Supplement A: Construction and properties of a reected diusion

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1 Supplement A: Construction and properties of a reected diusion Jakub Chorowski 1 Construction Assumption 1. For given constants < d < D let the pair (σ, b) Θ, where Θ : Θ(d, D) {(σ, b) C 1 (, 1]) C 1 (, 1]) : b σ σ < D, inf x,1] σ (x) d}. For (σ, b) Θ consider the following Skorokhod type stochastic dierential equation: dx t b(x t )dt + σ(x t )dw t + dk t, (1) X x, 1] and X t, 1] for every t, where (W t, t ) is a standard Brownian motion and (K t, t ) is some adapted continuous process with nite variation, starting form, and such that for every t holds t 1 (,1)(X s )dk s. By the Engelbert-Schmidt theorem the SDE (1) has a weak solution, see 13, hm. 4.1]. In this section, we will present an explicit construction of a strong solution. o that end we extend the coecients b, σ to the whole real line. Denition. Dene f : R, 1] by { x n : n x < n + 1 f(x) (n + 1) x : n + 1 x < n + and b, σ : R R by b(x) b(f(x))f (x) σ(x) σ(f(x)). heorem 3. For every initial condition x, 1], independent of the driving Brownian motion W, the SDE dy t b(y t )dt + σ(y t )dw t, () Y x, has a non-exploding unique strong solution. Dene X t f(y t ). he process (X t, t ) is a strong solution of the SDE (1). 1

2 Proof. he existence of a non-exploding unique strong solution (Y t, t ) of the SDE () follows from 8, Proposition 5.17]. Process Y is a continuous semimartingale, hence by 1, Chapter VI heorem 1.] admits a local time process (L Y t, t ). By the Itô-anaka formula (1, Chapter VI heorem 1.5]) process X satises X t x + x + b(ys )f (Y s )ds + σ(y s )f (Y s )dw s + L Y t (n) L Y t (n + 1) n Z n Z b(x s )ds + σ(x s )db s + K t, where B t t f (Y s )dw s and K t n Z LY t (n) n Z LY t (n+1). Note that for any > the path (X t, t ) is bounded, hence K is well dened. Using Lévy's characterization theorem we nd that B is a standard Brownian motion. From the properties of the local time L Y t follows that K is an adapted continuous process with nite variation, starting from zero and varying on the set n Z {Y t n} {Y t n + 1} {X t {, 1}}. Consequently, X is a strong solution of the SDE (1). Notation. We will write f g (resp. g f) when f C g for some universal constant C >. f g is equivalent to f g and g f. From now on we take the Assumption (1) as granted. We denote by P σ,b the law of the diusion X on the canonical space Ω of continuous functions over the positive axis with values in, 1], equipped with the topology of the uniform convergence on compact sets and endowed with its σ eld F. We denote by the corresponding expectation operator. Modulus of continuity In this Section we want to prove a uniform upper bound on the moments of the modulus of continuity of the reected diusion X. Denition 4. Denote by ω the modulus of continuity of the path (X t, t ), i.e. ω (δ) sup X t X s. s,t t s δ heorem 5. For every p 1 there exists constant C p > s.t. sup ω p ( )] C p p/ (1 ln(/ )) p (3) (σ,b) Θ Proof. Fischer and Nappo 3] proved the above bound for the standard Brownian motion. We will now generalize their result to diusions with boundary reection. Step 1. Consider a martingale M with dm t σ(x t )dw t. By Dambis, Dubins-Schwarz theorem M t B t σ (X u)du for some Brownian motion B. Consequently M t M s B t σ (X u)du B s σ (X u)du ω B ( t s σ ), where ω B is the modulus of continuity of B. hus (3) holds for the martingale M, with a constant that depends only on the upper bound on the volatility σ.

