Géraldine Walther
email: geraldine.walther@uzh.ch

Benoît Sagot
email: benoit.sagot@inria.fr

Speeding up corpus development for linguistic research: language documentation and acquisition in Romansh Tuatschin

In this paper, we present ongoing work for developing language resources and basic NLP tools for an undocumented variety of Romansh, in the context of a language documentation and language acquisition project. Our tools are designed to improve the speed and reliability of corpus annotations for noisy data involving large amounts of code-switching, occurrences of child speech and orthographic noise. Being able to increase the efficiency of language resource development for language documentation and acquisition research also constitutes a step towards solving the data sparsity issues with which researchers have been struggling.

Introduction

Contemporary linguistic research relies more and more heavily on the exploration of statistical patterns in language. The non-categorical distribution and variety of linguistic units has become a focus for understanding complex variations across patterns, such as dative alternations (Bresnan et al., 2007) or cases of optionality (Wasow et al., 2011). Studies like these require the availability of large consistently annotated corpora.

For lesser or undescribed languages however, such resources are not readily available. What is worse, the current rate of language extinction could lead to the disappearance of 20-90% of today's spoken languages by the end of the 21st century (Krauss, 1992). Documenting endangered languages will allow us to preserve traces of the current language diversity, a part of the world's cultural heritage. Building reliable linguistic resources will allow us to study them according to fast evolving research standards.

Similarly, studies on language acquisition are based on recorded, transcribed, and annotated data of parent-child interactions. Language acquisition research has produced significant databases, such as the CHILDES project (McWhinney, 2000), yet mainly manually and at enormous costs.

Relying solely on manual annotators is too costly an option. Minimising resource development costs is crucial. Manual language resource development for language documentation and language acquisition projects should be sped up, as soon as technically possible, by employing NLP tools such as spelling correction/normalisation tools and part-of-speech (POS) taggers. For instance, POS taggers used as pre-annotators have been shown to increase both annotation speed and annotation quality, even when trained on limited amounts of data (Marcus et al., 1993; Fort and Sagot, 2010).

Yet language acquisition and language documentation data presents specific challenges for automatic linguistic annotation. Firstly, such data usually consists of transcriptions of spontaneous speech. Secondly, previously undescribed languages are often not written and lack established orthographies, resulting in noisy transcriptions. Thirdly, acquisition data consists of recordings of child-parent interactions. The recorded target children's language production can differ dramatically from adult language, adding another layer of linguistic variation. Finally, as new data is usually still being collected, available raw and even more so annotated data is rare, which significantly limits the available training data for annotation tools.

In this paper, we show the interaction between manual resource development (morphological lexicon, spelling and POS-annotated corpus) and automatic tools on current annotation experiments for a language documentation and acquisition project on the undocumented and previously non-written Romansh dialect of Tuatschin.

Romansh Tuatschin

The term Romansh denotes a set of Romance languages with important Germanic lexical and grammatical influence, mostly spoken in the canton of the Grisons in South-Eastern Switzerland. Although Romansh is considered one of the four official national languages of Switzerland, the term Romansh covers in fact a variety of languages and dialects with significantly differing features. The dialect we focus on in present paper corresponds to a previously undocumented dialect of the Romansh Sursilvan variety called Tuatschin. It is spoken by approximately 1,500 speakers in the Val Tujetsch area. Contrary to the neighbouring main Sursilvan dialect, which is also the main language in local schools, Tuatschin is at this point an unwritten language. It is however still natively spoken and transmitted both in the local area and within families who have left and settled in larger cities within the country. Speakers are proud of their language and culture and promote it through a local cultural association and occasional, nonnormalised, publications in the local newspaper.

The development of the resources described here is part of a project which combines language documentation and acquisition research. One aim of this project is to gain better understanding of intergenerational language transmission in endangered language contexts, which might contribute to eventually slowing down, if not reversing, linguistic and cultural erosion of minority languages. The data used in this project is mostly original data from fieldwork in the Val Tujetsch, original recordings within five Tuatschin speaking families with at least one child aged between 2 and 3 years, and updated and normalised lexical data from the Tuatschin word-list by Caduff (1952).

Adult speakers of Tuatschin are usually multilingual, natively speaking at least two, if not more, Romansh dialects (Tuatschin and Sursilvan), as well as German (both High German and the local variety of Swiss German), and often a fair amount of Italian and French. Everyday conversations in the Tuatschin speaking area comprise a high amount of code-switching. Conversations between speakers of neighbouring dialects more often than not result in each speaker speaking their own variety, accommodating the other person's dialect to varying degrees. Insertion of German lex-ical items or full utterances is ubiquitous.

As a result, our recorded Tuatschin data comprises a high amount of German and standard Sursilvan. This is even more acute in the acquisition corpus data used within present experiments, as the children's families tend to be natively bilingual. While the children's mothers are all native Tuatschin speakers, the fathers are Sursilvan, Swiss German or Italian speakers. The children therefore produce a significant amount of mixed utterances. In addition to the high amount of codeswitching due to this particular language setting, language acquisition data also comprises intrinsic noise due to the differences between child and adult language, including nonce-words or specific child-speech. The variation observed in our corpus data ranges from language and dialectal variation to adult/child register differences. Developing automated tools for such a corpus requires coping with noisy, heterogeneous corpus data.1 3 Developing guidelines

Orthography

The first challenge in developing linguistic resources and automated tools for a previously unwritten language like Tuatschin consists in developing an orthography that can be used for transcribing recorded data, and training future transcribers (native speakers) in using this new orthography. We wanted this orthography to also be usable by native speakers outside the project. In collaboration with two native speakers within the Val Tujetsch, we developed a new orthography for Tuatschin, which is mainly based on the orthography for the neighbouring written dialect Sursilvan, but accommodates the phonetic and morphological differences of Tuatschin, from the pronunciation of specific vowels and diphthongs to diacritic marking of infinitive forms. Once the main principles of the orthography had been established, we started training our native corpus transcribers. However, without complete resources (such as a full lexicon or grammar) at their disposal, each one of them still had their own interpretation of the overall principles for the tran-scription of individual words, adding an additional transcriber-related layer to the variation within the data. Developing the new orthography also required several passes, based on feedback from our transcribers and progress in our own understanding of the language data through our ongoing field work. Subsequent changes contributed to variation even within an individual transcriber's orthography, however reducing the differences between different transcribers' orthographic strategies.

Annotation tagset inventories

For our corpus annotation, we developed two separate tagsets, one for part-of-speech (POS) annotation and one for morphosyntactic features. Just as for our orthographic conventions, our tagset evolved alongside field work progress while annotation was already ongoing, adding further noise to our corpus data. This kind of noise is a common problem in language documentation projects where data collection, annotation, and analysis are conducted in parallel. Without the availability of automatic tools, it normally requires several passes of manual post-cleaning and adds to the overall cost of language resource development.

Our POS tagset comprises a fine-grained and a coarse POS inventory. The full coarse-grained inventory, used for our POS tagging experiments (see Section 5.2) is the following: ADJ, ADV(erb), COMPL(ementiser), CONJ(unction), DET(erminer), INTER(jection), N(oun), PN (proper noun), PREP(osition), PRN (pronoun), PUNCT(uation), QW (question word), SOUND, V(erb). Fine-grained tags add information such as DET def for definite articles or PREP loc for locative prepositions. It also comprises a specific childspeech refinement of all POS for words that are specific to child-speech. The morphosyntactic features follow the Leipzig Glossing Rule conventions commonly used in language documentation projects and comprise distinctions for number and gender, as well as tense, mood and person.

Although we have designed this languagespecific inventory as a means to better model the morphological and morphosyntactic properties of Tuatschin, we recognise the relevance and importance of the Universal Dependency (UD) initiative, 2 whose aim is to "develo[p] crosslinguistically consistent treebank annotation for many languages, with the goal of facilitating mul-2 http://universaldependencies.org tilingual parser development, cross-lingual learning, and parsing research from a language typology perspective." In the current version (2.0), several dozen languages are already covered, sometimes with more than one treebank. What is more, the UD website announces a Romansch and a (distinct) Sursilvan Romansch treebank. Our initiative is not related to these treebank development efforts-we had independently decided to develop a deterministic mapping between our languagespecific label inventory and a UD-compatible annotation scheme, in order to pave the way for a future Romansch-Tuaschin UD corpus. Together with the dependency annotation of our data, this will be the focus of future work.

Manual ressource development 4.1 TuatLex

We manually developed a morphological lexicon for Tuatschin. For that, we devised an explicit grammatical description of Tuatschin nominal and verbal morphology based on our own field work data. We implemented this description within the Alexina P ARSLI lexical framework (Sagot and Walther, 2013). In addition to the implemented morphological description, our Tuatschin lexicon, TuatLex, comprises a list of 2,176 lemmas, based on an updated version of the Tuatschin word-list by Caduff (1952) complemented with our newly collected data, among which 780 verbs, 949 nouns, and 146 adjectives. The Alexina inflection tool uses the grammar to produce 46,089 inflected entries, among which 29,361 verbal, 15,137 nominal, and 762 adjectival forms.

Manual corpus annotation

After devising the tagset, we trained advanced linguistic undergraduate students to manually annotate our corpus data. The students were no native speakers of Romansh. They were asked to annotate each token for (fine-grained) POS and morphosyntactic information and to indicate an English translation for each token's base form. 3 Using the WebAnno annotation tool (Eckart de Castilho et al., 2014), annotators took approximately 15 to 25 hours for annotating files containing typically 1,000 to 2,000 words. Difficulties for annotators mainly came from orthographic vari-ation, variation in the representation of word tokens, 4 and dialectal variation. 5 Weekly meetings between two trained linguists, among whom one Sursilvan native speaker, and all annotators were organised to compare notes, discuss recurrent difficulties, and adapt the tagsets whenever necessary. 6

Manual corpus normalisation

In order to simplify the annotation task, we set up a procedure for systematic orthographic correction. We first asked one of the native speakers, who had been involved in the development of the orthographic conventions, to manually correct already transcribed corpus data for orthographic errors and individual-word-based code-switching in a separate normalised tier within the corpus. 7 We then manually introduced an additional tier indicating for each 'normalised' token whether it had been corrected for orthography, code-switching, child-speech, or actual pronunciation errors. This intermediate layer, in addition to being useful for subsequent acquisition or code-switching studies, is meant to help the development of an automatic spell-checker. Aside from variation in the usage of diacritics, some of the most frequent errors in the transcribed data involved the amalgamation of words meant to be written as separate tokens. 8 5 Automation and tools

Tokenisation and orthographic correction

In order to speed up the manual development of orthographically normalised corpora based on new transcriptions, but also to prepare the fully automatic processing of non-annotated text, we first developed an automatic tokenisation and spelling standardisation/correction tool. It is implemented 4 E.g. vegni instead of vegn i 'it comes' (lit. 'comes it'). 5 For example, unexpected morphological forms that prevented the recognition of inflected forms such as verbs ending in Sursilvan -el instead of Tuatschin -a in the first person singular. 6 The project being mainly a documentation project on a previously undescribed language, data collection, annotation, and data analysis are currently being carried out in parallel. Data annotation in particular has been performed as a collaborative task rather than as a task conducted by individual annotators, that would have to be evaluated for inter-annotator agreement. 7 The purpose of this normalised layer is solely to help automatic (and, to a lesser extent, manual) annotation of the data. It is not meant to replace the original transcription layer, which remains the relevant layer for subsequent linguistic analysis.

