Georg Müller

Mark Bergen

Shantanu Dutta

Daniel Levy
email: levyda@mail.biu.ac.il.

Allan ) Chen

Bob Chirinko

Hashem Dezhbakhsh

Akshay Rao

Sourav Ray

Private Label Price Rigidity during Holiday Periods

Keywords: E12, E31, L11, L20, L16, M21, M31 Price Rigidity, Holidays, Private Label, National Brand, Social Consumption

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Introduction

Although, the study of variation in price rigidity across markets, industries, and products has received some attention in the literature, numerous authors have called for more studies in this area, arguing that understanding the reasons for the variation "… is crucial for the theory of price adjustment" (Gordon (1981, p. 517). 1 In this paper we explore variation in price rigidity between private label and nationally branded products. 2 There is good reason to believe, a priori, that adjustment patterns may vary between private labels and national brands. For example, [START_REF] Barsky | What Can the Price Gap between Branded and Generic Products Tell Us about Markups? In Scanner data and price indexes[END_REF] find dramatically different markups for private labels than national brands across a wide variety of product categories. [START_REF] Dutta | Price Flexibility in Channels of Distribution: Evidence from Scanner Data[END_REF] document variation in cost shock pass-through patterns between private labels and national brands. 3 Holiday periods provide an interesting opportunity to study variation in price rigidity because studies have documented interesting pricing patterns during holiday periods. 4 For example [START_REF] Müller | Holiday Price Rigidity and Cost of Price Adjustment[END_REF] provide evidence that retail prices are more rigid during holiday periods. 5 Our discussions with retail managers also suggest that holiday periods are unusual for retailers in terms of their price adjustment activities.

We use store-level weekly time series data of actual retail and wholesale transaction prices for 4,532 products in 18 categories over a four-year period at a large Mid-western supermarket chain. We find greater price rigidity for private label products in comparison to national brands during holiday periods. We argue that this is due to the increased emphasis on social consumption during holiday periods and the corresponding substitution of branded products for private label items, which increases the value of nationally branded products relative to store brands.

The paper is organized as follows. In section 2, we describe the data. In section 3, we report the statistical test results. In section 4, we discuss the role of social consumption in determining the extent of holiday price rigidity. Section 5 concludes.

1 See also [START_REF] Mankiw | Small Menu Costs and Large Business Cycles: a Macroeconomic Model of Monopoly[END_REF], [START_REF] Carlton | The Rigidity of Prices[END_REF], [START_REF] Cecchetti | The Frequency of Price Adjustment: A study of the Newsstand Prices of Magazines[END_REF], [START_REF] Caplin | State-Dependent Pricing and the Dynamics of Money and Output[END_REF], [START_REF] Caplin | Individual Inertia and Aggregate Dynamics[END_REF], [START_REF] Sheshinski | Optimal Pricing, Inflation, and the Cost of Price Adjustment[END_REF], [START_REF] Basu | Intermediate Goods and Business Cycles: Implications for Productivity and Welfare[END_REF], [START_REF] Kashyap | Sticky Prices: New Evidence from Retail Catalogues[END_REF][START_REF] Blinder | Asking about Prices: A New Approach to Understanding Price Stickiness[END_REF], [START_REF] Slade | Optimal Pricing with Costly Adjustment: Evidence from Retail-Grocery Prices[END_REF][START_REF] Levy | Heterogeneity in Price Rigidity: Evidence from a Case Study Using Micro-Level Data[END_REF], and the references cited therein. 2 Private labels are also known as "store brands" and refer to the in-house brand, which is usually owned and sold by a particular retail supermarket chain. Nationally branded products are manufactured by big national producers and carried by most large supermarkets. Dominick's carries large number of private label products in nearly all categories we study. See [START_REF] Barsky | What Can the Price Gap between Branded and Generic Products Tell Us about Markups? In Scanner data and price indexes[END_REF] for more details. 3 There is also a growing literature in marketing, documenting differences between national brands and private labels. See, for example, [START_REF] Hoch | When Do Private Labels Succeed?[END_REF]. 4 See, for example, Pashigian (1994), [START_REF] Warner | The Timing and Magnitude of Retail Store Markdowns: Evidence from Weekends and Holidays[END_REF][START_REF] Warner | The Timing and Magnitude of Retail Store Markdowns: Evidence from Weekends and Holidays[END_REF][START_REF] Chevalier | Why Don't Prices Rise during Periods of Peak Demand? Evidence from Scanner Data[END_REF]. 5 They, however, do not study variation in the rigidity across private label and national brand products.

Data

The data come from Dominick's, a large Mid-western supermarket chain operating 94 stores in the Chicago area with a market share of about 25 percent. 6 The specific time series we use come from the scanner data of 6 stores of the chain and contain the retail prices, which are the actual transaction prices at the cash register, and information on the stores' promotional activity. The data are weekly, which parallels the chain's weekly pricing cycle, as documented by Slade (1998). 7 Our data include 4,532 products in 18 categories as listed in Table 1. The data cover a four-year period, from September 14-20 (1989 to September 16-22 (1993, yielding a total of 210 weekly observations.8

There are many holidays throughout the year, but few are as closely associated with consumption of food as Thanksgiving and Christmas. [START_REF] Warner | The Timing and Magnitude of Retail Store Markdowns: Evidence from Weekends and Holidays[END_REF] and [START_REF] Chevalier | Why Don't Prices Rise during Periods of Peak Demand? Evidence from Scanner Data[END_REF] also suggest that the two holidays are the busiest shopping periods. Indeed, our analysis shows that unit sales are 6% higher (with t = 3.22, significant at 1% level) during these holiday period. In fact, Dominick's managers describe the time just before Thanksgiving through the end of Christmas as "the holiday season." Thus, we define the 6-week period from the week before Thanksgiving through the week of Christmas as the holiday period.9

Findings

In Tables 2 and3 we report the average number of price changes per week, during holiday and non-holiday periods for national brand and private label products, respectively. A price change here is defined as a price difference in two successive weeks. The number of price changes is calculated across all six stores to accommodate possible variation in product selection across stores. Overall, the figures indicate that prices are more rigid during holiday weeks in comparison to the rest of the year, and that holds for both national brands and private labels. 10Comparing the extent of holiday price rigidity between national brand and private label products, we find that the phenomenon is more pronounced for the private label products. For example, according to Tables 2 and3, the frequency of price changes for national brand products declines by 13% on average (with t = -5.80, which is significant at 1% level). For private label products, in contrast, the frequency of price changes declines by 26% on average (with t = -4.23, which is significant at 1% level). In some categories, the size of the difference is particularly striking. These include Analgesics, 14% versus 58%, Cheeses, 3% versus 25%, Dish Detergents, 6% versus 38%, Frozen Juices, 9% versus 28%, Laundry Detergents, 20% versus 60%, Paper Towels, 21% versus 65%, Canned Fish, 17% versus 65%, and Tooth Paste, 17% versus 61%, respectively.

We considered the possibility that the finding may be related to changes in holiday period promotional practices. For this we computed the average number of promotions per week for national brands and private labels during holiday and non-holiday periods.

According to the results, which are reported in Tables 4 and5, promotional activity goes down for both national brands and private labels for most categories. On a percentage basis, however, private label promotions go down at least as much or more as national brands with the exception of two categories, Bottled Juices and Soft Drinks. An alternative explanation for the difference in the extent of holiday price rigidity between national brand and private label products may be related to differences in the extent of rigidity in their wholesale prices across the holiday/non-holiday periods. To examine this possibility, we calculated the average frequency of wholesale price changes during the holiday and non-holiday periods for both national brand and private label products. The wholesale price data also come from the chain's scanner database and is calculated from retail margins, which are included in the database. Despite their shortcomings, as [START_REF] Barsky | What Can the Price Gap between Branded and Generic Products Tell Us about Markups? In Scanner data and price indexes[END_REF] note, these wholesale price data are quite unique given that cost data are usually proprietary. According to Dominick's managers, they rely on these wholesale prices for making their pricing decisions. The results suggest that the increased rigidity of private label products relative to national brands is only partially accounted for by increased rigidity of wholesale prices in either case: the wholesale price changes decline only by 5% for national brands and by 8% for private labels. This finding holds for most of the individual categories as well. 11

The Role of Social consumption

We believe that at least some of the holiday price rigidity of private label products may be due to the increased emphasis on social consumption during holiday periods, which leads to an increase in the value of nationally branded products relative to store brand products. A large literature in marketing suggests that national brand products may be more valuable during holiday periods because they are consumed in social settings, with friends and family members. 12 People want to leave positive impressions, and show they care, by buying the "best," or at least do not want to appear cheap. Thus, social consumption that takes place during holiday periods decreases the value of private label products relative to nationally branded products. This makes a price change for private label products less effective during holiday periods. This will be particularly true if costs of price adjustment are indeed higher during holidays, as reported by [START_REF] Müller | Holiday Price Rigidity and Cost of Price Adjustment[END_REF]. We would expect, therefore, that it would be store brand products, i.e. those products whose price changes have lower marginal value, which would see fewer price changes relative to national brands. Indeed, this is what we find.

We considered alternate explanations but they seem inconsistent with our findings.

Consider Blinder et al.'s (1998) list of price rigidity theories. Theories based on the nature of contracts, imperfect information, or the nature of market interactions seem unable to explain these results because these factors are unlikely to vary between holiday and nonholiday periods and between national brand and private label products. Similarly, theories of price rigidity based on the nature of costs can be ruled out because cost factors are unlikely to vary between holiday and non-holiday periods. 13 Finally, demand based price adjustment during holiday periods. In this paper, our focus is on the differences between national brand and private label products. 11 Chevalier, et al. (2003) also find that changes in wholesale prices at this chain are small in absolute as well as in relative terms. 12 See, for example, [START_REF] Belk | It's the Thought that Counts: A Signed Digraph Analysis of Gift Giving[END_REF], [START_REF] Cheal | Showing them You Love Them: Gift Giving and the Dialectic of Intimacy[END_REF][START_REF] Otnes | Gift Selection for Easy and Difficult Recipients: a Social Roles Interpretation[END_REF]. 13 The costs of price adjustment theory, which, as Muller, et al. (2002) have argued, lead to the holiday price rigidity for both branded and private label products, is obviously an exception. rigidity theories are not likely to provide a satisfactory explanation because these factors do not vary between holiday and non-holiday weeks. For example, theories of pro-cyclical elasticity of demand actually predict greater price flexibility because during holiday periods, customers become more price-sensitive. Finally, quality or other non-price aspects of the products, as far as we know, do not vary between the holiday and nonholiday weeks.