3 Step. Consider a semimartingale X with dx t b(x t )dt + dm t. hen X t X s b(x u )du ˆ s b(x u )du + M t M s t s b + ω M ( t s ). Consequently (3) holds for semimartingales with a constant that depends only on the upper bounds on σ and b. Step 3. For (σ, b) Θ consider the reected diusion process X satisfying the SDE (1). Let dy t b(y t )dt + σ(y t )dw t, X t f(y t ), where b, σ and f are as in Denition. From Step follows that (3) holds for the semimartingale Y with a uniform constant on Θ. Since ω X ω Y, we conclude that the claim (3) holds for the reected diusion X. 3 Local time In this section we introduce some preliminary results regarding the local time of the reected diusion X. A standard reference is 1, Chapter VI]. Denition 6. Set t >. For any Borel set A, 1] we dene the occupation measure t (A) of the path (X s, s t), with respect to the quadratic variation of X, by t (A) 1 A (X s )σ (X s )ds. When t (A) is absolutely continuous with respect to the Lebesgue measure dx on the interval, 1], we dene the local time by the Radon-Nikodym derivative: L t (x) d t dx. heorem 7 (Itô-anaka formula). Let X be the solution of the SDE (1) with (σ, b) Θ. hen, the local time L exists and has a continuous version in both t > and x (, 1). For every x, 1] the process (L t (x), t ) is non-decreasing and increases only when X t x. Furthermore, if f is the dierence of two convex functions, we have f(x t ) f(x ) + f (X s )σ(x s )dw s + + f (X s )b(x s )ds + 1 ˆ 1 L t (x)f (dx)+ f (X s )dk s. (4) Proof. Reected diusion X is a continuous semimartingale. By 1, Chapter VI, heorem 1. and heorem 1.5] there exists a process (L t (x) : x (, 1), t ), continuous and nondecreasing in t, cadlag in x and such that (4) holds. Furthermore, by 1, Chapter VI, heorem 1.7] for every x (, 1) L t (x) L t (x ) 1 {Xsx}b(X s )ds + 1 {Xsx}dK s. he concentration of the associated measure dl t on the set {X t x} follows from 1, Chapter VI Proposition 1.3]. 3

4 Lemma 8. For every > and p 1 we have sup sup (σ,b) Θ x (,1) sup L p t (x)] <. t Proof. he usual way to bound the moments of the local time is to use the Itô-anaka formula for function f x (y) (y x) +, see e.g. 1, Chapter VI heorem 1.7]. Because of the additional reection term dk t, we make a less intuitive choice of the function f that guarantees f () f (1). Set >, p 1 and x (, 1/]. Let f x (y) 1(x y 3/4)(3y y ). By (4) 3 4x L t (x) f x (X t ) f x (X ) 1(x < X s 3 4 )(3 4X s)σ(x s )dw s 1(x < X s 3 4 )(3 4X s)b(x s )ds + 1(x < X s 3 4 )σ (X s )ds. Applying the uniform (on Θ) bounds on b and σ, together with the Burkholder-Davies-Gundy inequality, we conclude that for t sup sup (σ,b) Θ x (,1/) L t (x) p ] C p,, holds with some positive constant C p,. For x (1/, 1) we consider function f x (y) 1(1/4 y x)(y y ) and proceed similarly. heorem 9. For any >, p 1 and x, y (, 1) we have sup (σ,b) Θ sup L t (x) L t (y) p] C p, x y p. (5) t In particular, the family L of the local times can be chosen such that functions x L t (x) are almost surely Hölder continuous of order α for every α < 1/ and uniformly in t. Moreover, for every p 1 and t we have sup sup L p t (x)] <. (6) (σ,b) Θ x,1] Proof. he proof goes along the same lines as 1, Chapter VI heorem 1.7]. We will rst show the inequality (5). For x (, 1), by the Itô-anaka formula (4) 1 L t(x) (X t x) + (X x) + 1(X s > x)b(x s )ds 1(X s > x)σ(x s )dw s + 1(X s 1)dK s. Since the function x (X t x) + (X x) + t 1(X s 1)dK s is uniformly Lipschitz on Θ, we need only to consider the martingale term Mt x t 1(X s > x)σ(x s )dw s and the nite variation term Dt x t 1(X s > x)b(x s )ds. For x, y (, 1) Hölder's inequality and Lemma 8 yield ( ˆ sup Dt x D y y ) p ] t p] L (z)dz t x 4

