reflected diffusion processes is in Section 3. 2 LDP for perturbed diffusion processes In this section, we will give a large deviation principle of the perturbed diffusion processes 1.3. Let H denote the Cameron-Martin space, i.e. H = ht = ḣsds : [, 1] R; 1 } ḣs 2 ds < +, and ν, > } be the probability measure induced by X on C x [, 1], R, the space of all continuous functions f : [, 1] R such that f = x, equipped with the remum norm topology. For h C [, 1], R, define Ĩ : C [, 1], R [, ] by 1 1 Ĩh = 2 ḣs 2 ds, if h H, +, otherwise. The well known Schilder theorem states that the laws µ of B t, t [, 1]} satisfies a large deviation principle on C [, 1], R with the rate function Ĩ, that is, a for any closed subset C C [, 1], R, lim log µ C inf Ĩg, g C 3

5 b for any open subset G C [, 1], R, lim inf log µ G inf Ĩg. g G Let F h be the unique solution of the following deterministic perturbed equation: F ht = x + σf hsḣsds + bf hsds +α st F hs, h H, t [, 1]. 2.1 It is not difficult to obtain the existence and uniqueness of the solution to 2.1 by the same arguments of Theorem 2.1 in [6]. Then we have the following main result: Theorem 2.1 If α, 1, then the family ν, > } obeys a large deviation principle with the rate function: I x g = inf Ĩh, h H: F h=g} for g C x [, 1], R, where the inf over the empty set is taken to be. Before giving the proof of Theorem 2.1, recall the following lemma see e.g. Lemma 2.19 in Stroock [13]. Lemma 2.1 Let Z t := s CudB u + s Dudu be an Itô process, where s < t < and C, D : [, Ω R are F t t progressively measurable random processes. If C M 1 and D M 2, then for T > s and R > satisfying R > M 2 T s, we have P Z t R exp R M 2T s 2 stt 2M T s Proof of Theorem 2.1. For h H, let F h be defined as in 2.1. First we prove that for any R, δ, a >, there exist ρ > and > such that, for any h C [, 1], R satisfying Ĩh a and, P exp t1 R X t F ht δ, t1 B t ht ρ

6 By 1.3 and 2.1, X t F ht = + + +α σxs dbs ḣsds σx s σf hs ḣsds bx s bf hs ds Xs F hs st st. 2.4 Consequently, X t F ht σxs dbs ḣsds +L Xs F hs 1 + ḣs ds +α Xs F hs, 2.5 st where L > is the Lipschitz coefficient, and we also used the fact that us vs st st us vs, st for any two continuous functions u and v on R +. Thus it follows from 2.5 that, for t [, 1], Xu 1 F hu ut 1 α u σx s dbs ḣsds ut + L 1 α us Xu F hu 1 + ḣs ds. 2.6 By the Gronwall lemma this yields that Xt 1 F ht t1 1 α t σx s dbs ḣsds t1 1 L exp 1 α 1 + ḣs ds C 1 σxs dbs ḣsds, 2.7 t1 5

7 where C 1 := 1 1 α exp L1+ h H 1 α < with h H := 1 ḣs 2 ds 1 2 for h H. Thus to prove 2.3, it suffices to prove that, for any R, δ, a >, there exist ρ > and > such that, for any h C [, 1], R satisfying Ĩh a and, P exp t1 R σxs dbs ḣsds δ, B t ht ρ t For >, define a probability measure P on Ω by 1 1 dp = Z dp := exp ḣsdb s ḣs 2 ds dp. Then by Girsanov theorem, Bt := B t 1 ḣt, t [, 1]} is a standard Brownian motion on Ω, F, P. If we let A t = σx s dbs ḣsds δ, } B t ht ρ t1 t1 = σxs db s δ, } Bt ρ, t1 t1 then by the Hölder inequality, PA = Z 1 χ A ωp dω Ω Ω 1 Z 2 ωp 2 dω P A 1 2. Note that Bt, t [, 1]} is a standard Brownian motion under P, then [ Z 2 ωp dω = E P exp 2 1 ḣsdb s ] Ω ḣs 2 ds [ = E P exp 2 1 ḣsdb s 1 1 ] ḣs 2 ds [ = E P exp 2 1 ḣsdb s 2 1 ] ḣs 2 ds 1 1 exp ḣs 2 ds 1 = exp h 2 H. 6

