Large Deviations for Perturbed Reflected Diffusion Processes Lijun Bo & Tusheng Zhang First version: 31 January 2008
Research Report No. 4, 2008, Probability and Statistics Group School of Mathematics, The University of Manchester
Large deviations for perturbed reflected diffusion processes Lijun Bo1,2 and Tusheng Zhang3,∗ 1. Department of Mathematics, Xidian University, Xi’an 710071, China 2. School of Mathematical Sciences, Nankai University, Tianjin 300071, China 3. Department of Mathematics, University of Manchester Oxford Road, Manchester M13 9PL, England, U.K.
January 31, 2008
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: 60F10, 60J27
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: Xt = x +
Z t 0
σ(Xs )dBs +
and Yt = y + ∗
Z t 0
Z t 0
b(Xs )ds + α sup Xs , t ∈ [0, 1],
(1.1)
0≤s≤t
a(Ys )dBs + α sup Ys + Lt , t ∈ [0, 1], 0≤s≤t
Corresponding author. E-mail address:
[email protected]
1
(1.2)
where α ∈ (0, 1), x ∈ R, y ∈ R+ are deterministic, b, σ : R → R and a : R+ → R are bounded Lipschitz continuous function, {Lt , t ∈ [0, 1]} is non-decreasing with L0 = 0 and Z t 0
χ{Ys =0} dLs = Lt , t ∈ [0, 1].
We may think of {Lt , t ∈ [0, 1]} the local time of the semimartingale {Yt , t ∈ [0, 1]} at point zero. {Bt , t ∈ [0, 1]} is a 1-dimensional standard Brownian motion on a completed probability space (Ω, F, (Ft )t≥0 , 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): Xtε
Z t √ Z t ε =x+ ε σ(Xs )dBs + b(Xsε )ds + α sup Xsε , t ∈ [0, 1], 0
0
(1.3)
0≤s≤t
and Ytε = y +
√ Z t ε a(Ysε )dBs + α sup Ysε + Lεt , t ∈ [0, 1]. 0
(1.4)
0≤s≤t
The aim of this paper is to establish a large deviation principle (LDP) for the laws of X·ε and Y·ε 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 (Ei , di ) (i = 1, 2) be two Polish spaces and Φεi : Ω → Ei , ε > 0 (i = 1, 2) be two families of random variables. Assume that (1) {Φε1 , ε > 0} satisfies a large deviation principle with the rate function I1 : E1 → [0, ∞]. (2) There exists a map K : {I1 < ∞} → E2 such that for every a < ∞, K : {I1 ≤ a} → E2 is continuous. (3) For any R, δ, a > 0, there exist ρ > 0 and ε0 > 0 such that, for any h ∈ E1 satisfying I1 (h) ≤ a and ε ≤ ε0 , µ
P (d2 (Φε2 , K(h))
≥
δ, d1 (Φε1 , h)
¶
R ≤ ρ) ≤ exp − . ε
(1.5)
Then {Φε2 , ε > 0} satisfies a large deviation principle with the rate function I(g) = inf{I1 (h); K(h) = g}. 2
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 [10]. 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 sup0≤s≤t Xsε and α sup0≤s≤t Ysε 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 sup0≤s≤t Xsε and α sup0≤s≤t Ysε 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. ½
H = h(t) =
Z t 0
˙ h(s)ds : [0, 1] → R;
Z 1 0
¾ 2 ˙ |h(s)| ds < +∞ ,
and {νε , ε > 0} be the probability measure induced by X·ε on Cx ([0, 1], R), the space of all continuous functions f : [0, 1] → R such that f (0) = x, equipped with the supremum norm topology. For h ∈ C0 ([0, 1], R), define Ie : C0 ([0, 1], R) → [0, ∞] by ( e I(h) =
1 R1 ˙ 2 2 0 |h(s)| ds,
+∞,
if h ∈ H, otherwise.
√ The well known Schilder theorem states that the laws µε of { εBt , t ∈ [0, 1]} satisfies a large deviation principle on C0 ([0, 1], R) with the rate function e I(·), that is, (a) for any closed subset C ⊂ C0 ([0, 1], R), e lim sup ε log µε (C) ≤ − inf I(g), g∈C
ε→0
3
(b) for any open subset G ⊂ C0 ([0, 1], R), e lim inf ε log µε (G) ≥ − inf I(g). ε→0
g∈G
Let F (h) be the unique solution of the following deterministic perturbed equation: Z t
F (h)(t) = x +
0
˙ σ(F (h)(s))h(s)ds +
Z t 0
b(F (h)(s))ds
+α sup F (h)(s), h ∈ H, t ∈ [0, 1].