8 Cf. the vegni/vegn i example from previous footnote.

in the form of a Tuatschin-specific parametrisation of the SXPipe shallow processing architecture (Sagot and Boullier, 2008). Note that the spelling standardisation/correction tool is not meant to be used as a standalone tool. It has been designed and developed only for speeding up future corpus development and for serving as a cleaning step before applying the POS tagger, whose results are obviously better on (even only partially) normalised data.

Our spelling standardisation/correction tool relies on a list of deterministic rewriting patterns that we automatically extracted and manually selected based on the data described in Section 4.3. More precisely, we applied standard alignment techniques for extracting n-to-m correspondences between raw and normalised text. Among the extracted rules, 695 were deterministic, which means that there was a unique output in the corpus for a given input. Out of these 695 candidate rules, we manually selected 603, whose systematic application could not result in any over-correction. 9 Several others are non-deterministic. However, a careful manual examination of these candidate ambiguous rules showed that most of the ambiguity is an artifact of the rule extraction process, and that true ambiguities can be resolved in context. 10 As a result, contextual information was included in our standardisation/correction patterns, thus resulting in a fully deterministic standardisation/correction module.

POS tagger

In order to assess and improve the quality of the POS annotation, but also to have a POS preannotation tool for speeding up future manual annotation, we trained the MElt POS tagger (Denis and Sagot, 2012) on the manually POS-annotated data available so far, using coarse POS to reduce sparsity (see Section 3.2). The 2,571 already annotated sentences, containing 9,927 tokens, were divided in training, development and test sets by randomly selecting sentences with re- 9 A few examples: stù→stu, sèl→sè'l, schia→schéia. 10 For instance, vegnì and all other verbs from TuatLex's inflection class Vi can produce ambiguous tokens such as vegni, which must be changed into vegnì (infinitive) before pronouns such as ju, té, el, ella, el, i, nus, vus, els, ellas, ins, but, in most other contexts, must be rewritten as vegn i (V+PRN) with an expletive pronoun i. We applied the latter to those verb tokens likely to appear with expletive subjects like vegnì 'to come' while inserting the infinitive diacritic on all other verb instances (such as capì 'understand'). spective probabilities 0.8, 0.1 and 0.1. We also extracted from the TuatLex lexicon described above a set of 19,771 unique (form, POS) pairs, to be used by MElt as a complementary source of information, in addition to the training corpus.

We trained a first version of MElt, and applied the resulting POS tagger on the training data themselves. Annotation mismatches allowed us to identify several systematic errors in the training data. Some of them came from individual errors or annotator disagreement. But most were due to changes made in the annotation guidelines while manual annotation was already ongoing, and of which some had not yet been correctly retroactively applied to already annotated data. 11 We applied a series of POS correction rules to the whole corpus (train+dev+test) as a result of this training-corpus-based study, and re-trained MElt on the corrected data. The result is a POS tagger trained on 2,062 sentences (7,901 words) with a 91.7% accuracy overall and a 65.3% accuracy on words unknown to the training corpus. Interestingly, if trained without TuatLex lexical information, accuracy figures do not drop as much as usually observed (Denis and Sagot, 2012): respectively 91.6% and 62.5%. This suggests that lexical information might not be as important for improving POS taggers for child-related speech as it is for tagging standard text, a fact likely related to the more limited vocabulary size in such corpora.

Conclusion

We have described ongoing efforts for developing language resources (lexicon, annotated corpus)12 and basic NLP tools for the Romansh variety of Tuatschin, in the context of a project on language description and language acquisition dedicated to a previously non-written language. Our next step will consist in using our tools for pre-annotating new raw data, in order to speed up annotation while increasing its quality and consistency. They will also be used for creating automatically annotated data, which will complement the manually annotated corpus.

On a longer term, our tools are also meant to be used for automatically categorising tokens and sequences of tokens into occurrences of childspeech or various types of code-switching, relying on the information comprised within the intermediate tier introduced during the normalisation procedure (see Section 4.3). This intermediate layer is meant to be ultimately automatically generated by SXPipe. The richer and more accurate information that our tools will be able to provide will also facilitate subsequent quantitative linguistic studies on Romansh Tuatschin, its acquisition by children and its influences by surrounding languages, especially Sursilvan and (Swiss) German.

Acknowledgements

The authors gratefully acknowledge the SNF project number 159544 "The morphosyntax of agreement in Tuatschin: acquisition and contact" whose funding has been instrumental to the work presented here. We also thank our project collaborators and especially our annotators Nathalie Schweizer, Michael Redmond, and Rolf Hotz for their help in annotating the data, and Claudia Cathomas for her extremely valuable help and native expertise on Romansh data. For their help in developing a new Tuatschin orthography we would like to express our warmest thanks to Ciril Monn and Norbert Berther.

Note that while Sursilvan does have an established orthography and is used a a language of instruction in schools, there are no automatic tools available for the language. The existing online lexicon can only be queried online but is not freely available. Despite the languages' similarity there are no existing resources that could be leveraged to facilitate automatic work on Tuatschin at this point.

For example, for the Tuatschin token nûrsas, they would have provided the following annotations: POS = N, Morphosyn = pl, and the English translation 'EWE'.

For instance, the previously defined POS "V particle" had been discarded at some point during corpus development, yet it still had a number of occurrences in the training data.

The lexicon and tools will all be made freely available.


Robert Eymard

Sophie Mercier

Michel Roussignol

Importance and Sensitivity Analysis in Dynamic Reliability

Keywords: Dynamic reliability, Sensitivity analysis, Importance factor, Piecewise deterministic Markov process, Chapman-Kolmogorov equation

In dynamic reliability, the evolution of a system is governed by a piecewise deterministic Markov process, which is characterized by different input data. Assuming such data to depend on some parameter p ∈ P, our aim is to compute the first-order derivative with respect to each p ∈ P of some functionals of the process, which may help to rank input data according to their relative importance, in view of sensitivity analysis. The functionals of interest are expected values of some function of the process, cumulated on some finite time interval [0, t], and their asymptotic values per unit time. Typical quantities of interest hence are cumulated (production) availability, or mean number of failures on some finite time interval and similar asymptotic quantities. The computation of the first-order derivative with respect to p ∈ P is made through a probabilistic counterpart of the adjoint state method, from the numerical analysis field. Examples are provided, showing the good efficiency of this method, especially in case of a large P.

Introduction

In dynamic reliability, the time-evolution of a system is described by a piecewise deterministic Markov process (PDMP) (I t , X t ) t≥0 introduced by [START_REF] Davis | Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models[END_REF]. The first component I t is discrete, with values in a finite state space E. Typically, it indicates the state (up/down) for each component of the system at time t. The second component X t , with values in a Borel set V ⊂ R d , stands for environmental conditions, such as temperature, pressure, and so on. Both components of the process interact one in each other: the process jumps at countably many isolated random times; by a jump from (I t -, X t -) = (i, x) to (I t , X t ) = ( j, y) (with (i, x), ( j, y) ∈ E × V), the transition rate between the discrete states i and j depends on the environmental condition x just before the jump and is a function x -→ a (i, j, x). Similarly, the environmental condition just after the jump X t is distributed according to some distribution μ (i, j,x) (dy), which depends on both components just before the jump (i, x) and on the after jump discrete state j. Between jumps, the discrete component I t is constant, whereas the evolution of the environmental condition X t is deterministic, solution of a set of differential equations which depends on the fixed discrete state: given that I t = i for all t ∈ [a, b ], we have d dt X t = v(i, X t ) for all t ∈ [a, b ], where v is a mapping from E × V to V. Contrary to the general model from [START_REF] Davis | Markov models and optimization, monographs on statistics and applied probability[END_REF], we here assume that the eventual reaching of the frontier of V does not entail jumps for the process (I t , X t ) t≥0 . Now, let (I t , X t ) t≥0 be a PDMP for which the jump rates a(i, j, x), the jump distribution μ (i, j,x) (dy) and the velocity fields v (i, x) are assumed to depend on some family of parameters P ∈ R k , where k ∈ N can be quite large. Our aim here is to provide information about the sensitivity with respect to the elements of P, of expressions with the following form: R ρ0 (t) = E ρ0 t 0 h(I s , X s ) ds , where ρ 0 is the initial distribution of the process and h is some bounded measurable function which can also depend on p ∈ P. Such expressions include e.g. cumulative availability or production availability on some [0, t], mean number of failures on [0, t], mean time spent by (X s ) 0≤s≤t between two given bounds. We are also interested in the sensitivity with respect to p ∈ P of the corresponding asymptotic quantities per unit time, namely in quantities of the form lim t→+∞ R ρ0 (t) t .

This sensitivity analysis can be guided by the knowledge of the first-order derivatives of R ρ0 (t) with respect to p, where p ∈ P. More specifically, we consider the following normalized derivative:

I F p (t) = p R ρ0 (t) ∂ R ρ0 (t) ∂ p , ( 1
)
for t ≤ +∞ and p ∈ P, which we call the importance factor of parameter p in R ρ0 (t). Such a dimensionless expression may help to rank the different parameters p ∈ P according to their relative importance in R ρ0 (t). This kind of sensitivity analysis was already studied by [START_REF] Gandini | Importance and sensitivity analysis in assessing system reliability[END_REF] and by [START_REF] Cao | Perturbation realization, potentials, and sensitivity analysis of Markov processes[END_REF] for pure jump Markov processes with countable state space, and extended to PDMPs by [START_REF] Mercier | Sensitivity estimates in dynamic reliability[END_REF], with more restrictive a model than in the present paper however.