5 y x p 1 ˆ y x L p (z)]dz Cp, y x p, for some constant C p, >. o bound the increments of the martingale M x we use the Burkholder-Davies-Gundy inequality together with Hölder's inequality, obtaining ( ˆ sup Mt x M y y ) p ] t p] C p L (z)dz C p, y x p. t x We nished the proof of the bound (5). From the Kolmogorov continuity criterion (see 1, Chapter I, heorem.1]) follows that there exists a modication L of the family of local times L, such that functions x L t (x) are almost surely Hölder continuous of order α for every α < 1/ and uniformly in t. Furthermore, for any α < 1/ and p we have ( L t (x) sup sup L t (y) ) p ] (σ,b) Θ x y x y α <. Fix x (, 1). By the bound above and Lemma 8 we conclude that sup sup L p t (x)] (σ,b) Θ x,1] sup (σ,b) Θ ( L t (x) sup L t (x ) x x x x α + L ) p ] t (x ) heorem 1. Set > and dene the (chronological) occupation density µ by µ (x) L (x) σ (x). <. hen, for any bounded Borel measurable function f, the following occupation formula holds: 1 f(x s )ds ˆ 1 f(x)µ (x)dx. Furthermore, the occupation density µ inherits the regularity properties of the local time L. In particular, for every p 1 and > we have sup sup µ p (x)] <. (7) (σ,b) Θ x,1] sup µ (x) µ (y) p] C p, x y p. (8) (σ,b) Θ Proof. he existence and form of the occupation density follow from heorem 7 and Denition 6. Given that σ 1 inequality (7) follows directly from (6). Finally, (5) and the uniform Lipschitz property of σ imply (8). 4 Estimation of the occupation time of an interval Estimation of the local time from nite data observations has been extensively studied and is nowadays well established, see e.g. 6, 7]. Nevertheless, until recently, much less was known about the estimation of the occupation time of a given Borel set. In the breakthrough paper Ngo and Ogawa 1] authors considered Riemann sum approximations of the occupation 5

6 time of a half-line. 1, heorem.] stated a convergence rate 3 4 for diusion processes with bounded coecients, which was dened as normalization required for tightness of the estimation errors. It is important to note that 3 4 is a better rate than could be obtained using only the regularity properties of the local time. In the special case of the Brownian occupation time of the positive half-line, 3 4 was shown to be the upper bound of the root mean squared error (see 1, heorem.3]) and to be optimal. Further development was done in Kohatsu-Higa et al. 9]. By means of the Malliavin calculus 9, heorem.3] proved that for any suciently regular scalar diusion X and an exponentially bounded function h inequality 1 h(x s )ds 1 N n h(x ) p ] C,X,h p+1/ (9) holds with constant C,X,h > depending on the time horizon, diusion X and function h. In what follows, we extend (9), for h being characteristic function, to reected diusions with coecients in Θ. heorem 11. For any > and α (, 1) we have 1 sup 1(X < α) 1 (σ,b) Θ N n with some positive constant C. 1(X s < α)ds ] 1 C 3, Remark 1. As the right hand side of (9) does not scale linearly in p, it is not optimal to use the Girsanov theorem to generalize the Brownian bound to diusions with bounded coecients. Nevertheless, we proceed with this approach, as the suboptimal rate /3 is sucient for our purposes. Proof. Fix >. he proof is divided in several steps, generalizing the result from a standard Brownian motion process to reected diusions. Step 1. Let W t be a standard Brownian motion. We will show that there exists a constant C > such that for any α R we have 1 E 1(W < α) 1 N n 1(W s < α)ds p ] C p+1/. (1) Set α R and h α (x) 1(x < α). Following 9, proof of Proposition.1], for M N, denote ρ M (x) c ρ Me (Mx) 1 1 ( 1,1) (x), ˆ h α,m (x) h α (x y)ρ M (y)dy, R ( 1 where the constant c ρ 1 e 1 1 y 1 dy) is such that ρm integrates to 1. Direct calculations show that A(i) : h α,m h α in L 1 A(ii) : sup M sup x R h α (x) + h α,m (x) A(iii) : sup M sup u h α,m (x) e x 1 u dx c ρ 1 e 1 y 1 e (α+y/m) u dy

