Andreu Cecilia
email: andreu.cecilia@upc.edu

Daniele Astolfi
email: daniele.astolfi@univ-lyon1.fr

Michelangelo Bin
email: michelangelo.bin@unibo.it

Ramon Costa-Castelló

Cancelling output disturbances in observer design through internal model filters

This work proposes a redesign method for nonlinear observers to reduce the effect of a particular class of output disturbances. Specifically, it is considered a disturbance composed by a term generated from a known system and an unstructured term. The proposed approach does not require modifying the original observer and is based on adding a simple filter that includes an internal model of the disturbance generator. Sufficient conditions for stability of the proposed filter-observer architecture are given. Moreover, the approach is validated through numerical simulations.

Introduction

Many control and monitoring problems require an estimate of the state variables from the available measurements. In such scenarios, it is common to design and implement an observer [START_REF] Bernard | Observer design for continuous-time dynamical systems[END_REF]. One of the major limiting factors of the performances of an observer is the presence of sensor noise [START_REF] Goodwin | A critique of observers used in the context of feedback control[END_REF], which yields a well-known trade-off between convergence rate and noise attenuation.

The effect of noise can be attenuated through a suitable tuning of the observer parameters as, for instance, in the Kalman filter approach [START_REF] Kalman | New results in linear filtering and prediction theory[END_REF], or by combining a minimization of the H ∞ /L 2 input-output gain and the bounded real lemma [START_REF] Duan | State observation and filtering[END_REF]. Alternatively, one can couple the observer with a system that filters-out some spectral components of the output error signal. An example of this methodology is the Proportional-Integral observer [START_REF] Beale | Robust control system design with proportional integral observer[END_REF][START_REF] Shafai | LTR design of discrete-time proportional-integral observers[END_REF][START_REF] Busawon | Disturbance attenuation using proportional integral observers[END_REF], which combines the output-estimation error with its integral, see also the extensions [START_REF] Astolfi | On the use of low-pass filters in high-gain observers[END_REF][START_REF] Farza | Improved high gain observer design for a class of disturbed nonlinear systems[END_REF] to more general low-pass filters. Recently, [START_REF] Astolfi | Stubborn and dead-zone redesign for nonlinear observers and filters[END_REF] proposed the use of nonlinear dynamic dead-zones or dynamic saturations.

The mentioned approaches only achieve a reduction of the effect of output disturbances. A stronger result can be obtained if there is a known generating model of the output disturbance [START_REF] Goodwin | Fundamental performance properties of a general class of observers for linear systems having predictable disturbances[END_REF]. In such case, the generating model can be included in the observer in order to exactly cancel the disturbance. In the presence of a generating model, here referred to as the exosystem, it is common to consider an extended system composed by the original system and the exosystem, and to design an observer that jointly estimates the states of the original system and exosystem [START_REF] Goodwin | Fundamental performance properties of a general class of observers for linear systems having predictable disturbances[END_REF]. However, this approach is not easily extendable to nonlinear systems, where the existence of an observer for the original system does not guarantee that of an observer for the extended system. With this in mind, the objective of this work is to interconnect an existing observer with a system that perfectly cancels the output disturbance without requiring any redesign of the original observer.

There exists a large literature on design methodologies to compensate disturbances with a known generating model affecting the measured output. Some remarkable examples are [START_REF] Marino | Adaptive stabilization of linear systems with outputs affected by unknown sinusoidal disturbances[END_REF][START_REF] Pyrkin | Adaptive controller for linear system with input delay and output disturbance[END_REF][START_REF] Vau | Adaptive rejection of narrow-band disturbances in the presence of plant uncertainties-a dual youla-kucera approach[END_REF], where adaptive controllers are proposed to solve a stabilization problem in the presence of harmonic disturbances. Alternatively, [START_REF] Aranovskii | Adaptive observer of an unknown sinusoidal output disturbance for linear plants[END_REF] presents an adaptive observer to estimate the output disturbance of a linear system. Nonetheless, to the best of the authors' knowledge, this work considers a different problem that has not been previously studied in the literature. That is, the case where a (possibly nonlinear) observer is already given and the objective is to compensate a particular disturbance without modifying the observer structure or gain tuning. Such an extra constraint is motivated by possible applications that go beyond the mere scope of observers design, such as distributed multi-agent systems [START_REF] Chopra | Passivity-based control of multi-agent systems[END_REF] or cyber-security problems [START_REF] Cecilia | A masking protocol for private communication and attack detection in nonlinear observers[END_REF], in which completely redesigning the given feedback controller could be critical or not allowed due to practical constraints.