8 Therefore if Ĩh a, then P A exp a P A Note that under the probability P, X satisfies the following stochastic differential equation: U t = x + Therefore, P A = P = P σu s db s + t1 t1 bu s + σu s σxs dbs ḣsds + α Us.2.1 st δ, Bt ρ t1 σus db s δ, B t ρ t1 Thus, in view of 2.9, to prove 2.8, the proof of 2.8 is reduced to show that, for any R, δ, a >, there exist ρ > and > such that, for any h C [, 1], R satisfying Ĩh a and, P t1 σus db s δ, B t ρ t1 For n N fixed, set = k n Then for δ 1 >, A t1 for k, 1, 2,, n}. Define U,n t := U, if t < +1, k, 1,, n 1}. t1 t1 σus σus,n db s δ 2, } Ut U,n t > δ 1 := B C D. On the set t1 Ut U,n t δ 1 }, exp R t1 σus,n db s δ 2, B t ρ t1 σus σus,n 2 L 2 δ1. 2 s1 7 } Ut U,n t δ 1 }.

9 By Lemma 2.1, it follows that if δ 1 PB exp δ 2L 2R. On the other hand, on the set δ2 8L 2 δ1 2 exp R, 2.12 t1 } B t ρ, for any t [, 1], σus,n db s = n 1 σu t j B tj+1 t B tj t j= 2 B t σut j t1 j j,,n 1}:t j t} 2nM ρ, where M > is a common bound of b and σ. Therefore if ρ < D =. δ 4nM, then To treat C, we note that for t [, +1, k, 1,, n 1}, Ut U,n t t σus db s + bu s + σus ḣs ds t k +α st U s s U s Consider two possible cases: The case I: st Us = stk Us. We have Ut U,n t σus db s + bu s + σus ḣs ds The case II: st U s > stk U s. Then there exists u, t] such that U u = st U s, which yields that U t U σus db s + t k bu s + σus ḣs ds +α Uu Us s σus db s + bu s + σus ḣs ds 8

10 +α U u Ut k σus db s + bu s + σus ḣs ds +α Ut Ut k. t<+1 This implies Ut U,n t<+1 = Ut Ut k t<+1 t<+1 C 2 t σu s db s + bu s + σus ḣs ds, 2.15 where C 2 := 1 1 α. Since tk s<+1 σu s 2 M 2, tk s<+1 bu s M and t<+1 σus ḣsds 2a M n, it follows from 2.15 and Lemma 2.1 that for n n 1 := [ 8C 2 ] 2 M 2 a + 1, δ 2 1 Therefore, [ 2MC2 δ 1 ] + 1 n 2 := P Ut U,n t > δ 1 t<+1 P σus db s + bu s ds > δ 1 t<+1 2C 2 +P σus ḣsds > δ 1 t<+1 2C 2 exp P C = P δ 2 1 n 8C2 2M 2 max k,1,,n 1} Ut U,n t > δ 1 t<+1