Conclusion

Using unique data set on actual retail transaction prices for thousands of products at a major retail chain, we demonstrate that prices are more rigid for private labels than for national brands during holiday periods in comparison to non-holiday periods. We find that social consumption theory provides the best explanation for the phenomenon. This explanation lends credence to the importance of customer considerations in determining price change activity as suggested by [START_REF] Okun | Prices and Quantities: a Macroeconomic Analysis[END_REF] customer market theory, and more recently by [START_REF] Blinder | Asking about Prices: A New Approach to Understanding Price Stickiness[END_REF], [START_REF] Rotemberg | Customer Anger at Price Increases, Time Variation in the Frequency of Price Changes and Monetary Policy[END_REF], [START_REF] Ball | Inflation and the Informativeness of Prices[END_REF][START_REF] Ball | Inflation and the Informativeness of Prices[END_REF][START_REF] Zbaracki | Managerial and Customer Costs of Price Adjustment: Direct Evidence from Industrial Markets[END_REF], who emphasize the importance of customer considerations in firm-level price setting decisions. More generally, this study underscores the importance of studying price adjustment behavior during holiday periods, and the value of holiday periods as a "natural laboratory" for economists for studying various issues.

Table 1

 1 Product categories and the number of products in each category The number of products is calculated using the data for all six stores combined.

	No. Product Category	Number of Products
	1	Analgesics	1,362
	2	Bottled Juices	1,578
	3	Cereals	1,740
	4	Cheeses	2,262
	5	Crackers	822
	6	Canned Soups	1,824
	7	Dish Detergents	1,086
	8	Frozen Entrees	3,306
	9	Frozen Juices	702
	10	Fabric Softeners	1,176
	11	Laundry Detergents	2,160
	12	Paper Towels	510
	13	Refrigerated Juices	672
	14	Soft Drinks	3,666
	15	Snack Crackers	1,368
	16	Canned Fish	1,008
	17	Tooth Pastes	1,530
	18	Toilet Tissues	420
		Total	27,192
	Note:

Table 2

 2 Average number of price changes per week for national brand products Note: Superscripts a, b, and c indicate statistical significance at 1%, 5%, and 10%, respectively. The corresponding critical values are 2.33, 1.64, and 1.28, respectively.

	Category	Non-Holiday	Holiday	% Change	t-statistic
	Analgesics	74	64	-14%	-1.33 c
	Bottled Juices	152	125	-18%	-1.90 b
	Cereals	126	81	-36%	-2.69 a
	Cheeses	229	221	-3%	-0.36
	Crackers	87	72	-17%	-1.25
	Canned Soups	165	161	-2%	-0.27
	Dish Detergents	63	59	-6%	-0.56
	Frozen Entrees	336	210	-38%	-6.05 a
	Frozen Juices	93	85	-9%	-0.80
	Fabric Softeners	62	47	-24%	-2.12 b
	Laundry Detergents	109	87	-20%	-2.44 a
	Paper Towels	42	33	-21%	-1.93 b
	Refrigerated Juices	106	94	-11%	-1.47 c
	Soft Drinks	655	607	-7%	-1.55 c
	Snack Crackers	146	185	27%	2.04 b
	Canned Fish	81	67	-17%	-1.90 b
	Tooth Pastes	111	92	-17%	-1.27
	Toilet Tissues	51	39	-24%	-2.39 a
	Total	2,688	2,329	-13%	-5.80 a

Table 3

 3 Frequency of price changes per week for private label products

	Category	Non-Holiday	Holiday	% Change	t-statistic
	Analgesics	7.6	3.2	-58%	-3.23 a
	Bottled Juices	13.0	9.1	-30%	-2.07 b
	Cereals	11.0	5.7	-48%	-2.98 a
	Cheeses	56.0	42.0	-25%	-1.97 b
	Crackers	4.5	4.6	2%	0.08
	Canned Soups	8.3	8.3	0%	0.00
	Dish Detergents	7.4	4.6	-38%	-1.83 b
	Frozen Entrees	There are no private label products in this category
	Frozen Juices	13.0	9.4	-28%	-2.28 b
	Fabric Softeners	5.4	2.3	-57%	-4.19 a
	Laundry Detergents	4.0	1.6	-60%	-4.01 a
	Paper Towels	2.0	0.7	-65%	-3.81 a
	Refrigerated Juices	8.0	6.2	-23%	-1.39 c
	Soft Drinks	72.0	62.0	-14%	-0.91
	Snack Crackers	4.9	3.1	-37%	-1.73 b
	Canned Fish	3.7	1.3	-65%	-3.64 a
	Tooth Pastes	1.8	0.7	-61%	-2.29 b
	Toilet Tissues	3.0	1.5	-50%	-1.88 b
	Total	225.6	166.3	-26%	-4.23 a

Note: Superscripts a, b, and c indicate statistical significance at 1%, 5%, and 10%, respectively. The corresponding critical values are 2.33, 1.64, and 1.28, respectively.

Table 4

 4 Frequency of promotions per week for national brand products

	Category	Non-Holiday	Holiday	% Change	t-statistic
	Analgesics	18	32	78%	2.89 a
	Bottled Juices	79	61	-23%	-2.41 a
	Cereals	66	40	-39%	-4.11 a
	Cheeses	85	96	13%	0.77
	Crackers	39	59	51%	4.96 a
	Canned Soups	57	100	75%	1.67 b
	Dish Detergents	32	26	-19%	-1.64 b
	Frozen Entrees	172	75	-56%	-4.73 a
	Frozen Juices	46	48	4%	0.27
	Fabric Softeners	34	20	-41%	-4.67 a
	Laundry Detergents	69	42	-39%	-7.12 a
	Paper Towels	26	25	-4%	-0.43
	Refrigerated Juices	56	44	-21%	-2.88 a
	Soft Drinks	357	314	-12%	-2.07 b
	Snack Crackers	58	106	83%	2.15 b
	Canned Fish	23	92	300%	18.45 a
	Tooth Pastes	83	56	-33%	-3.16 a
	Toilet Tissues	28	28	0%	0.00
	Total	1,328	1,264	-5%	-1.26

Note: Superscripts a, b, and c indicate statistical significance at 1%, 5%, and 10%, respectively. The corresponding critical values are 2.33, 1.64, and 1.28, respectively.

Table 5

 5 Frequency of promotions per week for private label products

	Category	Non-Holiday	Holiday	% Change	t-statistic
	Analgesics	8.2	14.0	71%	1.47 c
	Bottled Juices	7.7	6.2	-19%	-1.23
	Cereals	5.3	2.8	-47%	-2.64 a
	Cheeses	25.0	28.0	12%	0.55
	Crackers	4.8	4.0	-17%	-0.67
	Canned Soups	1.8	2.9	61%	0.73
	Dish Detergents	2.6	3.7	42%	0.67
	Frozen Entrees	There are no private label products in this category
	Frozen Juices	8.9	7.0	-21%	-1.20
	Fabric Softeners	1.4	0.8	-43%	-1.16
	Laundry Detergents	0.9	0.3	-67%	-2.16 b
	Paper Towels	2.2	0.3	-86%	-5.47 a
	Refrigerated Juices	9.2	7.0	-24%	-1.49 c
	Soft Drinks	50.0	48.0	-4%	-0.22
	Snack Crackers	0.4	0.5	25%	0.28
	Canned Fish	3.1	0.5	-84%	-4.13 a
	Toothpastes	1.4	0.0	-100%	-5.50 a
	Toilet Tissues	1.1	0.0	-100%	-5.26 a
	Total	134.0	126.0	-6%	-0.67

Note: Superscripts a, b, and c indicate statistical significance at 1%, 5%, and 10%, respectively. The corresponding critical values are 2.33, 1.64, and 1.28, respectively.

For more details see[START_REF] Müller | Holiday Price Rigidity and Cost of Price Adjustment[END_REF],[START_REF] Barsky | What Can the Price Gap between Branded and Generic Products Tell Us about Markups? In Scanner data and price indexes[END_REF],[START_REF] Chevalier | Why Don't Prices Rise during Periods of Peak Demand? Evidence from Scanner Data[END_REF]. The data are available at www.gsb.uchicago.edu/research/mkt/MarketingHomePage.html.

Dominick's has three price zones, and the 6 stores in our sample are in the mid-price zone. Prices for all stores within the chain are set centrally at corporate headquarters. During data collection, the chain was conducting pricing experiments, but the stores in our sample come from the "control" group, which followed the same pricing strategy. We also analyzed data for three other stores that faced a greater competition but the results we obtained were nearly identical to what we report here.

Due to product additions and deletions, not all products have a full 210-week sales history.

We also ran the analyses for other combinations of holiday weeks, including two weeks before Christmas and two weeks after Christmas, or focusing on each holiday individually. Our results were similar for all the alternative combinations we ran. In addition, we run a similar analysis by including other holiday periods such as Memorial Day, 4 th of July, and Labor Day, but found that the holiday period price rigidity results primarily hold for the Thanksgiving and the Christmas holidays.

This finding, however, is not the focus of this paper, because it is reported and discussed in[START_REF] Müller | Holiday Price Rigidity and Cost of Price Adjustment[END_REF]. Their explanation for the greater holiday price rigidity is higher opportunity costs of price


L I Galtchouk

S M Pergamenshchikov
email: serge.pergamenshchikov@univ-rouen.fr

Non asymptotic sharp oracle inequalities for high dimensional ergodic diffusion models. *

Keywords: High dimensional diffusion models, Sharp oracle inequality, Discrete data, Model selection, Quadratic risk, Sequential estimation AMS 2000 Subject Classifications: primary 62G08, secondary 62G05, 62G20

In this paper we consider high dimensional ergodic diffusion models in nonparametric setting on the basis of discrete data, when the diffusion coefficients are unknown. For this problem, by using efficient sequential point-wise estimators we construct a model selection procedure and then we show sharp oracle inequalities, i.e. the inequalities in which the main term coefficient is closed to one. This means that the proposed sequential model selection procedure is optimal in this sense. Particularly, we show that the constructed procedure is the best in the class of weighted least square estimators with the Pinsker coefficients which provide the efficient estimation in the minimal asymptotical quadratic risk sense.