More precisely, this work proposes an alternative route to embed the exosystem in the observer with a modular approach. First, we assume that an observer for the system in nominal conditions, i.e. in the absence of the measurement disturbance, is known. Second, the observer is connected in cascade with a filter that includes the generating model of the disturbances. The design of the filter follows from an internal model based philosophy [START_REF] Francis | The internal model principle for linear multivariable regulators[END_REF] in order to exactly cancel the output disturbance. We consider nonlinear plants and observers [START_REF] Bernard | Observer design for continuous-time dynamical systems[END_REF] satisfying an incremental passivity assumption [START_REF] Pavlov | Incremental passivity and output regulation[END_REF]. Additionally, this work relies on the property of incremental input-to-state stability (δISS) [START_REF] Angeli | A Lyapunov approach to incremental stability properties[END_REF]. A definition of these properties is given at the end of this section.

Notation: |x| denotes the Euclidean norm; I k denotes the identity matrix of dimension k; sup s∈[0,t] |u(s)| denotes the supremum norm of u(s) in the interval [0, t]; rank(A) and σ(A) denote the rank and the spectrum of matrix A, respectively. By blckdiag(•) we denote block-diagonal concatenations. R denotes the set of real numbers, R ≥0 := [0, ∞) C denotes the of complex numbers, C -:= {s ∈ C : ℜ(s) ≤ 0} C + := {s ∈ C : ℜ(s) ≥ 0}, and ⋆ denotes the off-diagonal elements of a symmetric matrix. For the definition and properties of class K and KL functions, we refer to [START_REF] Kellett | A compendium of comparison function results[END_REF].

Consider a system of the form

ẋ = f (x, d), (1)
where x ∈ R nx is the state, d ∈ D ⊂ R nu is a known measurable and locally essentially bounded signal taking values in a compact set D, and f is locally Lipschitz in its first argument. We have the following definition [START_REF] Angeli | A Lyapunov approach to incremental stability properties[END_REF].

Definition 1 (Incremental input-to-state stability). System (1) is δISS with respect to the input d if there exist β ∈ KL and ρ ∈ K such that, for every two solutions x and x ′ to (1) subject to inputs d and d ′ , the following holds for all t ≥ 0

|x(t) -x ′ (t)| ≤ β(|x(0) -x ′ (0)|, t) + sup s∈[0,t) ρ |d(s) -d ′ (s)| . Consider now the system ẋ = f (x) + Bu y = Cx, (2)
where x ∈ R nx is the state, y ∈ R ny is the measured output, and u ∈ U ⊂ R nu is a bounded input. The function f is sufficiently smooth in its argument. We have the following definition [START_REF] Pavlov | Incremental passivity and output regulation[END_REF].

Definition 2 (Incremental passivity). System (2) is incrementally passive from u to y if there exists a C 1 storage function S(x, x ′ ) such that for every pair of inputs u, u ′ ∈ U and any pair (x, x ′ ) of solutions of (2) corresponding to these inputs, with corresponding outputs y, y ′ , the following is satisfied

Ṡ ≤ (y -y ′ ) ⊤ (u -u ′ ).
For systems of the form (2), there exists a sufficient condition to verify incremental passivity in the form of a matrix inequality [START_REF] Pavlov | Incremental passivity and output regulation[END_REF]Lemma 3]. Specifically, if the following holds for some symmetric positive definite matrix P ∈ R nx×nx and for all x ∈ R nx P ∂f ∂x (x) + ∂f ∂x (x) ⊤ P ⪯ 0

P B = C ⊤ ,
then, the system is incrementally passive from u to y with a storage function S = (x -x ′ ) ⊤ P (x -x ′ ).