The starting point for this study is the Markov property of the process (I t , X t ) t≥0 , which allows to write the associated Chapman-Kolmogorov equations fulfilled by its marginal distributions, see [START_REF] Davis | Markov models and optimization, monographs on statistics and applied probability[END_REF] or Cocozza-Thivent et al. (2006b). Such equations appear as weak forms of linear first order hyperbolic equations, see [START_REF] Eymard | An implicit finite volume scheme for a scalar hyperbolic problem with measure data related to piecewise deterministic markov processes[END_REF] e.g. The expressions for the derivatives of interest can then be obtained by solving the dual problem, as suggested by [START_REF] Lions | Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles. Avant propos de P. Lelong[END_REF] for a wide class of partial differential equations. The Chapman-Kolmogorov equations and its dual problem are here solved using finite volume methods as provided by Cocozza-Thivent et al. (2006a) or [START_REF] Eymard | An implicit finite volume scheme for a scalar hyperbolic problem with measure data related to piecewise deterministic markov processes[END_REF], which here prove to be well adapted.

An alternate way to compute the marginal distributions of the PDMP might be to start from the Markov renewal equations they fulfill, as proposed in [START_REF] Chiquet | A method to compute the transition function of a piecewise deterministic Markov process with application to reliability[END_REF] for a specific case of PDMPs, or to use Monte-Carlo simulations.

In order to study the asymptotic quantities, one must put assumptions on the process to ensure its positive Harris recurrence. These assumptions may be proved to be true for Markov processes on general state space using techniques like those in [START_REF] Davis | Markov models and optimization, monographs on statistics and applied probability[END_REF] and Meyn and Tweedie (1993a, b), as is done for both examples at the end of this paper. Alternately, one may use specific results for piecewise deterministic Markov processes, as provided in [START_REF] Davis | Markov models and optimization, monographs on statistics and applied probability[END_REF] and Costa andDufour (1999, 2003).

The paper is organized as follows: technical assumptions are set up in Section 2. An existence result for the derivatives of the transient quantities is provided in Section 3 and a computable expression is given for them in Section 4, using duality. Asymptotic quantities are studied in Section 5. Numerical procedures are exposed in Section 6, using finite volume methods. Two numerical studies close the paper in Sections 7 and 8.

Assumptions

All the paper is written under the following general assumptions:

-for all i, j ∈ E, the function x -→ a(i, j, x) is continuous and bounded on V, -for all i, j ∈ E and all continuous and bounded function f on V, the function

x -→ V f (y) μ (i, j,x) (dy) is continuous and bounded on V, -for all i ∈ E, the velocity field v(i, x) is such that the function x -→ v(i, x) is Lipschitz continuous and bounded by V 1 = max i∈E v (i, •) ∞ > 0.
These assumptions guarantee the existence and uniqueness of the solution to the set of differential equations dx dt = v (i, x); we denote by g (i, x, t) the single solution (defined on

E × V × R) such that g (i, x, 0) = x.
The jump rates a(i, j, x), the jump distribution μ (i, j,x) , the velocity field v(i, x) and the function h are assumed to depend on some parameter p, where p belongs to an open set O ⊂ R or R k . All the results are written in the case where O ⊂ R but extension to the case O ⊂ R k is straightforward. We add exponent ( p) to each quantity depending on p, such as h ( p) .

We then have:

R ( p) ρ 0 (t) = t 0 ρ ( p) s h ( p) ds = i∈E V t 0 h ( p) (i, x) ds ρ ( p) s (i, dx) = t 0 ρ 0 P ( p) s h ( p) ds = i∈E V t 0 P ( p) s h ( p) (i, x) ds ρ 0 (i, dx)
In order to prove existence and to calculate derivatives of the functional R ( p) ρ 0 , we must give a sense to the derivatives of the transition probability distributions. With that aim, we need the following additional assumptions that we denote as assumptions H 1 (resp. H 2 ):

For each p in O, there is some neighborhood N( p) of p in O such that, for all i, j ∈ E × E:

-the function (x, p) -→ a ( p) (i, j, x) is bounded on V × N( p), once (resp. twice) continuously differentiable on V × O, with all partial derivatives uniformly bounded on V × N( p), -for all function f ( p) (x) bounded and once (resp. twice) continuously differentiable on V × O, with all partial derivatives uniformly bounded on V × N( p), the function

(x, p) -→ V f ( p) (y) μ ( p)
(i, j,x) (dy) is bounded and once (resp. twice) continuously differentiable on V × O, with all partial derivatives uniformly bounded on

V × N( p), -the function (x, p) -→ v ( p) (i, x) is bounded on V × N( p), once (resp. twice) continuously differentiable on V × O, with all partial derivatives uniformly bounded on V × N( p), -the function (x, p) -→ h ( p) (i, x) is bounded on V × N( p), once (resp. twice) continuously differentiable on V × O with uniformly bounded partial derivatives on V × N ( p).
The third point implies (see e.g. Cartan 1967) that, for all i ∈ E, the function (x, p) -→ g ( p) (i, x, s), solution of the differential equation dg dt = v ( p) (i, g) with g(i, x, 0) = x, is once (resp. twice) continuously differentiable on V × N( p), with all partial derivatives uniformly bounded on V × N( p).

Throughout the paper, under assumptions H 1 or H 2 , for each p in O, we shall refer to a N( p) fulfilling the four points of the assumptions without any further notice.

We may now give a sense to the derivatives of the transition probability distributions, which is done in next section.

Existence Result

We shall use the infinitesimal generators of both Markov processes (I t , X t ) t≥0 and (I t , X t , t) t≥0 :

Definition 1 Let D H0 be the set of functions f (i, x) from E × V to R such that for all i ∈ E the function x -→ f (i, x) is bounded, continuously differentiable on V and such that the function x -→ v ( p) (i, x) • ∇ f (i, x) is bounded on V.
For f ∈ D H0 , we define

H ( p) 0 f (i, x) = j∈E a ( p) (i, j, x) V f ( j, y) μ ( p) (i, j,x) (dy) + v ( p) (i, x) • ∇ f (i, x)
where we set

a ( p) (i, i, x) = -j =i a ( p) (i, j, x) and μ ( p) (i,i,x) = δ x . Let D H be the set of functions f (i, x, s) from E×V ×R + to R such that for all i ∈ E the function (x, s) -→ f (i, x, s) is bounded, continuously differentiable on V × R + and such that the function (x, s) -→ ∂ f ∂s (i, x, s) + v ( p) (i, x) • ∇ f (i, x, s) is bounded on V × R + .
For f ∈ D H , we define

H ( p) f (i, x, s) = j a ( p) (i, j, x) V f ( j, y, s) μ ( p) (i, j,x) (dy) + ∂ f ∂s (i, x, s) + v ( p) (i, x) • ∇ f (i, x, s) (2)
We then have for all f ∈ D H0 :

P ( p) t f = f + t 0 H ( p) 0 P ( p) u f du (3) ρ ( p) t f = ρ 0 f + t 0 ρ ( p) u H ( p) 0 f du (4)
and for all f ∈ D H :

P ( p) t f (•, •, t) = f (•, •, 0) + t 0 P ( p) u H ( p) f (•, •, u) du (5) ρ ( p) t f (•, •, t) = ρ 0 f (•, •, 0) + t 0 ρ ( p) u H ( p) f (•, •, u) du (6)
These are Chapman-Kolmogorov equations.

Thanks to these equations and to Theorem 4 in the Appendix, we get the following result: Proposition 1 Let f (i, x) be a function (independent of p) from E × V to R such that for all i ∈ E, the function x -→ f (i, x) is bounded, continuously differentiable on V with uniformly bounded partial derivatives on V. Under assumptions H 1 , for all i ∈ E, the function (x, p) -→ P H 2 , if for all i ∈ E the function x -→ f (i, x) is bounded, twice continuously differentiable on V with uniformly bounded partial derivatives on V, then for all i ∈ E, the function (x, p) -→ P ( p) t f (i, x) is twice continuously differentiable on V × O, with all partial derivatives uniformly bounded on V × N ( p) for all p ∈ O. Proof Let p 0 ∈ P and N( p 0 ) be a neighborhood of p 0 . Setting ϕ (s, i, (x, p)) = P ( p) t f (i, x) for (i, x, p, s) ∈ E × V × N ( p 0 ) × R + , the Chapman-Kolmogorov Eq. 3 can be written as:

ϕ (t, i, (x, p)) = f (i, x) + t 0 ⎛ ⎝ j∈E a ( p) (i, j, x) V ϕ(u, j, (y, p)) μ ( p) (i, j,x) (dy) + v ( p) (i, x) • ∇ x ϕ(u, i, (x, p)) ⎞ ⎠ du (7)
where ∇ x stands for the gradient with respect of x. Using a similar method as in Cocozza-Thivent et al. (2006b), we introduce the function φ defined by:

φ(t, i, (x, p)) = ϕ(t, i, (g ( p) (i, x, -t), p))
Noting that

ϕ(t, i, (x, p)) = φ(t, i, (g ( p) (i, x, t), p)),
Equation 7 may now be written as:

φ(t, i, (x, p)) = f (i, x) + t 0 ⎛ ⎝ j∈E a ( p) (i, j, g ( p) (i, x, -u)) V φ(u, j, (g ( p) ( j, y, u), p))μ ( p) (i, j,g ( p) (i,x,-u)) (dy) ⎞ ⎠ du
Using the notations of Theorem 4, we set z = (x, p), φ0 (i, z) = f (i, x) and

F (s, ψ) (i, z) = j∈E a ( p) (i, j, g ( p) (i, x, -u)) V ψ( j, (g ( p) ( j, y, s), p))μ ( p) (i, j,g ( p) (i,x,-s)) (dy) for (i, x, p, s) ∈ E × V × N ( p 0 ) × R + and ψ : E × V × N ( p 0 ) → R
bounded and measurable. The function φ then satisfies the equations:

φ(t, i, z) = φ0 (i, z) + t 0 F(u, φ(u, ., .))(i, z) du.
We now check that assumptions of Theorem 4 are fulfilled and we set I to be an interval such that [0, T] ⊂ I.

Symbols W = V × N( p 0 ), C b 1 (E, W), C b 1 (I, E, W)
and J (F ) are as in the Appendix. With such notations, the function φ0 belongs to C b 1 (E, W) and assumptions H 1 imply that:

-if ∈ C b 1 (I, E, W), then F(s, (s, ., .))(i, (x, p)) is in C b 1 (I, E, W), -if ∈ C b
1 (E, W), then for all s ∈ I, F(s, ) and ∇F (s, ) are uniformly Lipschitz with respect to .