1 Introduction

Problem and Motivations

In this paper we consider the following diffusion model dy t = q j=1 θ j ψ j (y t ) dt + b(y t ) dW t , 0 ≤ t ≤ T , (1.1) where the functions (ψ j ) 1≤j≤q are known linear independent functions, (W t ) t≥0 is a standard Wiener process and the diffusion coefficient b(•) is an unknown function. Moreover, it is assumed that the observations are accessible only at discrete times, i.e. (y t j ) 1≤j≤N , t j = jδ ,

where the frequency δ = δ T ∈ (0, 1) is some function of T which will be specified later and the sample size N = N (T ) = [T /δ]. Here [a] denotes the integer part of a. It should be noted that in this case the diffusion coefficient is considered as a nuisance parameter. The problem is to estimate the unknown parameters (θ j ) 1≤j≤q in the "high dimensional" setting, i.e. when the number of parameters q > N . Usually, for such models one uses one of two methods: Lasso algorithm proposed in [START_REF] Tibshirani | Regression shrinkage and selection via the Lasso[END_REF] for i.i.d. regression models and modified later for diffusion regression models (1.1) in [START_REF] De Gregorio | Adaptive LASSO-type estimation for multivariate diffusion processes[END_REF] and the Dantzig selector method proposed in [START_REF] Candés | The Dantzig selector: statistical estimation when p is much larger than n[END_REF] and applied to diffusion models in [START_REF] Fujimori | The Danzing selector for a linear model of diffusion processes[END_REF]. But in all these papers the number of parameters q is known and, therefore, unfortunately, these methods can't be applied to estimate the number of parameters in regression models. In this paper we study this problem in the nonparametric estimation framework, i.e. we study the diffusion process defined as dy t = S(y t ) dt + b(y t ) dW t (1.3) and the problem is to estimate the unknown function S(•) on the basis of the observations (1.2). The nonparametric setting allows to consider the models (1.1) with unknown q or with q = +∞. Note that, the case when the number of parameters q is unknown is one of challenging problems in the signal processing theory (see, for example, [START_REF] Beltaief | Model selection for the robust efficient signal processing observed with small Levy noise[END_REF]). The models (1.3) are very important for various fields in the theory of stochastic processes such as optimal control, identification problem, filtration, financial markets, insurance etc (see, for example, [START_REF] Kabanov | Two Scale Stochastic Systems: Asymptotic Analysis and Control[END_REF][START_REF] Karatzas | Methods of Mathematical Finance[END_REF][START_REF] Kutoyants | Statistical Inferences for Ergodic Diffusion Processes[END_REF][START_REF] Lamberton | Introduction to stochastic calculus applied to finance[END_REF][START_REF] Liptser | Statistics of a random process[END_REF]). Nonparametric estimation problems of the drift S were studied in a number of papers in the case of complete observations, i.e. when the whole trajectory (y t ) 0≤t≤T is observed. In [START_REF] Galtchouk | Sequential nonparametric adaptive estimation of the drift coefficient in diffusion processes[END_REF][START_REF] Galtchouk | Nonparametric sequential minimax estimation of the drift coefficient in diffusion processes[END_REF][START_REF] Galtchouk | Asymptotically efficient sequential kernel estimates of the drift coefficient in ergodic diffusion processes[END_REF] a sequential approach was proposed for the point-wise efficient estimation. In [START_REF] Dalalyan | Sharp Adaptive Estimation of the Trend Coefficient for Ergodic Diffusion[END_REF][START_REF] Dalalyan | Asymptotically efficient trend coefficient estimation for ergodic diffusion[END_REF][START_REF] Galtchouk | Nonparametric sequential estimation of the drift in diffusion via model selection[END_REF][START_REF] Galtchouk | Adaptive sequential estimation for ergodic diffusion processes in quadratic metric[END_REF] this problem was studied for integral risks when the diffusion coefficient is known. In practice, usually only discrete time observations are accessible. A natural question arises about proprieties and the behavior of estimates based on discrete time observations for such models. Nonparametric estimation based on discrete time observations for models (1.3) was considered firstly for estimating the squared unknown diffusion coefficient b 2 (•) on a fixed interval [0, T ] (see, for example, [START_REF] Florens-Zmirou | On estimating the diffusion coefficient from discret observations[END_REF][START_REF] Genon-Catalot | Nonparametric estimation of the diffusion coefficient by wavelets methods[END_REF][START_REF] Hoffmann | Adaptive estimation in diffusion processes[END_REF][START_REF] Jacod | Non-parametric kernel estimation of the coefficient of a diffusion[END_REF] and the references therein). Later, in [START_REF] Gobet | Nonparametric estimation of scalar diffusions based on low frequency data[END_REF] kernel estimates for the drift and diffusion coefficients were studied for reflecting ergodic (1.3), i.e. for the processes with values in [0, 1]. Concerning the estimation in the ergodic case, a sequential procedure was proposed in [START_REF] Hoffmann | Adaptive estimation in diffusion processes[END_REF] for nonparametric estimating the drift coefficient in an integral metric and in [START_REF] Galtchouk | Efficient pointwise estimation based on discrete data in ergodic nonparametric diffusions[END_REF] a special sequential method was developed to provide the efficient estimation for the point-wise risks, i.e. sequential kernel procedures for which normalized minimax risks attain a minimal value. This property is very important for nonparametric estimation, since usually the convergence rate is very slow, therefore, the influence of the value of the normalized risks in the estimation accuracy is much more important than in the parametric cases.

Main tool

In this paper for the problem (1.3) we develop a new model selection method which is an adaptive selection rule λ of an estimator S * = S λ in the family of weighted least square estimators ( S λ ) λ∈Λ . Our goal is to show that the constructed procedure is optimal in the sense of the following oracle inequality: for any small ρ > 0 and any T > 0

E S * -S 2 ≤ (1 + ρ) min λ∈Λ E S λ -S 2 + B T ρT , (1.4)
where f 2 = b a f 2 (x)dx for some a < b and the term B T is some slowly varying function of T , i.e. for any δ > 0 lim

T →∞ B T T δ = 0 . (1.5)
It should be noted that for the first time, such inequalities for non-Gaussian regression models in discrete time were obtained in [START_REF] Galtchouk | Sharp non-asymptotic Oracle inequalities for nonparametric heteroscedastic regression models[END_REF], and for continuous processes in [START_REF] Konev | Efficient robust nonparametric estimation in a semimartingale regression model[END_REF]. Later, such inequalities were called sharp (since the main coefficient 1 + ρ arbitrarily close to one). Moreover, it should be noted that such inequalities allow us to prove asymptotic efficiency without knowing the regularity of the estimated function (see, for example, [START_REF] Galtchouk | Adaptive sequential estimation for ergodic diffusion processes in quadratic metric[END_REF]).

Our approach is based on the sequential estimation method developed in [START_REF] Galtchouk | Efficient pointwise estimation based on discrete data in ergodic nonparametric diffusions[END_REF] for the efficient nonparametric point-wise estimation in a special functional class. Similarly to [START_REF] Galtchouk | Adaptive sequential estimation for ergodic diffusion processes in quadratic metric[END_REF] we will use these estimators to pass to a heteroscedastic regression model and, then, using the methods developed in [START_REF] Galtchouk | Sharp non-asymptotic Oracle inequalities for nonparametric heteroscedastic regression models[END_REF][START_REF] Galtchouk | Adaptive asymptotically efficient estimation in heteroscedastic nonparametric regression[END_REF] for such regression we construct the sequential model selection procedure. To obtain the oracle inequalities we use a special uniform concentration inequality from [START_REF] Galtchouk | Uniform concentration inequality for ergodic diffusion processes observed at discrete times[END_REF] obtained for discrete time observations.

It should be noted also that in this paper we construct the model selection procedure for arbitrary orthonormal basis. We recall that in [START_REF] Galtchouk | Adaptive sequential estimation for ergodic diffusion processes in quadratic metric[END_REF] the model selection procedure was constructed only for the trigonometric basis. In practice, this basis is not well appropriate sometimes to signal processing problems (see, for example, [START_REF] Chernoyarov | Restoration of deterministic and interference distorted signals and images with use of the generalized spectra based on orthogonal polynomials and functions[END_REF][START_REF] Debnath | Wavelet Transforms and Their Applications[END_REF] and the references therein). So in the paper, we develop a new analytical tool to provide sharp oracle inequalities for any orthonormal basis on the grid.

Plan of the paper

The paper is organized as follows. In Section 2 we describe functional classes that will be used. Sequential estimators of the drift coefficient are constructed in Section 3. In Section 4 we introduce a regression model based on the sequential estimators. In Section 5 we construct the model selection procedure based on the sequential pointwise estimators. Main results are announced in Section 6 and their proofs are given in Section 8. In Section 7 we study some properties of the model selection. In Appendix we give all auxiliary results.

Main Conditions

In the paper we consider the estimation problem for the drift S on the interval [x 0 , x 1 ], where x 0 < x 1 are some arbitrary fixed points. In order to obtain a reliable estimator of S, it is necessary to impose some conditions on this function which are similar to the periodicity of the deterministic signal in the white noise model. One of conditions which is sufficient for this purpose is the assumption that the process (y t ) t≥0 in (1.3) returns to any vicinity of each point x ∈ [x 0 , x 1 ] infinite times. The ergodicity provides this property, when the coefficients of equation are known. In the case of unknown coefficients, one needs to impose the uniform ergodicity property. To obtain the uniform ergodicity property for the process (1.3) we use the functional class introduced in [START_REF] Galtchouk | Efficient pointwise estimation based on discrete data in ergodic nonparametric diffusions[END_REF], i.e. for any fixed L ≥ 1, M > 0 and

x * > |x 0 | + |x 1 | we set Σ L,M = S ∈ C 1 (R) : sup |x|≤x * |S(x)| + | Ṡ(x)| ≤ M , -L ≤ inf |x|≥x * Ṡ(x) ≤ sup |x|≥x * Ṡ(x) ≤ -1/L . (2.1)
Here and in the sequel we denote by ḟ and f the correspoding derivatives. Moreover, for some fixed parameters 0

< b min ≤ b max < ∞ we denote by B the class of functions b from C 2 (R) such that b min ≤ inf x∈R |b(x)| ≤ sup x∈R max |b(x)| , | ḃ(x)| , | b(x)| ≤ b max . (2.2)
Now we set

Θ = Σ L,M × B = (S, b) : S ∈ Σ L,M and b ∈ B . (2.3)
It is easy to see that the functions from Σ L,M are uniformly bounded on [x 0 , x 1 ], i.e. s * = sup

x 0 ≤x≤x 1 sup S∈Σ L,M S 2 (x) < ∞ . (2.4)
It should be noted that, for any ϑ ∈ Θ, there exists an invariant density for the process (1.3) which is defined as

q ϑ (x) = R b -2 (z) e S(z) dz -1 b -2 (x) e S(x) , (2.5)
where S(x) = 2

x 0 b -2 (v)S(v)dv (see,e.g., [START_REF] Gihkman | Stochastic differential equations[END_REF], Ch.4, 18, Th2). It is easy to see that this density is uniformly bounded on the class (2.3) , i.e.

q * = sup x∈R sup ϑ∈Θ q ϑ (x) < +∞ (2.6)
and bounded away from zero on the interval [-x * , x * ], i.e.

q * = inf |x|≤x * inf ϑ∈Θ q ϑ (x) > 0 . (2.7) For any q ϑ -integrable R → R function f , i.e. R |f (x)| q ϑ (x) dx < ∞, we set m ϑ (f ) = R f (x) q ϑ (x) dx . (2.8)
We need the following condition for the observation frequency.