7 Let S N,α 1 N S N,α,M 1 N n n h α (W ) 1 h α,m (W ) 1 Arguing as in 9, proof of Eq. 3.1], we obtain that for any α Hence, it is sucient to show that lim S p M N,α,M ] S p N,α ]. h α (W s )ds, h α,m (W s )ds. S p N,α,M ] C N (p+1/), (11) for some constant C independent of M and α. But, given uniform bounds A(ii) and A(iii), inequality (11) follows by the same calculations as in 9, proof of heorem 3.]. Step. Consider diusion Y satisfying dy t h(y t )dt+dw t, with uniformly bounded drift h. We will show that Denote 1 E 1(Y < α) 1 N n ( Z t exp 1(Y s < α)ds ] 1 C, h 17/4. (1) h(y s )dw s 1 ) h (Y s )ds. Since h is bounded, by Novikov's condition Z is a martingale. Dene the probability measure Q by the Radon-Nikodym derivative: dq dp F t Z t. By Girsanov's theorem the process Y is a standard Brownian motion under the probability measure Q. From Hölder's inequality, for 1 p + 1 q 1, follows that 1 E 1(Y < α) 1 N n 1(Y s < α)ds ] 1 1 E Q 1(Y < α) 1 N n 1 E Q 1(Y < α) 1 N n Since the drift function h is uniformly bounded, E Q Z q 1(Y s < α)ds Z 1 ] 1 1(Y s < α)ds p ] 1 p E Q Z q] 1 q. ] (q 1) ] ( E Z E exp (q 1) h(y s )dw s + q 1 )] h (Y s )ds 7.

8 ( q(q 1) ) exp h C q q, h. Hence, by (1), with p 6/5, inequality (1) holds. Step 3. Consider diusion Y satisfying dy t b(y t )dt + σ(y t )dw t, with bounded drift b and positive, Lipschitz continuous σ. Let S(x) x σ 1 (y)dy, dz t S(Y t ). It follows from Itô's formula that dz t g(y t )dt + dw t, where g(x) b(x) σ(x) 1 σ (x). Since σ is a strictly positive function, S is increasing and invertible. Denote h(x) g(s 1 (x)). We have with dz t h(z t )dt + dw t, h b inf σ + 1 σ. From (1) follows, that there exists a constant C, depending only on the bounds on the diusion coecients, such that 1 E 1(Y < α) 1 N n 1(Y s < α)ds ] 1 1 E 1(Z < S(α)) 1 N n Step 4. Fix (σ, b) Θ. Let dy t b(y t )dt + σ(y t )dw t, X t f(y t ), 1(Z s < S(α))ds ] 1 C (13) where b, σ and f are as in Denition. Recall that by heorem 3 process X is the reected diusion with coecients (σ, b). By denition of the function f, for any α (, 1) and s >, we have {X s < α} m Z{Y s (m α, m + α)}. (14) Denote Γ N (m) 1 N n 1(Y (m α, m + α)) 1 1(Y s (m α, m + α))ds. By (13) there exists a uniform on Θ constant C >, such that for any m Z, we have Let Y t ] 1 Γ N(m) sup s t Y s. From (14) follows that C (X < α) 1(X s < α)ds Γ N (m) Γ N (m)1(y m 1). N n m Z m Z 8

9 Since Γ N (m), for any M, using (14) we obtain Γ N (m)1(y m + 1) ] m Z M m M C M ] Γ N(m) + m,k>m m>m P σ,b (Y (m k) 1). 1(Y m 1) ] By the Burkholder-Davies-Gundy inequality, together with uniform on Θ bounds on diusion coecients, for any p 1, (Y ) p] ( Consequently, by the Markov inequality P σ,b (Y (m k) 1) m,k>m σ (Y s )ds ) ] p + ( b ) p C p,. M<k m C p, C p, (m 1) p (m 1) (p 1) M (p ). m>m We conclude that Γ N (m)1(ys m + 1) ] m Z Hence the claim follows for M 3 8 and any p 4. 5 Upper bounds on the transition kernel M M (p ). In this section we prove a uniform Gaussian upper bound on the transition kernel of the reected diusion X with coecients in Θ. Under the assumption of smooth coecients, existence of Gaussian o-diagonal bounds follows from the general theory of partial dierential equations, see 4, 9] c.f. 5, Chapter 9]. Sharp upper bounds are also established for diusion processes in the divergence form, Chapter VII] and more recently were derived for multivariate diusions with the innitesimal generator satisfying Neumann boundary conditions, see 14]. Nevertheless, as demonstrated by 11, heorem ] Gaussian upper bounds do not hold in general for scalar diusions with bounded measurable drift. heorem 13. he transition kernel p t of the reected diusion X satises for all x, y, 1], t and c, C >. sup p t (x, y) C e (x y) ct, (σ,b) Θ t 9