Problem Formulation

Framework

This work considers the multi-output nonlinear system ẋ = f (x, u)

y = Cx + d, (3)
where x ∈ R nx is the state, u ∈ R nu is a known control input, y ∈ R ny is the measured output, d ∈ R ny is an unknown output disturbance, and f is a locally Lipschitz function in its first argument. A standing assumption in this work is that the states of the system and the disturbances are bounded.

Assumption 1. The disturbance d and the control input u are Lebesgue measurable and bounded. In particular, d(t) ∈ D and u(t) ∈ U for all t ≥ 0, for some compact sets D ⊂ R ny and U ⊂ R nu .

Assumption 2. There exists a compact set X 0 ⊂ R nx that is forward invariant for system (3) for every input u satisfying Assumption 1.

In this work, we restrict ourselves to initial conditions of the system (3) in X 0 ⊂ R nx .

To ease certain parts of the presentation, some conditions will be particularized to the case of linear systems. In such a context, system (3) takes the form

ẋ = Ax + Bu y = Cx + d, (4)
for some matrices A, B, C of suitable dimension.

We suppose that we are given an observer of the form

ẋ = f (x, u) + κ(x, y -C x), (5)
in which x ∈ R nx is the estimation of the state, and κ denotes the output injection term, satisfying κ(x, 0) = 0 for all x ∈ R nx . Under Assumptions 1 and 2, we assume that the observer ensures global exponential stability of the error variable x -x, uniformly in u, and satisfies an input-to-state stability property with respect to the output disturbance d, namely the following holds

|x(t) -x(t)| ≤ ke -λt |x(0) -x(0)| + sup s∈[0,t] ρ(|d(s)|)
for all t ≥ 0, for every pair of initial conditions (x(0), x(0)) ∈ X 0 × R nx , for some class K function ρ, and for some constants k, λ > 0. Typical examples of observers that fit such a framework are the high-gain observer [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF][START_REF] Wang | High-gain observers with limited gain power for systems with observability canonical form[END_REF], observers derived from LMIs or circle-criterion approaches [START_REF] Arcak | Nonlinear observers: a circle criterion design and robustness analysis[END_REF][START_REF] Zemouche | Circle criterion-based H∞ observer design for lipschitz and monotonic nonlinear systemsenhanced lmi conditions and constructive discussions[END_REF] and observers derived from a dissipativity condition [START_REF] Moreno | Observer design for nonlinear systems: A dissipative approach[END_REF]. See also [START_REF] Bernard | Observer design for continuous-time dynamical systems[END_REF]Section 4] for more references. We assume that the plant's dynamics and the observer dynamics are expressed in the same set of coordinates. However, this is not always the case, as often the observer dynamics lives in a space of larger dimension. If this is the case, one can suppose that the plant's dynamics are expressed in the same coordinates of the observer and then develop the following result in such a coordinates. See, e.g., [START_REF] Bernard | Observer design for continuous-time dynamical systems[END_REF]Section 8].

We suppose that the output disturbance d is modeled as the sum of a signal composed by a finite sum of sinusoids and a term v representing unstructured bounded noise, namely

d j (t) = N d i=1 d j,i sin(ω i t + φ j,i ) + v j (t), j = 1, . . . , n y ,
where d j and v j denote the j-th components of d and v, respectively. The disturbance can be therefore assumed to be generated by an exosystem of the form