A 1 ) The frequency δ in the observations (1.2) has the following form

δ = δ T = 1 (T + 1)l T , (2.9)
where the function l T is such that, lim

T →∞ l T ln T = +∞ . (2.10)
For example, one can take l T = (ln T ) 1+ι for some ι > 0.

In this paper we consider the quadratic risk defined for any estimator S as

R( S, S) = E ϑ x 1 x 0 | S(x) -S(x)| 2 dx , (2.11)
where E ϑ is the expectation with respect to the distribution of the process (1.3) under the parameter ϑ ∈ Θ.

Remark 2.1. Note that we consider the estimation problem only for the drift function S, i.e. in this case the diffusion coefficient b is considered as a nuisance parameter.

Sequential point-wise estimation

In order to obtain a reliable estimator of the function S on the interval [x 0 , x 1 ] we need some efficient point-wise estimators of this function. To give such estimators, we begin with the partition of the interval [x 0 , x 1 ] by points (z k ) 0≤k≤n defined as

z k = x 0 + k(x 1 -x 0 ) n , (3.1)
where n = n(T ) is an integer-valued function of T such that lim

T →∞ n(T ) √ T = 1 . (3.2)
To construct an efficient procedure for S(z k ), at any point z k , we need to use some estimators of the invariant density q ϑ (•) and the squared diffusion coefficient b 2 (•). To this end, we will estimate these functions by making use of the first N 0 observations, i.e. (y t j ) 1≤j≤N 0 . We begin with estimating the function S. We set

N 0 = [N γ (T )] with 3/4 < γ < 1 . (3.3)
Now we estimate S(z k ), at every point z k , by making use of the observations (y t j ) N 0 +1≤j≤N . To this end we use the sequential kernel estimator from [START_REF] Galtchouk | Asymptotically efficient sequential kernel estimates of the drift coefficient in ergodic diffusion processes[END_REF] with the indicator kernel function χ(y) = 1 {|y|≤1} and the sample size given by the stopping time

τ k = inf    l ≥ N 0 + 1 : l j=N 0 +1 χ j,k (h) ≥ H k    , (3.4)
where

χ j,k (h) = χ y t j-1 -z k h , h = x 1 -x 0 2n
and H k is some positive random threshold which will be specified later. On the set

Γ k = {τ k ≤ N } (3.5)
we define the correction coefficient 0 < κ k ≤ 1 from the equation

τ k -1 j=N 0 +1 χ j,k (h) + κ k χ τ k ,k (h) = H k . (3.6)
Moreover, on the Γ c k we set κ k = 0. Using this definition we introduce the weight sequence

κ j,k = 1 {j<τ k } + κ k 1 {j=τ k } . (3.7)
One can check directly that, for any j ≥ 1, the coefficients κ k,j are G j-1

measurable, where G j = σ y t l , 0 ≤ l ≤ j . Now we define the sequential estimator for S(z k ) as

S * k = 1 δH k τ k j=N 0 +1 κ j,k χ j,k (h)∆y t j 1 Γ k , (3.8)
where ∆y t j = y t j -y t j-1 .

To specify the threshold H k we need a truncated estimator q(z k ) for the invariant density q ϑ (z k ) given in (3.12) which must be greater than some positive function υ T . We impose the next condition on this function.

A 2 ) Assume, that lim

T →∞ υ T + ln T T (υ T ) 2 + ln T l T (υ T ) 5 = 0 . (3.9)
For example, one can take υ T = ln -ι (T +1) and l T ≥ ln 1+6ι T , for some ι > 0. Finally, as in [START_REF] Galtchouk | Efficient pointwise estimation based on discrete data in ergodic nonparametric diffusions[END_REF], we set

H k = h(N -N 0 )(2 q(z k ) -υ T ) . (3.10)
Note that in [START_REF] Galtchouk | Sequential nonparametric adaptive estimation of the drift coefficient in diffusion processes[END_REF] it has been shown that such form of the threshold H k provides the optimal convergence rate.

Remark 3.1. It should be noted that it is not necessary to have a good estimator for the invariant density. As we will see later it suffices to construct some rough estimators, just to correct corresponding coefficients.

Remark 3.2. In fact, our procedure uses only the observations belonging to the interval

[z k -h , z k + h].
This implies that the sample size equals asymptotically to 2N h q(z k ). This is related with the choice of the estimator kernel that is an indicator function. It is easy to verify (see [START_REF] Galtchouk | Nonparametric sequential minimax estimation of the drift coefficient in diffusion processes[END_REF]) that this kernel minimizes the variance of stochastic term in the kernel estimator.

Ultimately, the last result provides efficiency of the procedure.

Estimation of the invariant density

To estimate the density q ϑ we will make use of the following kernel estimator

q(z k ) = 1 2 N 0 h 0 N 0 j=2 χ j,k (h 0 ) , (3.11)
where h 0 (T ) = 1 T 0 and T 0 = δN 0 .

We set

q(z k ) =          (υ T ) 1/2 , if q(z k ) < (υ T ) 1/2 ; q(z k ) , if (υ T ) 1/2 ≤ q(z k ) ≤ (υ T ) -1/2 ; (υ T ) -1/2 , if q(z k ) > (υ T ) -1/2 .
(3.12)

The properties of this estimator are studied in [START_REF] Galtchouk | Efficient pointwise estimation based on discrete data in ergodic nonparametric diffusions[END_REF].

Estimation of the squared diffusion coefficient

To estimate the squared diffusion coefficient b 2 (z k ) we use the following sequential procedure. First we define the sample size for this procedure. To do this we set

τ * ,k = inf{j ≥ 1 : j l=1 χ l,k (h 0 ) ≥ H * } ∧ N 0 , (3.13)
where

h 0 = 1 T 0 and H * = 1 ln(T + 1) h 0 N 0 .
Then we set

b k = τ * ,k j=1 χ j,k (h 0 )(y t j -y t j-1 ) 2 δ H * 1 Γ * ,k , (3.14)
where Γ * ,k = { N 0 j=1 χ j,k (h 0 ) ≥ H * }. This estimator satisfies the following property. Proposition 3.1. For any a > 0, lim

T →∞ sup ϑ∈Θ max 1≤k≤n T γ-1/2-a E ϑ | b k -b 2 (z k )| = 0 . ( 3

Regression model

In order to obtain an oracle inequality for discrete time data, we shall pass to a regression model by the same way as in [START_REF] Galtchouk | Adaptive sequential estimation for ergodic diffusion processes in quadratic metric[END_REF]. From (1.3) and (3.8) it follows that, for any k on the set Γ k ,

S * k = S(z k ) + B k + ζ k , (4.1)
where the approximation term

B k = 1 δH k τ k j=N 0 +1 κ j,k χ j,k (h) t j t j-1 S(y u ) du -S(z k ) (4.2)
and the stochastic term

ζ k = 1 δH k τ k j=N 0 +1 κ j,k χ j,k (h) t j t j-1 b(y u ) dW u .
Therefore, if we set

ξ k = 1 δH k τ k j=N 0 +1 κ j,k χ j,k (h)∆W t j and σ k = b(z k ) δH k , (4.3)
then we can rewrite the stochastic term as

ζ k = B k + σ k ξ k ,
where

B k = 1 δH k τ k j=N 0 +1 κ j,k χ j,k (h)
t j t j-1 (b(y s ) -b(z k ))dW s . (4.4)
So, setting

G * = ∩ n k=1 Γ k and Y k = S * k 1 G * , (4.5)
we can represent the equality (4.1) on the set G * as nonparametric regression model, i.e. for any 1

≤ k ≤ n, Y k = S(z k ) + g k + σ k ξ k , g k = B k + B k . (4.6)
Note that the coefficients (σ l ) 1≤l≤n are random variables and using their definitions one can obtain the following bounds

σ 0, * ≤ min 1≤l≤n σ 2 l ≤ max 1≤l≤n σ 2 l ≤ σ 1, * , (4.7)
where

σ 0, * = υ T b min δN h and σ 1, * = b max υ T δ(N -N 0 )h . Now we set g * T = T max 1≤k≤n sup ϑ∈Θ E ϑ g 2 k 1 G * (4.8)
Proposition 4.1. Assume that the observation frequency δ and the bandwidth h are defined in (2.9) and (3.4), respectively. Then, for any a > 0, lim

T →∞ g * T T a = 0 . (4.9)
Concerning the random variables (ξ k ) 1≤k≤n , we can show the following property.

Proposition 4.2. The random variables (ξ k ) 1≤k≤n are N (0, 1) i.i.d. conditionally to G N 0 .

Similarly to the proof of Proposition 4.5 in [START_REF] Galtchouk | Efficient pointwise estimation based on discrete data in ergodic nonparametric diffusions[END_REF] and using Theorem A.1 we can obtain the following property for the set G * in (4.5).

Proposition 4.3. Assume that the conditions A 1 ) -A 2 ) hold. Then, for any a > 0, lim

T →∞ T a sup ϑ∈Θ P ϑ (G c * ) = 0 . (4.10)
We estimate the parameter σ 2 l as follows:

σ l = b l δH l , (4.11)
where b l is the estimator of the squared diffusion coefficient b 2 (z l ) defined in (3.14).

In order to write the oracle inequality, we need to study the properties of the last estimator. To this end we set *

T = n max 1≤l≤n E ϑ | σ l -σ 2 l | . (4.12)
Proposition 4.4. Assume that the conditions A 1 ) -A 2 ) hold. Then, for any a > 0, lim

T →∞ T γ-1/2-a * T = 0 . (4.13)
Remark 4.1. Note, that the main technical difference of the considering case from the estimation problem on the complete observations considered in [START_REF] Galtchouk | Adaptive sequential estimation for ergodic diffusion processes in quadratic metric[END_REF] is such that in the heteroscedastic regression model (4.6) the noise variances tend to zero as n → ∞ since the number of the point is proportionally to √ T , but not to T as in [START_REF] Galtchouk | Adaptive sequential estimation for ergodic diffusion processes in quadratic metric[END_REF]. By this reason we need to normalize the difference in (4.12) by n.

Model selection

First we choose a basis (φ j ) j≥1 in L 2 ([x 0 , x 1 ]) such that, for any 1 ≤ i, j ≤ n,

(φ i , φ j ) n = x 1 -x 0 n n l=1 φ i (z l )φ j (z l ) = 1 {i=j} . (5.1)
For example, one can take the trigonometric basis defined as Tr 1 (x) ≡ 1/ √ x 1 -x 0 and, for j ≥ 2,

Tr j (x) = 2 x 1 -x 0    cos(2π[j/2] l 0 (x)) for even j ; sin(2π[j/2]l 0 (x)) for odd j , (5.2)
where [a] denotes the integer part of a and l 0 (x) = (x -x 0 )/(x 1 -x 0 ). Note that if n is odd, then this basis is orthonormal for the empirical inner product, i.e. satisfies the property (5.1). By making use of this property we define the discrete Fourier representation for S on the sieve (3.1), i.e.,

S(z

k ) = n j=1 θ j,n φ j (z k ) , 1 ≤ k ≤ n , (5.3)
where

θ j,n = (S, φ j ) n = x 1 -x 0 n n l=1 S(z l )φ j (z l ) .
Moreover, using the regression model (4.6) we estimate these coefficients as

θ j,n = (Y, φ j ) n = x 1 -x 0 n n l=1
Y l φ j (z l ) .