(2.1)
0≤s≤t
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 ∈ (0, 1), then the family {νε , ε > 0} obeys a large deviation principle with the rate function: Ix (g) =
inf
{h∈H: F (h)=g}
e I(h),
for g ∈ Cx ([0, 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]). R
R
Lemma 2.1 Let Zt := st C(u)dBu + st D(u)du be an Itˆ o process, where 0 ≤ s < t < ∞ and C, D : [0, ∞) × Ω → R are (Ft )t≥0 progressively measurable random processes. If |C(·)| ≤ M1 and |D(·)| ≤ M2 , then for T > s and R > 0 satisfying R > M2 (T − s), we have Ã
P
!
sup |Zt | ≥ R
s≤t≤T
Ã
(R − M2 (T − s))2 ≤ exp − 2M12 (T − s)
!
.
(2.2)
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 > 0, there exist ρ > 0 and ε0 > 0 such that, for e any h ∈ C0 ([0, 1], R) satisfying I(h) ≤ a and ε ≤ ε0 , Ã
P
!
√ sup |Xtε − F (h)(t)| ≥ δ, sup | εBt − h(t)| ≤ ρ
0≤t≤1
µ
≤ exp −
0≤t≤1
¶
R . ε
(2.3) 4
By (1.3) and (2.1), Xtε
− F (h)(t) =
Z t 0
+ +
σ(Xsε )
Z t
´
˙ εdBs − h(s)ds
˙ (σ(Xsε ) − σ(F (h)(s))) h(s)ds
0
Z t 0
³√
(b(Xsε ) − b(F (h)(s))) ds
Ã
+α
!
sup 0≤s≤t
Xsε
− sup F (h)(s) .
(2.4)
0≤s≤t
Consequently, |Xtε
− F (h)(t)|
¯Z t ³√ ´¯¯ ¯ ε ˙ ¯ ≤ ¯ σ(Xs ) εdBs − h(s)ds ¯¯ 0 Z t ³
+L
0
´
˙ |Xsε − F (h)(s)| 1 + |h(s)| ds
+α sup |Xsε − F (h)(s)|,
(2.5)
0≤s≤t
where L > 0 is the Lipschitz coefficient, and we also used the fact that ¯ ¯ ¯ ¯ ¯ ¯ ¯ sup u(s) − sup v(s)¯ ≤ sup |u(s) − v(s)|, ¯0≤s≤t ¯ 0≤s≤t 0≤s≤t
for any two continuous functions u and v on R+ . Thus it follows from (2.5) that, for t ∈ [0, 1], ¯
sup |Xuε − F (h)(u)| ≤
0≤u≤t
¯
Z ´¯ ³√ ¯ u 1 ˙ ¯ sup ¯¯ εdBs − h(s)ds σ(Xsε ) ¯ (1 − α) 0≤u≤t 0 Z t ³ ´ L ˙ + sup |Xuε − F (h)(u)| 1 + |h(s)| ds. (1 − α) 0 0≤u≤s (2.6)
By the Gronwall lemma this yields that ¯
sup 0≤t≤1
|Xtε
¯
Z ´¯ ³√ ¯ t 1 ˙ ¯ ¯ − F (h)(t)| ≤ sup ¯ σ(Xsε ) εdBs − h(s)ds ¯ (1 − α) 0≤t≤1 0 ¶ µZ 1 L ˙ (1 + |h(s)|)ds × exp 0 (1 − α) ¯Z t ´¯¯ ³√ ¯ ε ˙ ¯ ≤ C1 sup ¯ σ(Xs ) εdBs − h(s)ds ¯¯ , (2.7) 0
0≤t≤1
5
³
´
³
´1
R 1 H) 2 ds 2 for ˙ where C1 := 1−α exp L(1+khk < ∞ with khkH := 01 |h(s)| 1−α h ∈ H. Thus to prove (2.3), it suffices to prove that, for any R, δ, a > 0, there exist ρ > 0 and ε0 > 0 such that, for any h ∈ C0 ([0, 1], R) satisfying e I(h) ≤ a and ε ≤ ε0 , Ã
! ¯Z t ´¯¯ ³√ ¯ √ ε ˙ ¯ ≥ δ, sup | εBt − h(t)| ≤ ρ P sup ¯¯ σ(Xs ) εdBs − h(s)ds ¯ 0≤t≤1 0 0≤t≤1 µ ¶
≤ exp −
R . ε
(2.8)
For ε > 0, define a probability measure Pε on Ω by µ
1 dP = Zε dP := exp √ ε ε
Z 1 0
1 ˙ h(s)dB s− 2ε
Then by Girsanov theorem, {Btε := Bt − Brownian motion on (Ω, F, Pε ). If we let
Z 1 0
¶ 2 ˙ |h(s)| ds dP.
