Time-dependent weak rate of convergence for functions of generalized bounded variation

Let $W$ denote the Brownian motion. For any exponentially bounded Borel function $g$ the function $u$ defined by $u(t,x)= \mathbb{E}[g(x{+}\sigma W_{T-t})]$ is the stochastic solution of the backward heat equation with terminal condition $g$. Let $u^n(t,x)$ denote the corresponding approximation generated by a simple symmetric random walk with time steps $2T/n$ and space steps $\pm \sigma \sqrt{T/n}$ where $\sigma>0$. For quite irregular terminal conditions $g$ (bounded variation on compact intervals, locally H\"older continuous) the rate of convergence of $u^n(t,x)$ to $u(t,x)$ is considered, and also the behavior of the error $u^n(t,x)-u(t,x)$ as $t$ tends to $T$


Introduction
The objective of this article is to study the rate of convergence of a finite-difference approximation scheme for the backward heat equation. The error analysis is carried out for a large class of exponentially bounded terminal condition functions having bounded variation on compact intervals.
During the past decades, convergence rates of finite-difference schemes for parabolic boundary value problems have been studied with varying assumptions on the regularity of the initial/terminal condition, the domain of the solution, properties of the possible boundary data, etc. (see, e.g., [1][2][3][4][5]). In order to study the convergence, several techniques have been applied. Our approach is probabilistic: The solution of the PDE is represented in terms of Brownian motion, and the approximation scheme is realized using an appropriately scaled sequence of simple symmetric random walks in the same probability space, in the spirit of Donsker's theorem. The possible discontinuities of the terminal function produce error bounds which are not uniform over the time-nets under consideration, and hence the time dependence of the error is of particular interest here.
To explain our setting in more detail, fix a finite time horizon T > 0, a constant r > 0, and consider the backward heat equation @ @t u þ r 2 2 @ 2 @x 2 u ¼ 0, ðt, xÞ 2 0, TÞ Â R, uðT, xÞ ¼ gðxÞ, x 2 R: The terminal condition g : R ! R is assumed to belong to the class GBV exp consisting of exponentially bounded functions that have bounded variation on compact intervals (the precise description is given in Definition 2.3). The stochastic solution to the problem (1) is given by where ðW t Þ t!0 denotes the standard Brownian motion. To approximate the solution (2), we proceed as follows. Given an even integer n 2 2N, a level z 0 2 R, and time and space step sizes d > 0 and h > 0, respectively, define The finite-difference scheme we will consider is given by the following system of equations defined on grids G n z 0 :¼ T n Â S n z 0 & ½0, T Â R, v n ðt n k , xÞ À v n ðt n kÀ1 , xÞ t n k À t n kÀ1 þ r 2 2 v n ðt n k , x þ 2hÞ À 2v n ðt n k , xÞ þ v n ðt n k , x À 2hÞ ð2hÞ 2 ¼ 0, v n ðT, Á Þ ¼ g: Letting d :¼ T n and h :¼ r ffiffi ffi T n q , the system (4) can be rewritten in an equivalent form as v n ðt n kÀ1 , xÞ ¼ 1 4 v n ðt n k , x þ 2hÞ þ 2v n ðt n k , xÞ þ v n ðt n k , x À 2hÞ Â Ã , v n ðT, Á Þ ¼ g: This scheme is explicit: Given the set of terminal values gðxÞ j x 2 S n z 0 È É , the solution v n of (5) is uniquely determined by a backward recursion. We extend the function v n in continuous time by letting v n ðt, xÞ :¼ v n ðt n k , xÞ for t 2 ½t n k , t n kþ1 Þ, 0 k < n 2 , (6) and study the error of approximation e n ðt, xÞ on ðt, xÞ 2 ½0, TÞ Â S n z 0 , where e n ðt, xÞ :¼ v n ðt, xÞ À uðt, xÞ: Theorem 2.5, the main result of this article, states that for a constant C > 0 depending only on g, e n ðt, xÞ j j CwðxÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nðT À tÞ p 1 t6 ¼t n k f g þ CwðxÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nðT À t n k Þ p , ðt, xÞ 2 ½t n k , t n kþ1 Þ Â S n z 0 , 0 k < where the function wðxÞ ¼ wðjxj, g, r, TÞ > 0 is given explicitly in Section 2.