(5.4)

By the model (4.6), we obtain on the set G *

θ j,n = θ j,n + ζ j,n , ζ j,n = g j,n + x 1 -x 0 n ξ j,n , (5.5)
where

ξ j,n = x 1 -x 0 n n l=1 σ l ξ l φ j (z l ) , g j,n = x 1 -x 0 n n l=1 g l φ j (z l ) .
We estimate the values S(z k ), 1 ≤ k ≤ n, by the weighted least squares estimators

S λ (z k ) = n j=1 λ(j) θ j,n φ j (z k ) , 1 ≤ k ≤ n , (5.6)
where the weight vector λ = (λ(1), . . . , λ(n)) belongs to some finite set Λ from [0, 1] n . In the sequel, we denote by ν the cardinal number of the set Λ, ν = card(Λ), which is a function of T , i.e. ν = ν T . Moreover, we set the norm for Λ as

Λ * = max λ∈Λ n j=1 λ(j) (5.7)
which can be a function of T , i.e. Λ * = Λ * (T ). We assume that the basis functions and the weights set Λ satisfy the following condition.

A 3 )For any a > 0, lim

T →∞ φ * T + ν T T a = 0 and lim T →∞ Λ * (T ) T 1/6+a = 0 , (5.8)
where φ * T = max 1≤j≤n max x 0 ≤x≤x 1 |φ j (x)|. To estimate the function S on the interval x ∈ [x 0 , x 1 ], we use the stepfunction approximation, i.e.,

S λ (x) = n l=1 S λ (z l )1 {z l-1 <x≤z l } .
(5.9)

Now one needs to choose a cost function in order to define an optimal weight λ ∈ Λ. A best candidate for the cost function should be the empirical squared error given by the relation

Err n (λ) = S λ -S 2 n → min .
In our case, the empirical squared error is equal to

Err n (λ) = n j=1 λ 2 (j) θ 2 j,n -2 n j=1 λ(j) θ j,n θ j,n + n j=1 θ 2 j,n . (5.10)
Since coefficients θ j,n are unknown, we need to replace the term θ j,n θ j,n by some estimator which we choose as

θ j,n = θ 2 j,n - x 1 -x 0 n σ j,n and σ j,n = x 1 -x 0 n n l=1 σ l φ 2 j (z l ) , (5.11)
where σ l is the estimator for σ 2 l defined in (4.11). Note that if the diffusion is known, then we take in (5.11) σ j,n = σ j,n and

σ j,n = x 1 -x 0 n n l=1 σ 2 l φ 2 j (z l ) .
(5.12)

It is clear that the inequalities (4.7) imply

σ 0, * ≤ min 1≤l≤n σ l,n ≤ max 1≤l≤n σ l,n ≤ σ 1, * . (5.13)
Now, for using the estimator (5.11) instead of θ j,n θ j,n one needs to add to the cost function a corresponding penalty term that we take as

P n (λ) = x 1 -x 0 n n j=1 λ 2 (j) σ j,n (5.14)
if the diffusion is unknown and as

P n (λ) = x 1 -x 0 n n j=1 λ 2 (j) σ j,n (5.15)
when the diffusion is known. Finally, we use the following cost function

J n (λ) = n j=1 λ 2 (j) θ 2 j,n -2 n j=1 λ(j) θ j,n + ρ P n (λ) , (5.16)
where the positive coefficient 0 < ρ < 1 will be specified later. We define the model selection procedure as λ = argmin λ∈Λ J n (λ) and S * = S λ .

(5.17)

Remark 5.1. It should be emphasized that if in the model (1.3) the diffusion coefficient b(•) in known, then the model selection procedure (5.17) is defined through the minimazing the cost function J n (λ) with the penalty term (5.15).

Main results

First we set

U T = ν υ 3 T 1 + g * T + Λ * * T + T /υ T sup ϑ∈Θ P ϑ (G c * ) , (6.1)
where the function υ T is given in (3.10). Now we give the oracle inequality for the discrete norm (5.1).

Theorem 6.1. Assume that the conditions A 1 ) -A 3 ) hold. Then there exists a positive constant ľ > 0 such that for any T ≥ 1, 0 < ρ ≤ 1/8 and ϑ ∈ Θ, the following oracle inequality holds for the model selection procedure (5.17) :

E ϑ S * -S 2 n ≤ 1 + 5ρ 1 -6ρ min λ∈Λ E ϑ S λ -S 2 n + ľU T ρ T . (6.2)
Moreover, for any a > 0, lim

T →∞ U T T a = 0 . (6.3)
Note that, if the diffusion coefficient b(•) is known, then the term * T = 0 in the function U T . Using Lemma A.3 with ε = ρ we obtain now the oracle inequality for the risk (2.11). Theorem 6.2. Assume that the conditions A 1 ) -A 3 ) hold. Then, for any T ≥ 1, 0 < ρ ≤ 1/8 and ϑ ∈ Θ, the estimation procedure S * defined in (5.17) satisfies the following inequality

R( S * , S) ≤ (1 + ρ) 2 (1 + 5ρ) 1 -6ρ min λ∈Λ R n ( S λ , S) + B T ρ T , (6.4)
where the rest term B T satisfies the limit the property (6.3) for any a > 0.

In the sequel to obtain the efficient properties for this procedure we will use the special weight coefficients introduced in [START_REF] Galtchouk | Sharp non-asymptotic Oracle inequalities for nonparametric heteroscedastic regression models[END_REF]. To this end consider a numerical grid of the form

A = {1, . . . , k * } × {r 1 , . . . , r m * } ,
where r i = iε and m * = [1/ε 2 ]. The both parameters k * ≥ 1 and 0 < ε ≤ 1 are some functions of T , i.e. k * = k * T and ε = ε T , such that, for any γ > 0, lim

T →∞ ε T + 1 T γ ε T + 1 k * T + k * T ln T = 0 . (6.5)
One can take, for example,

ε T = 1 ln(T + 1)
and k * = k + ln(T + 1), for some fixed k ≥ 0. For each α = (β, r) ∈ A we introduce the weight sequence λ α = (λ j (α)) j≥1 as

λ j (α) = 1 {1≤j≤j 0 } + 1 -(j/ω α ) β 1 {j 0 <j≤ωα} , (6.6)
where 2β+1) and

j 0 = j 0 (α) = [ω α / ln(n + 1)], ω α = ω * (r n) 1/(
ω * = (x 1 -x 0 ) (β + 1)(2β + 1) π 2β β 1/(2β+1)
.

We set Λ = (λ α ) α∈A . (6.7)

Note that, in this case, the cardinal ν of the set Λ is the function of T , i.e. ν = ν T = k * m * and the conditions (6.5) imply that for any a > 0 lim

T →∞ ν T T a = 0 . (6.8)
Moreover, from the definition (6.6) we can obtain that, for any α ∈ A,

n j=1 λ j (α) ≤ ω α ≤ ω * n ε T 1/3 .
Therefore, for any a > 0, lim

T →∞ Λ * T 1/6+a = 0 . (6.9)
Hence, the condition A 3 ) holds and we obtain the following theorem.

Theorem 6.3. Assume that the conditions A 1 ) -A 2 ) hold. Then, the model selection procedure (5.16) with the weights (6.7) satisfies the oracle inequality (6.4) with the rest term satisfying the property (6.3) for any a > 0.

Remark 6.1. It should be note that similarly to [START_REF] Galtchouk | Adaptive sequential estimation for ergodic diffusion processes in quadratic metric[END_REF], we will use the inequality (6.4) to provide the efficiency property in the adaptive setting. This means that without using the regularity properties of the unknown function S we can estimate from above the risk for the model selection procedure by the risk for the efficient estimation procedure constructed on the regularity properties of the function S. The upper bound for the risk of the model selection procedure follows from the sharp oracle inequality.

7 Properties of the model (5.5)

In order to prove the oracle inequalities, we need to study the conditions introduced in [START_REF] Konev | Efficient robust nonparametric estimation in a semimartingale regression model[END_REF] for the general semimartingale model. To this end, for any λ ∈ R n , we define the functions

Ξ(λ) = x 1 -x 0 n n j=1 λ(j) ξ j,n and B(λ) = x 1 -x 0 √ n n j=1 λ(j) ξ j,n , (7.1)
where ξ j,n = ξ 2 j,n -σ j,n , the variables ξ j,n are from (5.5), σ j,n = Eξ 2 j,n .

Proposition 7.1. For any n ≥ 1 and any λ

= (λ 1 , . . . , λ n ) ∈ [0, 1] n , E ϑ Ξ 2 (λ) ≤ σ 1, * (x 1 -x 0 ) n |λ| 2 . (7.2)
Proof. From Proposition 4.2, (5.1), (5.5) and (4.7) we can obtain directly that

E ϑ Ξ 2 (λ) = (x 1 -x 0 ) 2 n 2 E ϑ n l=1 σ 2 l   n j=1 λ(j)φ j (z l )   2 ≤ σ 1, * (x 1 -x 0 ) 2 n 2 n j,k=1 λ(j)λ(k) n l=1 φ j (z l )φ k (z l ) = σ 1, * (x 1 -x 0 ) n n j=1 λ 2 (j) (x 1 -x 0 ) n n l=1 φ 2 k (z l ) .
Hence Proposition 7.1.