ẇ = Φw, d = Γ ⊤ w + v. (6)
where w ∈ R nw is the exosystem state, the matrix Φ is such that σ(Φ) = {±jω i : i = 1, . . . , N d }, and the pair (Φ, Γ ⊤ ) is observable. Since the particular representation of Φ, Γ does not play any role for the forthcoming results, without loss of generality we can assume that Φ and Γ have the following form Φ = blkdiag(ϕ, . . . , ϕ ny times

), Γ = blkdiag(γ, . . . , γ ny times

), [START_REF] Busawon | Disturbance attenuation using proportional integral observers[END_REF] with ϕ, γ such that

ϕ = blkdiag(ϕ 1 , . . . , ϕ N d ), ϕ i = 0 ω i -ω i 0 , γ = g ⊤ . . . g ⊤ ⊤ , g = 0 1 .
Note that if one considers ω i = 0, then, the corresponding block dimension is 1, and in this case

ϕ i = 0, g = 1.
The objective of this article is to augment the observer (5) with a disturbance compensator to guarantee that the state estimation error converges asymptotically to a bound that only depends on the unstructured term of the noise, namely, for some class K function ρ, we have lim sup

t→∞ |x(t) -x(t)| ≤ sup s∈[0,∞) ρ(|v(s)|). (8)
As anticipated in the introduction, a direct method to solve this problem would be to consider an extended sys-tem that includes the disturbance dynamics

ẋ = f (x, u) ẇ = Φw y = Cx + Γ ⊤ w + v, (9)
and design an observer for the resulting system [START_REF] Farza | Improved high gain observer design for a class of disturbed nonlinear systems[END_REF]. Although this is a perfectly valid approach for linear systems, it cannot be easily extended to the nonlinear case. First, the existence of an observer for the original system (3) does not guarantee, in general, the existence of an observer for the extended dynamics [START_REF] Farza | Improved high gain observer design for a class of disturbed nonlinear systems[END_REF]. Indeed, nonlinear observer design commonly relies on structural assumptions that may be lost in the extended dynamics [START_REF] Farza | Improved high gain observer design for a class of disturbed nonlinear systems[END_REF]. Therefore, even if it is possible to implement a certain nonlinear observer technique in the original system, it is not guaranteed that the same technique can be implemented for the extended system. Second, this approach does not take particular advantage of the previous existence of an observer. To overcome these limitations, this work proposes a solution following a modular perspective. Specifically, we follow a strategy based on an internal model approach [START_REF] Francis | The internal model principle for linear multivariable regulators[END_REF]. The main idea is to connect in cascade the already designed observer with a filter that includes an internal model of the disturbance generator. Similar filters can be found in previous literature of control and signal processing. Some remarkable examples are the "notch filter" [START_REF] Marino | Adaptive notch filters are local adaptive observers[END_REF] or the "washout filter" in the context of pre-processing output regulation [START_REF] Wang | Pre-processing nonlinear output regulation with non-vanishing measurements[END_REF]. Nonetheless, to the best of our knowledge, these results have not been implemented in the observer case. It should be remarked that this work follows a modular philosophy. That is, we assume that the already designed observer [START_REF] Beale | Robust control system design with proportional integral observer[END_REF], and in particular the structure of its gain κ cannot be modified.

Filter design

The proposed filter is designed as

η = Ψη + Γ(y -C x) z = (y -C x) -Γ ⊤ η (10)
where η(t) ∈ R nη is the state of the filter, partitioned as η = (η 1 , . . . , η ny ), with η i (t) ∈ R 2N d , z(t) ∈ R ny is the filter's output and the matrix Ψ is selected as

Ψ = Φ -ΓΓ ⊤ = blkdiag(ϕ -γγ ⊤ , . . . , ϕ -γγ ⊤ ny times
). ( 11)