In the 1950s, Juncosa and Young [2] considered a finite difference approximation of the (forward) heat equation on a semi-infinite strip ½0, 1Þ Â ½0, 1, where the initial condition was assumed to have bounded variation. Using Fourier methods, they proved [2,Theorem 7.1] that the error is O n À 1 2 ð Þ uniformly on ½t, 1Þ Â ½0, 1 for any fixed t > 0. However, they did not study the order of the blow-up of the error as t # 0 (which translates to t " T in the case of the backward equation (1)). Indeed, the bound (8) suggests that the convergence is not uniform in (t, x). Nevertheless, one obtains the rate n À 1 2 on any compact subset of ½0, TÞ Â R, and this rate is also sharp for the class GBV exp : The blow-up in (8) vanishes if g has more regularity: For a-H€ older continuous g with a 2 ð0, 1 and r ¼ 1, it was shown in [6,Corollary 4] that je n ðt, xÞj Cn À a 2 holds uniformly in (t, x), where C ¼ CðTÞ > 0 is a constant.
The main result is derived using the following probabilistic approach. Let ðn i Þ i¼1, 2, ::: be a sequence of i.i.d. Rademacher random variables, and define u n ðt, xÞ : where ðW n t Þ t2½0, T is the random walk given by (dÁe denotes the ceiling function). The key observation is that the function u n , when restricted to G n z 0 , is the unique solution of (5) for every z 0 2 R: Relation (6) also holds for u n by definition. Moreover, since the random walk ðW n t Þ t2½0, T affects the value of u n only through its distribution, we may consider a special setting where the Rademacher variables n 1 , n 2 , ::: are chosen in a suitable way. Defining these variables as the values of the Brownian motion ðW t Þ t!0 sampled at certain first stopping times (see Section 2.1) enables us to apply techniques from stochastic analysis for the estimation of the error (7) where v n ¼ u n : The above procedure was applied in Walsh [7] (cf. Rogers and Stapleton [8]) in relation to a problem arising in mathematical finance. More precisely, the weak rate of convergence of European option prices given by the binomial tree scheme (Cox-Ross-Rubinstein model) to prices implied by the Black-Scholes model is analyzed in [7] (cf. Heston and Zhou [9]). A detailed error expansion is presented in [7,Theorem 4.3] for terminal conditions belonging to a certain class of piecewise C 2 functions.
Using similar ideas, we complement this result by considering a large class of functions containing the class considered in [7]. Moreover, instead of studying the error at time t ¼ 0 only, we derive a time-dependent error bound. Finally, a gap is closed in the proof done in [7]. It concerns the estimate [7, Proposition 11.2(iv)] for which a detailed proof is given in Section 5.2.
It is argued in [7,Sections 7 and 12] that the rate remains unaffected if the geometric Brownian motion is replaced with a Brownian motion, and the binomial tree is replaced with a random walk. It seems plausible that also our time-dependent results in the Brownian setting can be transferred into the geometric setting with essentially the same upper bounds.
The article is organized as follows. In Section 2, we introduce the notation, recall the construction of a simple random walk using first hitting times of the Brownian motion, and formulate the main result Theorem 2.5. Using this sequence of first hitting times, the error (7) will be split into three parts. Estimates for the adjustment error and the local error are derived in Section 3, and the global error is treated in Section 4. Section 5 contains the result for the sharpness of the rate and the key moment estimates applied in Section 4. The remaining auxiliary results and estimates can be found in the appendix, where also the construction of the terminal function class and its properties are briefly discussed.