˙ √1 h(t), t ε
∈ [0, 1]} is a standard
(
) ¯Z t ´¯¯ ³√ ¯ √ ε ˙ ¯ ¯ = sup ¯ σ(Xs ) εdBs − h(s)ds ¯ ≥ δ, sup | εBt − h(t)| ≤ ρ 0≤t≤1 0 0≤t≤1 ( ) ¯Z t ¯ ¯ ¯ √ ε √ ε ε = sup ¯¯ σ(Xs ) εdBs ¯¯ ≥ δ, sup | εBt | ≤ ρ ,
ε
A
0≤t≤1
0
0≤t≤1
then by the H¨older inequality, Z
P(Aε ) =
Ω
Zε−1 χAε (ω)Pε (dω)
µZ
≤ Note that Z Ω
{Btε , t
¶1 Ω
Zε−2 (ω)Pε (dω)
2
1
(Pε (Aε )) 2 .
∈ [0, 1]} is a standard Brownian motion under Pε , then
Zε−2 (ω)Pε (dω)
·
= = =
=
µ
Z
Z
¶¸
1 2 1 1 ˙ ˙ EPε exp − √ h(s)dB |h(s)|2 ds s+ ε 0 ε 0 · µ ¶¸ Z 1 Z 2 1 1 ˙ ε 2 ˙ ε √ EP exp − h(s)dBs − |h(s)| ds ε 0 ε 0 · µ ¶¸ Z 1 Z 2 2 1 ˙ 2 ε ˙ EPε exp − √ |h(s)| ds h(s)dBs − ε 0 ε 0 ¶ µ Z 1 1 2 ˙ |h(s)| ds × exp ε 0 µ ¶ 1 exp khk2H . ε
6
e Therefore if I(h) ≤ a, then µ ¶
P (Aε ) ≤ exp
1 a (Pε (Aε )) 2 . ε
(2.9)
Note that under the probability Pε , X ε satisfies the following stochastic differential equation: Z t √ Z t ˙ + α sup Usε .(2.10) σ(Usε )dBs + (b(Usε ) + σ(Usε )h(s))ds ε
Utε = x +
0
0
0≤s≤t
Therefore, Ã
ε
! ¯ ¯Z t ¯ ¯ √ ε ε √ ε¯ ¯ = P sup ¯ σ(Xs ) εdBs ¯ ≥ δ, sup | εBt | ≤ ρ 0≤t≤1 0≤t≤1 0 à ! ¯Z t ¯ ¯ ¯ √ √ ε = P sup ¯¯ σ(Us ) εdBs ¯¯ ≥ δ, sup | εBt | ≤ ρ .
ε
ε
P (A )
0≤t≤1
0
0≤t≤1
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 > 0, there exist ρ > 0 and ε0 > 0 such that, for any e h ∈ C0 ([0, 1], R) satisfying I(h) ≤ a and ε ≤ ε0 , Ã
P
! ¯Z t ¯ µ ¶ ¯ ¯ √ √ R ε sup ¯¯ σ(Us ) εdBs ¯¯ ≥ δ, sup | εBt | ≤ ρ ≤ exp − . (2.11) ε 0
0≤t≤1
0≤t≤1
For n ∈ N fixed, set tk =
k n
for k ∈ {0, 1, 2, · · · , n}. Define
Utε,n := Utεk , if tk ≤ t < tk+1 , k ∈ {0, 1, · · · , n − 1}. Then for δ1 > 0, ) ¯Z t ¯ ¯ ¯ δ ε,n ε ε ε,n √ ¯ ¯ sup ¯ (σ(Us ) − σ(Us )) εdBs ¯ ≥ , sup |Ut − Ut | ≤ δ1 2 0≤t≤1 0≤t≤1 0 ( )
( ε
A
⊂
sup |Utε − Utε,n | > δ1
∪
0≤t≤1
(
) ¯Z t ¯ ¯ ¯ √ √ δ ∪ sup ¯¯ σ(Usε,n ) εdBs ¯¯ ≥ , sup | εBt | ≤ ρ 2 0≤t≤1 0≤t≤1 0
:= B ε ∪ C ε ∪ Dε . o
n
On the set sup0≤t≤1 |Utε − Utε,n | ≤ δ1 , sup ε|σ(Usε ) − σ(Usε,n )|2 ≤ L2 εδ12 .
