Treating Small getdoc71a6. 257KB Jun 04 2011 12:04:31 AM

Proposition 4.4 formula H. We have for q ≥ q that X i jq t =q Z t X i jq−1 s X k ∂ k F i ds , x s ∂ j x k s D x s − ∂ k F i ds , y s ∂ j y k s D y s − q Z t X i jq−1 s X k ,l,m   ∂ m F k ds , x s ∂ l x m s ∂ l x k s ∂ j x i s D x s 3 − ∂ m F k ds , y s ∂ l y m s ∂ l y k s ∂ j y i s D y s 3   + β L − β N 4 q 2 − d − 1 2 β N q − 1 2 β L q Z t X i jq s ds + qq − 1 2 β N Z t X i jq−2 s X k X k j2 s ds − q β N + β L 2 Z t X i jq−1 s X k ,l   ∂ j x k s ∂ l x i s ∂ l x k s D x s 3 − ∂ j y k s ∂ l y i s ∂ l y k s D y s 3   ds + qq − 1 β N + β L 4 Z t X i jq−2 s X k ,n   X l ∂ l x n s ∂ l x k s ∂ j x i s D x s 3 − ∂ l y n s ∂ l y k s ∂ j y i s D y s 3   2 ds − qq − 1 β N + β L 2 Z t X i jq−2 s X k ,l X k j s   ∂ l x k s ∂ l x i s ∂ j x i s D x s 3 − ∂ l y k s ∂ l y i s ∂ j y i s D y s 3   ds + 3 4 q β N + β L Z t X i jq−1 s X k ,l,m,n ∂ l x k s ∂ l x n s ∂ m x k s ∂ m x n s ∂ j x i s D x s 5 ∂ l y k s ∂ l y n s ∂ m y k s ∂ m y n s ∂ j y i s D y s 5 ds + qq − 1 Z t X i jq−2 s X k ,l ∂ j x k s ∂ j y l s D x s D y s ˜b i ,i k ,l x s − y s ds − qq − 1 Z t X i jq−2 s X k ,l,m,n   ∂ j x k s ∂ m y n s ∂ m y l s ∂ j y i s D x s D y s 3 + ∂ j y k s ∂ m x n s ∂ m x l s ∂ j x i s D y s D x s 3   ˜ b i ,l k ,n x s − y s ds + qq − 1 Z t X i jq−2 s X k ,l,m,n,p,r ∂ l x m s ∂ l x k s ∂ j x i s ∂ p y r s ∂ p y n s ∂ j y i s D x s 3 D y s 3 ˜b k ,n r ,m x s − y s ds. 33 Proof: There is nothing left to show. ƒ Proposition 4.4 will be useful to estimate the expectation in Lemma 2.3 on the event {T ≤ τ} but since this requires some additional preparations we will first consider the reversed case in the following intermezzo.

4.3 Treating Small

|x − y| And Large T Since we obviously have from Schwarz’ inequality that E – sup ≤t≤T |ψ t x − ψ t y| q 11 {T ≥τ} ™ 1 q ≤ E – sup ≤t≤T |ψ t x − ψ t y| 2q ™ 1 2q P [T ≥ τ] 1 2q 34 it seems reasonable to compute some useful estimate for the tails of τ. We will also immediatly specify conditions on r and ˜r. Assume first with Lemma 1.3 that ˜r ¯r is small enough to ensure that for any 0 ≤ r ≤ ˜r we have p 2 1 − B L r ≤ 2 p β L r and d − 1 1 − B N r r − cr ≤ 2d − 1β N − c r. 35 2343 Lemma 4.5 tails of τ . If we assume q ≥ q then we have for T ≤ log ˜r |x− y| 128β L q ∧ log ˜r |x− y| 4 p β L [ 2d −1β N −c−2β L ] = log ˜r |x− y| 128β L q that P [T ≥ τ] ≤ 2 ˜r 2q |x − y| 2q ∧ 2 ˜r 4q |x − y| 4q . Proof: Let X t = |x − y| + R t 2 p β L X s dW s + R t 2d − 1β N − c X s ds i.e. X t := |x − y| exp n 2 p β L W s + 2d − 1β N − c − 2β L t o for some BM W t t ≥0 . Then we may start with see 3 and 35 P [T ≥ τ] =P – sup ≤t≤T |x t − y t | ≥ ˜r ™ ≤ P – sup ≤t≤T X t ≥ ˜r ™ 36 =P   sup ≤t≤T W t ≥ 1 2 p β L log ˜r |x − y| − 2d − 1β N − 2β L − c + T   ≤P    sup ≤t≤1 W t ≥ log ˜r |x− y| 2 p β L T − 2d − 1β N − 2β L − c + p T    =: I X . Let first 2d − 1β N − 2β L − c ≤ 0. Then we have using 1 − Φt ≤ e −12t 2 that I X ≤2   1 − Φ    log ˜r |x− y| 2 p β L T       ≤ 2 e − 1 8  log ˜r |x− y| ‹ 2 βL T =2 |x − y| ˜r 1 8βL T log ˜r |x− y| ≤ 2 ˜r 2q |x − y| 2q 37 for T ≤ log ˜r |x− y| 16β L q remember |x − y| r ˜r . Let now 2d − 1β N − 2β L − c 0. In this case we can use for T ≤ log ˜r |x− y| 4 p β L [ 2d −1β N −c−2β L ] ∧ log ˜r |x− y| 128β L q that I X =P    sup ≤t≤1 W t ≥ log ˜r |x− y| 2 p β L T − 2d − 1β N − 2β L − c p T    ≤P    sup ≤t≤1 W t ≥ log ˜r |x− y| 4 p β L T    = 2   1 − Φ    log ˜r |x− y| 4 p β L T       ≤2 exp ¨ − 1 2 1 16β L T log ˜r |x − y| 2 « = 2 |x − y| ˜r log ˜r |x− y| 32βL T ≤2 |x − y| 2q ˜r 2q ∧ 2 |x − y| 4q ˜r 4q . 38 The proof is complete. ƒ 2344 Lemma 4.6 estimate for T after τ. For |x − y| ≤ r ≤ e −1 ˜r and arbitrary q ≥ q and T 0 we have E – sup ≤t≤T |ψ t x − ψ t y| q 11 {T ≥τ} ™ 1q ≤ ¯c 2 |x − y| 2 e Λ 2 + 1 2 q σ 2 2 T 39 with Λ 2 := Λ 1 ∨ 1, σ 2 := p 2σ 1 ∨ 2 p 128β L ∨ 2 and ¯c 2 := p 2 ¯ C 1 ˜r ∨ 1 ˜r β N +β L 128β L + p 2π˜ce − 5 12 p 16β L ∨ 2β L +β N ˜r ∨ 2e − 5 12 p 256˜cπβ L ˜r . Proof: If T ≤ log ˜r |x− y| 128β L q ∧ log ˜r |x− y| 4 p β L [ 2d −1β N −c−2β L ] = log ˜r |x− y| 128β L q then we just have to combine Lemmas 4.2 and 4.5 with 34 to obtain E – sup ≤t≤T |ψ t x − ψ t y| q 11 {T ≥τ} ™ 1q ≤ |x − y| ¯ C 1 ˜r 2 1 2q e € Λ 1 + 1 2 σ 2 1 2q Š T which means we only have to consider the case T ≥ log ˜r |x− y| 128β L q . By Lemma 4.2 we first observe for T := log ˜r |x− y| 128β L q that E – sup ≤t≤T |ψ t x − ψ t y| q ™ 1q ≤ β N + β L log ˜r |x− y| 128β L q + 2e − 5 12 p 2π˜c p q + 1 s log ˜r |x− y| 128β L q ≤   β N + β L 128β L + p 2π˜ce − 5 12 p 16β L   ˜r |x − y| ≤   β N + β L 128β L + p 2π˜ce − 5 12 p 16β L   ˜r |x − y| Λ2+ 1 2 σ2 2 q 128βL q −1 40 = 1 ˜r   β N + β L 128β L + p 2π˜ce − 5 12 p 16β L   |x − y|e Λ2+ 1 2 σ2 2 q 128βL q log ˜r |x− y| ≤ ¯c 2 |x − y|e Λ 2 + 1 2 q σ 2 2 T . Since f : T 7→ ¯c 2 |x − y|e Λ 2 + 1 2 q σ 2 2 T is a convex function and g : T 7→ β N + β L T + 2e − 5 12 p 2π˜c p q + 1 p T is concave one we may just check that f ′ T ≥ g ′ T . g ′ T = β N + β L + e − 5 12 p 256π˜cβ L p qq + 1 1 log ˜r |x− y| ≤ ¯c 2 ˜r 2 Λ 2 + 1 2 q σ 2 2 ˜r |x − y| Λ2+ 1 2 σ2 2 q 128βL −1 + ¯c 2 ˜r 2 Λ 2 + 1 2 q σ 2 2 ˜r |x − y| Λ2+ 1 2 σ2 2 q 128βL −1 =¯c 2 Λ 2 + 1 2 q σ 2 2 ˜r |x − y| Λ2+ 1 2 σ2 2 q 128βL |x − y| = f ′ T . 41 Thus the proof of Lemma 4.6 is complete. ƒ To treat 33 we finally need the following proposition. 2345 Proposition 4.7 expectation of zero-mean-martingales on rare events. For i , j ∈ {1, . . . , d} we define ˇ M t = ˇ M i j t := R t X i jq−1 s d V I I t − R t X i jq−1 s d V I I I t . Then we have 1. ˇ M t is a zero-mean-martingale and we have any t ≥ 0 that ¬ ˇ M ¶ t ≤ ˇct a.s. for ˇc := 4d 2 + 2d 4 + d 6 max k ,l,m,n sup z ∈R d ∂ k ∂ l b m ,n z. 2. For all ≤ t ≤ T and q ≥ q we get that E ” ˇ M t 11 {T ≤τ} — ≤ |x − y| 2q ¯c 3 ˜r 2q e Λ 3 + 1 2 σ 2 3 q+ σ 2 4 q 2 T =: f T for ¯c 3 := p 2ˇc ∨ q ˇc 128β L ∨ 1, Λ 3 := 1 2e , σ 3 := q 64β L e ∨ 128β L ˇc 1 4 and σ 4 := p 256β L . Proof: 1. is clear from the definition of ˇ M except for the value of ˇc. This value follows from 30, 31 and 32 since ¬ ˇ M ¶ t ≤ 〈V I I〉 i + 〈V I I I〉 t + 2| 〈V I I, V I I I〉 t |. For the proof of 2. first assume T ≤ log ˜r |x− y| 128β L q . From Lemma 4.5 we know that P [T τ] ≤ |x − y| 4q 2 ˜r 4q . This combined with Schwarz’ inequality and 1. yields E ” ˇ M t 11 {T ≤τ} — = E ” ˇ M t 11 {T τ} — ≤ q E ”¬ ˇ M ¶ T —p P [T τ] ≤ p 2ˇcT |x − y| 2q ˜r 2q ≤ p 2ˇc |x − y| 2q ˜r 2q e T 2e 42 which proves Proposition 4.7 in this case. We always have as above that E ” ˇ M t 11 {T ≤τ} — ≤ p ˇcT =: gT and so we can conclude for the same T as before gT = È ˇc 128β L r log ˜r |x − y| ≤ È ˇc 128β L ˜r |x − y| 1 2e ≤¯c 3 ˜r |x − y| σ2 3 128βL ≤ ¯c 3 ˜r |x − y| Λ3+ σ 2 3 q+ σ2 4 q2 128βL q |x − y| ˜r 2q =|x − y| 2q ¯c 3 ˜r 2q e Λ 3 + 1 2 σ 2 3 q+ σ 2 4 q 2 T =: f T 43 so now with Lemma 1.4 we only have to establish the inequality g ′ T ≤ f ′ T to complete the proof of Proposition 4.7. The proof of this is as follows. g ′ T = 1 2 s 128β L q ˇc log ˜r |x− y| ≤ 1 2 p 128β L q ˇc 44 ≤ |x − y| ˜r 2 Λ 3 + 1 2 σ 2 3 q + σ 2 4 q 2 ˜r |x − y| Λ3+ 1 2 σ2 3 q+ σ2 4 q2 128βL = f ′ T . The proof is complete. ƒ 2346

4.4 Evaluation Of Formula H