2. The setting and the main result 2.1. Notation related to the random walk Consider a standard Brownian motion ðW t Þ t!0 on a stochastic basis ðX, F , P, ðF t Þ t!0 Þ, where ðF t Þ t!0 stands for the natural filtration of ðW t Þ t!0 : Let ðX t Þ t!0 :¼ ðrW t Þ t!0 , where r > 0 is a given constant. By s ðÀh, hÞ we denote the first exit time of the process ðX t Þ t!0 from the open interval ðÀh, hÞ, In order to represent the error (7), we construct a random walk on the space ðX, F , P, ðF t Þ t!0 Þ: Following [7], we define recursively for k ¼ 1, 2, :::: Then s k is a P-a.s. finite ðF t Þ t!0 -stopping time for all k ! 0, and the process ðX s k Þ k¼0, 1, ::: is a symmetric simple random walk on Z h :¼ mh : m 2 Z f g : For every integer k ! 1, we also let The strong Markov property of ðX t Þ t!0 implies that ðDs k , DX s k Þ k¼1, 2, ::: is an i.i.d. process such that for each k ! 1, we have PðDX s k ¼ 6hÞ ¼ 1=2, ðDs k , DX s k Þ ¼ d ðs ðÀh, hÞ , X s ðÀh, hÞ Þ, and ðDs k , DX s k Þ is independent of F s kÀ1 þ : Moreover, as shown in [8,Proposition 1], the increments DX s 1 and Ds 1 are independent. Consequently, the processes ðDs k Þ k¼1, 2, ::: and ðDX s k Þ k¼1, 2, ::: are independent (see also [7,Proposition 11.1] and [10, Proposition 2.4]). We deduce, in particular, that for all N ! 1 the random variable X s N is distributed as h P N k¼1 n k , where ðn k Þ k¼1, 2, ::: is an i.i.d. sequence of Rademacher random variables. Therefore, for W n TÀt defined in (10), we have the equality in law Note that in this case the sequence of stopping times ðs k Þ k¼0, 1, ::: (11) depends on n via h ¼ hðnÞ: The error (7) will be split into three parts, where each of these parts will take into account different properties of the given function g. For this purpose, let us introduce some more notation. Let h n denote the smallest multiple of 2T=n greater than or equal to T À t: That means, for given n 2 2N and t 2 ½0, TÞ, % 2 2, 4, :::, n f g : It is clear that 0 h n À ðT À tÞ 2T n and h n # T À t as n ! 1: Note also that the connection between lattice points t n k ¼ 2kT=n 2 T n introduced in (3) and the time instant h n 2 ð0, T is explained as follows:

The class of terminal functions
The approximation error will be estimated for functions g belonging to the class GBV exp introduced below. This class is contained in the class of exponentially bounded Borel functions.
The class of all Borel functions with the above property will be denoted by B exp : The function class GBV exp generalizes functions of bounded variation (which are bounded) by allowing exponential growth. While these functions have bounded variation on each compact interval, their total variation may be unbounded (or undefined) over unbounded intervals. See [11] and Appendix A.1 for more information on this topic. that can be written as a difference of two measures l 1 , l 2 : BðRÞ ! ½0, 1 such that l 1 ðKÞ < 1 nad l 2 ðKÞ < 1 for all compact sets K 2 BðRÞ: Below it is understood that ½a, bÞ ¼ ; whenever a ! b: Definition 2.3 (The class GBV exp ). Denote by GBV exp the class of functions g : R ! R which can be represented as where c 2 R is a constant, l 2 M, and J ¼ ða i , x i Þ i¼1, 2, ::: & R 2 is a countable set such that x i 6 ¼ x j whenever i 6 ¼ j: In addition, we require that for some constant b ! 0, ð R e Àbjxj djljðxÞ þ To give some examples of classes of functions contained in GBV exp , we have the remark below. See Appendix A.1 for the proof. (i) Every polynomial belongs to the class GBV exp : (ii) Each increasing (resp. decreasing) function g 2 B exp belongs to GBV exp : (iii) Each convex (resp. concave) function g 2 B exp belongs to GBV exp : (iv) K exp & GBV exp , where K exp is the class of functions g : R ! R considered in Walsh [7] (pp. 340, 345-346, and 348), i.e. they satisfy the below criteria: (v) g, g 0 , and g 00 belong to B exp (vi) g, g 0 , and g 00 have at most finitely many discontinuities and no oscillatory discontinuities (vii) gðxÞ ¼ 1 2 ðgðxþÞ þ gðxÀÞÞ at each point x 2 R:

The main result
The main result of this article, Theorem 2.5, describes the approximation error between the solution of the backward heat equation (1) and its finite-difference approximation (9) for terminal functions belonging to the class GBV exp : Theorem 2.5. Let n 2 2N, and let u and u n be the functions introduced in (2) and (9). Suppose that g 2 GBV exp is a function given by (15) and that b ! 0 is as in (16). Then, for all ðt, xÞ 2 ½0, TÞ Â R, ðiÞ u n ðt, xÞ À uðt, xÞ ðiiÞ u n ðt n k , xÞ À uðt n k , xÞ C b, r, T ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nðT À t n k Þ p e bjxj , 0 k < n 2 , where C b, r, T :¼ C ffiffiffi ffi T p e 3b 2 r 2 T and C > 0 is a constant depending only on g.