The proposed internal model filter can be understood as a "bank" of identical filters, each one acting on a different output component. Specifically, consider the decomposition y = (y 1 , y 2 , . . . , y ny ), where y i ∈ R for i = 1, . . . , n y . Then, [START_REF] Astolfi | Stubborn and dead-zone redesign for nonlinear observers and filters[END_REF] reads as ηi = (ϕ -γγ ⊤ )η i + γ(y i -ŷi )

z i = (y i -ŷi ) -γ ⊤ η i , ∀i = 1, . . . , n y ,
where ŷi is the estimation of the ith component of the output produced by the observer [START_REF] Beale | Robust control system design with proportional integral observer[END_REF]. Note that, by construction, the matrix Ψ is Hurwitz. The latter is a direct consequence of the fact that ϕ is skewsymmetric and (Φ, Γ) is controllable. To interconnect the filter [START_REF] Astolfi | Stubborn and dead-zone redesign for nonlinear observers and filters[END_REF] and the observer ( 5), we substitute in the output injection term of the observer (5) the error signal ỹ with the filtered output z. This gives the modified observer

ẋ = f (x, u) + κ(x, z). ( 12
)
To better understand the motivation of the internal model filter [START_REF] Astolfi | Stubborn and dead-zone redesign for nonlinear observers and filters[END_REF], notice that such a filter is a relative degree zero linear system that has the zeros placed at the eigenvalues of Φ. Intuitively, the presence of a zero in the filter implies that the signal z cannot have any spectral component at such a frequency. Therefore, as the zeros of the filter are placed at the eigenvalues of Φ, the modelled part of the output disturbance d does not have any effect on the steady-state estimation. Hence, the desired disturbance rejection problem is solved if the overall observer ( 10),( 12) still possesses good convergence properties. This result is formalized in the next sections.

Asymptotic Analysis

The overall filtered observer [START_REF] Astolfi | Stubborn and dead-zone redesign for nonlinear observers and filters[END_REF], [START_REF] Marino | Adaptive stabilization of linear systems with outputs affected by unknown sinusoidal disturbances[END_REF]

reads ẋ = f (x, u) + κ(x, Cx -C x -Γ ⊤ η + d) η = Ψη + Γ(Cx -C x + d). ( 13
)
The objective of this section is to prove that, under an additional assumption given below, the observer (13) rejects the disturbance generated by the exosystem (6).

Assumption 3. Under Assumptions 1 and 2, system (13) is δISS with respect to the disturbance d, uniformly in u and x. Namely, there exist β ∈ KL and ρ ∈ K such that, for every input u and every two inputs d and d ′ satisfying Assumption 1, every solution x to (3) originating in X 0 and corresponding to u, and every two solutions (x, η) and (x ′ , η ′ ) of (13) corresponding to (u, x, d) and (u, x, d ′ ), respectively, the following holds

|(x(t), η(t)) -(x ′ (t), η ′ (t))| ≤ β(|(x(0), η(0)) -(x ′ (0), η ′ (0))|, t) + sup s∈[0,t) ρ |d(s) -d ′ (s)| (14)
for all t ≥ 0.

Assumption 3 is a strong incremental stability condition requiring that the gains of the filter [START_REF] Astolfi | Stubborn and dead-zone redesign for nonlinear observers and filters[END_REF] have been selected so as to strengthen the ISS properties of the original observer (5) to a stricter δISS property (see Definition 1) with respect to the disturbance d. While Assumption 3 may not be satisfied for all filter-observer interconnections [START_REF] Angeli | A Lyapunov approach to incremental stability properties[END_REF],

Section 4 provides sufficient and constructive conditions to design a filter such that Assumption 3 holds.

We now present the main result of the article, showing that the solutions to the filtered observer (13) satisfy the asymptotic bound property [START_REF] Astolfi | On the use of low-pass filters in high-gain observers[END_REF].

Theorem 1. Consider system (3) and the filtered observer (13) subject to the input d satisfying [START_REF] Shafai | LTR design of discrete-time proportional-integral observers[END_REF]. Suppose that Assumptions 1-3 hold. Then, every solution of (3), ( 6), ( 13) with x originating in X 0 satisfies [START_REF] Astolfi | On the use of low-pass filters in high-gain observers[END_REF], in which ρ is the same function for which Assumption 3 holds.