Proposition 7.2. For any n ≥ 1 and any λ

= (λ 1 , . . . , λ n ) ∈ [0, 1] n , E ϑ [B 2 (λ)| G N 0 ] ≤ 6σ 1, * (x 1 -x 0 ) P n (λ) . (7.3)
Proof. We begin with presentation of the variables ξ j,n and B(λ) from (7.1) as follows

ξ j,n = (x 1 -x 0 ) n n l=1 σ 2 l φ 2 j (z l )η l + 21 l≥2 ξ l R j,l ,
where η l = ξ 2 l -1, R j,l = σ l φ j (z l ) l-1 r=1 σ r φ j (z r )ξ r , and

B(λ) = (x 1 -x 0 ) 2 n 3/2 n l=1 η l γ 1,l + 2ξ l γ 2,l with γ 1,l = σ 2 l n j=1 λ(j)φ 2 j (z l ), γ 2,l = n j=1 λ(j)R j,l 1 l≥2 . In view of Proposition 4.2, one has E B 2 (λ)|G N 0 = (x 1 -x 0 ) 4 n 3 n l=1 2γ 2 1,l + 4E [γ 2 2,l |G N 0 ] := M 1,1 + M 1,2 .
Due to the Buniakovski-Cauchy-Schwartz inequality and the property (5.1), one has

γ 2 1,l = σ 4 l   n j=1 λ(j)φ 2 j (z l )   2 ≤ σ 4 l   n j=1 λ 2 (j)φ 2 j (z l )     n j=1 φ 2 j (z l )   ≤ σ 1, * n (x 1 -x 0 ) σ 2 l n j=1 λ 2 (j)φ 2 j (z l ) .
This implies for the term M 1,1 :

M 1,1 ≤ 2σ 1, * (x 1 -x 0 ) 3 n 2 n l=1 σ 2 l n j=1 λ 2 (j)φ 2 j (z l ) = 2σ 1, * (x 1 -x 0 ) 3 n 2 n j=1 λ 2 (j) n l=1 σ 2 l φ 2 j (z l ) = 2σ 1, * (x 1 -x 0 ) P n (λ) .
In order to bound the terme M 1,2 we have

E [γ 2 2,l |G N 0 ] = σ 2 l l-1 r=1 σ 2 r   n j=1 λ(j)φ j (z l )φ j (z r )   2 ≤ σ 1, * σ 2 l n r=1 n j,k=1 λ(j)λ(k)φ j (z l )φ k (z l )φ j (z r )φ k (z r ) = σ 1, * σ 2 l n j,k=1 λ(j)λ(k)φ j (z l )φ k (z l ) n r=1 φ j (z r )φ k (z r ) = σ 1, * n (x 1 -x 0 ) σ 2 l n j=1 λ 2 (j)φ 2 j (z l ) ,
where the last equality follows from the property (5.1). Therefore, we obtain that the term M 1,2 can be estimated as

M 1,2 ≤ 4σ 1, * (x 1 -x 0 ) 3 n 2 n l=1 σ 2 l n j=1 λ 2 (j)φ 2 j (z l ) = 4σ 1, * (x 1 -x 0 ) P n (λ) .
The upper bounds for M 1,1 and M 1,2 imply

E B 2 (λ)|G N 0 = M 1,1 + M 1,2 ≤ 6σ 1, * (x 1 -x 0 ) P n (λ) .
Hence Proposition 7.2.

Proofs

8.1 Proof of Theorem 6.1

First of all, note that on the set G * we can represent the empirical squared error Err n (λ) in the form

Err n (λ) = J n (λ) + 2 n j=1 λ(j) θj,n + S 2 n -ρ P n (λ) (8.1)
with θj,n = θ j,n -θ j,n θ j,n . From (5.5) and (5.11) one obtains

θj,n = θ j,n ζ j,n + x 1 -x 0 n ( ξ j,n -σ j,n ) + 2 x 1 -x 0 n ξ j,n g j,n + g 2 j,n ,
where ξ j,n = ξ 2 j,n -σ j,n and σ j,n = σ j,n -σ j,n . Setting now

M (λ) = n j=1 λ(j) θ j,n ζ j,n and D(λ) = n j=1 λ(j) σ j,n , (8.2)
we have from (8.1)

Err n (λ) = J n (λ) + 2M (λ) + 2M 1 (λ) - 2(x 1 -x 0 ) n D(λ) + S 2 n -ρ P n (λ) -ρ Pn (λ) , (8.3)
where Pn (λ) = P n (λ) -P n (λ) =

x 1 -x 0 n n j=1 λ 2 (j) σ j,n , M 1 (λ) = 1 √ n B(λ) + ∆(λ) with ∆(λ) = ∆ 1 (λ) + ∆ 2 (λ) . (8.4)
The function B(λ) is given in (7.1) and

∆ 1 (λ) = n j=1 λ(j) g 2 j,n and ∆ 2 (λ) = 2 x 1 -x 0 n n j=1 λ(j) ξ j,n g j,n .
In view of Proposition 7.2, for any λ

∈ [0, 1] n , E ϑ B 2 (λ)| G N 0 ≤ 6σ * (x 1 -x 0 ) P n (λ) . (8.5)
To estimate the second term in (8.4) note that sup

λ∈[0,1] n ∆ 1 (λ) ≤ n j=1 g 2 j,n = g 2 n . (8.6)
To study the function ∆ 2 (λ) we recall that, for any ε > 0 and any x, y ∈ R ,

2xy ≤ εx 2 + ε -1 y 2 . (8.7)
Therefore, for any 0 < ε < 1,

|∆ 2 (λ)| ≤ ε x 1 -x 0 n n j=1 λ 2 (j) ξ 2 j,n + g 2 n ε = εP n (λ) + ε |B(λ 2 )| √ n + g 2 n ε ,
where the vector λ 2 = (λ 2 (j)) 1≤j≤n . Thus, for any λ

∈ [0, 1] n , |∆(λ)| ≤ εP n (λ) + ε |B(λ 2 )| √ n + 2ε -1 g 2 n .
This implies

2|M 1 (λ)| ≤ 2 |B(λ)| √ n + 2 |B(λ 2 )| √ n + 2εP n (λ) + 4ε -1 g 2 n . (8.8)
Then, taking into account here that P n (λ 2 ) ≤ P n (λ), one gets, for any 0 < ε < 1 and any λ ∈ Λ,

2 |B(λ)| √ n + 2 |B(λ 2 )| √ n ≤ εP n (λ) + 2 ε n B 2 (λ) P n (λ) + B 2 (λ 2 ) P n (λ 2 ) (8.9)
Note that the inequalities (4.7) imply that

P 0,n (λ) ≤ P n (λ) ≤ P 1,n (λ) , (8.10)
where

P 0,n (λ) = σ 0, * (x 1 -x 0 )|λ| 2 n and P 1,n (λ) = σ 1, * (x 1 -x 0 )|λ| 2 n .
From the inequalities (8.9)-(8.10) it follows

2 |B(λ)| √ n + 2 |B(λ 2 )| √ n ≤ εP n (λ) + 2 ε σ 0, * (x 1 -x 0 ) B * (λ) , (8.11)
where

B * (λ) = B 2 (λ) |λ| 2 + B 2 (λ 2 ) |λ 2 | 2 .
By choosing ε = ρ/3 one has from (8.8)

2|M 1 (λ)| ≤ ρP n (λ) + 6 ρ Υ n (λ) , (8.12)
where

Υ n (λ) = B * (λ) σ 0, * (x 1 -x 0 ) + 2 g 2 n .
To obtain an upper bound for the empirical risk we evaluate the maximal value of the term D(•), i.e.

D * = max λ∈Λ |D(λ -λ 0 )| .
One can check directly that

E ϑ D * ≤ 2 λ∈Λ E ϑ |D(λ)| ≤ 2ν Λ * n * T , (8.13)
where * T is defined in (4.12). Moreover, we need to study the term

Pn (λ) = P n (λ) -P n (λ) = x 1 -x 0 n n j=1 λ 2 (j) σ j,n . (8.14)
To this end, denoting

P * = sup λ∈Λ | Pn (λ)| ,
we obtain that

E ϑ P * ≤ λ∈Λ E ϑ | Pn (λ)| ≤ λ∈Λ x 1 -x 0 n n j=1 λ 2 (j)E ϑ | σ j,n -σ j,n | ≤ λ∈Λ x 1 -x 0 n 2 |λ| 2 * T ≤ x 1 -x 0 n 2 Λ * ν * T . (8.15)
From ( 8.3) we obtain that, for some fixed λ 0 ∈ Λ,

Err n ( λ) -Err n (λ 0 ) = J n ( λ) -J n (λ 0 ) + 2 M ( µ) - 2(x 1 -x 0 ) n D( µ) + 2(M 1 ( λ) -M 1 (λ 0 )) -ρ(P n ( λ) -P n (λ 0 )) -ρ( Pn ( λ) -Pn (λ 0 )) ,
where µ = λ -λ 0 . By the definition of λ in (5.17) we obtain on the set G *

Err n ( λ) ≤ Err n (λ 0 ) + 2 M ( µ) + 6 ρ Υ n ( λ) + 2(x 1 -x 0 ) n D * + 2ρP n (λ 0 ) -ρ( Pn ( λ) -Pn (λ 0 )) . (8.16)
To study the terme Υ n (λ) in the previous inequality, we begin with the terme B * (λ). From (8.5) and (8.10) it follows that

E ϑ 1 G * B * (λ) ≤ E ϑ B 2 (λ) |λ| 2 + B 2 (λ 2 ) |λ 2 | 2 ≤ 6σ 1, * (x 1 -x 0 ) E ϑ P n (λ) |λ| 2 + P n (λ 2 ) |λ 2 | 2 ≤ 12 σ 2 1, * (x 1 -x 0 ) 2 n .
To estimate the norm g 2 n note that g l g k = 0 for l = k. Therefore,

E ϑ 1 G * g 2 n = E ϑ 1 G * n j=1 g 2 j,n ≤ (x 1 -x 0 ) 2 n 2 E ϑ n j=1 n l=1 g l φ j (z l ) 2 = (x 1 -x 0 ) 2 n 2 E ϑ n j=1 n l=1 g 2 l φ 2 j (z l ) ≤ (x 1 -x 0 ) g * T T ,
where g * T is given by (4.8). This implies

E ϑ 1 G * Υ n = E ϑ 1 G * B * (λ) σ 0, * (x 1 -x 0 ) + 2 g 2 n ≤ 2(x 1 -x 0 ) 6σ 2 1, * nσ 0, * + g * T T . (8.17)
Let us study now the term M in (8.3). For any λ ∈ Λ, we represent it as

M (µ) = Z(µ) + V (µ) and µ = λ -λ 0 , (8.18)
where

Z(µ) = x 1 -x 0 n n j=1 µ(j) θ j,n ξ j,n and V (µ) = n j=1 µ(j) θ j,n g j,n .
We begin with the weighted discrete Fourier transformation, i.e. we set

Šµ = n j=1 µ(j) θ j,n φ j . (8.19)
Due to definition of ξ j,n in (5.5), we can estimate the term Z(µ) as

E ϑ 1 G * Z 2 (µ) ≤ σ 1, * (x 1 -x 0 ) n Šµ 2 n . (8.20)
Moreover, using the inequality (8.7) with ε = ρ, we obtain

2V (µ) = 2 n j=1 µ(j) θ j,n g j,n ≤ ρ Šµ 2 n + g 2 n ρ . (8.21)
Therefore, on the set G * 2M (µ) ≤ 2ρ Šµ

2 n + Z * nρ + g 2 n ρ , (8.22)
where

Z * = sup µ∈Λ-λ 0 nZ 2 (µ) Šµ 2 n
.

It is clear, that the upper bound (8.20) yields

E ϑ 1 G * Z * ≤ µ∈Λ-λ 0 nE ϑ 1 G * Z 2 (µ) Šµ 2 n ≤ νσ 1, * (x 1 -x 0 ) . (8.23)
To estimate the norm Šµ 2 n note that in view of (5.5) on the set

G * Šµ 2 n -S µ 2 n = n j=1 µ 2 (j)(θ 2 j,n -θ 2 j,n ) ≤ -2 n j=1 µ 2 (j) θ j,n ζ j,n = -2Z 1 (µ) -2V 1 (µ) , (8.24)
where

Z 1 (µ) = x 1 -x 0 n n j=1 µ 2 (j)θ j,n ξ j,n and V 1 (µ) = n j=1 µ 2 (j) θ j,n g j,n .
Taking into account that |µ(j)| ≤ 1, similarly to inequality (8.20), we find

E ϑ 1 G * Z 2 1 (µ) ≤ σ 1, * (x 1 -x 0 ) n Šµ 2 n .
Moreover, for the random variable

Z * 1 = sup µ∈Λ-λ 0 nZ 2 1 (µ) Šµ 2 n
, we obtain the same upper bound as in (8.23), i.e.

E ϑ Z * 1 1 G * ≤ νσ 1, * (x 1 -x 0 ) . (8.25)
Furthermore, similarly to (8.21) we estimate the second term in (8.24) as

2|V 1 (µ)| ≤ ρ Šµ 2 n + g 2 n ρ . Therefore, on the set G * Šµ 2 n ≤ S µ 2 n + 2ρ Šµ 2 n + Z * 1 nρ + g 2 n ρ , i.e. Šµ 2 n ≤ 1 1 -2ρ S µ 2 n + 1 (1 -2ρ)ρ Z * 1 n + g 2 n . (8.26)
Using this inequality in (8.22) and putting

Z * 2 = Z * + Z * 1 yield on the set G * 2M ( µ) ≤ 2ρ 1 -2ρ S µ 2 n + 1 ρ(1 -2ρ) Z * 2 n + g 2 n ≤ 4ρ(Err n ( λ) + Err n (λ 0 )) 1 -2ρ + 1 ρ(1 -2ρ) Z * 2 n + g 2 n .
Using this bound in (8.16), we obtain that

Err n ( λ) ≤ 1 + 2ρ 1 -6ρ Err n (λ 0 ) + 2ρ(1 -2ρ) 1 -6ρ P n (λ 0 ) + 1 ρ(1 -6ρ) Z * 2 n + g 2 n + 1 -2ρ 1 -6ρ 6 ρ Υ n ( λ) + 4(x 1 -x 0 ) n D * -ρ( Pn ( λ) -Pn (λ 0 )) .
Using here (8.13), (8.15), (8.17), (8.23), (8.25), we get

E ϑ Err n ( λ)1 G * ≤ 1 + 2ρ 1 -6ρ E ϑ Err n (λ 0 )1 G * + 2ρ(1 -2ρ) 1 -6ρ E ϑ 1 G * P n (λ 0 ) + (x 1 -x 0 ) ρ(1 -6ρ) (74 -144ρ)σ 1, * (ν + σ * ) n + (13 -24ρ)g * T T + (x 1 -x 0 ) (1 -2ρ)(8 + 2ρ) (1 -6ρ)n 2 νΛ * * T ,
where σ * = σ 1, * /σ 0, * . From Proposition A.4 with ε = 2ρ it follows

E ϑ Err n ( λ)1 G * ≤ K 0 E ϑ Err n (λ 0 )1 G * + (x 1 -x 0 ) K 1 σ 1, * ρ(1 -6ρ)n + K 2 g * T T + K 3 νΛ * * T n 2 + 2 S n (x 1 -x 0 )σ 1, * n P ϑ (G c * ) , (8.27)
where

K 0 = 1 -12ρ 2 (1 -4ρ)(1 -6ρ) , K 1 = ρ(1 -2ρ) + (74 -144ρ)(ν + σ * ), K 2 = 13 -75ρ + 94ρ 2 ρ(1 -4ρ)(1 -6ρ) and K 3 = (1 -2ρ)(8 + 2ρ) (1 -6ρ) .
It is easy to see that, for 0 < ρ ≤ 1/8,

1 + 4ρ 1 -6ρ < K 0 ≤ 1 + 5ρ 1 -6ρ , K 1 < 1 + 74(ν + σ * ), K 2 < 39 ρ(1 -6ρ) and K 3 < 8 1 -6ρ .
Using these bounds and replacing in (8.27)

E ϑ Err n ( λ)1 G * and E ϑ Err n (λ 0 )1 G * by E ϑ S * -S 2 n -S 2 n P ϑ (G c * ) and E ϑ S λ 0 -S 2 n -S 2 n P ϑ (G c * )
, respectively, we come to the inequality (6.2):

E ϑ S * -S 2 n ≤ 1 + 5ρ 1 -6ρ E ϑ S λ 0 -S 2 n + ľ U T ρ T , (8.28)
where

U T = ν υ 3 T (1 + g * T + Λ * * T ) + (T /υ T ) sup ϑ∈Θ P ϑ (G c * ) , ľ = sup T >0 max 4(x 1 -x 0 ) max b max (υ 2 T (1 + 74ν) + 74b) ν , 39υ 3 T ν , 1 , B with b = b max /b min , B = 2M b max √ x 1 -x 0 .
The weight λ 0 ∈ Λ being arbitrary in (8.28), we come to the oracle inequality (6.2). The property (6.3) follows directly from asymptotic behaviour of the right-hand part components of the function U T . Hence Theorem 6.1.

Proof of Proposition 4.1

First, note that from (4.2) we can represent the term B k in the following form

B k = B 1,k + B 2,k + B 3,k
where

B 1,k = 1 δH k τ k j=N 0 +1 κ j,k χ j,k (h) j , j = t j t j-1
(S(y u ) -S(y t j-1 ))du ,

B 2,k = 1 H k τ k j=N 0 +1 κ j,k χ j,k (h) (S(y t j-1 ) -S(z k )) , B 3,k = 1 H k τ k j=N 0 +1
(1κ j,k ) κ j,k χ j,k (h)S(y t j-1 ) .

In Appendix we show that there exists some constant C * > 0 such that max

1≤k≤n sup ϑ∈Θ E ϑ B 2 1,k ≤ C * δ . (8.29)
Moreover, using the definition (2.1), we obtain that

|B 2,k | ≤ M h H k and |B 3,k | ≤ M H k .
Taking into account here the property (2.9), the definition of the bandwidth h in (3.4) and that for sufficiently large T the threshold H k ≥ hN √ υ T /2, we obtain that lim

T →∞ T sup ϑ∈Θ E ϑ B 2 k = 0 .
Moreover, the term B k defined in (4.4) we can represent as

B k = B 1,k + B 2,k ,
where

B 1,k = 1 δH k τ k j=N 0 +1 κ j,k χ j,k (h) * j , * j = t j t j-1 (b(y u ) -b(y t j-1 ))dW u , B 2,k = 1 δH k τ k j=N 0 +1 κ j,k χ j,k (h) b(y t j-1 ) -b(z k ) ∆W t j .
In Appendix we show that there exists some constant C * > 0 such that for sufficiently large T

max 1≤k≤n sup ϑ∈Θ E ϑ B 2 1,k ≤ 8C * b 2 max δ hT √ υ T . ( 8

A Appendix A.1 Concentration inequalities

In this section we remind the concentration inequalities from [START_REF] Galtchouk | Uniform concentration inequality for ergodic diffusion processes observed at discrete times[END_REF] which will be used to study non asymptotic properties of the sequential estimation procedures. We set

D N,h (z) = N j=1 1 h χ y t j -z h -q ϑ,h (z) δ , (A.1)
where N = [T /δ], the frequency δ is defined in (2.9) -(2.10) and q ϑ,h (z) = 1 -1 q ϑ (z + uh)du.

Theorem A.1. Assume that the condition (3.9) holds. Then, for any z * > 0 and a > 0, lim

T →∞ T a max 1≤n≤N sup h≥T -1/2 sup |z|≤z * sup ϑ∈Θ P ϑ |D n,h (z)| ≥ υ T T = 0 . (A.2)
The proof of this result follows directly from Theorem 2.3 in [START_REF] Galtchouk | Uniform concentration inequality for ergodic diffusion processes observed at discrete times[END_REF].

If we consider the deviation problem for the first N 0 observations with the N 0 defined in (3.3), we have to check the conditions (3.9) for the time duration

T 0 = δN 0 ≈ T 2γ-1 l 1-γ T as T → ∞ .
Therefore, the frequency has be represented as

δ = 1 T 0 l 0 (T ) and l 0 (T ) = T l T T 0 ≈ T 2-2γ l 2-γ T as T → ∞ .
So, we need to check the conditions (3.9) by replacing T with T 0 and l T with l 0 (T ), respectively. It is clear that these conditions hold if we take , for example, υ T = ln -ι (T + 1) and l T = ln 1+6ι T , for some ι > 0.

Corollary A.2. Assume that the condition (3.9) holds. Then, for any z * > 0 and a > 0, lim

T →∞ T a sup h≥T -1/2 0 sup |z|≤z * sup ϑ∈Θ P ϑ |D N 0 ,h (z)| ≥ υ T T 0 = 0 , (A.3)
where N 0 is defined in (3.3) and T 0 = δN 0 .

A.2 Proof of Proposition 3.1

First note that Corollary A.2 implies directly that, for any a > 0, lim

T →∞ T a P(Γ c * ,k ) = 0 .
Using the model (1.3) in (3.14), we obtain that

y t j -y t j-1 = √ δb(y t j-1 )η j + U j ,
where η j = (W t j -W t j-1 )/ √ δ ∼ N (0, 1) and

U j = t j t j-1 S(y u )du + t j t j-1 b(y u ) -b(y t j-1 ) dW u .
Due to the bound (A.6), there exists some positive constant C * such that, for any j ≥ 1,

sup ϑ∈Θ E ϑ U 2 j |F t j-1 ≤ C * δ 2 1 + y 2 t j-1 . (A.4)
So, the squared difference can be represented as

(y t j -y t j-1 ) 2 = δb 2 (y t j-1 )η 2 j + U j and U j = U 2 j + 2 √ δb(y t j-1 )η j U j .
From (A.4) we obtain that, for some positive constant C * and for any j ≥ 1,

sup ϑ∈Θ E ϑ | U j | | F t j-1 ≤ C * δ √ δ 1 + y 2 t j-1 . (A.5)
Therefore, on the set Γ * ,k we obtain that

b k = τ * ,k j=1 χ j,k (h 0 ) b 2 (y t j-1 ) H * + Υ 1,T + Υ 2,T , where Υ 1,T = τ * ,k j=1 χ j,k (h 0 ) b 2 (y t j-1 ) η j H * and Υ 2,T = τ * ,k j=1 χ j,k (h 0 ) U j δ H * , and
η j = η 2 j -1. Note now, that on the set Γ * ,k 0 ≤ τ * ,k j=1 χ j,k (h 0 ) -H * ≤ χ τ * ,k ,k (h 0 ) ≤ 1 and on Γ c * ,k τ * ,k j=1 χ j,k (h 0 ) ≤ H * . 29
Hence,

τ * ,k j=1 χ j,k (h 0 ) ≤ H * + 1 a.s.
Therefore, for some constant C * > 0

E ϑ | b k -b 2 (z k )| ≤ C * h 0 + 1 H * + E ϑ |Υ 1,T | + E ϑ |Υ 2,T | + b 2 (z k ) P(Γ c * ,k ) .
Note here, that

E ϑ Υ 2 1,T = 1 H 2 * E ϑ τ * ,k j=1 χ j,k (h 0 ) b 4 (y t j-1 ) η 2 j ≤ 2b 4 max H * + 1 H 2 * ≤ 4 H * b 4 max and E ϑ |Υ 2,T | ≤ 1 δ H * N 0 j=1 E ϑ 1 {j≤τ * ,k } χ j,k (h 0 ) E ϑ | U j | | F t j-1 ≤ C * √ δ H * N 0 j=1 E ϑ 1 {j≤τ * ,k } χ j,k (h 0 ) 1 + y 2 t j-1 ≤ C * √ δ .
Finally, for some constant C * > 0 and for sufficiently large T → ∞,

E ϑ | b k -b 2 (z k )| ≤ C * h 0 + 1 H * + √ δ + P(Γ c * ,k ) .
Hence Proposition 3.1.

A.3 Proof of Proposition 4.4

First of all note that

E ϑ | σ l -σ 2 l | ≤ 1 υ T δ(N -N 0 )h E ϑ | b l -b 2 (z l )| .
Taking into account that according to the definition N 0 in (3.3) we obtain through Proposition 3.1 the equality (4.13). Hence Proposition 4.4.

A.4 Technical results

Lemma A.3. Let f be an absolutely continuous [x 0 , x 1 ] → R function with ḟ < ∞ and g be [x 0 , x 1 ] → R a step-wise function of the form

g(t) = n j=1 c j χ (z j-1 ,z j ] (t),
where c j are some constants and the sequence (z j ) 0≤j≤n is given in (3.1).

Then for all ε > 0, the function ∆ = f -g satisfies the following inequalities

∆ 2 ≤ (1 + ε) ∆ 2 n + 1 + 1 ε ḟ 2 n 2 (x 1 -x 0 ) 2 ,
and

∆ 2 n ≤ (1 + ε) ∆ 2 + 1 + 1 ε ḟ 2 n 2 (x 1 -x 0 ) 2 .
The proof is given in Lemma A.2 from [START_REF] Konev | Robust model selection for a semimartingale continuous time regression from discrete data[END_REF].

A.5 Proof of the upper bound (8.29)

First, note that

τ k j=N 0 +1 κ j,k χ j,k (h) = τ k j=N 0 +1 κ j,k χ j,k (h) + τ k j=N 0 +1 κ j,k -κ j,k χ j,k (h) = H k + ( √ κ k -κ k ) χ τ k ,k (h) ≤ H k + 1 .
So, by the Bunyakovskii -Cauchy -Schwarz inequality,

E ϑ B 2 1,k ≤ E ϑ 1 δ 2 H k τ k j=N 0 +1 κ j,k χ j,k (h) E ϑ 1 H k τ k j=N 0 +1 κ j,k χ j,k (h) 2 j ≤ 2 δ 2 E ϑ 1 H k τ k j=N 0 +1 κ j,k χ j,k (h)E ϑ 2 j |F t j-1 .
Moreover, taking into account that the function S from the class (2.1) is lipschitzian with the constant L 1 = M ∨ L, we obtain that

E ϑ 2 j |F t j-1 ≤ δL 2 1 t j t j-1
E ϑ (y u -y t j-1 ) 2 |F t j-1 du .

Note here that, for t j-1 ≤ u ≤ t j , E ϑ ((y u -y t j-1 ) 2 |F t j-1 ) ≤ 2δ κ j,k χ j,k (h) ≤ C * δ .

Hence the upper bound (8.29).

A.6 Proof of the upper bounds (8.30)- (8.31) Due to orthogonality of stochastic integral increaments, we obtain that

E ϑ B 2 1,k = 1 δ 2 E ϑ 1 H 2 k τ k j=N 0 +1 κ j,k χ j,k (h) E ϑ ( * j ) 2 |F t j-1 ≤ 2b 2 max δ 2 E ϑ 1 H 2 k τ k j=N 0 +1
κ j,k χ j,k (h)

t j t j-1
E ϑ (y u -y t j-1 ) 2 |F t j-1 du .

By using the bound (A.7) for the last conditional expectation we get

E ϑ B 2 1,k ≤ 4C * b 2 max E ϑ 1 H k .
This inequality and (3.12), (3.4) provide the inequality (8.30). Moreover, one has

E ϑ B 2 2,k = 1 δ 2 E ϑ 1 H 2 k τ k j=N 0 +1 κ j,k χ j,k (h) b(y t j-1 ) -b(z k ) 2 (∆ W t j ) 2 ≤ 2b 2 max δ E ϑ 1 H 2 k τ k j=N 0 +1 κ j,k χ j,k (h)(y t j-1 -z k ) 2 ≤ 4b 2 max h 2 δ E ϑ 1 H k .
Hence the upper bound (8.31).

A.7 Proof of Proposition A.4 (1 -λ(j)) 2 θ 2 j,n .

Taking into account here that ζ 2 j,n = g 2 j,n +

x 1 -x 0 n ξ 2 j,n + 2

x 1 -x 0 n g j,n ξ j,n , we obtain

Err n (λ) ≥ x 1 -x 0 n n j=1 λ 2 (j)ξ 2 j,n + 2

x 1 -x 0 n I 1 -2 x 1 -x 0 n I 2 ,
where I 1 = n j=1 λ 2 (j)g j,n ξ j,n and I 2 = n j=1 (1-λ(j))λ(j)θ j,n ξ j,n . Moreover, note that, for any 0 < ε < 1,

2 x 1 -x 0 n I 1 ≤ 1 ε g 2 n + ε(x 1 -x 0 ) n n j=1
λ 2 (j)ξ 2 j,n .

Therefore

Err n (λ 0 ) ≥ (1 -ε)(x 1 -x 0 ) n n j=1 λ 2 (j)ξ 2 j,n - 2 √ x 1 -x 0 √ n I 2 - 1 ε g 2 n and E ϑ 1 G * Err n (λ 0 ) ≥ (1 -ε)(x 1 -x 0 ) n E ϑ 1 G * n j=1 λ 2 (j)ξ 2 j,n - 2 √ x 1 -x 0 √ n E ϑ 1 G * I 2 - 1 ε E ϑ 1 G * g 2 n .
Taking into account here the definition of B(•) in (7.1) and that E ϑ I 2 = 0 we can rewrite the last inequality as

E ϑ 1 G * Err n (λ 0 ) ≥ (1 -ε)E ϑ 1 G * P n (λ) + (1 -ε) √ n E ϑ 1 G * B(λ 2 ) + 2 √ x 1 -x 0 √ n E ϑ 1 (G * ) c I 2 - 1 ε E ϑ 1 G * g 2 n .
Now Propositions 7.1 -7.2 imply that

E ϑ 1 G * Err n (λ 0 ) ≥ (1 -2ε)E ϑ 1 G * P n (λ) - 2(x 1 -x 0 )σ 1, * nε - 2 S n (x 1 -x 0 )σ 1, * √ n P ϑ (G c ) - 1 ε E ϑ 1 G * g 2 n .
Hence Proposition A.4.

. 15 ) 3 . 3 .

 1533 Remark Note that in the case of known diffusion coefficient b(•), we can take b k = b 2 (z k ) .

Proposition A. 4 .( 1

 41 For any 0 < ε < 1/2,E ϑ 1 G * P n (λ 0 ) ≤ 1 1 -2ε E ϑ Err n (λ 0 )1 G * + (x 1 -x 0 )g * T ε(1 -2ε)T + 2(x 1 -x 0 )σ 1, * nε + 2 S n (x 1 -x 0 )σ 1, * √ n P ϑ (G c * ) ,where the term g * T is given in (4.6). Proof. Note that on the setG * Err n (λ) = n j=1 (λ(j) θ j,n -θ j,n ) 2 = -λ(j))λ(j)θ j,n ζ j,n + n j=1

  Acknowledgements. This work was partially supported by the research project no. 2.3208.2017/4.6 (the Ministry of Science and Higher Education of the Russian Federation). The work of the last author was partially supported by the Russian Federal Professor program, project no. 1.472.2016/1.4 (the Ministry of Science and Higher Education of the Russian Federation).

							.30)
	and	max 1≤k≤n	sup ϑ∈θ	E ϑ B 2 2,k ≤	8b 2 max h T √ υ T	.	(8.31)
	From here we obtain the limit equation (4.9). Hence Proposition 4.1.

  E ϑ (S 2 (y v )|F t j-1 ) dv + b 2 (1 + E ϑ (y 2 v |F t j-1 ) )dv + b 2 max .Due to Proposition A.6 from[START_REF] Galtchouk | Efficient pointwise estimation based on discrete data in ergodic nonparametric diffusions[END_REF], one has fort j-1 ≤ v ≤ t j , E ϑ (y 2 v |F t j-1 ) ≤ D * L + y 2 Lx * + 2x * ) 2 (L + M ) + b 2 max . So, for t j-1 ≤ u ≤ t j and 0 < δ < 1, E ϑ y u -y t j-1 2 |F t j-1 ≤ 2δ c 1 + c 2 y 2 L + 1) and c 2 = 2L 2 1 . Taking into account that χ j,k (h)y 2 t j-1 ≤ 2(h 2 + z 2

			u
						max
			t j-1
					u
		≤ 2δ 2L 2 1
				t j-1
					t j-1
					t j-1	+ b 2 max ,	(A.6)
	where c 1 = 2L 2 1 (D (A.7)
	and, therefore,
	sup ϑ∈Θ	E ϑ B 2 1,k ≤ C * δ sup ϑ∈Θ	E ϑ	1 H k	τ k j=N 0 +1

, where D * = (M + * k ) , we obtain that, for some constant C * > 0, sup

t j-1 ≤u≤t j sup ϑ∈Θ χ j,k (h) E ϑ (y u -y t j-1 ) 2 |F t j-1 ≤ C * δ

* This work is supported by the RSF Grant no. 17-11-01049, National Russian Research