Remark 2.6. The error bounds in Theorem 2.5 grow exponentially as functions of the variable x. If the terminal condition g satisfies (16) with b ¼ 0 one obtains bounds which are uniform in x.
Proof of Theorem 2.5. The function u n ðt, xÞ is constant in t on intervals of length 2T=n, while t 7 ! uðt, xÞ is continuous. Therefore, for t 2 ½t n k , t n kþ1 Þ, we will split the error and write u n ðt, xÞ À uðt, xÞ ¼ ðu n ðt n k , xÞ À uðt n k , xÞÞ þ ðuðt n k , xÞ À uðt, xÞÞ, where e adj n ðt, xÞ :¼ uðt n k , xÞ À uðt, xÞ will be called the adjustment error. Next, we exploit the construction of the random walk ðX s k Þ k!0 by Skorokhod embedding from the process ðX t Þ t!0 and let J n ðxÞ :¼ inff2m 2 2N : s 2m ðxÞ > h n g, x 2 X, which is the index of the first stopping time s 0 , s 2 , s 4 , ::: exceeding the value h n : Consequently, by construction, X s Jn will be 'rather close' to X h n for large n. Therefore, we will write xÞ þ e loc n ðt, xÞ, where the first term on the right-hand side e glob n ðt, xÞ : is referred to as the global error, and second term e loc n ðt, xÞ : denotes the local error. The local error is influenced by the smoothness properties of the terminal condition g, while for the global error only integrability properties of g are needed.
Since g 2 GBV exp , there exists a constant A ¼ AðbÞ ! 0 such that gðxÞ j j Ae bjxj for all x 2 R: Indeed, relation (14) is satisfied for a function g given by (15) by letting b :¼ b and A to be equal to the sum of jcj and the left-hand side of (16).
Consequently, by Theorems 3.1, 3.8, and 4.3 (the bounds for the error terms e adj n , e loc n , and e glob n , respectively), there exists a constant C > 0 such that It remains to observe that since ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The adjustment error and the local error

The adjustment error
The purpose of this subsection is to derive an upper bound for the modulus of the adjustment error defined in (17) for a terminal function g belonging to GBV exp : Theorem 3.1. Let n 2 2N, and suppose that g 2 GBV exp and b ! 0 is as in (16). Then, for all ðt, xÞ 2 ½0, TÞ Â R, Proof. Denote by p s the density of X s ¼ rW s for s > 0, and consider the function Fix n 2 2N and suppose that t n k ¼ 2kT=n is the lattice point such that (13) implies that h n ¼ T À t, and thus e adj n t, x ð Þ ¼ 0 by (17). Suppose then t 2 t n k , t n kþ1 À Á and use the representation (29) for the function z 7 ! g x þ z ð Þ in order to rewrite Since g is exponentially bounded, one may apply Fubini's theorem to rewrite The mean value theorem and the fact ffiffiffiffiffiffiffiffiffiffiffi were applied. Consequently, it follows by (22) that and an analogous computation for the integral I 2 yields Since e adj n t, as a function of h, h ð Þ, where g 2 GBV exp : The random variable J is given by where s k was defined in (11). Afterward, upper bounds for the error (25) are derived in the dynamical setting, where the step size h and the level h will depend on n.
Observe that J ¼ J n holds for J n defined in (18) when one substitutes h, Let us start by introducing the following notation: (o refers to "odd" and e refers to "even"); then In addition, we will abbreviate As in [7], we project functions onto piecewise linear functions in order to compute the conditional expectation E g X s J ð ÞjF h Â Ã : Definition 3.2. Define operators P o and P e acting on functions u : R ! R by The key ingredient in the estimation of the error e loc h, h g ð Þ is the following result, which was proposed in [7,Section 9]. For the convenience of the reader, a sketch of the proof is given below. Recall Definition 2.1 for the class B exp , and denote by N 0 :¼ 0, 1, 2, ::: f g the set of non-negative integers.