0≤s≤1
7
By Lemma 2.1, it follows that Ã
δ2 P(B ) ≤ exp − 2 2 8L εδ1
!
ε
if δ1 ≤
¶
µ
R , ≤ exp − ε
(2.12)
δ √ . 2L 2R
n o √ On the other hand, on the set sup0≤t≤1 | εBt | ≤ ρ , for any t ∈ [0, 1], ¯ ¯
¯ ¯Z t ¯ ¯ ε,n √ ¯ σ(Us ) εdBs ¯¯ ¯
¯ ¯
X ¯ √ ¯n−1 = ε ¯¯ σ(Utεj )(Btj+1 ∧t − Btj ∧t )¯¯ ¯ j=0
0
¯
X
√ ≤ 2 sup | εBt | 0≤t≤1
|σ(Utεj )|
j∈{j∈{0,···,n−1}:tj ≤t}
≤ 2nM ρ, where M > 0 is a common bound of b and σ. Therefore if ρ
sup0≤s≤tk Usε . Then there exists u ∈ (tk , t] such that Uuε = sup0≤s≤t Usε , which yields that ¯ ¯ ¯ ε ¯ ¯Ut − Utεk ¯
¯Z t ¯
≤ ¯¯
tk
Z t³ ´ ¯¯ √ ˙ b(Usε ) + σ(Usε )h(s) ds¯¯ σ(Usε ) εdBs + t
!k
Ã
+α Uuε − sup Usε 0≤s≤tk
¯Z t Z t³ ´ ¯¯ ¯ √ ˙ b(Usε ) + σ(Usε )h(s) ds¯¯ ≤ ¯¯ σ(Usε ) εdBs + tk
tk
8
³
+α Uuε − Utεk
´
¯Z t Z t³ ´ ¯¯ ¯ ε √ ε ε ˙ ¯ b(Us ) + σ(Us )h(s) ds¯¯ ≤ ¯ σ(Us ) εdBs + tk tk ¯ ¯ ¯ ¯ +α sup ¯Utε − Utεk ¯ . tk ≤t 0 e and ε0 > 0 such that, for any h ∈ C0 ([0, 1], R) satisfying I(h) ≤ a and ε ≤ ε0 , ! ¯ ¯ √ sup |Ytε − G(h)(t)| + sup |Lεt − ηt | ≥ δ, sup ¯ εBt − h(t)¯ ≤ ρ
Ã
P
0≤t≤1
µ
≤ exp −
¶
0≤t≤1
0≤t≤1
R . ε
Proof. The proof is similar to that of Theorem 2.1. We only highlight the differences. Note that for f1 , f2 ∈ Cy ([0, 1], R) and t ∈ [0, 1], sup |(Γf1 )(s) − (Γf2 )(s)| ≤ 2 sup |f1 (s) − f2 (s)|.
0≤s≤t
0≤s≤t
Then by (3.1) and (3.3), sup |Ytε − G(h)(t)| + sup |Lεt − ηt | ≤ 3 sup |Ztε − V (h)(t)|.