10 Proof. We will generalize the bound on the transition kernel from diusions with bounded drift and unit volatility to reected processes with coecients (σ, b) Θ. Step 1. Consider diusion Z satisfying dz t g(z t )dt + dw t, where g is a bounded measurable function. hen by 11, heorem 1] the transition kernel p Z of the process Z satises p Z t (x, y) 1 ˆ ze (z g t) / dz. t Using the inequality 1, Formula ]: we obtain that hus ˆ a/ t ˆ x ze (z b t) / dz e z dz ˆ a t b t x y / t p Z t (x, y) C, g e x x + x + 4/π (w + b ( t)e w / dw e (a bt) t π e x, (15) 1 + b πt ). 1 t e (x y) t + g x y. (16) Step. Consider diusion Y satisfying dy t b(y t )dt + σ(y t )dw t. Let S(x) x σ 1 (y)dy and Z t S(Y t ). It follows from Itô's formula that dz t g(y t )dt + dw t, where g(x) b(x) σ(x) 1 σ (x). For any x, y we have Hence, from (16) follows that σ 1 1 x y S 1 (x) S 1 (y) σ x y. p Y t (x, y) p Z t (S 1 (x), S 1 (y)) C, g t C, g t ( exp (S 1 (x) S 1 (y)) t ( (x y) exp t σ 1 + g σ x y ) + g S 1 (x) S 1 (y) ). (17) Step 3. For (σ, b) Θ let b, σ and f be as in Denition and Y as in Step. Note rst, that by (17) there exist uniform on Θ constants c, c 1, c > such that p Y t (x, y) c ( (x ) y) exp + c x y. (18) t c 1 t By heorem 3 X t f(y t ) is the reected diusion process corresponding to the coecients (σ, b). For y, 1] let (y m ) m Z be such that y m m, m + 1] and f(y m ) y. hen p X t (x, y) m Z p Y t (x, y m ). Since for any m Z we have x y + ( m ) + x y m m +, from (18) follows that p X t (x, y) c ( exp (x y m) ) + c x y m t c 1 t m Z c e (x y) c 1 t t m Z ( exp ( m ) +] ) + c ( m + ). c 1 1

11 6 Mean crossings bounds hroughout this section we consider xed time horizon, for simplicity we assume 1. Denition 14. For α (, 1) and n,..., N 1 denote heorem 15. For every α (, 1) we have χ(n, α) 1,α) (X (n+1) ) 1,α) (X ). ( n χ(n, α) (X (n+1) X ) ) ] 1 1/. Proof. Fix α (, 1). Since χ(n, α) 1 if and only if the increment (X, X (n+1) ) crosses the level α, the claim is equivalent to the inequalities: ( n ( n 1(X < α)1(x (n+1) > α)(x (n+1) X ) ) ] 1 1(X > α)1(x (n+1) < α)(x (n+1) X ) ) ] 1 1/, 1/. Below, we only prove the rst inequality. he second one can be obtained in a similar way or by a time reversal argument. Denote We have ( n η n 1(X < α)1(x (n+1) > α)(x (n+1) X ). 1(X < α)1(x (n+1) > α)(x (n+1) X ) ) ] n η n]+ n<m η n η m ]. Denote by p t the transition kernel of the diusion X. From heorem 13 and the inequality (15) follows that ˆ α ˆ 1 α p (x, y)(y x) 4 dydx ˆ α ˆ 1 α ˆ α ˆ 1 α e (y x) c (y x) 4 dydx ˆ 1 x c α x c e z dzdx ˆ α ˆ 1 x c α x c e z z 4 dzdx e (α x) c dx 5/. (19) Similarly ˆ α ˆ 1 α p (x, y)(y x) dydx 3/. () For simplicity we will use the stationarity of X, which is granted by the assumption x d µ.using more elaborated arguments the result could be obtained for an arbitrary initial condition. By stationarity, for any t, the one dimensional margin X t is distributed with respect 11