(s L is equal to the largest of the stopping times s 0 , s 1 , ::: less than h). Then, for a function g 2 B exp , Proof. If g 2 B exp , then also P e g 2 B exp and P o P e g 2 B exp : The expectations on the right-hand side of (28) thus exist and are finite. Using the Markov property of the process X t ð Þ t!0 , it can be shown that If g 2 GBV exp is a function given by (15) and Using the representation (29) and linearity, the estimation of the error e loc h, h g x ð Þ essentially reduces to the estimation of integrals whose integrands consist of indicator functions or their linear approximations given by the operators P e and P o (introduced in Definition 3.2). The following lemma enables us to interchange the order of integration or summation with the application of these operators.
and that g 2 GBV exp admits the representation (15). Then, for all x 2 R, Idea of the proof. The representations in (i) and (ii) follow by using the representation (29), linearity of the operations f 7 ! P e f , f 7 ! P o f , and f 7 ! Ð fdjlj, and relation (83).
Suppose that g 2 GBV exp admits the representation (15) and that b ! 0 is as in (16). Then, for all x 2 R, Proof. Let us denote by p ¼ p Á , h ð Þ the density of X h : By Lemma 3.4, we may decompose the expectation on the left-hand side of (30) as follows: where q ¼ q z, h ð Þ is the function introduced in (57) which satisfies q z ð Þ ¼ P L evenjX h ¼ z ð Þ(Leb-a.e.). To show (30), we derive upper estimates for the quantities jE i ð Þ j, 1 i 6, in the following steps. Step In fact, it also holds that since jP e 1 À1, yÀx which is a direct consequence of the relation Step Moreover, by (34) and by the linearity of P o , we obtain : Therefore, since it also holds (for each x i ) that jx i j 2h þ jxj þ jzj whenever jz À x i À x ð Þ j 2h, Step 4: E 6 ð Þ : If n 2 Z h e , relations (83), (88), and the linearity of P o imply that P o P e 1 fng z ð Þ À P e 1 fng z ð Þ Therefore, since jx i j 3h þ jxj þ jzj whenever jz À x i À x ð Þ j 3h, we get The proof is completed by combining relation (31) with the bounds (32)-(33) and (35)-(38). w Before presenting the main result of this subsection, Theorem 3.8, we provide an auxiliary convention regarding the notation. It enables us to distinguish between the general setting h, h ð Þ and the specific n-dependent setting h n , h n ð Þ also in the later sections.
Assumption 3.6. For given t 2 0, T ½ Þ and n 2 2N, we substitute h, h ð Þ ¼ h n , h n ð Þ, where % as in (12). For notational convenience, we will drop the subscript n from h n .
Theorem 3.8. Let n 2 2N. Suppose that the function g 2 GBV exp admits the representation (15) and that b ! 0 is as in (16). Then, under Assumption 3.6, there exists a constant C > 0 such that for all t, x ð Þ 2 0, T ½ ÞÂ R, where the coefficient C b, r, T > 0 implied by (30) can be estimated as follows: Ce 3b 2 r 2 T for a constant C > 0. Since n h T ¼ n T À t n k ð Þ for t 2 t n k , t n kþ1 Â Á by (13), we obtain the desired result.

The global error
Our aim is to derive an upper bound for the modulus of the global error defined in (19), that is where the function g is an exponentially bounded Borel function and X s k ð Þ k¼0, 1, ::: is the random walk considered in Section 2.1. For this purpose, we need a collection of estimates related to the behavior of the random walk X s k ð Þ and the random variable J n . A part of these estimates are given in this section, while the more involved ones are presented later in Section 5.2 and Appendix A.2.
Note: Assumption 3.6 is taken as a standing assumption throughout Section 4.
Recall the definitions of n h and h n given in (12), and that J n x ð Þ ¼ inff2m 2 2N : s 2m x ð Þ > h n g as was defined in (18). A result similar to the lemma below was proved in [7,Corollary 11.4].
since cosh y ð Þ e y 2 =2 holds for any y 2 R: (ii) Firstly, observe that by the definition of J n we have jX s Jn À X h n j 2h: Secondly, The following upper bounds are later needed for the estimation of the global error.

Proposition 4.2.
(i) Suppose that p ! 0, g 2 B exp , and that b ! 0 is as in (14). Then there exists a constant C p > 0 such that for all x 2 R, C p e bjxjþb 2 r 2 T : Moreover, for every p > 0 there exists a constant C p > 0 such that iii ð Þ sup where n i ð Þ i¼1, 2, ::: is an i.i.d. Rademacher sequence (see Section 2.1). Hence, Consequently, by the symmetry of S n h and Markov's inequality, and thus, uniformly in (n, t), for p > 0, r pÀ1 e Àr 2 =2 dr :¼C p < 1: H€ older's inequality, (44), and (39) then imply that 1=2 2p e bjxjþb 2 r 2 T : This proves (41) for p > 0, and the case p ¼ 0 can be seen from the last line as well.