11 n 1 k= n exp P Ut U,n t > δ 1 t<+1 δ 2 1 n 8C2 2M Now given R >, choose first δ 1 > such that 2.12 holds. Then choose n large enough so that δ 2 1 n exp n 8C2 2M 2 exp R. Finally for the fixed n N, there exists ρ > such that D =. Combining above inequalities proves Next, we prove the map F : H C x [, 1], R is continuous on h : Ĩh a} for a [,. Let h n, h h C [, 1], R : Ĩh a} with h n h in C [, 1], R. Since h : Ĩh a} is weakly compact in H, we conclude that h n also weakly converges to h in H. By the Lipschitz continuity, F h n s F hs st ξ n t + α F h n s F hs +L st us F h n u F hu 1 + ḣns ds, where ξ n t = Applying the Gronwall s lemma, s σf huḣnu ḣudu st F h n t F ht t1 2M 1 α ξ L 1 + 2Ĩh n n1 exp 1 α 2M 1 α exp L 1 + 2a ξ n 1. 1 α 1

12 Since h n h weakly in H, for every s [, 1], s σf huḣnu ḣudu, as n. On the other hand, it is easy to see that there exists a constant C a > such that t σf huḣnu ḣudu C a t s 1 2. s Therefore, we conclude that ξ n 1, as n. Consequently, F h n t F ht, as n. t1 which proves the continuity of the mapping F. Now, letting Φ 1 = B and K = F, Theorem 2.1 follows from Proposition LDP for perturbed reflected diffusion processes In this section, we will prove the large deviation principle for the solution of the perturbed reflected diffusion equation 1.4. For y and f C y [, 1], R, define two operators Γ : C y [, 1], R C y [, 1], R + and K : C y [, 1], R C [, 1], R + by Γf = f + f and Kf = f, where ft := inf fs, t [, 1]. st By the reflection principle, the solution Y of 1.4 is given by Y t = ΓZ t and L t = KZ t, t [, 1], 3.1 where Z is a solution of the following stochastic equation: Zt = y + σγz sdb s + α ΓZ s, t [, 1]. st For h H, let Gh be the unique solution of the following equation: Ght = y + σghsḣsds + α Ghs + η t, t [, 1], 3.2 st where Gh is continuous, nonnegative, and η is an increasing continuous function satisfying η t = χ Ghs=}dη s. The existence and uniqueness of 11

13 the solution to 3.2 might be obtained by Theorem 3.1 and Theorems 4.2, 4.3 in [6]. Similar as 3.1, Gh can also be written as Ght = ΓV ht and η t = KV ht, t [, 1], 3.3 where V h satisfies V ht = y + σγv hsḣsds + α ΓV hs, t [, 1]. st Let ν 1 be the law of Y on C y [, 1], R +. Our main result in this section is Theorem 3.1 If α, 1 2, then the family ν1, > } obeys a large deviation principle with the rate function: I y g = inf Ĩh, for g C y [, 1], R +, h H: Gh=g} where the inf over the empty set is taken to be. By Proposition 1.1, Theorem 3.1 is the consequence of the following two propositions. Proposition 3.1 If α, 1 2, then for any R, δ, a >, there exist ρ > and > such that, for any h C [, 1], R satisfying Ĩh a and, P Yt Ght + L t η t δ, B t ht ρ t1 t1 t1 exp R. Proof. The proof is similar to that of Theorem 2.1. We only highlight the differences. Note that for f 1, f 2 C y [, 1], R and t [, 1], Γf 1 s Γf 2 s 2 f 1 s f 2 s. 3.4 st st Then by 3.1 and 3.3, Yt Ght + L t η t 3 Zt V ht. t1 t1 t1 12

14 Therefore, the proof of Proposition 3.1 is reduced to show that for any R, δ, a >, there exist ρ > and > such that, for any h C [, 1], R satisfying Ĩh a and, P Z t V ht δ, B t ht ρ t1 t1 Using 3.4 and the Gronwall s lemma, t1 Zt V ht C 3 t1 2L exp 1 2α h H σγz s dbs, exp R. 3.5 where C 3 := 1 2α. Thus as in Section 2, by the Girsanov s theorem, to prove 3.5, it suffices to prove P t1 σγ Z s db s δ, B t ρ exp R, t1 where Z satisfies Z t = y + σγ Z t sdb s + σγ Z sḣsds +α Γ Z s, t [, 1]. st For a function f, put Γf n t = Γf, if t < +1, k, 1,, n 1}. Then, Ã := t1 t1 t1 t1 t1 := B C D, σγ Z s db s δ, } B t ρ t1 σγ Z s σγ Z n s db s δ 2, Γ Z t Γ Z n t δ 1 } Γ Z t Γ Z n t > δ 1 } σγ Z n s db s δ 2, B t ρ t1 13 }