12 to the invariant measure µ(x)dx. Conditioning on X, from (19) and uniform bounds on the density µ follows Hence η n ] ˆ α ˆ 1 n he Cauchy-Schwarz inequality implies N n η n η n+1 ] α p (x, y)(y x) 4 dyµ(x)dx 5/. η n] N 5 3. N n Finally, using (), for m > n + 1, we obtain η n η m ] Consequently ˆ α ˆ 1 ˆ α ˆ 1 α ˆ α ˆ 1 α α η n] 1 Eσ,b η n+1] 1 N 5 3. p (x, y)(y x) p (m n 1) (z, x)(z w) p (w, z)µ(w)dydxdzdw ˆ p (x, y)(y x) 1 α dydx (m n 1) 5/ 1 m n 1. N 3 n mn+ Denition 16. Dene the event N 3 η n η m ] 5/ n N 5/ n1 N n k1 ˆ 1 α (z w) p (w, z)dzdw N 3 1 5/ N n k n n 5/ N 3/. R 1 : {ω 1 ( ) µ 1 5/11 v}. Using Markov's inequality together with heorem 5 and (6) we obtain that heorem 17. For any α 1 J, 1 1 J ] holds 1 R1 n P σ,b (Ω \ R 1 ) /3. ( 1,α) (X (n+1) ) 1,α) (X ) )( (X (n+1) α) (X α) ) ] 1 /3. Proof. Fix α 1 J, 1 1 J ]. On the event R 1, whenever 1,α) (X (n+1) ) 1,α) (X ) we must have X α, X (n+1) α ω( ) < 4/9. Consider function d :, 1] R given by d(x) (x α) 1( x α 4/9 ). 1

13 We have ( 1,α) (X (n+1) ) 1,α) (X ) )( (X (n+1) α) (X α) ) Step 1. We will rst show that Note that 1 R1 ( 1,α) (X (n+1) ) 1,α) (X ) )( d(x (n+1) ) d(x ) ). n 1,α) (X ) ( d(x (n+1) ) d(x ) ) ] 1 d (x) (x α)1( x α 4/9 ), 1 d (x) 4/9 δ {α 4/9 } + 1( x α 4/9 ) 4/9 δ {α+ 4/9 }, /3. (1) where the second derivative must be understood in the distributional sense. Since we xed α separated from the boundaries, d () d (1) for small enough. Denote by L s,t (x) : L t (x) L s (x), the local time of the path fragment (X u, s u t). From the Itô-anaka formula (4) follows that d(x (n+1) ) d(x ) + d (X s )σ(x s )dw t + d (X s )b(x s )ds+ σ (X s )1( X s α 4/9 )ds 4/9 L,(n+1) (α 4/9 ) 4/9 L,(n+1) (α + 4/9 ) : d (X s )σ(x s )dw t + D n. First, we will bound the sum of the martingale terms. Since martingale increments are uncorrelated, using Itô isometry, we obtain that n n 1,α) (X ) 1,α) (X ) d (X s )σ(x s )dw t ] ] ˆ (d (X s )) σ (X s )ds 8 1 ] 9 Eσ,b 1( X s α 4 9 )ds 8 9 ˆ α+ 4 9 α 4 9 µ 1 (x)]dx 4 3, where the last inequality follows from heorem 1. Now, we will bound the sum of the nite variation terms: n 1,α)(X )D n. Note rst, that since b is uniformly bounded, we have n 1,α) (X ) d (X s )b(x s )ds ˆ 1 4/9 1( x α 4/9 )µ 1 (x)dx 8/9 µ 1. 13