, by Markov's inequality and (44) we obtain for all q > 0. Choose q ! 10p and multiply both sides of (45) by n p h to obtain (42). (ii) For every K > 0, Markov's inequality and Proposition 5.9 imply that for some constant C K > 0: For given p > 0, it remains to choose K ! 10p and multiply both sides of (46) by n Proof. The rough idea behind the estimation of the global error is to decompose it into a sum of a part, which corresponds to certain moments of the random variables X s n h and J n À n h , and to a part, which can be bounded by a term which is "of the order" n Àp h for some p > 1. Define a set and decompose the error e glob n t, x ð Þ into the sum of expectations E 1 ð Þ and E 2 ð Þ , where Using the estimates of Lemma 4.1 and Proposition 4.2, it can be shown that for some constantC 0 > 0: This is done in Lemma A.2(i). Estimation of jE 1 ð Þ j requires more subtlety. Denote the probability mass functions of X s n h þk =h and J n À n h by P n h þk y ð Þ :¼ P X s n h þk ¼ yh À Á and P J n h y ð Þ :¼ P J n À n h ¼ y ð Þ , y 2 Z: By Lemma A.2(ii), there exists a constantC 1 > 0 such that Next, we use relation (78) in order to rewrite the double sum on the right-hand side of (50) as By Proposition 5.7, there exist constants c 1 , c 2 > 0 such that jE J n À n h ½ À 4 3 j c 1 n À 1 2 h and E J n Àn h ffiffiffi ffi h : In particular, the following inequalities hold: ffiffiffiffi ffi n h p : Consequently, by (51) and (41), there exist constantsC 2 ,C 3 > 0 such that To complete the proof, it remains to observe that n  (27). In this subsection we derive a representation for the function based on first exit time probabilities of a Brownian bridge. This representation (60) together with the associated estimates derived in [12] is applied in order to prove Proposition 5.5, the main result of this subsection. x, l, y t Cov B x, l, y s , B x, l, y is called a (generalized) Brownian bridge from x to y of length l.
Remark 5.2. By comparing mean and covariance functions, it is easy to verify that a Brownian bridge ðB x, l, y t Þ t2 0, l ½ is equal in law with the transformed processes below: A continuous version of a Brownian bridge ðB x, h, y t Þ t2 0, h ½ can be thought as a random function on the canonical space C 0, h ½ , B C 0, h ½ ð Þ, P x, h, y À Á , where P x, h, y denotes the associated probability measure. In the following proposition we give different characterizations for the function (53) in terms of hitting times. For all c 2 R, a < b, and x 2 C 0, h ½ , we let Then, for all k 2 Z, Þand L x ð Þ is even, the path t 7 ! X t x ð Þ does hit 2kh at s L x ð Þ and afterwards, i.e. on s L x ð Þ, h ½ Þ , it does not hit any other mh (m 6 ¼ 2k) and hence stays inside 2k À 1 ð Þ h, 2k þ 1 ð Þh À Á : Therefore, the last entry of this path into 2kh, 2k þ 1 ð Þh ð Þoccurs via 2kh, and thus where P 0 denotes the Wiener measure on C 0, h ½ , where we used relations (54), (59), and the fact that P Á jX h ¼ y Þ (see e.g. [13,Chapter 1] In addition, from (59) we deduce that : Then z 6 2 a, b ð Þ, a < 0 < b, b À a ¼ h=r, and hence by (61), (62), and d o y Before we proceed to prove the sharpness result for the class GBV exp , Proposition 5.5, we list the assertions of [12] which are needed for the proof.  The expression n 1 2 e loc n 0, 0 ð Þ is bounded from above by Theorem 3.8. For the lower bound, we note that by Definition 3.2, (which was introduced in (18)), which are applied in Section 4. We begin with a proposition which generalizes [7, Proposition 11.2(ii)-(iii)] to the time-dependent setting t 6 ¼ 0 ð Þ: For the proof, the reader is referred to [14, Section 6.1].
Proposition 5.7. Suppose that Assumption 3.6 holds. Then there exists a constant C > 0 such that for all n, t ð Þ 2 2N Â 0, T ½ Þ, To derive an estimate for EjJ n À n h j K for arbitrary K > 0, we recall (see, e.g. [15,Theorem 14.12]) a version of the Azuma-Hoeffding inequality.