0≤t≤1
0≤t≤1
0≤t≤1
12
(3.4)
Therefore, the proof of Proposition 3.1 is reduced to show that for any R, δ, a > 0, there exist ρ > 0 and ε0 > 0 such that, for any h ∈ C0 ([0, 1], R) e satisfying I(h) ≤ a and ε ≤ ε0 , Ã
P
sup 0≤t≤1
! µ ¶ ¯√ ¯ R − V (h)(t)| ≥ δ, sup ¯ εBt − h(t)¯ ≤ ρ ≤ exp − . (3.5)
|Ztε
ε
0≤t≤1
Using (3.4) and the Gronwall’s lemma, sup 0≤t≤1
|Ztε
¯ ¯Z t ¯ ¯ √ ε ε¯ ¯ − V (h)(t)| ≤ C3 sup ¯ σ((ΓZ )(s)) εdBs ¯ , 0≤t≤1
0
2L exp( 1−2α khkH ) where C3 := . Thus as in Section 2, by the Girsanov’s theorem, 1−2α to prove (3.5), it suffices to prove
Ã
P
! ¯ ¯Z t µ ¶ ¯ ¯ ¯√ ¯ √ R ε ¯ ¯ ¯ ¯ ¯ , εBt ≤ ρ ≤ exp − sup ¯ σ((ΓZ )(s)) εdBs ¯ ≥ δ, sup ε
0≤t≤1
0
0≤t≤1
where Z¯ ε satisfies Z¯tε = y +
Z t √ Z t ˙ ε σ((ΓZ¯ ε )(s))dBs + σ((ΓZ¯ ε )(s))h(s)ds 0
+α sup (ΓZ¯ ε )(s), t ∈ [0, 1].
0
0≤s≤t
For a function f , put Γ(f )(n) (t) = Γ(f )(tk ), if tk ≤ t < tk+1 , k ∈ {0, 1, · · · , n− 1}. Then, (
Aeε
:=
) ¯ ¯Z t ¯ ¯ ¯ ¯√ √ ε sup ¯¯ σ((ΓZ¯ )(s)) εdBs ¯¯ ≥ δ, sup ¯ εBt ¯ ≤ ρ
0≤t≤1
(
⊂
0
0≤t≤1
¯ ¯Z t ¯ ¯ √ δ ε ε (n) ¯ ¯ ¯ sup ¯ (σ((ΓZ )(s)) − σ((ΓZ ) (s))) εdBs ¯¯ ≥ , 2 0≤t≤1 0 )
sup |Γ(Z¯ ε )(t) − Γ(Z¯ ε )(n) (t)| ≤ δ1
0≤t≤1
(
)
sup |Γ(Z¯ ε )(t) − Γ(Z¯ ε )(n) (t)| > δ1
∪
0≤t≤1
(
) ¯Z t ¯ ¯ ¯ √ √ δ ε (n) ¯ ¯ ¯ ∪ sup ¯ σ((Γ(Z ) (s)) εdBs ¯ ≥ , sup | εBt | ≤ ρ 2 0≤t≤1 0≤t≤1 0 eε ∪ C eε ∪ D e ε, := B
13
Let us just sketch the proof of µ ¶ R ε e P(C ) ≤ exp − .
(3.6)
ε
For t ∈ [tk , tk+1 ) , it is easy to see that ¯
¯
¯ ¯ sup ¯Γ(Z¯ ε )(s) − Γ(Z¯ ε )(n) (s)¯ tk ≤s≤t
¯
¯
sup ¯Γ(Z¯ ε )(s) − Γ(Z¯ ε )(tk )¯
=
tk ≤s≤t
¯
¯
≤ 2 sup ¯Z¯ ε (s) − Z¯ ε (tk )¯ .
(3.7)
tk ≤s≤t
But, ¯ ¯ ¯ ¯ε ¯tε ¯¯ ¯Z t − Z k
≤
≤
¯Z t ¯ Z t ¯ ¯ √ ε ε ˙ ¯ ¯ ¯ σ((ΓZ )(s)) εdBs + σ((ΓZ )(s))h(s)ds¯¯ ¯ tk tk ¯ ¯ ¯ ¯ ¯ ¯ ε ε ¯ ¯ +α ¯ sup (ΓZ )(s) − sup (ΓZ )(s)¯ ¯0≤s≤t ¯ 0≤s≤tk ¯ ¯Z t Z t ¯ ¯ √ ε ε ˙ ¯ ¯ ¯ σ((ΓZ )(s))h(s)ds¯¯ σ((ΓZ )(s)) εdBs + ¯ tk tk ¯ ¯ ε ε ¯ ¯ ¯ +α sup Γ(Z )(s) − Γ(Z )(tk )¯ tk ≤s≤t
≤
¯ ¯Z t Z t ¯ ¯ √ ˙ ¯ ε )(s))h(s)ds ¯ ε )(s)) εdBs + ¯ ¯ σ((Γ Z σ((Γ Z ¯ ¯ tk tk ¯ ε ¯ +2α sup ¯Z¯ (s) − Z¯ ε (tk )¯ . (3.8) tk ≤s≤t
Therefore, sup tk ≤t