15 Let us just sketch the proof of P C exp R. 3.6 For t [, +1, it is easy to see that But, Γ Z s Γ Z n s st = Γ Z s Γ Z st 2 Z s Z. 3.7 st Z t Z t k σγ Z s db s + σγ Z sḣsds +α Γ Z s Γ Z s st s σγ Z s db s + σγ Z sḣsds +α Γ Z s Γ Z st σγ Z s db s + σγ Z sḣsds +2α Z s Z. 3.8 st Therefore, Z t Z t<+1 1 t 1 2α t<+1 σγ Z s db s + σγ Z sḣsds. The rest of the proof is the same as the proof of 2.17 of Theorem 2.1. We omit it. Proposition 3.2 If α, 1 2, then G : H C y[, 1], R + and η : H C [, 1], R + defined by 3.3 are continuous on the compact set h C [, 1], R : Ĩh a} for any a [,. 14

16 Proof. By 3.4, it suffices to prove that the conclusion holds for V : H C y [, 1], R. Let h n, h h C [, 1], R : Ĩh a} with h n h in C [, 1], R. Then we have V h n t V ht st 2L 1 2α us V h n s V hs ḣns ds α ξ nt, where ξ n t := s σγv hsḣns ḣsds. st The rest of the proof is similar to the last part of the proof of Theorem 2.1. Acknowledgement. discussions. We thank Professor Yongjin Wang for stimulating References [1] R. Azencott 198: Grandes déviations et applications. Ecole d Eté de Probabilitiés de Saint-Flour VIII, Lecture Notes in Math Springer, New York. [2] P. Carmona, F. Petit and M. Yor 1994: Some extentions of the arcsine law as partial consequences of the scaling property of Brownian motion. Probab. Theory Relat. Fields [3] P. Carmona, F. Petit and M. Yor 1998: Beta variables as times spent in [, by certain perturbed Brownian motions. J. London Math. Soc [4] L. Chaumont and R. Doney 1999 Pathwise uniqueness for perturbed versions of Brownian motion and reflected Brownian motion. Probab. Th. Relat. Fields [5] R. Doney 1998: Some calculations for perturbed Brownian motion. Sém. Probab. XXXII, Lecture Notes Math Springer, New York. 15

17 [6] R. Doney and T. Zhang 25: Perturbed Skorohod equations and perturbed reflected diffusion processes. Ann. I. H. Poincaré-PR [7] H. Doss and P. Priouret 1983: Petites perturbations de systems dynamiques avec reflexion. Lecture Notes in Math. No. 986, Springer, New York, [8] J.F. Le Gall and M. Yor 1986: Excursions browniennes et carrés de processus de Bessel. C. R. Acad. Sci. Paris Sér. I [9] J.F. Le Gall and M. Yor 199: Enlacements du mouvement brownien autour des courbes de l espace. Trans. Amer. Math. Soc [1] A. Millet, D. Nualart and M. Sanz 1992: Large deviations for a class of anticipating stochastic differential equations. Ann. Probab [11] A. Mohammed and T. Zhang 26: Large deviations for stochastic systems with memory. Disc. and Contin. Dyn. Sys.-Series B [12] M. Perman and W. Werner 1997: Perturbed Brownian motions. Probab. Th. Relat. Fields [13] W. Stroock 1983: Some applications of stochastic calculus to partial differential equations. Ecole d Eté de Probabilités de Saint-Flour XI, Lecture Notes in Math. 976 Springer, New York. 16