Proposition 5.8 (Azuma-Hoeffding inequality). Suppose that M j ð Þ j¼0, 1, ::: is a martingale started for which M 0 ¼ 0 holds. In addition, assume that for all i ! 1 there exists a constant a i > 0 such that jM i À M iÀ1 j a i a.s. Then, for all k 2 N and every s > 0, The following result, proved below in the time-dependent setting, can be found in [7, Proposition 11.2(iv)] for t ¼ 0. The original proof, however, does not cover the case corresponding to the inequality (67) for the set A 3 .
Proposition 5.9. Suppose that Assumption 3.6 holds, and let K > 0. Then there exists a constant C K > 0 depending at most on K such that Proof. It suffices to prove the claim for K ! 2, since the case K 2 0, 2 ð Þ then follows by Jensen's inequality. Since jJ n À n h j is a non-negative random variable, We show that there exist constants C 1 Step 1: Since K ! 2 and n h ! 2, we have that Step 2: Suppose that n h > 2 and define d n h u ð Þ :¼ 2 n h b n h u 2 c: Then The idea here is to estimate the tail probability inside the integral on the last line of (68) with the help of Lemma A.6. To proceed, fix a constant a 2 0, 1 ð small enough such that for every m 2 N, where the function H is defined below in (91). Depending on the value of n h , we split the right-hand side of (68) into the sum of the integrals If a 2 2=n h , 1 ð Þ, by (69) and the fact that Þare even integers, we may apply Lemma A.6 and estimate By the properties of the floor function, for u 2 0, 1 ð it holds that and thus the right-hand side of (70) can be bounded from above by Here we used the fact that u 7 ! d n h u ð Þ is nondecreasing, that n h 1 þ d n h a=3 ð Þ À Á and n h 1 À d n h a=3 ð Þ À Á are (even) integers, condition (69), and inequality (71). Notice that the right-hand side converges to zero as n h ! 1: Consequently, there exists a constant C 2, 2 ð Þ K > 0 such that I 2, 2 n h ð Þ C 2, 2 ð Þ K for all n h , and (67) for k ¼ 2 follows.
Step 3: To estimate I 3 n h ð Þ , we apply the Azuma-Hoeffding inequality to the tail distribution of the random variable J n ¼ inf 2m 2 2N : s 2m > h n f g : Recall that s i À s iÀ1 , i ¼ 1, 2, :::, are i.i.d. (see Section 2.1) and that n T h n ¼ n h according to (4). Let f i :¼ n T s i À s iÀ1 ð Þ , i ! 1: Then, for all m 2 N, we have where N 2 N is chosen such that 3=4 < c N :¼ E f i Ù N ½ < E f i ½ ¼ 1: Then jE f i Ù N ½ À f i Ù Nj N for all i ! 1, and by (73) and the Azuma-Hoeffding inequality (Proposition 5.8), Since c N 2 3=4, 1 ð Þ, there exist constants c, c 0 > 0 such that Under Assumption 3.6, let us recall from (49) the notation P n h þk y ð Þ ¼ P X s n h þk ¼ hy À Á and P J n h y ð Þ ¼ P J n À n h ¼ y ð Þ , y 2 Z: Notice also that for all k 2 2N, P k y ð Þ ¼ k kþy 2 ! 2 Àk , y 2 2Z, jyj k: As in [7], we define the "effective order" of a monomial k p y q n r with p, q, r 2 N 0 to be Ô k p y q n r :¼ p þ q 2 À r: We will use the following result from [7] in the proof of Lemma A.2. Then there exists a constant C 0 > 0, an integer n 0 , and a finite sum R 2 ð Þ of monomials of effective order at most À3=2 such that for all n, k, y ð Þ2 D R ð Þ with n > n 0 , jR n, k, y ð ÞÀ 1 À R 1 ð Þ n, k, y ð Þþ R 2 ð Þ n, k, y ð Þ h i j C 0 n À3=2 : The lemma below presents upper estimates which are applied in the proof of Theorem 4.3.
Lemma A.2. Suppose that g 2 B exp and that b ! 0 is as in (14). Suppose also that R 1 ð Þ is as in (75) and that C n h is given by (47). Then there exists a constant C > 0 such that for all x 2 R and n h 2 2N,