Assumptions H 2 and φ0 ∈ C b 2 (E, W) imply that:

-if ∈ C b 2 (I, E, W), then F(s, (s, ., .))(i, (x, p)) is in C b 2 (I, E, W), -if ∈ C b 2 (E, W)
, then for all s ∈ I, F(s, ), ∇F (s, ) and J (F ) (s, ) are uniformly Lipschitz with respect to .

All required assumptions are then checked, which provides the result.

Remark 1 Using the explicit form of transition probabilities (see Cocozza-Thivent et al. 2006b)

, the functional f -→ ∂ ∂ p P ( p) t f (i, x) appears to be a continuous linear functional of f which involves f and ∂ f ∂ xk for 1 ≤ k ≤ d (
and possibly other derivatives). Limiting this functional to the set of functions f such that x -→ f (i, x) is infinitely differentiable with a compact support for all i ∈ E, it may then be seen as a distribution. We actually define this functional on the greater space of bounded continuously differentiable functions with uniformly bounded partial derivatives and we use the following notation:

∂ ∂ p P ( p) t f (i, x) = ∂ P ( p) t ∂ p , f (i, x)
Next corollary is straightforward.

Corollary 1 Under assumptions H 1 , the function (x, p) -→ P ( p) t h ( p) (i, x) is con- tinuously differentiable on V × O for all i ∈ E, with all partial derivatives uniformly bounded on V × N ( p) for all p ∈ O. Besides: ∂ ∂ p P ( p) t h ( p) (i, x) = ∂ P ( p) t ∂ p , h ( p) (i, x) + P ( p) t ∂h ( p) ∂ p (i, x) for all (i, x, t, p) ∈ E × V × R + × O.
We derive the following theorem:

Theorem 1 Under assumptions H 1 , the function p -→ R ( p) ρ 0 (t) is continuously differentiable with respect to p and

∂ ∂ p R ( p) ρ 0 (t) = t 0 i∈E V ∂ P ( p) u ∂ p , h ( p) (i, x) ρ 0 (i, dx) du + t 0 i∈E j∈E V V ∂ ∂ p h ( p) ( j, y)P ( p) u (i, x, j, dy)ρ 0 (i, dx) du = t 0 ∂ρ ( p) u ∂ p , h ( p) du + t 0 ρ ( p) u ∂h ( p) ∂ p du ( 8
)
where we set:

∂ρ ( p) u ∂ p , h ( p) = i∈E V ∂ P ( p) u ∂ p , h ( p) (i, x) ρ 0 (i, dx). ( 9
)
Remark 2 Assumption on h ( p) in H 1 may be generalized to the case where h ( p) (i, x) is continuously differentiable only for almost all (a.a.) x ∈ V, where a.a. means with respect to Lebesgue measure.

Our purpose now is to compute this derivative. The marginal distribution ρ ( p) u ( j, dy) may be estimated by different methods, as indicated in the introduction. However, we do not know how to compute directly the derivative of the marginal distribution which appear in the above expression. We now transform it in order to make it easier to compute.

Results in the Transient Case

Let us first define the notion of importance functions.

Definition 2 We say that a function ϕ ( p) t

∈ D H is the importance function associated to the function h ( p) and to t if:

-ϕ ( p) t is solution of the differential equation H ( p) ϕ ( p) t (i, x, s) = h ( p) (i, x) for all (i, x, s) ∈ E × V × [0, t[, -ϕ ( p) t (i, x, t) = 0 for all (i, x) in E × V.
Proposition 2 Let t > 0. Under assumptions H 1 , the importance function associated to h ( p) , t exists, is unique, is continuously differentiable on V × O, bounded with all partial derivatives uniformly bounded on V × N( p) for all p ∈ O. Under assumptions H 2 , the importance function associated to h ( p) , t is moreover twice continuously differentiable on V × O with all partial derivatives uniformly bounded on V × N( p) for all p ∈ O.

Proof Let ϕ ( p) ∈ D H (eventually depending on p) and let us set:

φ(p) (i, z, s) = ϕ ( p) (i, g (i, z, -s) , t -s) (10) for all (i, z, s) ∈ E × V × [0, t].
The function φ(p) is bounded, continuously differentiable on V with partial derivative with respect of time given by:

∂ φ(p) ∂s 1 (i, z, s) = - ∂ϕ ( p) ∂s 1 i, g ( p) (i, z, -s) , t -s -∇ϕ ( p) i, g ( p) (i, z, -s) , t -s • v ( p) i, g ( p) (i, z, -s) (11)
The function ϕ ( p) and φ(p) are in one-to-one correspondence with:

ϕ ( p) (i, z, t -s) = φ(p) i, g ( p) (i, z, s) , s (12) for all (i, z, s) ∈ E × V × [0, t].
Using Eqs. 10-11, we get:

H ( p) ϕ ( p) i, g ( p) (i, z, -s) , t -s = j a ( p) i, j, g ( p) (i, z, -s) V ϕ ( p) ( j, y, t -s)μ ( p) (i, j,g ( p) (i,z,-s)) (dy) - ∂ φ(p) ∂s 1 (i, z, s) = H(p) φ(p) (i, z, s)
where

H(p) φ(p) (i, z, s) = j∈E a ( p) i, j, g ( p) (i, z, -s) V φ(p) j, g ( p) ( j, y, s) , s μ ( p) (i, j,g ( p) (i,z,-s)) (dy) - ∂ φ(p) ∂s 1 (i, z, s) .
The problem now resumes to show existence, uniqueness and regularity of φ(p) such that

H(p) φ(p) (i, z, s) = h ( p) i, g ( p) (i, z, -s) (13) for all (i, z, s) ∈ E × V × [0, t] with φ(p) (i, z, 0) = ϕ ( p) (i, z, t) = 0 for all (i, z) ∈ E × V.
The o.d.e. Eq. 13 may be written as:

φ(p) (i, z, t) = t 0 R ( p) φ(p) (•, •, s) , s (i, z) ds with R ( p) φ(p) (•, •, s) , s (i, z) = j∈E a ( p) i, j, g ( p) (i, z, -s) V φ(p) j, g ( p) ( j, y, s) , s μ ( p) (i, j,g ( p) (i,z,-s)) (dy) -h ( p) i, g ( p) (i, z, -s) .
Theorem 4 from the Appendix then provides the result in the same way as for Proposition 1.

Next lemma provides a duality result based on the definition of ϕ ( p) t , which transports the differentiation with respect of p from the marginal distribution ρ ( p) s to operator H ( p) .

Lemma 1 Under assumptions H 2 , we have:

t 0 ∂ρ ( p) s ∂ p , H ( p) ϕ ( p) t (•, •, s) ds = - t 0 ρ ( p) s ∂ H ( p) ∂ p ϕ ( p) t (•, •, s) ds
where we set:

∂ H ( p) ∂ p ϕ (i, x, s) := j∈E ∂a ( p) ∂ p (i, j, x) V ϕ( j, y, s) μ ( p) (i, j,x) (dy) + j∈E a ( p) (i, j, x) ∂ ∂ p V ϕ( j, y, s) μ ( p) (i, j,x) (dy)) + ∂v ( p) ∂ p (i, x) • ∇ϕ(i, x, s) for all ϕ ∈ D H and all (i, x, s) ∈ E × V × R + .
Proof Thanks to Proposition 2, functions ϕ ( p) t (i, x, s) and ∂ϕ ( p) t ∂ p (i, x, s) are in D H . We first know from the Chapman-Kolmogorov Eq. 6 applied to ϕ ( p) t that:

t 0 ρ ( p) s H ( p) ϕ ( p) t (•, •, s) ds = ρ ( p) t ϕ ( p) t (•, •, t) -ρ 0 ϕ ( p) t (•, •, 0) = -ρ 0 ϕ ( p) t (•, •, 0) due to ϕ ( p) t (•, •, t) = 0.
By differentiating this expression with respect to p, we derive:

t 0 ∂ρ ( p) s ∂ p , H ( p) ϕ ( p) t (•, •, s) ds + t 0 ρ ( p) s ∂ H ( p) ∂ p ϕ ( p) t (•, •, s) ds + t 0 ρ ( p) s H ( p) ∂ϕ ( p) t ∂ p (•, •, s) ds = -ρ 0 ∂ϕ ( p) t ∂ p (•, •, 0) . ( 14
)
Chapman-Kolmogorov Eq. 6 applied to ∂ϕ ( p) t ∂ p gives:

t 0 ρ ( p) s H ( p) ∂ϕ ( p) t ∂ p (•, •, s) ds = ρ ( p) t ∂ϕ ( p) t ∂ p (•, •, t) -ρ 0 ∂ϕ ( p) t ∂ p (•, •, 0) = -ρ 0 ∂ϕ ( p) t ∂ p (•, •, 0) .
Hence the result, substituting this last outcome in Eq. 14.

We easily derive the following result, using the definition of

R ( p) ρ0 (t) and of ϕ ( p) t (•, •, s).
Theorem 2 Under assumptions H 2 , we have:

∂ R ( p) ρ 0 ∂ p (t) = t 0 ρ ( p) s ∂h ( p) ∂ p ds - t 0 ρ ( p) s ∂ H ( p) ∂ p ϕ ( p) t (•, •, s) ds, ( 15
)
where ϕ ( p) t

is the importance function associated to h ( p) , t .

Remark 3 In case that R ρ0 (t) depends on a family of parameters P, the computation of all ∂ R (P) ρ 0

∂ p (t) for each p ∈ P consequently requires only one single computation of both ρ (P) s and ϕ (P) t , and simple summations on [0, t] for each value of p. This is to be compared to the usual finite differences method, which requires one first evaluation of ρ (P) s and another evaluation of ρ (Pp,ε) s for each value of p ∈ P and some ε > 0, where P p,ε stands for the family P in which parameter p has been changed into p + ε. In case of a large P, the present methods, which only requires two effective computations, consequently appears as cheaper than finite differences.

In applications, the importance function will generally be computed numerically. An analytical form is however available, which is also useful for the asymptotic study:

Lemma 2 Let t > 0. Under assumptions H 1 , the importance function associated to h ( p) , t is given by:

ϕ ( p) t (i, x, s) = ⎧ ⎨ ⎩ - t-s 0 P ( p) u h ( p) (i, x) du if 0 ≤ s ≤ t 0 otherwise (16) for all (i, x) ∈ E × V. Proof Let ϕ ( p) t
be the function defined by Eq. 16. It is clear that ϕ ( p) t (i, x, t) = 0 for all (i, x) ∈ E × V and thanks to Proposition 1, the function ϕ ( p) t is in D H . Besides, for 0 ≤ s < t, we have:

H ( p) ϕ ( p) t (•, •, s) = -H ( p) t-s 0 P ( p) u h ( p) du = -H ( p) 0 t-s 0 P ( p) u h ( p) du - ∂ ∂s t-s 0 P ( p) u h ( p) du = - t-s 0 H ( p) 0 P ( p) u h ( p) du + P ( p) t-s h ( p) = h ( p)
due to the Chapman-Kolmogorov Eq. 3, which completes the proof.

We easily derive the following Corollary from Theorem 2 and the previous lemma.

Corollary 2 Under assumptions H 2 , we have:

∂ R ( p) ρ 0 ∂ p (t) = t 0 ρ ( p) s ∂h ( p) ∂ p ds + t 0 ρ ( p) s ∂ H ( p) ∂ p t-s 0 P ( p) u h ( p) du ds (17)
Equation 17 is an extension of the results of [START_REF] Gandini | Importance and sensitivity analysis in assessing system reliability[END_REF] for pure jump Markov processes with countable state space.

Asymptotic Results

We are now interested in asymptotic results and we need to assume the process (I t , X t ) t≥0 to be uniformly ergodic, according to the following assumptions H 3 :

-the process (I t , X t ) t≥0 is positive Harris-recurrent with π ( p) as unique stationary distribution, -for each p ∈ O, there exists a function f ( p) such that

+∞ 0 f ( p) (u) du < +∞, +∞ 0 u f ( p) (u) du < +∞, lim u→+∞ f ( p) (u) = 0, ( 18
)
and

P ( p) u h ( p) (i, x) -π ( p) h ( p) ≤ f ( p) (u) (19)
for all (i, x) ∈ E × V and all u ≥ 0.

As already noted in the introduction, such assumptions may be proved to be true by using results for Markov processes on general state spaces or specific results for piecewise deterministic Markov processes (see the introduction for references).

In order not to give too technical assumptions difficult to check in practice, we constraint our asymptotic study to the special case where only the jump rates a ( p) (i, j, x) and the function h ( p) (i, x) depend on the parameter p. The quantities μ (i, j,x) and v(i, x) are consequently assumed to be independent of p. Assumptions H 2 are then substituted by assumptions H 2 , where conditions on μ (i, j,x) and on v(i, x) are removed.

We now transform Eq. 17 in view of studying its asymptotic expression.

Lemma 3 Under assumptions H 2 and H 3 , we have:

1 t ∂ R ( p) ρ0 ∂ p (t) = 1 t t 0 ρ ( p) s ∂h ( p) ∂ p ds + 1 t t 0 ρ ( p) s ∂ H ( p) ∂ p t-s 0 P ( p) u h ( p) -π ( p) h ( p) du ds (20)
Proof The first term is clear. Besides, setting 1 to be the constant function equal to 1, we have:

∂ H ( p) ∂ p 1 = 0 since H ( p) 1 = 0. As π ( p) h ( p) is a constant (independent of (i, x)), we derive ∂ H ( p) ∂ p π ( p) h ( p) = π ( p) h ( p) ∂ H ( p) ∂ p 1 = 0
and consequently:

∂ H ( p) ∂ p t-s 0 π ( p) h ( p) du = (t -s) ∂ H ( p) ∂ p π ( p) h ( p) = 0.
Whence the result.

We may now prove existence and provide an asymptotic expression for

1 t ∂ R ( p) ρ 0 ∂ p (t).
Theorem 3 Let us assume that μ (i, j,x) and v(i, x) are independent of p and that H 2 , H 3 are true. Then:

Uh ( p) (i, x) := +∞ 0 P ( p) u h ( p) (i, x) -π ( p) h ( p) du
exists for all (i, x) ∈ E × V and Uh ( p) is element of D H0 . Besides:

lim t→+∞ 1 t ∂ R ( p) ρ 0 ∂ p (t) = π ( p) ∂h ( p) ∂ p + π ( p) ∂ H ( p) 0 ∂ p Uh ( p) ( 21
)
where we set:

∂ H ( p) 0 ∂ p ϕ 0 (i, x) := j∈E ∂a ( p) ∂ p (i, j, x) V ϕ 0 ( j, y) μ (i, j,x) (dy)
for all ϕ 0 ∈ D H0 and all (i, x) ∈ E × V.

Proof Due to H 3 , the quantity Uh ( p) (i, x) is clearly defined for all (i, x) ∈ E × V and Uh ( p) is element of D H0 . Under H 3 , we know that:

lim t→+∞ 1 t t 0 ρ ( p) s ϕ ( p) ds = π ( p) ϕ ( p) (22)
for each measurable and bounded ϕ ( p) . The first term in right side of Eq. 20 consequently converges to the first term in Eq. 21. For the second term, setting p) du, we have:

U s := +∞ s P ( p) u h ( p) -π ( p) h (
1 t t 0 ρ ( p) s ∂ H ( p) ∂ p t-s 0 P ( p) u h ( p) -π ( p) h ( p) du ds = 1 t t 0 ρ ( p) s ∂ H ( p) ∂ p Uh ( p) ds - 1 t t 0 ρ ( p) s ∂ H ( p) ∂ p U t-s ds.
By assumption, the function ∂ H ( p) ∂ p Uh ( p) is bounded and independent of time, and consequently: p) .

lim t→+∞ 1 t t 0 ρ ( p) s ∂ H ( p) ∂ p Uh ( p) ds = π ( p) ∂ H ( p) ∂ p Uh ( p) = π ( p) ∂ H ( p) 0 ∂ p Uh (
It now remains to prove that

lim t→+∞ 1 t t 0 ρ ( p) s ∂ H ( p) ∂ p U t-s ds = 0.
We have:

∂ H ( p) ∂ p U t-s (i, x) ≤ j∈E ∂a ( p) ∂ p (i, j, x) V U t-s ( j, y)μ (i, j,x) (dy) As |U s (i, x)| ≤ +∞ s f ( p) (u) du due to H 3 , we have: ∂ H ( p) ∂ p U t-s (i, x) ≤ K +∞ t-s f ( p) (u) du and 1 t t 0 ρ ( p) s ∂ H ( p) ∂ p U t-s ds ≤ K 1 t t 0 +∞ t-s f ( p) (u) du ds ≤ K 1 t +∞ 0 u f ( p) (u) du.
This ends the proof.

The previous theorem provides an extension of the results given in Cao and Chen (1997) for pure jump Markov processes with countable state space.

Next proposition gives a tool to compute the function Uh ( p) .

Proposition 3 Let us assume that μ (i, j,x) and v(i, x) are independent of p and that H 2 , H 3 are true. Then the function U h ( p) is the unique solution of the following ordinary differential equation:

H ( p) 0 Uh ( p) (i, x) = π ( p) h ( p) -h ( p) (i, x) (23) for all (i, x) ∈ E × V such that π ( p) Uh ( p) = 0.
Proof Under assumptions H 3 , we clearly have π ( p) Uh ( p) = 0 by Fubini's theorem. We now check that Uh ( p) is solution of Eq. 23: under H 3 , we may write

H ( p) 0 Uh ( p) (i, x) = H ( p) 0 +∞ 0 P ( p) u h ( p) (i, x) -π ( p) h ( p) du = +∞ 0 H ( p) 0 P ( p) u h ( p) (i, x) du since H ( p) 0 π ( p) h ( p) = π ( p) h ( p) H ( p)
0 1 = 0. We derive:

H ( p) 0 Uh ( p) (i, x) = lim t→+∞ t 0 H ( p) 0 P ( p) u h ( p) (i, x) du = lim t→+∞ P ( p) t h ( p) (i, x) -h ( p) (i, x) (Chapman-Kolmogorov equation) = π ( p) h ( p) -h ( p) (i, x) ,
which shows that Uh ( p) is solution of Eq. 23. Now, let ϕ be the difference between two solutions ϕ 1 , ϕ 2 of Eq. 23 such that π ( p) ϕ 1 = π ( p) ϕ 2 = 0. We then have: Davis 1993, e.g.) and Eq. 3, we get P t ϕ = ϕ and:

H ( p) 0 ϕ = 0 and consequently: P t H ( p) 0 ϕ = 0 for all t > 0. Using P t H ( p) 0 ϕ = H ( p) 0 P t ϕ (see
ϕ (i, x) = lim t→+∞ 1 t t 0 (P t ϕ) (i, x) = π ( p) ϕ for all (i, x) ∈ E × V. Whence the result, using π ( p) ϕ = π ( p) ϕ 1 -π ( p) ϕ 2 = 0.
Remark 4 Just as for the transitory results, in the case when the data depend on some family of parameters P, the evaluation of all lim t→+∞ 1 t ∂ R (P) ρ 0 ∂ p (t) only requires one single computation of both π (P) and Uh (P) , which seems cheaper than finite differences methods in case of a large P.

Numerical Procedure

In this section, we propose some method for the numerical evaluation of

∂ R ( p) ρ 0 ∂ p (t)
and of its asymptotic rate per unit time when t → +∞, based on the implicit finite volume scheme given in [START_REF] Eymard | An implicit finite volume scheme for a scalar hyperbolic problem with measure data related to piecewise deterministic markov processes[END_REF]. These methods are obtained by translating the continuous procedure described above into the discrete setting of finite volume methods, and are provided without proof.

Principle of the Numerical Scheme in the Transient Case

Here, we consider the general case where any quantity (except ρ 0 ) may depend on p. We consider a mesh (or partition) M of V ⊂ R d which satisfies some regularity hypotheses (details are given in [START_REF] Eymard | An implicit finite volume scheme for a scalar hyperbolic problem with measure data related to piecewise deterministic markov processes[END_REF], and some time step δt > 0.

Realizing a discrete version of the Chapman-Kolmogorov Eq. 4 and starting from ρ (0) = ρ 0 , the scheme resumes to compute the family of real values

ρ (n) i,K (i,K,n)∈E×M×N , such that ρ (n) i,K is an approximation of K ρ ( p) t (i, dx) for t ∈ [nδt, (n + 1) δt[.
Introducing an infinite matrix A = A (i,K)( j,L) (i,K),( j,L)∈E×M , the scheme can be written as

( j,L)∈E×M A (i,K)( j,L) ρ (n+1) j,L + ρ (n+1) i,K -ρ (n) i,K δt = 0
for all (i, K, n) ∈ E × M × N (see the quoted reference for the detailed coefficients), or equivalently

Aρ (n+1) + ρ (n+1) -ρ (n) δt = 0 (24) for all n ∈ N, setting ρ (n) := ρ (n) i,K (i,K)∈E×M . For t = Mδt with M ≥ 1, a discrete approximation of R ( p) ρ0 (t) then is R(p) M with: R(p) M = M n=1 i∈E K∈M δt ρ (n) i,K hi,K (25)
and

hi,K = 1 m(K) K h ( p) (i, x) dx (26)
for all (i, K) ∈ E × M, where m(K) stands for the d-dimensional Lebesgue measure of K.

In order to mimic the procedure used to evaluate

H (θ) (n) = A t θ (n) + θ (n) -θ (n+1) δt (27)
for all bounded families θ = θ (n) i,K (i,K,n)∈E×M×N .

We next introduce a discrete approximation of the importance functions ϕ t as φ = φ(n)

i,K (i,K,n)∈E×M×N
, single bounded family solution of:

H ( φ) (n) = h for all n ∈ {0, ..., M -1} (28)
with φ(n) i,K = 0 for all (i, K) ∈ E × M and all n ≥ M. The mathematical study of the well-posedness of Eq. 27 and of the resolution of Eq. 28 might be driven under classical hypothesis.

Following the same calculation steps as in the continuous case, we now get:

∂ ∂ p R(p) M = δt i∈E K∈M M n=1 ρ (n) i,K ⎛ ⎝ ∂ ∂ p hi,K - ∂ H ∂ p ( φ) (n-1) i,K ⎞ ⎠ (29)
where, for all

(i, K, n) ∈ E × M × N: ∂ H ∂ p ( φ) (n) i,K = ( j,L)∈E×M ∂ A ( j,L)(i,K) ∂ p φ(n) j,L .
In the case when the data depends on a family P of parameters, the numerical computation of ∂ ∂ p R (P) M for all p ∈ P hence requires to solve two implicit volume schemes provided by Eqs. 24 and 28, which are independent on the choice of p in P (see Remark 3). Due to the definition of H (see Eq. 27), such schemes appear as dual schemes, which clearly simplifies their implementation. Also, the different summations required for each p ∈ P for the estimation of Eq. 29 are made simultaneously by solving the schemes, which helps saving CPU time and memory stocking size too.

Asymptotical Numerical Procedure

Following the discretization technique introduced in the transient case, a discrete solution of the asymptotic problem is a family of real values πi,K (i,K)∈E×M , such that πi,K is an approximation of the quantity K π ( p) (i, dx). Using again the infinite matrix A = A (i,K)( j,L) (i,K),( j,L)∈E×M , the asymptotic scheme can then be written as:

( j,L)∈E×M A (i,K)( j,L) π j,L = 0 (30)
for all (i, K) ∈ E × M, under the constraint

( j,L)∈E×M π j,L = 1. (31)
Equivalently, the scheme writes:

A π = 0 and π • 1 = 1, ( 32
)
where 1 is the constant family with the required dimension and generic term equal to 1, and where • stands for the dot product (x • y = (i,K)∈E×M x i,K y i,K for all x = x i,K (i,K)∈E×M and y = y i,K (i,K)∈E×M ). This scheme is assumed to have a unique solution.

An approximation R(p) π of lim t→+∞

1 t R ( p) ρ 0 (t) now is: R(p) π = i∈E K∈M πi,K hi,K = π • h,
where hi,K is provided by Eq. 26 and h = hi,K

(i,K)∈E×M .
In the same way as in the transient case, we mimic the continuous procedure and we consider the discrete version H0 of H 0 , given by:

H0 (θ) = A t θ for all bounded families θ = θ i,K (i,K)∈E×M .
We next introduce a discrete version Ū of the potential function Uh ( p) , solution to

H0 Ū = π • h 1 -h and π • Ū = 0, (33)
which is assumed to have a unique solution. We finally derive the discrete approxi-

mation ∂ R(p) π ∂ p of lim t→+∞ 1 t ∂ R ( p) ρ 0
∂ p (t), which is given by:

∂ R(p) π ∂ p = π • ∂ h ∂ p + π • ∂ H0 ∂ p Ū with ∂ H0 ∂ p Ū i,K = ( j,L)∈E×M ∂ A ( j,L)(i,K) ∂ p Ū j,L for all (i, K) ∈ E × M.
The same remarks as in the transitory case are still valid here and in case when the data depends on a family P of parameters, the numerical computation of all ∂ R(p) π ∂ p for each p ∈ P requires to solve exactly two dual volume schemes, provided by Eqs. 32 and 33.

A First Example

Presentation-Theoretical Results

A single component is considered, which is perfectly and instantaneously repaired at each failure. The distribution of the life duration of the component (T 1 ) is absolutely continuous with respect of Lebesgue measure, with E (T 1 ) > 0. The successive life durations make a renewal process. The time evolution of the component is described by the process (X t ) t≥0 where X t stands for the time elapsed at time t since the last instantaneous repair (the backward recurrence time). There is one single discrete state so that component I t is here unnecessary. The failure rate for the component at time t is λ (X t ) where λ (•) is some non negative continuous and bounded function. The process (X t ) t≥0 is "renewed" after each repair so that μ (x) (dy) = δ 0 (dy) and the evolution of (X t ) t≥0 between renewals is given by g (x, t) = x + t.

We are interested in the rate of renewals on [0, t], namely in the quantity Q (t) such that:

Q (t) = R (t) t = 1 t E 0 t 0 λ (X s ) ds
where R (t) is the renewal function associated to the underlying renewal process. The function λ ( x) is assumed to depend on some parameter p > 0 and to meet with H 2 (= H 2 ) requirement. (Here, both μ (x) (dy) and v (x) are independent of p).

Assuming E T ( p) 1 < +∞, the process is known to have a unique stationary distribution π ( p) which has the following probability density function (p.d.f.):

f ( p) π (x) = P T ( p) 1 > x E T ( p) 1 = e -x 0 λ ( p) (u)du E T ( p) 1 . ( 34
)
Besides:

Q ( p) (∞) = 1 E 0 T ( p) 1 = 1 +∞ 0 e -v 0 λ ( p) (u)du dv , ( 35
)
which is a direct consequence from the key renewal theorem. We now provide conditions which ensure the process to be uniformly ergodic (H 3 ):

Proposition 4 Let us assume that E e δT1 < +∞ for some 0 < δ < 1 and that T 1 is new better than used (NBU), namely such that for all x, t ≥ 0, we have F (x + t) ≤ F (x) F (t), where F is the survival function F (t) = P (T 1 > t). Then, there are some C < +∞ and 0 < ρ < 1 such that:

P ( p) t λ ( p) (x) -π ( p) λ ( p) ≤ Cρ t for all x ∈ R + .
Proof We use the following result by [START_REF] Konstantopoulos | On the use of Lyapunov function methods in renewal theory[END_REF]: assume that E e δT1 < +∞ for some δ > 0. Let η > 0 be such that η < δ or η ≤ δ ∧ 1 and define

W δ,η (x) = 1 + e (δ-η)x +∞ 0 e δt F (x + t) F (x) dt ≥ 1.
Then there are positive constants C 1 < +∞ and ρ < 1 such that

P ( p) t (x, •) -π ( p) Wδ,η ≤ C 1 W δ,η (x) ρ t where ν Wδ,η = sup |g|≤W δ,η |νg| .
Now, under the assumption E e δT1 < +∞ for some 0 < δ < 1, let us choose η = δ.

As T 1 is NBU, we know that F(x+t) F(x) ≤ F (t) for all x, t ≥ 0. Setting f to be the p.d.f. of T 1 , we derive:

W δ,η (x) ≤ 1 + +∞ 0 e δt F (t) dt = 1 + +∞ 0 e δt +∞ t f (u) du dt = 1 + 1 δ E e δT1 -1
< +∞ using Fubini's theorem. From the quoted result, we now derive existence of C 2 < +∞ and ρ < 1 such that

P ( p) t (x, •) -π Wδ,η ≤ C 2 ρ t
which easily provides the result due to λ ( p)

λ ( p) ∞ ≤ 1 ≤ W δ,η .

Numerical Results

We assume that T 1 is distributed according to some Weibull distribution, which is slightly modified to meet with our assumptions:

λ (α,β) (x) = ⎧ ⎨ ⎩ αβx β-1 if x < x 0 P α,β,x0 (x) if x 0 ≤ x < x 0 +2 αβ (x 0 + 1) β-1 if x 0 +2 ≤ x where (α, β) ∈ O =]0, +∞[×]2, +∞[, x 0 is chosen such that T 1 > x 0 is a rare event (i.e. P 0 (T 1 > x 0 ) = e -αx β
0 is small) and P α,β,x0 (x) is some smoothing function which makes x -→ λ (α,β) (x) continuous and non decreasing on R + . For such a failure rate, it is easy to check that assumptions H 2 and H 3 are true, using Proposition 4.

Taking (α, β) = 10 -5 , 4 and x 0 = 100 (which ensures

P 0 (T 1 > x 0 ) 5 × 10 -435 ), we compute IF p (t) = p Q ( p) (t) ∂ Q ( p) (t) ∂ p
and the asymptotic value IF p (∞) for p ∈ {α, β} by the finite volume methods (EMR) from Section 6.

For comparison purpose, we also compute such quantities by finite differences (FD) using:

∂ Q (t) ∂ p 1 ε Q ( p+ε) (t) -Q ( p) (t)
for some small ε. As for the transitory FD quantities, we use the algorithm from [START_REF] Eymard | An implicit finite volume scheme for a scalar hyperbolic problem with measure data related to piecewise deterministic markov processes[END_REF] which provides an estimate for the renewal function R ( p) (t) and hence for

Q ( p) (t) = R ( p) (t) t .
For the asymptotic FD quantities, we use the exact formula Eq. 35. 8.212×10 -2 6.7 × 10 -1 2.8214 1.2 × 10 -4 10 -6

2.411×10 -1 3.5 × 10 -2 2.8214 1.1 × 10 -4 10 -8

2.499×10 -1 3.0 × 10 -4 2.8214 1.1 × 10 -4 10 -10 2.500×10 -1 7.5 × 10 -5 2.8214

1.1 × 10 -4 EMR - 2.500×10 -1 - 2.8211 -
The asymptotic results are gathered in Table 1.

The comparison between EMR and FD results for small ε clearly validate the method. The results for IF β (∞) by FD are good and very stable when choosing different values for ε. The approximation for IF α (∞) by FD however requires smaller ε to provide correct results.

The transitory results are next plotted in Figs. 1 and 2 for t ∈ [0, 50] and different values of ε. We observe that FD provides similar results as EMR for IF β (t) but requires smaller ε for IF α (t) to get similar results as EMR, just as in the transitory case. Finally, comparing IF α (t) and IF β (t) for t ≤ ∞, we may note that, for a Weibull distribution, the shape parameter β is much more influent on the rate of renewals than the scale parameter α.

A Second Example

Presentation-Theoretical Results

The following example is very similar to that from [START_REF] Boxma | On/off storage systems with state-dependent input, output, and switching rates[END_REF]. The main difference is that we here assume X t to remain bounded (X t ∈ [0, R]) whereas X t takes its values in R + in the quoted paper.

A tank is considered, which may be filled in or emptied out using a pump. This pump may be in two different states: "in" (state 0) or "out" (state 1). The level of liquid in the tank goes from 0 up to R. The state of the system "pump-tank" at time t is (I t , X t ) where I t is the discrete state of the pump (I t ∈ {0, 1}) and X t is the continuous level in the tank (X t ∈ [0, R]). The transition rate from state 0 (resp. 1) to state 1 (resp. 0) at time t is λ 0 (X t ) (resp. λ 1 (X t )). The speed of variation for the liquid level in state 0 is v 0 (x) = r 0 (x) with r 0 (x) > 0 for all x ∈ [0, R[ and r 0 (R) = 0: the level increases in state 0 and tends towards R. Similarly, the speed in state 1 is v 1 (x) = -r 1 (x) with r 1 (x) > 0 for all x ∈]0, R] and r 1 (0) = 0: the level of liquid decreases in state 1 and tends towards 0. For i = 0, 1, the function λ i (respectively r i ) is assumed to be continuous (respectively Lipschitz continuous) and consequently bounded on [0, R]. The level in the tank is continuous so that μ (i, 1i, x) (dy) = δ x (dy) for i ∈ {0, 1} and x ∈ [0, R].

In order to ensure uniform ergodicity of the process (H 3 ), we make the following additional assumptions, where, for i ∈ E = {0, 1} and x, y ∈ [0, R], τ (i) x,y is the deterministic time for reaching y following the curve (t, g (i, x, t)):

λ 1 (0) > 0 and λ 0 (R) > 0, (36)
τ (0) x0,R = R x0 1 r 0 (u) du = +∞ (37) τ (1) x 0 ,0 = y0 0 1 r 1 (u) du = +∞ (38)
for all x 0 ∈ [0, R[ and all y 0 ∈]0, R]. This ensures the first jump time T 1 to be finite almost surely:

P (0,x0) (T 1 < +∞) = 1 -e -R x 0 λ 0 (u) r 0 (u) du = 1 P (1,y0) (T 1 < +∞) = 1 -e -y 0 0 λ 1 (u) r 1 (u) du = 1
We get the following result:

Proposition 5 Under assumptions (36-38), the process (I t , X t ) t≥0 is positive Harris recurrent with single invariant distribution π given by:

π (i, dx) = f i (x) dx for i = 0, 1 and f 0 (x) = K π v 0 (x) e - x R/2 λ 1 (u) v 1 (u) + λ 0 (u) v 0 (u) du = K π r 0 (x) e x R/2 λ 1 (u) r 1 (u) - λ 0 (u) r 0 (u) du (39) f 1 (x) = - K π v 1 (x) e - x R/2 λ 1 (u) v 1 (u) + λ 0 (u) v 0 (u) du = K π r 1 (x) e x R/2 λ 1 (u) r 1 (u) - λ 0 (u) r 0 (u) du (40)
for all x ∈]0, R[, where K π > 0 is a normalization constant.

Remark 5 Though such results are very similar to some special case from [START_REF] Boxma | On/off storage systems with state-dependent input, output, and switching rates[END_REF], we have better give here a quick proof due to a few differences in the results, such as some eventual masses for π at the bounds of the interval in the quoted paper.

Proof Under our assumptions, one can first prove that the process (I t , X t ) t≥0 with values in F := {0, 1} × [0, R] is ϕ-irreducible (see Meyn and Tweedie 1993b), with maximal irreducibility measure ϕ = c {0,1} × l where c {0,1} is the counting measure on {0, 1} and l is the Lebesgue measure on [0, R]. Besides, the process (I t , X t ) t≥0 is nonevanescent, due to values in a compact set. One can also prove that it is a T-process, namely such that there is some probability measure a (dt) and some kernel T such that for all i, j ∈ E :

+∞ 0 P t ((i, x) , ( j, A)) a (dt) ≥ T ((i, x) , ( j, A))
for all x ∈ [0, R] and for all Borel set A ⊂ [0, R], where T ((i, •) , ( j, A)) is lower semicontinuous (l.s.c.) for all (i, j, A) and where T ((i, x) , E × [0, R]) > 0 for all (i, x). Indeed, let t 0 > 0 be fixed. For any Borel set A ⊂ [0, R] with interior Å, and any

(i, j, x) ∈ E 2 × [0, R], we set T ((i, x) , ( j, A)) = 1 {i= j} P (i,x) (T 1 > t 0 ) 1 Å (g(i, x, t 0 )) = 1 {i= j} c (i,x) (t 0 ) 1 Å (g(i, x, t 0 ))
where c (i,x) (t 0 ) = e -t 0 0 λi(g(i,x,s))ds .

We then have:

P t0 ((i, x) , ( j, A)) = E (i,x) I t0 = j, X t0 ∈ A ≥ 1 {i= j} c (i,x) (t 0 ) 1 A (g(i, x, t 0 )) ≥ T ((i, x) , ( j, A))
where T ((i, •) , ( j, A)) is l.s.c. because Å is an open set, so that (I t , X t ) t≥0 is a T-process. Using Theorem 3.3 from Meyn and Tweedie (1993b), the process (I t , X t ) t≥0 now is Harris recurrent and admits a unique invariant measure π up to some multiplicative constant. Besides, π and ϕ = c {0,1} × l are mutually absolutely continuous (see [START_REF] Davis | Markov models and optimization, monographs on statistics and applied probability[END_REF] and there is some positive measurable function f i such that:

π (i, dx) = f i (x) dx Using the fact that π (•, dx) is such that π H ( p) 0 ϕ = 0 for all ϕ continuously differen- tiable on E × [0, R] with H 0 ϕ (1, x) = -r 1 (x)ϕ (1, x) -λ 1 (x)ϕ (1, x) + λ 1 (x)ϕ (0, x) H 0 ϕ (0, x) = r 0 (x)ϕ (0, x) -λ 0 (x)ϕ (0, x) + λ 0 (x)ϕ (1, x) one easily finds that λ 1-i (x) f 1-i (x) -λ i (x) f i (x) - d dx (v i (x) f i (x)) = 0
for i = 0, 1. Solving this system of o.d.e. provides ( f 0 , f 1 ) of the form Eqs. 39-40.

Checking that, under our assumptions, R 0 f i (x) dx < ∞ for i = 0, 1, we derive that π is a finite measure which can then be normalized in a single way into a probability measure. Consequently, (I t , X t ) t≥0 is a positive Harris recurrent process, which ends the proof.

We now prove that the process (I t , X t ) t≥0 is uniformly ergodic (H 3 ).

Proposition 6 Under assumptions (36)(37)(38), assumption H 3 is true, namely there is some function f fulfilling Eq. 18 such that:

|(P t h) (i, x) -π h| ≤ f (t)
for all (i, x) ∈ E × [0, R] and all t ≥ 0.

Proof According to Theorem 8.1 by Meyn and Tweedie (1993a), if (I t , X t ) t≥0 is bounded in probability on average (condition 1), if there is some t 0 ≥ 0 such that the kernel n∈N P t0n ((i, x) , •) possesses an everywhere non trivial continuous component (condition 2) and if the state space is compact (condition 3), then there is some ρ ∈ [0, 1[ such that:

lim t→+∞ sup (i,x)∈E×[0,R] ρ -t P t ((i, x) , •) -π TV = 0 (41)
where . TV stands for the norm in total variation. Now, according to Theorem 3.2 from Meyn and Tweedie (1993a), as (I t , X t ) t≥0 is a ϕ-irreducible T-process which is positive Harris recurrent (see Proposition 5 and its proof), the process (I t , X t ) t≥0 is bounded in probability on average and condition 1 is true. The second condition is clear too because setting T 2 to be the second jump time, we have for any t 0 > 0 :

P t0 ((i, x) , ( j, A)) ≥ P (i,x) T 1 ≤ t 0 < T 2 , I t0 = j, X t0 ∈ A = 1 { j=1-i} t0 0 c (i,x) (s) λ (i, g (i, x, s)) c ( j,g(i,x,s)) (t 0 -s) 1 A (g ( j, g (i, x, s) , t 0 -s)) ds
for any (i, j, x) ∈ E 2 × [0, R] and for any Borel set A (see Cocozza-Thivent et al. 2006b). The kernel P t0 ((i, x) , •) then possesses an everywhere non trivial continuous component, and n∈N P t0n ((i, x) , •) does too. As the state space is compact, the three conditions are verified and we derive that Eq. 41 is true. Consequently, there is some t 1 such that, for all t > t 1 , we have:

sup (i,x)∈E×[0,R] ρ -t P t ((i, x) , •) -π TV ≤ 1
and hence:

|(P t h) (i, x) -π h| ≤ h ∞ ρ t for all (i, x) ∈ E × [0, R] and all t > t 1 . Setting f (t) = sup (i,x)∈E×[0,R] u≤t1 |(P t h) (i, x) -π h| ift ≤ t 1 h ∞ ρ t ift > t 1
easily provides the result.

Q 1 (t) = 1 t E ρ0 t 0 1 { R 2 -a≤Xs≤ R 2 +b } ds = 1 t 1 i=0 t 0 R 2 +b R 2 -a ρ s (i, dx) ds = 1 t t 0 ρ s h 1 ds (42) with h 1 (i, x) = 1 R 2 -a, R
2 +b (x). The second quantity of interest is the mean number of times the pump is turned off, namely turned from state "in" (0) to state "out" (1) by unit time, namely:

Q 2 (t) = 1 t E ρ0 0<s≤t 1 {I s-=0 and Is=1} = 1 t E ρ0 t 0 λ 0 (X s ) 1 {I s =0} ds = 1 t t 0 R 0 λ 0 (x) ρ s (0, dx) ds = 1 t t 0 ρ s h 2 ds (43) with h 2 (i, x) = 1 {i=0} λ 0 (x).

Numerical Example

We assume that the system is initially in the state (I 0 , X 0 ) = (0, R/2). Besides, we take:

λ 0 (x) = x α0 ; r 0 (x) = (R -x) ρ0 ; λ 1 (x) = (R -x) α1 ; r 1 (x) = x ρ1
for x ∈ [0, R] with α i > 1 and ρ i > 1. All conditions for irreducibility are here achieved. Our aim is to compute the importance factors with respect to p for p ∈ {α 0 , α 1 , r 0 , r 1 , a, b } both in Q 1 (t) and Q 2 (t), except for parameters a and b which intervenes only in Q 1 (t). We take the following numerical values: α 0 = 1.05; ρ 0 = 1.2; α 1 = 1.10; ρ 1 = 1.1; R = 1; a = 0.2; b = 0.2.

Similarly as for the first method, we test our results using finite differences (FD). For FD, the transitory results are computed via the finite volume scheme from [START_REF] Eymard | An implicit finite volume scheme for a scalar hyperbolic problem with measure data related to piecewise deterministic markov processes[END_REF] and the asymptotic results via Eqs. 39-40. The results are here rather stable choosing different values for ε and they are provided for ε = 10 -2 in case p ∈ {α 0 , α 1 , r 0 , r 1 } and for ε = 10 -3 in case p ∈ {a, b }. EMR results are computed using the finite volume methods from Section 6. The asymptotic results are given in Tables 2 and3, and the transitory ones are given in Tables 4 and5 for t = 2.

The results are very similar by FD and EMR both for the asymptotic and transitory quantities, which here again validate the method. Besides, we may observe that the asymptotic results coincides by both methods, even in the case when the velocity field v (i, x) depends on the parameter (here ρ i ), which however does not fit with our technical assumptions from Section 5. Due to that (and to other examples where the same remark is valid), one may conjecture that the results from Section 5 are valid under less restrictive assumptions than those given in that section. -6.80 × 10 -2 -6.79 × 10 -2 2, 12 × 10 -3 ρ 0 -1.25 × 10 -1 -1.24 × 10 -1 4, 27 × 10 -3 ρ 1 -4.11 × 10 -3 -4.03 × 10 -3 2, 00 × 10 -2

As for the respective importance of the different parameters in Q i (t), one may note that the importance factors at t = 2 of α 0 and ρ 0 in Q i (t) (i = 1, 2) are clearly higher than the importance factors of α 1 and ρ 1 in Q i (t) (i = 1, 2). This must be due to the fact that the system starts from state 0, so that on [0, 2], the system spends more time in state 0 than in state 1. The parameters linked to state 0 hence are more important than the ones linked to state 1. Similarly, the level is increasing in state 0 so that the upper bound b is more important than the lower one a.

In long-time run, the importance factors of α 0 and α 1 in Q i (t) (i = 1, 2) are comparable, which is conform with intuition. The same remark is valid for ρ 0 and ρ 1 , as well as for a and b .

Finally, parameters ρ 0 and ρ 1 are more important than parameters α 0 and α 1 in Q 1 (t), conversely to what happens in Q 2 (t). This seems coherent with the fact that quantity Q 1 (t) is linked to the level in the tank, and consequently to its evolution, controlled by ρ 0 and ρ 1 , whereas quantity Q 2 (t) is linked to the transition rates, and consequently to α 0 and α 1 . and ∇ϕ (n+1) (t, ., .) -∇ϕ (n) (t, ., .) ∞ ≤ C t t0 ϕ (n) (s, ., .) -ϕ (n-1) (s, ., .) C b 1 (E,W) ds so that ϕ (n+1) (t, ., .) -ϕ (n) (t, ., .) C b 1 (E,W) ≤ C t t0 ϕ (n) (s, ., .) -ϕ (n-1) (s, ., .) C b 1 (E,W) ds

Iterating this inequality, we obtain for all t in I: Hence for all n in N:

ϕ 1 -ϕ 2 C b 1 (I,E,W) ≤ K C n (b -a) n n!
The uniqueness follows. We can prove the second part of theorem in the same way.

Fig. 1 Fig. 2

 12 Fig.1IF α (t) by finite differences (FD) for ε = 10 -i with i ∈ {2, ..., 7} and by the present method (EMR)

  ϕ(n) (t, ., .) -ϕ(n-1) (t, ., .) C b 1 (E,W) ≤ K C n (tt 0 ) n n! Hence ϕ (n) -ϕ (n-1) C b 1 (I,E,W) ≤ K C n (ba) n n!Then the sequence ϕ(n) is a Cauchy sequence in the Banach space C b 1 (I, E, W) and it converges to a function ϕ in C b 1 (I, E, W) as n tends to infinity. Taking the limit in Eq. 44, we obtain:ϕ(t, i, z) = ϕ 0 (i, z) + t t0 F(s, ϕ(s, ., .)(i, z) dsIn order to prove the uniqueness of the solution, let ϕ 1 , ϕ 2 ∈ C b 1 (I, E, W) be two different solutions of the equation. We have:ϕ 1 (t, i, z) -ϕ 2 (t, i, z) = t t0(F (s, ϕ 1 (s, ., .)) -F(s, ϕ 2 (s, ., .))) ds and∇ϕ 1 (t, i, z) -∇ϕ 2 (t, i, z) = t t0 (∇F (s, ϕ 1 (s, ., .)) -∇F (s, ϕ 2 (s, ., .))) ds Then ϕ 1 (t, ., .) -ϕ 2 (t, ., .) C b 1 (E,W) ≤ C t t0 ϕ 1 (s, ., .) -ϕ 2 (s, ., .) C b 1 (E,W) ds and by iteration ϕ 1 (t, ., .) -ϕ 2 (t, ., .) C b 1 (E,W) ≤ K C n (tt 0 ) n n!

  Under the above technical assumptions, I

				( p) t , X t ( p)	t≥0	is known to be a Markov
	process with general state space E × V, see Davis (1993), Cocozza-Thivent et al.
	(2006b), with strong Feller transition semi-group. We denote by ρ t ( j, dy) the distri-( p)
	bution of the process I	( p) t , X t ( p)	t≥0	at time t with initial distribution ρ 0 (independent
	of p) and by P ( p) t (i, x, j, dy) the transition probability distribution of I t , X ( p) t ( p)	t≥0

or R

Table 1

 1 IF α (∞) and IF β (∞) by finite differences (FD) and by the present method (EMR)

		ε	IF α (∞)	Relative error	IF β (∞)	Relative error
				between FD		between FD
				and EMR		and EMR
	FD	10 -2	4.625×10 -3	9.8 × 10 -1	2.8242	1.1 × 10 -3
		10 -4

Table 2 IF

 2

	(1) p (∞) by	p	FD	EMR	Relative error
	FD and EMR	α 0 α 1 ρ 0 ρ 1 a	-3.59 × 10 -2 -4.45 × 10 -2 3.19 × 10 -1 2.80 × 10 -1 4.98 × 10 -1	-3.57 × 10 -2 -4.43 × 10 -2 3.17 × 10 -1 2.78 × 10 -1 4.98 × 10 -1	5, 40 × 10 -3 3, 65 × 10 -3 6, 95 × 10 -3 7, 19 × 10 -3 1, 06 × 10 -7
		b	5.09 × 10 -1	5.09 × 10 -1	1, 53 × 10 -7
	8.2 Quantities of Interest
	We are interested in two quantities: first, the proportion of time spent by the level in
	the tank between two fixed bounds R 2 -a and R 2 + b with 0 < a, b < R 2 and we set:

Table 3 IF

 3

	(2) p (∞) by FD	p	FD	EMR	Relative error
	and EMR	α 0 α 1 ρ 0 ρ 1	-1.81 × 10 -1 -1.71 × 10 -1 -6.22 × 10 -2 -6.05 × 10 -2	-1.81 × 10 -1 -1.71 × 10 -1 -6.19 × 10 -2 -6.01 × 10 -2	1, 67 × 10 -4 1, 30 × 10 -4 5, 21 × 10 -3 5, 58 × 10 -3

Table 4 IF

 4

	(1) p (t) for t = 2	p	FD	EMR	Relative error
	by FD and EMR	α 0 α 1 ρ 0 ρ 1 a	-8.83 × 10 -2 -9.10 × 10 -3 4.89 × 10 -1 1.97 × 10 -1 2.48 × 10 -1	-8.82 × 10 -2 -9.05 × 10 -3 4.85 × 10 -1 1.97 × 10 -1 2.48 × 10 -1	1, 08 × 10 -3 5, 29 × 10 -3 7, 51 × 10 -3 4, 04 × 10 -3 4, 89 × 10 -4
		b	7.11 × 10 -1	7.11 × 10 -1	7, 77 × 10 -6

Table 5 IF

 5

	(2) p (t) for t = 2	p	FD	EMR	Relative error
	by FD and EMR	α 0	-2.06 × 10 -1	-2.06 × 10 -1	9, 12 × 10 -4
		α 1

( p) ρ 0 (t).

( p) t f (i, x) is continuously differentiable on V × O, with all partial derivatives uniformly bounded on V × N ( p) for all p ∈ O. Under assumptions

Acknowledgements

The authors would like to thank Anne Barros, Christophe Bérenguer, Laurence Dieulle and Antoine Grall from Troyes Technological University (Université de Technologie de Troyes) for having drawn their attention to the present subject.

Appendix

This appendix gives a result of existence, uniqueness and regularity for the solution of the equation:

F (s, ϕ(s, ., .)

)(i, z) ds

We shall use the notations:

2 (E, W) and such that sup s∈I ϕ(s, ., .

Theorem 4 Let us consider the equation

F(s, ϕ(s, ., .))(i, z) ds and let us suppose that:

1 (E, W) there exists a positive constant C such that for all s in I :

Then there exists a unique solution which belongs to C b 1 (I, E, W) and is continuously differentiable with respect to the first variable. If, in addition, we have

2 (E, W) there exists a positive constant C such that for all s in I:

then there exists a unique solution which belongs to C b 2 (I, E, W) and is continuously differentiable with respect to the first variable.

Proof Let us prove the first part of the theorem. To prove the existence of a solution, starting from an initial function ϕ (0)

Thanks to theorem of differentiation under integral, we can prove that these functions are in C b 1 (I, E, W) and that the vector of partial derivatives with respect to variables in W satisfies: ∇ϕ (n+1) (t, i, z) = ∇ϕ 0 (i, z) + t t0 ∇F (s, ϕ (n) (s, ., .))(i, z) ds We have ϕ (n+1) (t, ., .) -ϕ (n) (t, ., .) ∞ ≤ C t t0 ϕ (n) (s, ., .) -ϕ (n-1) (s, ., .) C b 1 (E,W) ds