14 Since by the inequality (7) µ 1 has all moments nite, the root mean squared value of this sum is of smaller order than /3. Now, note that since on the event R 1 ω( ) < 4/9, condition X < α implies L,(n+1) (α + 4/9 ). On the other hand, whenever L,(n+1) (α 4/9 ) we must have X < α. Hence n 1,α) (X )( 4/9 L,(n+1) (α 4/9 )+ 4/9 L,(n+1) (α+ 4/9 )) 4/9 L 1 (α 4/9 ). Using rst the Cauchy-Schwarz inequality and then heorem 9 we obtain ˆ α 4/9 L 1 (α 4/9 ) L 1 (x)dx ] ˆ α 4/9 L 1 (x) L 1 (α 4/9 ) ]dx α 4/9 α 4/9 4/3. Consequently, to prove (1) we just have to argue that the root mean squared error of ˆ α α 4/9 L 1 (x)dx n n n 1,α) (X ) σ (X s )1( X s α 4/9 )ds (1(X s < α) 1(X < α))σ (X s )1( X s α 4/9 )ds (1(X s < α) 1(X < α))σ (X s )ds () is of order /3. From the Lipschitz property of σ follows that n (1(X s < α) 1(X < α))(σ (X s ) σ (α))ds n hus, by (7), we reduced () to ˆ 1 ˆ 1( X s α α+ 4/9 9 ) 9 ds 9 µ 1 (dx) 8 9 µ1. α 4/9 1(X s < α)ds 1 (1(X < α), N which is of the right order by heorem 11. We conclude that (1) holds. Step. Consider the time reversed process Y t X 1 t. Since X is reversible, the process Y, under the measure P σ,b, has the same law as X. Furthermore, the occupation density and the modulus of continuity of processes Y and X are identical, hence R 1 is a good event also for Y. Inequality (1) is equivalent to 1 R1 m Substituting n N m we obtain m 1,α) (Y m )(d(y (m+1) ) d(y m )) n 1,α) (Y m )(d(y (m+1) ) d(y m )) ] 1 3. n 1,α) (X (n+1) )(d(x (n+1) ) d(x )). 14

15 References 1] Abramowitz, M. and Stegun, I. A. (197). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical ables. Dover, New York, 9th edition. ] Bass, R. (1997). Diusions and Elliptic Operators. Springer-Verlag, New York. 3] Fischer, M. and Nappo, G. (1). On the moments of the modulus of continuity of Itô processes. Stochastic Analysis and Applications, 8(1). 4] Florens-Zmirou, D. (1993). On estimating the diusion coecient from discrete observations. J. Appl. Prob., 3. 5] Friedman, A. (1964). Partial dierential equations of parabolic type. Prentice-Hall Inc., Englewood Clis, N.J., nd edition. 6] J. M. Azaïs (1989). Approximation des trajectoires et temps local des diusions. Ann. Inst. Henri Poincaré Probab. Statist., 5(): ] Jacod, J. (1998). Rates of convergence of the local time of a diusion. Anns. Inst. Henri Poincaré Probab. Statist., 34(4): ] Karatzas, I. and Shreve, S. E. (1991). Brownian motion and stochastic calculus. Springer- Verlag, New York. 9] Kohatsu-Higa, A., Makhlouf, A., and Ngo, H. L. (14). Approximations of non-smooth integral type functionals of one dimensional diusion processes. Stochastic Processes and their Applications, 14: ] Ngo, H.-L. and Ogawa, S. (11). On the discrete approximation of occupation time of diusion processes. Electronic Journal of Statistics, 5: ] Qian, Z. and Zheng, W. (). Sharp bounds for transition probability densities of a class of diusions. C. R. Acad. Sci. Paris, I 335: ] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, volume I, II. Springer, New York, 3 edition. 13] Rozkosz, A. and Sªomi«ski, L. (1997). On stability and existence of solutions of SDEs with reection at the boundary. Stochastic processes and their applications, 68: ] Yang, X. and Zhang,. (13). Estimates of Heat Kernels with Neumann Boundary Conditions. Potential Analysis, 38: