Clemens Rohde
email: clemens.rohde@isi.franhofer.de

Jan Rosenow
email: jan.rosenow@ouce.ox.ac.uk

Nick Eyre
email: nick.eyre@ouce.ox.ac.uk

Louis-Gaëtan Giraudet
email: raudet@centre-cired.fr

Energy Saving Obligations Cutting the Gordian Knot of leverage? Pre-print version to appear in 'Energy Efficiency'

Keywords: Energy Savings Obligation, Leverage, Financing

Better leverage of public funding is essential in order to trigger the investment needed for energy efficiency. In times of austerity governments increasingly look at policy instruments not funded by public expenditure and Energy Savings Obligations represent one option. Because Energy Savings Obligations are paid for by all energy customers, the degree to which they are able to raise additional private capital for energy efficiency investments is crucial with regard to the financial burden on consumers. In this paper, we systematically assess how successful Energy Savings Obligations were in levering capital from parties other than the obligated entities including private investors and other public bodies. We analyse three countries with substantial experience with Energy Savings Obligations, identify the main design differences and the effect this has on the degree of leverage. We conclude that the design of Energy Savings Obligations largely determines the degree of leverage and that that there appears to be a trade-off between high leverage and additionality.

INTRODUCTION

Energy Savings Obligation (ESOs) are a prominent policy instrument for increasing the energy efficiency of the economy -they have been successfully implemented in a growing number of countries all over the world. Within the US, 34 states have implemented different variations of ESOs and 6 EU member states now have on-going ESOs. Also Australia has significant experience with the policy instrument [START_REF] Rap | Energy Savings ObligationsGlobal Lessons & the "Efficiency Utility" Model[END_REF]. The systems differ very much in design, often a result of different regulatory frameworks. In this paper, we will focus on the European ESOs.

ESOs already featured in the EU Energy End-Use Efficiency and Energy Services Directive of 2006 (2006/32/EC) as a possible market-based instrument for realising energy savings. However, the Directive only recommended ESOs and there was no requirement for member states to implement the instrument. This has now changed: The new Energy Efficiency Directive of 2012 (2012/27/EU) includes a requirement in Article 7 that all member states must introduce ESOs. Exemptions from this obligation can be made if member states provide verification of equivalent savings resulting from alternative instruments. So far, the UK, Italy, France, Denmark, the Flanders region, and Poland have introduced ESOs, although in very different forms. Slovenia is about to implement them [START_REF] Bertoldi | Energy supplier obligations and white certificate schemes: Comparative analysis of experiences in the European Union[END_REF].

One reason for the increasing prominence of ESOs is that they are widely considered as an option of using public funding sources (either paid by the taxpayer or via energy bills) to lever private capital at a time of constrained public budgets (Eichhammer 2012). This leverage effect describes the ratio of public funding to private (and in some cases other public) capital. In a scheme that obliges profit-oriented organisations such as energy companies to deliver a certain amount of energy savings it can be expected that the obligated parties utilise approaches with high leverage effects to minimise their cost. Achieving high leverage rates with public funds can be considered the Gordian knot of leverage in policy design.

However, so far no comprehensive analysis exists that compares different ESOs concerning the amount of leverage they create. In this paper, we systematically review three prominent ESOs, namely the UK, France, and Denmark, and compare their results in terms of leverage. The restriction of the paper to the three cases is a result of data availability -in some cases evaluations exist, but those are not available to the public.

The structure of the paper is as follows: First, we analyze different cost allocation paths within the obligation schemes. Second, we define what we mean by leverage and how it applies to ESOs. Third, we provide background information on how ESOs work in general and more specifically in the three countries under investigation. Fourth, we calculate and compare the degree of leverage achieved in the three countries. Finally, we summarise the findings from the different countries and draw conclusions regarding the impact of design differences on leverage.

COST ALLOCATION

One common aspect of obligation schemes and other policy instruments is that most schemes do not subsidise the energy efficiency investment by 100 percent and a remaining share of the investment has to be covered by other capital sources. The interest and repayment should be covered by the reduced energy costs resulting from the energy savings. Depending on the investor's financial situation, the capital may come from own capital sources (e.g. personal savings) or may be refinanced by a commercial bank. Thus, the measures themselves are also financed by the same target group as they are implemented by energy consumers. So the main question of cost allocation is how the money is shifted between the different consumers and how the other actors contribute to this mechanism1 .

Within the figures below, the capital flows between the actors are shown. Costs that are not refinanced by another actor, such as administrative costs of the investors, are not shown in the graphs. Figure 1 shows the cost allocation paths within ESOs. The different actors with their corresponding costs are shown and the cost allocation paths are indicated by arrows (whereas the direction of the arrow indicates the flow of capital).

Figure 1: Financing paths of Efficiency Obligations

For energy efficiency obligations five parties have to be considered, namely investors themselves, the energy customers, the obligated parties and a supervising authority. The fifth actor is the commercial banks, which contribute to the financing of the investment. Normally, the obligated party supports the investors by some sort of subsidy to initiate the investment. For the administration of the efficiency scheme, they may have to pay some administrative fees. They refinance themselves through energy prices (or reduced profit) [START_REF] Schlomann | Which Role for Market-Oriented Instruments for Achieving Energy Efficiency Targets in Germany?[END_REF].

Thus, the costs are finally allocated to the energy customers, who normally will pay according to their energy use giving the system a sort of polluter-pays principle. Nevertheless, the energy customers who profit most from the subsidies are not necessarily the ones who pay the final bill, as only those who receive the subsidies profit from lower energy costs. For the obligated parties, it may be easier and cheaper to spend their money on groups with better equity (Rosenow et al. 2013). This could increase leverage, for the need for subsidies is lower than in groups with more difficult access to capital (see section 4.1). To face these issues, ESOs in France and the UK incorporate a social component to ensure that some of the investments are made in deprived groups.

To illustrate the specific mechanism of obligation schemes, the contrasting example of a classical funding scheme is shown in Figure 1.

Figure 2: Financing paths of funding schemes

Within a tax based funding scheme the situation is different from an efficiency obligation. Instead of an obligated party, a funding body grants the subsidies, using capital provided by the state, which refinances itself via taxes. The funding bodylike the obligated party in the case of efficiency obligationsinitiates the investment by granting some form of subsidy.

The costs of the subsidies are finally borne by taxpayers, making it more like an abilityto-pay system. The funding body does not necessarily follow a logic of delivering the most (short-term) savings for their investment, but may follow a more long term investment strategy, making the system less cost-effective on the short term. It may also consider social issues in program design.

Cost allocation is an important issue to discuss when looking at leverage. The main difference between the schemes in terms of cost allocation is the endpoint of the allocation chain. Within a saving obligation it is the energy consumer, within the (tax based) funding scheme it is the taxpayer. The social and political implications of this potential burden to energy customers or tax payers require responsible use of the funds and thus should be a motivation to increase leverage.

DEFINING LEVERAGE

In this paper we assume leverage means delivering more investment in energy efficiency by multiplying the finance available from policy induced third party funding sources (subsequently called "public funding", including Government funding through tax reve-nues and consumer-funded expenditure such as ESOs and public benefits charges)2 . Regardless of the instrument chosen, most financing mechanisms rely on the consumer or taxpayer as the one who finally pays for the implementation of the energy efficiency measures. As these are all constrained resources, optimal use of the money should be secured.

It is important to appreciate the different types of leverage with regard to energy efficiency programmes as they use different sources of finance and result in potentially different degrees of leverage. The following four types of leverage are most prominent for energy efficiency measures. Options 1-3 fall into the category of continuous support of measures whereas option 4 only provides funding over a short time period assuming this is sufficient to stimulate the market sufficiently.

1. Using public sources to part-fund a project, e.g. 50% consumer-funded grant typical in past ESOs in the UK, but also in some Clean Development Mechanism (CDM) funded projects. The remaining proportion of the investment is provided by the beneficiaries themselves. 2. Using public funds to reduce or remove loan interest rates, e.g. CO 2 -Building Rehabilitation Programme in Germany. The Property Assessed Clean Energy (PACE) mechanism in the US also fits into this category, but uses property tax reductions rather than direct support. The cost of the investment is entirely covered by the beneficiaries. 3. Using public funds to grant tax breaks, e.g. reduced VAT rate on some products and income tax reductions as in France and Italy. The tax breaks provided constitute a subsidy toward the overall investment costs. 4. Using public funds to kick-start markets, which then become self-sustaining (or at least sustaining at lower rates of support). This could be as support for specific new technologies (e.g. the 100,000 roof programme in Germany supporting solar power) or in support for new business models (such as the Green Deal in the UK). In both cases this requires a dynamic economic analysis in contrast to the other types which are essentially static.

The additional investment levered is typically provided by the direct beneficiary of the projectusually the building owner and/or occupant, but in principle from other sources as well. For type 2 of the above list, this would generally be banks and other lenders, but this then usually requires the primary beneficiary to repay the loan and any reduced rate interest. For type 1, third parties may include municipalities or social housing providers who part-subsidise energy efficiency measures in order to reach a larger number of participants.

In order to measure leverage we define a leverage factor that is capable of representing the degree of leverage achieved by a given energy efficiency programme. For the purpose of this paper, we measure leverage in form of a leverage factor defined as: It follows that leverage is zero for a leverage factor value of 1.0. A leverage factor of 2.0, for example, suggests that contributions from the obligated party were matched with 100% funding from the beneficiary and third parties.

L =
Our computation of leverage factors is based on the cost estimates available for the different countries: [START_REF] Lees | Evaluation of the Energy Efficiency Commitment[END_REF][START_REF] Lees | Evaluation of the Energy Efficiency Commitment[END_REF] for the UK; [START_REF] Giraudet | The costs and benefits of White Certificates schemes[END_REF] for France; Togeby ( 2012) for Denmark. The methodology used to derive these estimates is similar across countries.

Regarding how much beneficiaries spend on a given measure, [START_REF] Lees | Evaluation of the Energy Efficiency Commitment[END_REF], [START_REF] Togeby | Evaluering af energiselskabernes energispareaktiviteter, EaEnergianalyse[END_REF] and [START_REF] Giraudet | The costs and benefits of White Certificates schemes[END_REF] only take into account the share of investment that is additional to what would have occurred in absence of the obligation. For instance, the full cost of insulation is considered, while only the cost differential between an efficient boiler and a standard one is considered.

Regarding additionality i.e., how many beneficiaries are induced by the obligation, Lees makes the assumption that 20% of the beneficiaries would have invested anyway in the

1 n O B Ti i Oa C C C L CI     
absence of the scheme. The figures he provides are net from any free-rider effects. Differently, the figures provided by [START_REF] Giraudet | The costs and benefits of White Certificates schemes[END_REF] assume full additionality of the obligation, as no assessment of free-rider effects could be performed. In Denmark, [START_REF] Bundgaard | Spending to save: evaluation of the energy efficiency obligation in Denmark, 2013 eceee summer study DECC[END_REF] assume a rather low additionality of less than 10% for measures in households (i.e. 90% free-riders), but an additionality of more than 50% in industry, which is the main focus of investment in Denmark.

Lastly, [START_REF] Lees | Evaluation of the Energy Efficiency Commitment[END_REF] and [START_REF] Giraudet | The costs and benefits of White Certificates schemes[END_REF] estimated the indirect cost borne by obligated parties for tasks induced by the obligation, such as marketing or development. These costs are not taken into account in the present study. Rather, the analysis focuses on direct costs, which constitute the main share of the overall costs. For the beneficiaries, the indirect costs are invisible and not relevant in terms of the economics of individual investments in energy efficiency improvements. Using this definition of the leverage factor has the benefit of showing the relative contribution to investments in energy efficiency measures required in order to mobilise private and/or third party capital. It does not, however, account for the total cost such as administration costs and search costs for example. While these costs are not insignificant, estimates for the British scheme show that they are only a small fraction of the total cost [START_REF] Lees | Evaluation of the Energy Efficiency Commitment[END_REF][START_REF] Lees | Evaluation of the Energy Efficiency Commitment[END_REF].

Further limitations of our definition of the leverage factor include the inability to account for non-finance measures such as informational measures that indirectly stimulate the uptake of energy efficiency measures. Further research could develop more sophisticated computations of leverage factors that include such programmes as well.

CHARACTERISTICS OF ENERGY SAVINGS OBLIGATIONS

The general concept of an energy saving obligation is to oblige a party to deliver a defined amount of energy savings within a certain time. Those parties are usually final energy suppliers or the distribution network operators, which is the case in all the European systems. But also independent entities can act as the obligated party. For example, the ESO in Vermont is designed in such a way, which resembles very much an energy efficiency funds with a fixed saving target and variable budget.

The energy saving measures that may be undertaken by the obligated parties or other actors have to be monitored. This is usually done by standard saving scores (also known as deemed savings), which rely on engineering estimates often corrected by direct measurements within a representative sample.

ESOs include a trading mechanism, which allows obligated parties and/or other actors to trade certified savings. If traded in an open market, a price for energy savings will be determined and obligated parties can decide to invest in energy savings themselves or buy (white) certificates from other actors.

To secure additionality, the baseline may be set according to existing regulations and policy instruments already in place. Some systems employ a different approach by deliberately allowing an overlap with existing instruments. The main intention of such a system is then to secure a certain level of energy savings and to promote existing instruments to maximise leverage.

We now proceed with a brief overview of the three ESOs discussed in this paper.

The design of Energy Savings Obligations in the UK

In the UK the ESOs are the most important instrument to deliver energy and carbon savings in the domestic sector (OFGEM, 2005) and both the 2004 and 2007 Energy Efficiency Action Plans highlight ESOs as the principal policy mechanism to deliver energy savings in the domestic sector (2004,2007). The basic concept of ESOs in Britain is that Government imposes an energy savings target on large energy suppliers (gas and electricity) that has to be achieved at the customer end, which may relate to energy consumption or carbon emissions. A distinct feature of the British scheme is the specific targeting of a priority group (PG) in the scheme, which focuses on households in receipt of specified state benefits and elderly residents.

In Britain, energy suppliers comply with their obligations by subsidising energy efficient technologies. There were various routes the energy suppliers used including [START_REF] Rosenow | Energy Savings Obligations in the UK -A History of Change[END_REF]):

 Subsidise installers of insulation: Energy suppliers contracted installers or delivery agents who contract installers of energy saving measures (for example cavity wall insulation) that carry out the work in homes.

 Subsidise products sold via retail: Suppliers have subsidised energy efficient products (for example insulation material) sold via 'do it yourself' (DIY) markets and other retailers.  Subsidise products sent via mail: Energy companies subsidised certain products, and, for example, promoted the use of compact fluorescent lamps (CFLs) via mass mail-outs of free light bulbs, although this is now prohibited.

The design of Energy Savings Obligations in France

In France, the first obligation period ran from 2006 to 2009. The second obligation period has been running since 2011 and will last until 2013. All energy suppliers above a certain sales threshold are subject to the obligation. Within the second obligation period, also the gasoline suppliers are among the obligated parties. Differing from the UK saving obligation, the French system covers all energy demand sectors, with eligible measures also in the transport sector.

Unlike in the UK, obligated parties in France do not systematically subsidise energy efficient technologies. They use a broader set of compliance strategies, including:

• Free energy audits and advice: these services are provided for the most part by contracted installers. They are also provided directly by the obligated parties via their call centres or websites.

• Direct subsidies: they tend to target specifically boilers and other measures on heating systems. They are by and large the main compliance strategy used by fuel oil suppliers. In contrast, they are little used by electricity suppliers. Department stores whose gasoline sales are covered by the obligation offer some vouchers for free purchases to customers who buy energy efficient products in their stores.

• Soft loans: Some major obligated parties have their own banks as subsidiaries and use them to offer reduced interest rate loans to consumers.

The French scheme does not exclude overlap with other instruments, so the investors usually combine the subsidies from the saving obligation with various incentives granted for efficiency products. This includes:

• Income tax reliefs for energy efficient products: This policy was implemented in 2005. All the energy efficient products covered are also eligible within the energy saving obligation.

• Zero interest rate loans granted by private banks on behalf of the Government: This policy was implemented in 2009. All the energy efficient products covered are also eligible to the energy saving obligation. However, the overlap between the two instru-ments is limited by the fact that soft loans can only be granted if several measures are combined, e.g., wall insulation together with an upgrade of the heating system.

The design of Energy Savings Obligations in Denmark

The Danish ESOs are considered very ambitious, even complying with the 1.5 % target of article 7 of the EED without major system changes. The Danish obligation is set on the distribution companies rather then energy suppliers. The target is set in a cooperative process between the obligated parties and the ministry.

The distribution companies implement their measures together with energy service companies and other actors like energy performance contractors to comply with the obligation.

For the accounting process deemed savings as well as individual assessments are possible depending on the type of measure. Measures from all sectors excluding transport are eligible for the ESOs. The accounting methodology of the savings in the Danish scheme is different from the British and French approach, which uses (discounted) lifetime savings. In Denmark, only the first year savings are taken into account. Some uplift factors are used to account for the different lifetimes of the measures. Nevertheless, the accounting methodology favours measures with shorter lifetimes.

An overlap of the scheme with other instruments is explicitly part of the design. Most of the measures are implemented in enterprises from the industrial and services sectorthe French and British ESOs focus mainly on domestic customers. In 2011 the share of savings delivered by enterprises was larger than 60 percent.

LEVERAGE -RESULTS FROM ENERGY SAVINGS OBLIGATION IN EUROPE

The example of the UK

The different routes entail varying degrees of subsidy and contributions from the recipients themselves. Detailed figures exist for the obligation periods 2002-2005 and 2005-2008. The following tables show the contributions for various measures and delivery routes made by energy suppliers, households, and third parties such as Local Authorities or Social Housing Providers. The leverage factors are provided as well (1.0 meaning no leverage i.e. 100% subsidy by supplier and 2.0 representing 50% contribution from the energy supplier and 50% from recipients and third parties). Note that the contribution is paid to the total cost of the investment and the percentages refer to the total direct cost excluding the administration costs. Overall, leverage during EEC 1 (2002)(2003)(2004)(2005) was about 1.9, i.e. for every euro spent by suppliers almost one euro was contributed by recipients and third parties with a larger share coming from recipients. Leverage was higher for ATP measures (2.3) compared to PG measures (1.5), but PG measures attracted more than twice as much third party finance. The highest leverage was achieved in retail schemes with leverage factors of up to 3.0. This was followed by subsidies to the installation of insulation showing leverage factors of up to 2.2. The lowest leverage factor was achieved in products sent via mail to households, mainly a result of distributing CFLs free of charge. During EEC 2 (2005)(2006)(2007)(2008), leverage was just half of what it was during EEC 1 (1.44 compared to 1.87), i.e. for every euro spent by suppliers 44 cents were contributed by recipients and third parties. However, leverage was higher in retail schemes than during EEC 1. The overall picture is similar to EEC 1 with highest leverage achieved in retail schemes and lowest in mail outs.

Note that the figures presented are direct cost contributions to the measures promoted and they do not include any indirect cost to the parties involved.

At the time of writing, no data on CERT (2008CERT ( -2012) ) was available. However, it is clear that in the last phase of CERT leverage for schemes subsidising insulation has declined and was even below 1.0 at the end of the scheme for some measures. This is where energy suppliers paid households the full cost of the measure plus an additional incentive.

As we discuss later in chapter 4, the cost allocation is important for the social acceptance of the instrument. Consumer groups with a rather low willingness to pay are less likely to be targeted by the obligated party if no specific provisions are included in the instrument. Therefore, in addition to the existing Priority Group, a Super-Priority Group (SPG) target was introduced in March 2011 as a sub-target within CERT. CERT originally required suppliers to deliver 40% of the targeted emission savings with households deemed to be in the 'priority group' (PG), which focused on people who are above 70 years old or receive certain benefits (DECC 2010). The SPG target required suppliers to meet 37.5 % of their PG target (15% of their total CERT target) by deliver-ing measures to a subset of households that were considered to be at high risk of fuel poverty. The SPG differed from the PG because of stricter eligibility requirements regarding recipient's household income and the benefits they receive.

Most suppliers struggled to achieve their SPG targets and because of this some failed to achieve their overall CERT targets within the obligation period, which ended in December 2012. This was the first time that a major supplier has failed to deliver on their obligations. OFGEM granted suppliers an extension period until 31 January to deliver their obligations (OFGEM 2012a). At the end of CERT, suppliers offered incentives for SPG recipients including vouchers toward their fuel bill and cash back schemes. Some suppliers also incentivised referrals by granting those who successfully referred a SPG household a lump sum. Because non-retail schemes for SPG households were fully funded in the last phase of CERT, suppliers were paying 100% + x% incentives resulting in a leverage factor of <1.0, i.e. for every euro suppliers spend they have to spend extra in order to achieve the required uptake.

Similar issues occur with insulation for non-SPG households: As part of the CERT extension in 2011 and for the first time since the inception of the ESO, Government decided to set a minimum share for insulation measures. Subsequently, suppliers were required to achieve 68% of the target under CERT by investing in insulation measures (DECC, 2010). Suppliers were also struggling to a target specifying they must achieve at least 68% of their carbon target from March 2011 to January 2013 via insulation measures (DECC, 2010). In June 2012, suppliers had achieved about 65% of their obligation, leaving 35% for the last 6 months. Energy suppliers offered free insulation to everyone, regardless of whether they were in the able-to-pay group the or PG. Hence, leverage for non-retail insulation can be assumed to be zero (i.e. leverage factor of 1.0) for the last phase of CERT.

The above shows that if the targets get tighter, leverage decreases because suppliers find it more difficult to achieve their obligations without fully subsidising the measures. Contributing factors are decreasing potentials for the various measures meaning that fewer opportunities exist leading to more intense competition and more attractive offers with lower leverage. As ESOs in Britain moved towards high cost measures mainly post-2012 [START_REF] Rosenow | The Green Deal and the Energy Company Obligation[END_REF], the issue of leverage becomes more and more important: Will households be willing to make a contribution to costly measures? If not, the cost for suppliers, which will be passed on to all customers, is going to increase.

There is anecdotal evidence that customers are much less willing to make contributions to the high-cost measures leading to lower leverage factors.

Overlap of instruments -The example of France

The French ESO is Europe's system with the broadest range of eligible measures. Still, the measures adopted in practice are only a few. During the first period, the installation of efficient boilers was the most prominent among them, accounting for more than half of the measures. Insulation measures were the second largest contributor in terms of energy savings, making the building domain the primary area that received measures as part of the ESOs. This breakdown has not changed much in the second period. As of April 2013, the share of boilers was 35% and insulation was 15% (DGEC 2013).

In the second period, residential buildings still represent 81% of the total energy savings. Compared to the other eligible sectors, residential buildings is the most costeffective potential for energy savings. This is why, for instance, gasoline suppliers primarily rely on the building sector to generate energy savings, instead of focusing on the transportation sector. Yet within the residential sector, [START_REF] Giraudet | White certificates schemes: the static and dynamic efficiency of an adaptive policy instrument[END_REF] have found that the measures taken in the first period were not necessarily the cheapest. In a competitive market, one would expect the obligated parties to address the most costeffective measures first, assuming that the required contribution is smaller compared to high-cost measures. This is the point where the interaction of the saving obligation scheme with other measures becomes important. Most prominent among them 4 is the tax credit scheme, which interacts closely with the saving obligation. During the first period of the obligation, the tax cuts granted by the tax credit scheme tended to be higher for measures on the heating system than for measures on the insulation, despite the fact that the latter tend to be more cost-effective.

Using cost figures estimated by [START_REF] Giraudet | The costs and benefits of White Certificates schemes[END_REF], leverage for the efficiency measuresset in relation to the additional cost of efficiency measures compared to the baseline, which is directly comparable to estimates found for the UKwas 1.37 in the first period. About 65 % of the additional investment was covered by the tax credit.

Considering the overall investment, the tax credits still coverednearly 35% of the total costs. Compared to these values, the direct contribution of the obligated parties to financing is rather limited. In relation to the additional investment, the obligated parties 4No data is available yet about the interaction between the obligation and the soft loan scheme, which was only implemented in 2009.

only contribute 4 % of the investments costs 5 . In relation to the total investment, this amount is even lower, accounting for 2 % only. Source: own calculations based on [START_REF] Giraudet | The costs and benefits of White Certificates schemes[END_REF] These figures show that the (direct) leverage effect of the saving obligation scheme is rather low, at least compared to what is observed in the UK. The obligated parties' contribution to the financing is rather low, compared to the costs borne by the other actors. Therefore the allocation of the leverage effect to the saving obligation scheme is questionable, for the main financial contribution comes from the tax credit scheme. Nevertheless, before the system was introduced, the demand for the tax credits was rather low and was heavily stimulated by the promotion of the obligated parties. Therefore the saving obligation scheme plays an important part in addressing the barriers, which are not only financial.

In the second period, preliminary analysis by ADEME (2013) found that overlap with the tax credit scheme has receded from nearly 100% to only 50%. At the same time, the amount of subsidies offered by obligated parties does not seem to have increased. This is reflected by the fact that the price of white certificates on the trading platform (www.emmy.fr) has been flat and confined between .325c€/kWh to .425c€/kWh over the course of the second period. This price proved to be an accurate proxy for the costs borne by obligated parties in the evaluation of the first period. This suggests that obligated parties have not yet moved towards high-cost energy saving potentials.

5Direct costs represent only 35% of the total costs incurred by obligated parties in France. In contrast, this share is 85% in the UK.

As a result of the lower contribution from the tax credit and the constant contribution from the obliged parties, the leverage factor should increase in the second period. This increase should however be temporary and followed by a decrease in the longer run, as obligated parties will need to tap harder-to-reach potentials in the future.

Lastly, analysis by ADEME suggests that advice and other information services offered by obligated parties provide consumers with much satisfaction. This suggests that the obligation is effective at inducing the obligated parties to address the information gap.

Denmark

As described in paragraph 3.4 the Danish obligation scheme iswhile comparable to the French scheme in terms of scopequite different in the actual mix of implemented measures. It is the only European scheme with a substantial share of measures undertaken in the commercial and industrial sector, rather than having a strong focus on domestic buildings. The Italian scheme has changed in that direction over the last few years, but data on that scheme is hard to access in an international context. Because these two groups are quite different in their economic behaviour, they are treated separately in the following considerations. In enterprises the leverage is rather low compared to the household sector, still the additional investment is rather high compared to the figures from France and UK. For households (where the main measures are deep renovation and boiler change) the leverage factor is very high compared to the other figures, so with very low capital investment by the obligated party large investments are triggered. About 10 percent of the measures are triggered without any subsidy, but by energy advice only.

However, additionality seems to be an important issue within the Danish obligation scheme. [START_REF] Bundgaard | Spending to save: evaluation of the energy efficiency obligation in Denmark, 2013 eceee summer study DECC[END_REF] showed in an evaluation of the scheme, that in households only a minority (<10 %) of the implemented measures are additional. The scheme then acts as a tool for measurement and verification of energy saving measures undertaken anyhow, but does not lead to significant additional investments. In contrast, capital going into such measures may even be withdrawn from more important investment opportunities. Therefore the high leverage factors of the Danish scheme have to be handled with care.

Country synthesis

The important question arising from the presented findings is what factors make the situation in Denmark so different from the British and French situation. The answer mainly lies in the different system designs. Although the general design of the efficiency obligations is very similar in the three countries, some features make a major difference.

The sheer number of measures carried out in the UK 6 shows that the measures are not undertaken in the regular maintenance cycle. For those out of schedule measures, the financial barrier is much more relevant than in case of regular maintenance where only the additional costs of the efficiency measure are relevant for the barriers because the investor/consumer has already decided to do something.

Investment within the regular maintenance cycle applies to the Danish scheme, and therefore the subsidy levels are considerably lower. Thus the leverage effect is much larger than in the UK and France. Within the French scheme there are large windfall gains, due to tax credits for installation of measures that may not be additional. This is nevertheless intended in the system design.

CONCLUSION

The required investment in energy efficiency is substantial if the multiple benefits of energy demand reduction are to be realised. However, government budgets are becoming increasingly constrained by austerity measures, making it increasingly difficult to provide the funds needed to finance energy efficiency. Equally, rising energy prices and the reluctance of regulators to add further cost to energy bills limit the ability of ESOs paid for by energy consumers to provide sufficient funding for energy efficiency. Better 6

The CERT Update gives a number of nearly 2.5 million wall insulation measures and 3.5 million loft insulation measures, which covers about 20 % of the British building stock leverage of public funding, whether provided in the form of grants (paid for by the exchequer) or through ESOs (paid for by energy consumers), is therefore essential.

We argued that ESOs have traditionally been seen as an instrument with a high leverage effect i.e. stimulating investments over and above the funds collected through energy bills from both private investors and other public bodies. However, so far there is no systematic analysis of the ability of ESOs to lever in private and public capital.

As a first step, we defined a leverage factor which can be used to appraise the degree of leverage of ESOs (and potentially other policy instruments). This leverage factor takes into account the amount of public funds provided, the additional private and public capital levered, but also situations where funding from ESOs is blended with further incentives (for example tax rebates). We applied this leverage factor to three prominent ESOs, namely the UK, France and Denmark. Our analysis suggests a wide range of leverage across ESOs and types of energy efficiency measures supported. This spread can largely be explained by the different design features of ESOs and the degree to which the measures supported are additional. In general we found that the higher the leverage factor, the lower the additionality of implemented actions.

ESOs in the UK historically focused on a large number of relatively simple measures that are in most cases not undertaken as part of the regular maintenance cycle requiring a higher contribution in the form of public funds. Furthermore, the UK system has a strong emphasis of social equity, allocating a large proportion of the activity to vulnerable households often on low incomes who cannot make substantial financial contributions. This leads to a lower leverage effect compared to a scheme that focuses entirely on the able-to-pay. The French ESOs display very similar leverage factors as the UK after the effect of tax rebates is taken into account. Without deducting the effect of tax rebates the French system would show a much higher leverage effect. ESOs in Denmark show high leverage factors, particularly in the household sector. However, there is considerable doubt regarding the additionality of some of the measures and further research is required to unravel the dynamics of the Danish scheme.

Future research should focus on how the trade-off between leverage and additionality can be minimised. More sophisticated methods to incorporate the degree of additionality into the leverage factor would allow for a systematic analysis of policy instruments showing the cost and leverage effect for additional energy savings which would be a more appropriate comparator.

  leverage factor C O = contribution from obligated party to total investment cost C B = contribution from beneficiary to total investment cost C T = contributions from any third parties to total investment cost I a = other incentives such as tax credits

Table 1 : Leverage during EEC 1 2002-05

 1

	Delivery route and promoted measure	Subsidy	Contribution from recipient	Contribution from 3rd party	Leverage factor
	Subsidise installers of insulation
	Cavity wall insula-
	tion
	Able to Pay Cus-tomers (ATP)	47%	44%	10%	2.2
	Priority Group (PG) 3	76%	6%	18%	1.3
	Loft insulation (top
	up)
	ATP	46%	29%	25%	2.2
	PG	62%	4%	33%	1.6
	Loft insulation
	(virgin)
	ATP	50%	42%	8%	2.0
	PG	87%	2%	11%	1.2
	Subsidise products sold via retail
	DIY loft insulation
	ATP	36%	64%	0%	2.8
	PG	34%	67%	0%	3.0
	CFLs
	PG	39%	57%	5%	2.6
	ATP	37%	59%	4%	2.7
	Subsidise products sent via mail
	CFLs
	ATP	81%	17%	2%	1.2
	3	The priority group covers households in receipt of specified state benefits or with elderly residents.

Table 2 : Leverage during EEC 2 2005-08

 2

	Delivery route and	Subsidy	Contribution	Contribution	Leverage factor
	promoted measure		from recipient	from 3rd party
	Subsidise installers of insulation
	Cavity wall insula-
	tion
	ATP	60%	37%	3%	1.7
	PG	92%	0%	7%	1.1
	Loft insulation (top
	up)
	ATP	50%	41%	9%	2.0
	PG	80%	3%	17%	1.3
	Loft insulation
	(virgin)
	ATP	65%	31%	4%	1.5
	PG	93%	0%	7%	1.1
	Subsidise products sold via retail
	DIY loft insulation
	ATP	30%	70%	0%	3.3

Source: own calculations based on

[START_REF] Lees | Evaluation of the Energy Efficiency Commitment[END_REF]

Table 3 : Leverage within the French obligation Scheme

 3

			Obligated	Investor's direct	Tax Credit Cost Sum	Leverage
			party's cost	costs			Factor
	Full Costs	M€	74	2341	1305
						3720
	Additional Costs			504
						1883
	Full Costs	%	2%	63%	35%	100%	2.70
	Additional Costs		4%	27%	69%	100%	1.37

Table 3

 3

	gives an overview of the triggered investments within the Danish obligation
	scheme.
		Obligated	Investor's	Obligated	Investor's	Leverage
		party's cost	direct costs	party's cost	direct costs	Factor
		DKK/kWh
	Enterprises 0,4	0,8	33%	67%	3
	Households 0,4	9,6	4%	96%	25
	Average	0,4	2,8	13%	88%	8
	Source: own calculations based on Togeby (2012)

For a more detailed description of saving obligation principles see[START_REF] Bertoldi | Tradable white certificate schemes: fundamental concepts[END_REF]

We exclude policy instruments like product regulation that require private investment with no matching public money.


P Cattiaux
email: cattiaux@math.univ-toulouse.fr

J León

C Prieur

Estimation for Stochastic Damping Hamiltonian Systems under partial observation. II. Drift term

Keywords: 2010 Mathematics Subject Classification. 60H10, 62M05, 62G08, 35H10 Hypoelliptic diffusion, Nonparametric drift estimation, Partial observations, High frequency estimation

. For some ergodic hamiltonian systems we obtained a central limit theorem for a non-parametric estimator of the invariant density, under partial observation (only the positions are observed). Here we obtain similarly a central limit theorem for a non-parametric estimator of the drift term. This theorem relies on the previous result for the invariant density.

INTRODUCTION.

Let Z t := (X t , Y t ) ∈ R 2d , t ≥ 0 be governed by the following Ito stochastic differential equation:

dX t = Y t dt dY t = σ dW t -(c(X t , Y t )Y t + ∇V (X t ))dt.
(1.1)

Each component Y i (1 ≤ i ≤ d)
is the velocity of a particle i with position X i . Function c is called the damping force and V the potential, σ is some (non-zero) constant and W a standard brownian motion. We shall assume that c and V are regular enough for the existence and uniqueness of a non explosive solution of (1.1). We shall also assume that the process is ergodic with a unique invariant probability measure µ, and that the convergence in the ergodic theorem is quick enough. Some sufficient conditions will be recalled in the next section.

In our previous work [START_REF] Cattiaux | Estimation for stochastic damping hamiltonian systems under partial observation-I. Invariant density[END_REF] we proposed a non-parametric estimator for the invariant density p s of the invariant measure µ. We refer to the introduction of [START_REF] Cattiaux | Estimation for stochastic damping hamiltonian systems under partial observation-I. Invariant density[END_REF] for some references on this problem, as well as short discussion of the physical interest of such models.

In the present paper we attack the problem of estimating the drift term g(x, y) = -(c(x, y)y + ∇V (x)) (1.2)

for incomplete but high-frequency data. We indeed observe only the first component of (Z t , t ≥ 0), and our asymptotic results are proved under the assumption that this component is available at arbitrarily short inter-observation time intervals.

As explained in [START_REF] Cattiaux | Estimation for stochastic damping hamiltonian systems under partial observation-I. Invariant density[END_REF], while there is an impressive literature on non-parametric estimation for the invariant density or the drift term, most of it deals with elliptic diffusion processes. Here we are looking at a fully degenerate process, but still hypo-elliptic. In addition we intend to propose an estimator based on the observation of the positions X only, at some discretized observation times. Actually some works have already been done, in the hypo-elliptic context, but in a parametric framework. We refer to the work by [START_REF] Pokern | Parameter estimation for partially observed hypoelliptic diffusions[END_REF] or to the recent work by [START_REF] Samson | A contrast estimator for completely or partially observed hypoelliptic diffusion[END_REF] and the bibliography therein. In this paper we focus on pointwise estimation, as it is a not straightforward issue in itself. There exist results for the non-parametric estimation of the drift of one-dimensional ergodic diffusions, see e.g. [START_REF] Comte | Penalized nonparametric mean square estimation of the coefficients of diffusion processes[END_REF] in the context of discretized observations, or [START_REF] Dalalyan | Sharp adaptive estimation of the drift function for ergodic diffusions[END_REF] in the context of full time-continuous observations. For partial observations, there are results in the parametric setting (see e.g. [START_REF] Gloter | Efficient estimation of drift parameters in stochastic volatility models[END_REF]. A natural development of our paper would be to extend this kind of results to our framework.

The main result of [START_REF] Cattiaux | Estimation for stochastic damping hamiltonian systems under partial observation-I. Invariant density[END_REF] reads as follows: if p s denotes the invariant density (see the next section for its existence), then one can find a discretization step h n , bandwidths b 1n and b 2n and kernels K such that, defining the estimator pn (x, y)

:= 1 nb d 1n b d 2n n i=1 K x -X ihn b 1n , y - X (i+1)hn -X ihn hn b 2n ,
corresponding to partial observation, it holds

nb d 1n b d 2n (p n (x, y) -p s (x, y)) D -----→ n→+∞ N 0, p s (x, y) K 2 (u, v)dudv ,
for all pairs (x, y). The previous convergence in distribution holds true under the stationary distribution. In the non stationary case we have to shift the summation. See Theorem 3.1 and the comment following the statement of the Theorem in section 3.

The intuitive explanation for the definition of our estimator is based on the Nadaraya-Watson kernel method for regression estimation. Assuming that we observe both coordinates (X t , Y t ) t≥0 on a grid ih n , i = 1, . . . , n, we deduce from the system (1.1) the following approximation

Y (i+1)hn -Y ihn ≈ σ(W (i+1)hn -W ihn ) + g(X ihn , Y ihn )h n .
The first term is interpreted as the noise part and the second one as the regression term. The Naradaya-Watson's method allows us to introduce the following estimator of the function g(x, y)p s (x, y) :

ǧn (x, y)p n (x, y) = 1 (n -1)b 1n b 2n n-1 i=1 K( x -X ihn b 1 n , y -Y ihn b 2 n ) Y (i+1)hn -Y ihn h 1/2 n = 1 (n -1)b 1n b 2n n-1 i=1 K( x -X ihn b 1 n , y -Y ihn b 2 n ) σ(W (i+1)hn -W ihn ) h 1/2 n + g(X ihn , Y ihn )h 1/2 n .
We then have

E ǧn (x, y)p n (x, y) h 1/2 n → g(x, y)p s (x, y).
The estimator ǧn (x, y) is an thus an asymptotically unbiased estimator of g(x, y).

In our work we assume that only the first coordinate X t is observed. Thus we define an estimator where Y ihn is replaced by X (i+1)hn -X ihn and Y (i+1)hn -Y ihn by X (i+1)hn -2X ihn + X (i-1)hn . Furthermore, to preserve some independence properties we need to have a certain lag between the observations used in the kernel and those defining the double increment. This last observation leads us to define X (i+ 1

3 )hn -X ihn for the first increment and X (i+1)hn -2X (i+ 2 3 )hn + X (i+ 1 3 )hn for the second one. We thus define the estimator ĝn as ĝn (x, y) pn (x, y)

:= 1 (n -1)b d 1n b d 2n n-1 i=1 K   x -X ihn b 1n , y - X (i+ 1 3 )hn -X ihn (hn/3) b 2n   D i,n (h n /3) 2 ,
where D i,n := X (i+1)hn -2X (i+ 2 3 )hn + X (i+ 1 3 )hn . Define Ĥn (x, y) := ĝn (x, y) pn (x, y). In Theorem 4.1 of the present paper we state that one can find h n and two different bandwidths b in (for the definition of Ĥn ) and c in (for the definition of pn ) (i = 1, 2) such that in the stationary regime

nb d 1n b d 2n h n (ĝ n (x, y) -g(x, y)) D -----→ n→+∞ N 0, ( 2 3 σ 2 p s (x, y) K 2 (u, v)dudv)I ,
where I denotes the identity matrix in R d . As for the invariant density we also prove such a central limit theorem in the non-stationary regime, shifting the summation (see section 5).

It should be very interesting to estimate separately c(x, y) and ∇V (x), who have different physical interpretations. Actually, there is no explicit relation between the invariant density and the drift term, unless c is constant, in which case

p s (x, y) = exp -2c σ 2 ( |y| 2 2 + V (x)
) . In that particular case, the estimation of the potential V and of its gradient ∇V can be deduced from the estimators of the invariant density and of the drift term. In full generality this will require some other ideas.

THE MODEL AND ITS PROPERTIES.

We are obliged to recall some facts on the model. A more detailed discussion is contained in [START_REF] Cattiaux | Estimation for stochastic damping hamiltonian systems under partial observation-I. Invariant density[END_REF].

We shall first give some results about non explosion and long time behaviour. In a sense, coercivity can be seen in this context as some exponential decay to equilibrium.

Let us first introduce some sets of assumptions: Hypothesis H 1 :

(i) the potential V is lower bounded, smooth over R d , V and ∇V have polynomial growth at infinity and there exists v > 0 such that

+∞ ≥ lim inf |x|→+∞ x.∇V (x) |x| ≥ v > 0 ,
the latter being often called "drift condition", (ii) the damping coefficient c(x, y) is smooth and bounded, and there exist c, L > 0 so that c s (x, y) ≥ cId > 0, ∀(|x| > L, y ∈ R d ), where c s (x, y) is the symmetrization of the matrix c(x, y), given by 1 2 (c ij (x, y)+c ji (x, y)) 1≤i,j≤d , These conditions ensure that there is no explosion, and that the process is positive recurrent with a unique invariant probability measure µ. We will denote by

P t f (z) = E z (f (Z t )
) which is well defined for all bounded function f , P t extends as a contraction semi-group on L p (µ) for all 1 ≤ p ≤ +∞. Further in the paper we add assumptions to ensure that the drift function g defined by (1.2) belongs to the domain of the infinitesimal generator L of P t (see Assumptions H 3 and H 4 ). Furthermore µ admits some exponential moment, hence polynomial moments of any order. Another key feature is that the process is actually α-mixing, i.e.

Proposition 2.1. There exist some constants C > 0 and ρ < 1 such that:

∀ g, f ∈ L ∞ (µ) , ∀ t ≥ 0, |Cov µ (f (Z t ), g(Z 0 ))| ≤ C ρ t/2 g -gdµ ∞ f -f dµ ∞ .
(2.1)

i.e., in the stationary regime, (Z t , t ≥ 0) is α-mixing with exponential rate.

As explained in section 2.2 of [START_REF] Cattiaux | Estimation for stochastic damping hamiltonian systems under partial observation-I. Invariant density[END_REF], the infinitesimal generator L is hypo-elliptic, which implies that µ(dz) = p s (z) dz with some smooth function p s . One can relax the C ∞ assumption on the coefficients into a C k assumption, for a large enough k, but this is irrelevant.

Furthermore it can be shown that p s is everywhere positive, for instance by using an extension of Girsanov theory which is available here.

One can relax some assumptions and still have the same conclusions:

Hypothesis H 2 :
(a) One can relax the boundedness assumption on c in H 1 , assuming that for all N > 0: sup |x|≤N,y∈R d c(x, y) H.S. < +∞, where H.S. denotes the Hilbert-Schmidt norm of a matrix; but one has to assume in addition conditions (3.1) and (3.2) in [START_REF] Wu | Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems[END_REF]. An interesting example (the Van der Pol model) in this situation is described in [START_REF] Wu | Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems[END_REF] subsection 5.3. (b) The most studied situation is the one when c is a constant matrix. Actually almost all results obtained in [START_REF] Wu | Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems[END_REF] or [START_REF] Bakry | Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré[END_REF] in this situation extend to the general bounded case.

Nevertheless we shall assume now that c is a constant matrix.

In this case a very general statement replacing H 1 (i) is given in Theorem 6.5 of [START_REF] Bakry | Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré[END_REF]. Tractable examples are discussed in Example 6.6 of the same paper. In particular one can replace the drift condition on

V by lim inf |x|→+∞ |∇V | 2 (x) > 0 and ∇ 2 V H.S. |∇V | .
Notice that one can relax the repelling strength of the potential, and obtain, no more exponential but sub-exponential or polynomial decay (see the discussion in [START_REF] Bakry | Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré[END_REF]).

From now on in the whole paper we will assume that Hypothesis H 1 (or H 2 ) is fulfilled. In all the proofs of the paper C denotes some constant which may vary from line to line.

Estimation of the invariant density.

In this section we recall the central limit theorem for a non-parametric estimator of the invariant density p s proposed in [START_REF] Cattiaux | Estimation for stochastic damping hamiltonian systems under partial observation-I. Invariant density[END_REF].

First we consider that one can observe the whole process Z . at discrete times with discretization step h n , i.e we consider pn (x, y)

:= 1 nb d 1n b d 2n n i=1 K x -X ihn b 1n , y -Y ihn b 2n . (3.1)
Second we consider the partially observed case, where only the position process X . can be observed, and we approximate the velocity, i.e. we consider pn (x, y)

:= 1 nb d 1n b d 2n n i=1 K x -X ihn b 1n , y - X (i+1)hn -X ihn hn b 2n . (3.2)
In both cases, the kernel K is some C 2 function with compact support A such that A K(x, y)dxdy = 1. We may also assume, without loss of generality that A is a bounded ball. Moreover, we assume that there exists m ∈ N * such that for all polynomials P (x, y) with degree between 1 and m, P (u, v)K(u, v)dudv = 0. That is, we assume the kernel K is of order m.

Let us state the main result in this section (Theorem 3.1 below).

Theorem 3.1 [START_REF] Cattiaux | Estimation for stochastic damping hamiltonian systems under partial observation-I. Invariant density[END_REF]). Assume Hypothesis H 1 or H 2 are fulfilled. Recall that p s denotes the density of the invariant measure µ. Assume that the bandwidths b 1n , b 2n and the discretization step h n tend to zero as n tends to infinity and satisfy the following assumptions:

(i) n b d 1n b d 2n → +∞, (ii) b1n b2n h 2 n → 0 , (iii) m is such that n b d 1n b d 2n max(b 1n , b 2n ) 2(m+1) → 0.
Then, in the stationary regime, one gets for any

(x, y) ∈ R 2d nb d 1n b d 2n (p n (x, y) -p s (x, y)) D -----→ n→+∞ N 0, p s (x, y) K 2 (u, v)dudv .
If in addition

(iv) nh n b d 1n b 2+d 2n → 0, (v) there exists p > 1 such that n h 2 n b d(2-p)/p 1n b 2+d 2n → 0.
Then, still in the stationary regime, one gets for any

(x, y) ∈ R 2d nb d 1n b d 2n (p n (x, y) -p s (x, y)) D -----→ n→+∞ N 0, p s (x, y) K 2 (u, v)dudv .
A similar statement holds true starting from any point z 0 ∈ R 2d . In this situation we have to slightly change the definition of our estimators replacing n i=1 by n+ln i=1+ln for some l n such that l n h n → +∞ as n → +∞.

Estimation of the drift term.

In this section we will consider an estimator of the drift function from

R 2d into R d , g(x, y) = -[c(x, y)y + ∇V (x)].
Let K : R 2d → R be a C 2 2d-dimensional kernel whose support is compact and such that there exists m ∈ N * such that for all non constant polynomial P (x, y) with degree less or equal than m, P (u, v)K(u, v)dudv = 0. The estimators of the invariant density, pn (x, y) and pn (x, y) are defined as in Section 3. However, for simplicity we will only use pn (x, y).

Define for i ∈ N * , n ∈ N * , D i,n := X (i+1)hn -2X (i+ 2 3 )hn + X (i+ 1 3 )hn (4.1) = (i+1)hn (i+ 2 3 )hn (Y s -Y (i+ 2 3 )hn ) ds + (i+ 2 3 )hn (i+ 1 3 )hn (Y (i+ 2 3 )hn -Y s ) ds .
Because one observes only the position of the particle and not its derivative, we must define our estimator by using only the position.

We introduce the estimator ĝn (x, y) of g(x, y) defined as ĝn (x, y) pn (x, y)

:= 1 (n -1)b d 1n b d 2n n-1 i=1 K   x -X ihn b 1n , y - X (i+ 1 3 )hn -X ihn (hn/3) b 2n   D i,n (h n /3) 2 , where pn (x, y) := 1 nb d 1n b d 2n n i=1 K x -X ihn c 1n , y - X (i+1)hn -X ihn hn c 2n .
Note that the the bandwidths b in , i = 1, 2 and c in , i = 1, 2 are constrained by (4.2) in Theorem 4.1. This constraint is needed for our proof of Theorem 4.1 from Proposition 4.3 which deals with the central limit theorem for Ĥn . The choice of our estimator ĝn has been motivated in the introduction of the paper. It is both based on a natural heuristic and on technical arguments needed for the proof of Proposition 4.2, which is a step towards the proof of our main result. This main result is stated in Theorem 4.1 below, it consists in a central limit theorem for ĝn (x, y).

During the proof we shall need an additional assumption namely:

Hypothesis H 3 : g belongs to the domain of the infinitesimal generator L, in all L p (µ) for 1 ≤ p < +∞.

According to the properties we recalled before, for H 3 to be satisfied it is enough that:

Hypothesis H 4 : the function c (resp. V ) and its first two derivatives (resp. its first three derivatives) have polynomial growth.

The aim of these assumptions is to ensure that the drift function g belongs to the domain of the infinitesimal generator L of P t .

Theorem 4.1. Assume that H 1 , H 2 and H 3 are satisfied and that the the bandwidths b 1n , b 2n , c 1n , c 2n and the discretization step h n tend to zero as n tends to ∞. Assume moreover that the following assumptions are satisfied respectively with r in = b in and r in = c in :

i) n h n r d 1n r d 2n → +∞, ii) m ∈ N * is such that nr d 1n r d 2n h n max(r 1n , r 2n ) 2(m+1) -----→ n→+∞ 0, iii) ∃ ε 1 > 0 such that n(r d 1n r d 2n ) 1-ε1 h 3 n -----→ n→+∞ 0, iv) ∃ ε 2 , ε 3 ∈ R * + , ε 2 , ε 3 < 1 such that h 2(1-ε 2 ) n (r1nr2n) ε 3 -----→ n→+∞ 0, v) there exist p > 1, p < +∞ and ε > 0, such that h 2 n r d( 1 p -1) 1n r -(2+d) 2n -----→ n→+∞ 0 and h n √ n r d( 1 p(1+ε) -1 2 ) 1n r -( d 2 +1) 2n -----→ n→+∞ 0.
Define ĝn = Ĥn (b)/p n (c) where the indication in brackets indicates the bandwidths we are using. Assume in addition that

h n (b, c) := h n b d 1n b d 2n c d 1n c d 2n → 0 as n → +∞. (4.2) Then nb d 1n b d 2n h n (ĝ n (x, y) -g(x, y)) D -----→ n→+∞ N 0, ( 2 3 σ 2 p s (x, y) K 2 (u, v)dudv)I .
For technical reasons, the proof of Theorem 4.1 is based on two preliminary results stated respectively in Proposition 4.2 and Proposition 4.3. We first introduce an intermediate kernel estimate of g(x, y) p s (x, y), denoted by Hn , and defined by:

Hn (x, y) = 1 (n -1)b d 1n b d 2n n-1 i=1 K x -X ihn b 1n , y -Y ihn b 2n D i,n (h n /3) 2 . (4.3)
Note that this estimate can not be the final estimate as the Y ihn s are not observed in practice. However, studying the convergence of Hn (x, y) is easier and is the first setp towards the proof of Theorem 4.1. Now, to see why the definition of Hn (x, y) is meaningful, let us first study the asymptotic bias of this estimator in the stationary regime.

Using stationarity we get

E[ Hn (x, y)] = 9 b d 1n b d 2n E K x -X 0 b 1n , y -Y 0 b 2n D 0,n h 2 n = 9 b d 1n b d 2n R 2d K x -u b 1n , y -v b 2n E D 0,n h 2 n |X 0 = u Y 0 = v p s (u, v)dudv.
Using (4.1), we may write

D 0,n h 2 n = 1 h 2 n σ[ hn 2 3 hn (W s -W 2 3 hn )ds + 2 3 hn hn 3 (W 2 3 hn -W s )ds] + I(h n ) , (4.4)
where

I(h n ) = hn 2 3 hn t 2 3 hn g(X s , Y s )dsdt + 2 3 hn hn 3 2 3 hn t g(X s , Y s )dsdt.
The independence of the increments of W and the semigroup properties yield

E[ D 0,n h 2 n |X 0 = u , Y 0 = v] = 1 h 2 n hn 2 3 hn t 2 3 hn P s g(u, v)dsdt + 2 3 hn hn 3 2 3 hn t P s g(u, v)dsdt =: G(h n , u, v). Notice that, G(h n , u, v) → 1 9 g(u, v) as n → +∞ since h n → 0. A change of variable entails E[ Hn (x, y)] = 9 b d 1n b d 2n R 2d K x -u b 1n , y -v b 2n G(h n , u, v)p s (u, v)dudv = 9 R 2d K(z 1 , z 2 )G(h n , x -b 1n z 1 , y -b 2n z 2 )p s (x -b 1n z 1 , y -b 2n z 2 )dz 1 dz 2 ,
which converges to g(x, y)p s (x, y) as n tends to infinity, according to the bounded convergence theorem and under Assumption H 4 . Indeed, Assumption H 4 ensures that g defined by (1.2) is regular enough to belong to the domain of the infinitesimal generator of the semi-group P t . The main argument to prove that is given in Lemma 1.1 in [START_REF] Wu | Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems[END_REF]. Hence Hn is an asymptotically unbiased estimator of gp s .

Starting from this consideration, we now state a central limit theorem for the estimator Hn (x, y). Let denote I the identity matrix in R d . Proposition 4.2. Assume that H 1 , H 2 and H 3 are satisfied and that the bandwidths b 1n , b 2n and the discretization step h n tend to zero as n tends to ∞. Assume moreover that the bandwidths b in , i = 1, 2, satisfy assumptions i) to iv) of Theorem 4.1. Then, in the stationary regime,

nb d 1n b d 2n h n Hn (x, y) -g(x, y)p s (x, y) D -----→ n→+∞ N 0, ( 2 3 σ 2 p s (x, y) K 2 (u, v)dudv)I .
Proof of Proposition 4.2: the proof is postponed to the Appendix.

We now deduce from Proposition 4.2 the following result on the asymptotic behaviour of Ĥn (x, y) = ĝn (x, y) pn (x, y): Proposition 4.3. Under the assumptions of Proposition 4.2 and assuming moreover assumption v) of Theorem 4.1 one gets

nb d 1n b d 2n h n Ĥn (x, y) -p s (x, y)g(x, y) D -----→ n→+∞ N 0, ( 2 3 σ 2 p s (x, y) K 2 (u, v)dudv)I .
Proof of Proposition 4.3: the proof is postponed to the Appendix.

Proof of the main result, Theorem 4.1: We have

ĝn -g = Ĥn pn -g = Ĥn -p s g p s + Ĥn 1 pn - 1 p s .
For the first term, we have according to Proposition 4.3,

nb d 1n b d 2n h n Ĥn -p s g p s (x, y) D -----→ n→+∞ N 0, ( 2 3 σ 2 p s (x, y) K 2 (u, v)dudv)I .
We shall show that the second one goes to 0 in probability and then conclude by using Slutsky theorem. We thus decompose

nb d 1n b d 2n h n Ĥn 1 pn - 1 p s = h n (b, c) Ĥn pn p s nc d 1n c d 2n (p s -pn ) .
The second term of this product converges in distribution according to Theorem 3.1. We shall show that the first term in the product goes to 0 in probability. Indeed, according to [START_REF] Cattiaux | Estimation for stochastic damping hamiltonian systems under partial observation-I. Invariant density[END_REF], pn -p s → 0 in probability as n → +∞, and as we previously saw, Ĥn is bounded in L 1 . Let a > 0. We have

P h n (b, c) Ĥn pn p s > a ≤ P (p n ≤ (p s /2)) + P Ĥn > (a p 2 s /2 h n (b, c)) ,
and both terms go to 0, using the Markov inequality for instance for the second one.

To conclude it remains to recall that if U n goes to 0 in probability and V n goes to V in distribution, the product U n V n goes to 0 in probability. .

We conclude this section by giving an explicit class of examples for the parameters h n , b i,n , c i,n to satisfy all the required assumptions in Theorem 4.1.

Proposition 4.4. Choose h n = n -γ , b in = n -αi and c in = n -βi for some γ, α i , β i > 0. If (1) α 2 = β 2 - ε d(4+4d) = 1-2ε d(4+4d) , (2) β 1 = 3-2ε+4d d(4+4d) , α 1 = β 1 + β2 2 -1 2d 2 , (3) 1 2d < γ < 1 2d + 3ε 4+4d , (4) m > 1+2d
1-ε . for some ε > 0 small enough, then Theorem 4.1 holds true. ).

Non-stationary case

In Section 3 we stated the central limit theorem for the estimate of the drift g(x, y) in the case where the process is in the stationary regime. Let us now define the new estimate

g n (x, y) = 1 (n-1)b d 1n b d 2n n+ln-1 i=ln+1 K x-X ihn b1n , y- X (i+ 1 3 )hn -X ihn (hn /3) b2n Di,n (hn/3) 2 1 nb d 1n b d 2n n+ln i=ln+1 K x-X ihn b1n , y- X (i+1)hn -X ihn hn b2n
(5.1)

We remark that given Z 0 ∼ µ(dz), g n (x, y)

L = ĝn (x, y) ∀ n ∈ N * .
Theorem 5.1 below states that we can estimate g(x, y) by using g n (x, y) with Z 0 = z 0 = (x 0 , y 0 ).

Theorem 5.1. Under the assumptions in Theorem 4.1, starting from any initial point z 0 = (x 0 , y 0 ), it holds

n h n b d 1n b d 2n (g n (x, y) -g(x, y)) D -----→ n→+∞ N 0, ( 2 3 σ 2 p s (x, y) K 2 (u, v)dudv)I ,
provided l n appearing in the definition (5.1) of g n (x, y) satisfies l n h n -----→ n→+∞ +∞.

Proof of Theorem 5.1: Recall that p s : R 2d → R + denotes the invariant density of Z and µ the associated invariant probability measure. Denote by C b (R) the set of bounded continuous functions h : R → R. It is only necessary to prove that, for any h ∈ C b (R), the difference

∆ n (h) = E h( h n nb d 1n b d 2n ĝn (x, y)|Z 0 ∼ µ) -h( h n nb d 1n b d 2n g n (x, y)|Z 0 = z 0 ) = E h( h n nb d 1n b d 2n g n (x, y)|Z 0 ∼ µ) -h( h n nb d 1n b d 2n g n (x, y)|Z 0 = z 0 ) goes to zero as n tends to infinity. Let h ∈ C b (R) and denote θ = h ∞ . To evaluate E (h(g n (x, y))|Z 0 ∼ µ) -E (h(g n (x, y))|Z 0 = z 0 ). Let us fix n ∈ N * .
We first make the computations conditionally to Z jhn , j > l n + 1.

One may write g n (x, y) = g n (x, y, Z (ln+1)hn , Z jhn , j > l n + 1) and h Z jhn , j>ln+1 (z ) = g n (x, y, z , Z jhn , j > l n + 1). Now, conditionally to Z jhn , j > l n + 1, one has:

|E (h(g n (x, y))|Z 0 ∼ µ) -E (h(g n (x, y))|Z 0 = z 0 )| = h Z jhn , j>ln+1 (z ) p s (z ) -q (ln+1)hn (z 0 , z ) dz ≤ θDρ (ln+1)hn Ψ(z 0 ) (5.2)
using Inequality (2.1) in [START_REF] Cattiaux | Estimation for stochastic damping hamiltonian systems under partial observation-I. Invariant density[END_REF].

Finally, as 0 < ρ < 1, we can conclude that ∆ n (h) goes to zero as n tends to infinity as soon as l n h n -----→ n→+∞ +∞, which concludes the proof of Theorem 5.1.

Examples and numerical simulation results

In this section, we consider examples of stochastic differential equations defined by (1.1) and implement the estimator on simulated data. However, the choice of the optimal bandwidths b 1n , b 2n , c 1n and c 2n as far as the choice of the optimal discretization step h n , and the optimal choice of the kernel K, although interesting, is not the purpose of this section nor this paper. This is a separate study to be addressed in future work.

To simulate sample paths, we use an approximate discrete sampling generated by an explicit Euler scheme (see Remark 6.1 at the end of this section). We consider three specific examples. The first one has been proposed in [START_REF] Pokern | Parameter estimation for partially observed hypoelliptic diffusions[END_REF]. It corresponds to a linear oscillator subject to noise and damping. The second example is one example of generalized Duffing oscillators described in [START_REF] Wu | Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems[END_REF] subsection 5.2. The last example is the Van der Pol oscillator whose damping force depends on both position and velocity coordinates. These three models are of type (1.1) and satisfy assumptions needed to apply our estimation results. Simulations are run with the Epanechnikov kernel.

6.1. Model I: harmonic oscillator. We consider an harmonic oscillator that is driven by a white noise forcing:

dX t = Y t dt dY t = σ dW t -(κY t + DX t )dt. (6.1)
with κ > 0 and D > 0. In the following we choose D = 2, κ = 2 and σ = 1. For this model we know that the stationary distribution is Gaussian, with mean zero and an explicit variance matrix given in e.g. [START_REF] Gardiner | Handbook of stochastic methods[END_REF]. With our choice of parameters, the Gaussian invariant density is

p s (x, y) = 2 √ 2 π exp -4x 2 -2y 2 .
And the drift is defined by g(x, y) = -2(y + x). In the following we make use of the explicit Euler scheme to simulate an approximated discrete sampling ( X i , Y i ) i∈N of (X t , Y t ) t∈R+ . For a given step δ > 0, the scheme is defined as

X i+1 -X i = Y i δ Y (i+1) -Y i = σ (W (i+1)δ -W iδ ) -(κ Y i + D X i )δ (6.2)
( X 0 , Y 0 ) = (0, 0). We take n = 5000, h = 0.28, b 1n = b 2n = 0.18, and the step for the explicit Euler scheme δ = 1 10 h/3. For some fixed value of x 0 , the drift g(x 0 , •) is estimated on a grid (z l ) l=1,...,L = (x 0 , y l ) l=1,...,L .

For some fixed value of y 0 , the drift g(•, y 0 ) is estimated on a grid (z l ) l=1,...,L = (x l , y 0 ) l=1,...,L .

On Figure 7.1 below we chose L = 40 and x 0 = 0.0230. On Figure 7.2 below we chose L = 40 and y 0 = 0.1878.

6.2.

Model II: generalized Duffing oscillator. We consider the noisy Duffing oscillator known as Kramers oscillator. The system (1.1) can now be written as

dX t = Y t dt dY t = σ dW t -(κY t + αX 3 t -βX t )dt (6.3) with σ, κ, α and β > 0. The potential is then V (x) = α x 4 4 -β x 2 2 . The invariant density is in that case p s (x, y) = √ κ √ πσC exp -2κ σ 2 ( α x 4 4 - β x 2 2 + y 2 2 ) ,
with C the normalizing constant.

And the drift is defined by g(x, y) = -(κy + αx 3 -βx).

Once more, we make use of the explicit Euler scheme to simulate an approximated discrete sampling. The choice for the parameters is σ = 1, κ = α = β = 1.

We take n = 10 4 , h n = 0.30, b 1n = b 2n = 0.30, and the step for the explicit Euler scheme δ = 1 10 h/3. The drift is estimated for some fixed value of x 0 , g(x 0 , •), on a grid (z l ) l=1,...,L = (x 0 , y l ) l=1,...,L . It is also estimated for some fixed value of y 0 , g(•, y 0 ), on a grid (z l ) l=1,...,L = (x l , y 0 ) l=1,...,L . On Figure 7.3 below we chose L = 40 and x 0 = 0.0230. On Figure 7.4 below we chose L = 40 and y 0 = 0.1878. 6.3. Model III: Van der Pol oscillator. We consider the Van der Pol oscillator defined by

dX t = Y t dt dY t = σ dW t -((c 1 X 2 t -c 2 )Y t + ω 2 0 X t )dt (6.4)
with σ, c 1 , c 2 and ω 2 0 > 0. In the following we choose σ = c 1 = c 2 = ω 0 = 1. The drift is then defined by g(x, y) = -((x 2 -1)y + 4x). The invariant density is unknown in that case. However, we know that it is the solution of the corresponding Fokker-Planck equation 1 2

∂ 2 p s (x, y)
∂y 2 -y ∂p s (x, y) ∂x + c(x, y)p s (x, y) + (c(x, y)y + V (x)) ∂p s (x, y) ∂y = 0 . (6.5)

Thus the invariant density p s (x, y) may be approximated by solving Equation (6.5) above, e.g. using a finite difference scheme. Doing so we remark that this density, although positive, is very small in many points. Therefore we plotted g(x, y) p s (x, y) to avoid numerical instabilities.

We made use of the explicit Euler scheme to simulate an approximated discrete sampling.

On Figures 7.5 and 7.6 below we took n = 10 5 , h n = 0.18, b 1n = b 2n = 0.10, and the step for the explicit Euler scheme δ = 1 10 h/3. The drift for some fixed value of x 0 , g(x 0 , •), is estimated on a grid (z l ) l=1,...,L = (x l , y 0 ) l=1,...,L .

On Figure 7.5 below we chose x 0 = 0.0230 and plotted g(x 0 , •) * p s (x 0 , •).

The drift for some fixed value of y 0 , g(•, y 0 ) is estimated on a grid (z l ) l=1,...,L = (x 0 , y l ) l=1,...,L .

L was chosen equal to 40. On Figure 7.6 below we chose y 0 = 0.1878 and plotted g(•, y 0 ) * p s (•, y 0 ). We remark that for this third model, we do not estimate the drift itself, but the drift multiplied by the invariant density. Indeed, the direct estimation of the drift required divisions by quantities close to zero. Even with a regularization term, the results were not convincing. Remark 6.1. In the simulations above we used the explicit Euler scheme, which may be unstable because the coefficients of the differential system 1.1 are unbounded for the models we considered as; see e.g. [START_REF] Talay | Stochastic Hamiltonian systems: exponential convergence to the invariant measure, and discretization by the implicit Euler scheme[END_REF]. We also implemented an implicit scheme, but it slowed too much the simulations, and the results were not better.

To conclude this section, note that non parametric estimation allows capturing the shape of a drift for which no parametric formula is available. For that reason, it is of real interest for practicians. Once the shape is captured, it might then become possible to propose a parametric model whose parameters should be estimated. To prove Proposition 4.2 we first prove that I 2n → 0 and then that

I 1n D -----→ n→+∞ N (0, ( 2 3 σ 2 p s (x, y) K 2 (x, y)dxdy)I).
Define

I i (h n ) := 9 (i+1)hn (i+ 2 3 )hn t (i+ 2 3 )hn g(X s , Y s )dsdt + (i+ 2 3 )hn (i+ 1 3 )hn (i+ 2 3 )hn t g(X s , Y s )dsdt , and
W i,n := 9 (i+1)hn (i+ 2 3 )hn (W s -W (i+ 2 3 )hn )ds + (i+ 2 3 )hn (i+ 1 3 )hn (W (i+ 2 3 )hn -W s )ds .
The vector (n -1)b d 1n b d 2n h n Hn (x, y) can be decomposed in two terms: the one driving the bias in the central limit theorem

S n,1 (x, y) := √ h n (n -1)b d 1n b d 2n n-1 i=1 K x -X ihn b 1n , y -Y ihn b 2n 1 h 2 n I i (h n ) ,
and the one driving the variance

S n,2 (x, y) := σ (n -1)b d 1n b d 2n n-1 i=1 K x -X ihn b 1n , y -Y ihn b 2n W i,n h 3/2 n .
Notice that ES n,2 (x, y) = 0. We thus have We define

I 2n = ES n,1 (x, y) -(n -1)b d 1n b d 2n h n g(x, y)p s (x, y) , ( 7
P i,n 9 := (i+1)hn (i+ 2 3 )hn t (i+ 2 3 )hn (P s g(X ihn , Y ihn ) -g(X ihn , Y ihn ))dsdt + (i+ 2 3 )hn (i+ 1 3 )hn (i+ 2 3 )hn t (P s g(X ihn , Y ihn ) -g(X ihn , Y ihn ))dsdt .
Thanks to stationarity, it holds

I 2n = (n -1)h n b d 1n b d 2n E K x -X 0 b 1n , y -Y 0 b 2n 1 h 2 n P 0,n + + (n -1)h n b d 1n b d 2n E K x -X 0 b 1n , y -Y 0 b 2n g(X 0 , Y 0 ) -b d 1n b d 2n g(x, y)p s (x, y) .
The second summand in the above expression can be treated as in a classical density estimation problem. More precisely, this term is equal to

(n -1)b d 1n b d 2n h n K(u, v) {g(x -ub 1n , y -vb 2n )p s (x -ub 1n , y, vb 2n ) -g(x, y)p s (x, y)} dudv .
Thus assuming there exists m ∈ N * such that for all polynomials P (x, y) with degree between 1 and m, P (u, v)K(u, v)dudv = 0, and performing a Taylor expansion, the above term converges to zero as n tends to infinity as soon as

nb d 1n b d 2n h n max(b 1n , b 2n ) 2(m+1) -----→ n→+∞ 0 .
Let us now study the first summand. As each of the coordinates of the drift function g belongs to the domain of the infinitesimal generator L according to

H 3 , ∀ 1 ≤ p < +∞ (P t g -g)/t is bounded in L p (µ) uniformly in t for t ∈ [0, 1], say by M p . Now write (n -1)h n b d 1n b d 2n E K x -X 0 b 1n , y -Y 0 b 2n 1 h 2 n P 0,n =: 9 (n -1)h n b d 1n b d 2n (A 1n + A 2n ) with A 1n = 1 h 2 n hn 2hn/3 t 2hn/3 (P s g(u, v) -g(u, v)) K x -u b 1n , y -v b 2n ds dt µ(du, dv)
and A 2n being similar just changing

hn 2hn/3 t 2hn/3 into 2hn/3 hn/3 2hn/3 t
. We thus only study A 1n . Using Fubini's theorem we may first integrate with respect to t and write

A 1n = 1 h 2 n hn 2hn/3 (h n -s) s P s g(u, v) -g(u, v) s K x -u b 1n , y -v b 2n ds µ(du, dv) .
Now we integrate with respect to µ, use Cauchy-Schwarz inequality and the remark we have made about P s g -g (assuming that h n ≤ 1). We thus have (|.| denoting the norm in R d )

|A 1n | ≤ M p h 2 n hn 2hn/3 (h n -s) s K r x -u b 1n , y -v b 2n µ(du, dv) 1/r ds ≤ CM p h n K r x -u b 1n , y -v b 2n µ(du, dv) 1/r with p > 1 , r < +∞ ∈ N * such that 1 p + 1 r = 1. It follows, using again the change of variables (n -1)h n b d 1n b d 2n A 1n ≤ C p (n -1)h 3 n (b 1n b 2n ) d( 1 r -1 2 ) .
Thanks to assumption iii), one can choose r such that this last term tends to zero as n tends to infinity.

Second step: Study of S n,2

We now consider the term driving the variance. To study the weak convergence of this sequence we adapt the proof of Theorem 3 in [START_REF] Beśka | Limit theorems for random sums of dependent d-dimensional random vectors[END_REF] and study the characteristic function of S n,2 . Let us recall that

S n,2 (x, y) := σ (n -1)b d 1n b d 2n n-1 i=1 K x -X ihn b 1n , y -Y ihn b 2n W i,n h 3/2 n and that ES n,2 (x, y) = 0.
The sketch of the proof of this step is the following. We first prove that the sequence of random variables {S n,2 (x, y) , n ≥ 1} is tight. We then prove that if S n k ,2 (x, y) is a subsequence of the original sequence, which converges in distribution, then it converges to Y ∼ N (0, 1). We then conclude that the sequence itself converges in distribution to Y .

We proceed now with the proof.

Define for any

u ∈ R + , F u := σ ((X l , Y l ) , 0 ≤ l ≤ u). We now introduce, for t ∈ R d , f n (t) := n-1 k=1 E[e i<t, σ √ (n-1)b d 1n b d 2n K( x-X khn b 1n , y-Y khn b 2n ) W k,n h 3/2 n > |F khn ] ,
where < •, • > denotes the usual scalar product in R d . Thanks to the independence of the Brownian increments we get

f n (t) = n-1 k=1 e - t 2 2 3 σ 2 d 2(n-1)b d 1n b d 2n K 2 ( x-X khn b 1n , x-Y khn b 2n ) = e - t 2 2 3 σ 2 d 2(n-1)b d 1n b d 2n n-1 k=1 K 2 ( x-X khn b 1n , x-Y khn b 2n
)

.

Now define

Z n = 1 (n -1)b d 1n b d 2n n-1 i=1 K 2 x -X ihn b 1n , y -Y ihn b 2n .
It satisfies

EZ n = 1 b d 1n b d 2n R 2d K 2 x -u b 1n , y -v b 2n p s (u, v)dudv → p s (x, y) K 2 (u, v)dudv = A. Furthermore E (|Z n -A|) ≤ 1 b d 1n b d 2n K 2 x -u b 1n , y -v b 2n p s (u, v) -A dudv ≤ 1 b d 1n b d 2n K 2 x -u b 1n , y -v b 2n |p s (u, v) -p s (x, y)| dudv ≤ K 2 (u, v) |p s (x -b 1n u, y -b 2n v) -p s (x, y)| dudv
and the latter goes to 0 by using the bounded convergence theorem and the continuity of p s .

Thus Z n → p s (x, y) K 2 (z 1 , z 2 )dz 1 dz 2 , in L 1 . Using the bounded convergence theorem, we deduce that

f n (t) P → e -t 2 2 3 σ 2 d 2 ps(x,y) K 2 (z1,z2)dz1dz2 =: φ(t).
Passing to subsequences if necessary, we can assume that this convergence holds almost everywhere i.e.

f n (t) → φ(t) a.e.

(7.3)

Let us now define for k = 1, . . . , n the sets

H nk = {e - t 2 2 3 σ 2 d 2(n-1)b d 1n b d 2n k-1 i=1 K 2 ( x-X ihn b 1n , y-Y ihn b 2n ) ≥ 1 2 φ(t)}.
Of course

H n,n ⊂ H n,n-1 ⊂ . . . ⊂ H n,2 ,
and using (7.3)

P{lim sup n→∞ H c n,n } = 0. (7.4)
Finally, introduce the random variables

ζ n,i = σ (n -1)b d 1n b d 2n K x -X ihn b 1n , y -Y ihn b 2n W i,n h 3/2 n 1 Hn,i ,
and also

f * n (t) = n-1 i=1 E[e i<t,ζn,i> |F ihn ] = n-1 i=1 e - t 2 2 3 σ 2 d 2(n-1)b d 1n b d 2n K 2 ( x-X ihn b 1n , y-Y ihn b 2n
) 1 H n,i .

It holds

f * n (t) ≥ 1 2 φ(t)
by definition and f * n (t) → φ(t) a.e. by (7.4) .

(7.5) So, we can assume that these two properties hold for the initial variables.

Let us now come back to the study of the weak convergence of S n,2 (x, y). By using Markov property and the independence of the Brownian increments, we obtain )

E e i<t,S n,2 (x,y)> fn(t) = E   n-1 i=1 e i<t, σ √ (n-1)b d 1n b d 2n K( x-X ihn b 1n , y-Y ihn b 2n ) W i,n h 3/2 n > e - t 2
F (n-1)h         = E   n-2 i=1 e i<t, σ √ (n-1)b d 1n b d 2n K( x-X ihn b 1n , y-Y ihn b 2n ) W i,n h 3/2 n > e - t 2 2 3 σ 2 d 2(n-1)b d 1n b d 2n K 2 ( x-X ihn b 1n , x-Y ihn b 2n )   = 1 using induction.
Now we are ready to prove the weak convergence E e i<t,Sn,2(x,y)> -φ(t) = E e i<t,Sn,2(x,y)> -E φ(t) e i<t,Sn,2(x,y)> f n (t)

= E e i<t,Sn,2(x,y)> 1 -

φ(t) f n (t) ≤ 2 φ(t) E|f n (t) -φ(t)| → 0.
This last equality is deduced from the L 1 -L ∞ Hölder inequality combined with the bound on f n (t) deduced from (7.5).

We may conclude the proof of the tightness of {S n,2 (x, y) , n ≥ 1}. To this end, let S n k ,2 (x, y) be a subsequence of the original sequence. We know that f n k (t) P → φ(t). Whence there exists another subsequence f n k j (t) a.e. → φ(t). By the above result S n k j converges weakly to a r.v. Y , moreover E[e i<t,Y > ] = φ(t). Thus the tightness.

All the limits of the convergent subsequences being the same, we directly conclude that the sequence {S n,2 (x, y) , n ≥ 1} converges weakly and that its limit is Y . It concludes the proof of the second step.

Third step: study of Z n := S n,1 (x, y) -ES n,1 (x, y)

Let us denote as before by P k i,n the k th coordinate of the vector P

i,n . Defin- ing Γ k n (i, x, y, X, Y ) = K( x-X b1n , y-Y b2n ) 1 h 2 n P k i,n , we write Z k n = √ h n (n -1)b d 1n b d 2n n-1 i=1 Γ k n (i, x, y, X ihn , Y ihn ), so that (n-1)b d 1n b d 2n hn V ar(Z k n ) = n-1 i=1 V ar(Γ k n (i, x, y, X ihn , Y ihn ))+ i =l Cov(Γ k n (i, x, y, X ihn , Y ihn ), Γ k n (l, x, y, X lhn , Y lhn )) .
To bound the above expression we first write as we did for the first step

E(Γ k n (i, x, y, X ihn , Y ihn )) 2 = E K( x -X b 1n , y -Y b 2n ) 1 h 2 n P k i,n 2 ≤ U 1n + U 2n with U 1n = 2 9 (i+1)hn (i+ 2 3 )hn t (i+ 2 3 )hn (P s g(X ihn , Y ihn ) -g(X ihn , Y ihn ))dsdt 2 and U 2n = 2 9 (i+ 2 3 )hn (i+ 1 3 )hn (i+ 2 3 )hn t (P s g(X ihn , Y ihn ) -g(X ihn , Y ihn ))dsdt 2 .
Then, using stationarity,

U 1n ≤ C h 4 n E   K 2 x -X 0 b 1n , y -Y 0 b 2n hn 2hn/3 (h n -s) (P s g(X 0 , Y 0 ) -g(X 0 , Y 0 )) ds 2   , U 2n being similar just replacing hn 2hn/3 (h n -s) by 2hn/3 hn/3 (s -(h n /3)). Using Cauchy-Schwarz inequality we get U 1n ≤ C h n hn 2hn/3 K 2 x -u b 1n , y -v b 2n (P s g(u, v) -g(u, v)) 2 dµ ds ≤ C h n hn 2hn/3 K 2 x -u b 1n , y -v b 2n P s g(u, v) -g(u, v) s 2 dµ ds .
We may argue as in the first step, this time using Hölder inequality for some conjugate pair (p, q) and H 3 in L 2q , to conclude that

U 1n ≤ C h 2 n (b 1n b 2n ) d/p . It follows h n (n -1)b d 1n b d 2n n-1 i=1 V ar(Γ k n (i, x, y, X ihn , Y ihn )) = O h 3 n (b 1n b 2n ) d( 1 p -1) . (7.6)
One can choose p such that the right hand term tends to zero thanks to assumption iv).

Let us now compute the covariances.

One has thanks to stationarity and mixing inequality (2.1)

i =l Cov Γ k n (i, x, y, X ihn , Y ihn ), Γ k n (l, x, y, X lhn , Y lhn ) ≤ Cn n-2 j=1 min ρ jhn/2 , Var(Γ k n (0, x, y, X 0 , Y 0 )) .
Then, using inequality (7.6) one gets the following bound

≤ Cn n-2 j=1 min ρ jhn/2 , (b 1n b 2n ) d p h 2 n .
Then using that for all x, y ≥ 0, and all a ∈ (0, 1), x ∧ y ≤ x 1-a y a one gets that the above quantity is in

O n(b 1n b 2n ) d p (1-a) h 1-2a
n for any 0 < a < 1.

We thus get

h n (n -1)b d 1n b d 2n i =l Cov Γ k n (i, x, y, X ihn , Y ihn ), Γ k n (l, x, y, X lhn , Y lhn ) = O (b 1n b 2n ) d p (1-a)-d h 2(1-a) n .
One can choose p and a such that the right hand term tends to zero as n tends to infinity thanks to assumption iv). This completes the proof.

(n -1)b d 1n b d 2n n-1 j=1   K x -X jhn b 1n , y -Y jhn b 2n -K   x -X jhn b 1n , y - X (j+ 1 3 )hn -X jh (hn/3) b 2n     D j,n h 3/2 . Let us define A j = K x-X jhn b1n , y-Y jhn b2n -K x-X jhn b1n , y- X (j+ 1 3 )hn -X jh (hn/3)

b2n

. We then write

D n = 1 (n -1)b d 1n b d 2n n-1 j=1 A j D j,n h 3/2 = 1 (n -1)b d 1n b d 2n n-1 j=1 A j (σ W j,n + I j (h n )) h 3/2 .
Using Hölder Inequality, we bound

E 1 √ (n-1)b d 1n b d 2n n-1 j=1 A j Ij (hn) h 3/2 by 1 (n -1)b d 1n b d 2n h -3/2 n-1 j=1 E|A j | 1+ε 1/(1+ε) E|I j (h n )| (1+ε)/ε ε/(1+ε) ,
with ε > 0.

We first consider E|I j (h n )| (1+ε)/ε = E|I 0 (h n )| (1+ε)/ε by stationarity. We use (4.4) and we get that (7.7) Recall that for all t > 0, and for all f ∈ L p , 1 ≤ p ≤ ∞, P t f (z) is defined as

E |I 0 (h n )| (1+ε)/ε |X 0 = u, Y 0 = v ≤
E z f (Z t ).
Thus for all t > 0 and all α > 0 |P t g(u, v)| α = g(x, y)p t (x, y; u, v)dxdy α ≤ |g(x, y)| α p t (x, y; u, v)dxdy = P t |g| α (u, v) with p t (z; z ) denoting the transition kernel for the Markov process Z.

We can thus bound the right hand side of (7.7), using L one gets EA 2 0 W 2 0,n = h 3 n EA 2 0 . We then conclude by using (7.8) for ε = 1 which yields EA 2 0 W 2 0,n = h 3 n h 2 n b d/p 1n b -2 2n for all 1 < p < +∞. This completes the proof, using v).

  Remark 4.5. Under assumptions of Proposition 4.4 we can reach a rate of convergence in n 3ε 4+4d .Proof of Proposition 4.4: the proof is postponed to the appendix.Remark 4.6. Remark that contrary to the case of Theorem 3.1, where it is possible to have m = 1 here m grows linearly with the dimension (m > 2+4d+6εd+2ε 2(1-ε)

  2n h n Hn (x, y) -g(x, y)p s (x, y) = (n -1)b d 1n b d 2n h n Hn (x, y) -E Hn (x, y) + E Hn (x, y) -g(x, y)p s (x, y) =: I 1n + I 2n .

  .1) while I 1n = (S n,1 -ES n,1 (x, y)) + S n,2 (x, y) . (7.2) First step: Study of I 2n

.

  n G ε (h n , u, v) with G ε (h n , u, v) -----→ n→+∞ |g| 1+ε ε (u, v).Using now similar arguments as inStep 2 in the proof of Theorem 3.3 in Cattiaux et al. (2014) (Theorem 3.1 of the present paper), one bounds E|A i | 1+ε by p < +∞. It converges to zero as we assumed (see Assumption v).It remains now to bound E 2 . The terms A j W j,n , 1 ≤ j ≤ n-1 are centered and uncorrelated. Thus by stationarity we have to bound First conditioning now on Z hn 3

Figure 7 . 1 .Figure 7 . 2 .Figure 7 . 3 .Figure 7 . 4 .Figure 7 . 5 .Figure 7 . 6 .

 717273747576 Figure 7.1. Estimation of the drift for the harmonic oscillator: theoretical g(x 0 , •) in plain line, estimated in dashed line for x 0 = 0.0230, 95% asymptotic confidence intervals (upper bounds with croices, lower bounds with stars)

Acknowledgements

The authors would like to thank both referees for their very careful reading of their work.

Research partially supported by the project ECOS NORD/FONACIT (France-Venezuela) V12M01 Equations différentielles stochastiques et analyse de sensibilité : applications environnementales.

7.3. Proof os Proposition 4.4. For Theorem 3.1 to be satisfied in this situation we must have (see Remark 3.4 in [START_REF] Cattiaux | Estimation for stochastic damping hamiltonian systems under partial observation-I. Invariant density[END_REF])

, so that we may find some γ sandwiched by both terms. For both (a) and (b) to be satisfied, it is necessary that

and then we can choose

Remark that our interest is to take β 2 as close as possible to its upper bound, β 1 as close as possible to 1 d -β 2 so that m can be chosen as small as possible (1 is possible). γ can then be chosen smaller than and close to 1/2d.

Now we look at Proposition 4.3 starting with conditions

We see that the latter will be automatically satisfied for ε 3 small enough. We have to add

, which is enough to furnish both p > 1 and ε > 0 in v) of Proposition 4.3.

Finally we have to add

Look at (1). The compatibility with (b) imposes

That is why we cannot take the same bandwidths b and c. We thus have a first necessary condition

which is satisfied as soon as

Now we choose for some small ε (say less than 10 -6 /d 2 ),

) .

(7.9) and the following inequalities are satisfied, as well as the final upper bound in (b). We have thus to choose γ such that, first γ > 1 2d for (7) to be satisfied, next for (1) and the first lower bound of (b) to be satisfied. Notice that the latter implies α 2 < α 1 . Notice that for ε > 0 we may choose 4+4d) . It remains to choose m for (2) to be satisfied.