Proof. System (3), ( 6), ( 13) reads as

ẇ = Φw ẋ = f (x, u) ẋ = f (x, u) + κ(x, Cx -C x -Γ ⊤ η + Γ ⊤ w + v) η = Ψη + Γ(Cx -C x + Γ ⊤ w + v). ( 15
)
Pick arbitrarily a solution pair ((w, x, x, η), v) to [START_REF] Aranovskii | Adaptive observer of an unknown sinusoidal output disturbance for linear plants[END_REF], and consider the signals

xss (•) := x(•), η ss (•) := w(•). (16)
Simple computations show that ((w, x, xss , η ss ), 0) (namely, obtained for the same (w, x) and with v = 0) is a solution pair of (15) as well. Indeed for such a choice one obtains κ(t, xss , 0) = 0 in the xss -dynamics, that is ẋss = f (x ss , u), while the η ss -dynamics reads ηss = Φη ss , due to the definition of Ψ in [START_REF] Goodwin | Fundamental performance properties of a general class of observers for linear systems having predictable disturbances[END_REF]. In other words, ( 16) is a solution to [START_REF] Aranovskii | Adaptive observer of an unknown sinusoidal output disturbance for linear plants[END_REF] for v = 0. As a consequence, direct application of the inequality [START_REF] Vau | Adaptive rejection of narrow-band disturbances in the presence of plant uncertainties-a dual youla-kucera approach[END_REF] in

which x′ = xss = x, η ′ = η ss = w, d = Γ ⊤ w + v and d ′ = Γ ⊤ w yields |(x(t), η(t)) -(x(t), w(t))| ≤ β(|(x(0), η(0)) -(x(0), w(0))|, t) + sup s∈[0,t) ρ |v(s)| ,
from which (8) directly follows.

To better understand the ideas underlying the proposed design, we now particularize the result to the linear case. In particular, we consider a linear system of the form (4). In this case, the observer (5) reads

ẋ = Ax + Bu(t) + L(y -C x) ( 17
)
where L ∈ R nx×ny is designed so that (A-LC) is Hurwitz.

In this linear context, the closed-loop system (13) reads

ẋ = Ax + Bu(t) + L(y -C x + d) -LΓ ⊤ η η = Ψη + Γ(y -C x + d). (18)
Note that due to linearity, a δISS property is equivalent to asymptotic stability. Thus, in this context, the result of Theorem 2 reads as follows.

Corollary 1. Consider the linear system (4) and the filtered observer [START_REF] Francis | The internal model principle for linear multivariable regulators[END_REF], and suppose that the following matrix

A cl = A -LC LΓ ⊤ ΓC Ψ
is Hurwitz. Then, every solution of (3), ( 6), ( 18) satisfies (8).

The proof is omitted for reasons of space. Note that for linear systems we do not need the boundedness conditions in Assumptions 1 and 2.

Sufficient passivity-based stability conditions

The previous asymptotic analysis is based on the assumption that the filter-observer architecture satisfies the δISS property of Assumption 3. The aim of this section is to develop constructive conditions for filter design such that the bound in Assumption 3 is satisfied.

Note that it is not easy to study the interconnection (10), ( 12) and standard small-gain arguments would in general fail to work, because the H ∞ gain of the filter [START_REF] Astolfi | Stubborn and dead-zone redesign for nonlinear observers and filters[END_REF] can not be tuned, and it is equal to 1. For instance, in the context of linear systems the small-gain condition would require the matrix A in (4) to be Hurwitz1 . As a consequence, we rely in this section on passivity arguments. To do so, it is convenient to include an additional set of assumptions. It should be remarked that the next assumptions are sufficient to obtain a constructive methodology, but are not necessary for the filter-observer stability.

Sufficient conditions

In the first assumption we provide sufficient condition for the observer [START_REF] Marino | Adaptive stabilization of linear systems with outputs affected by unknown sinusoidal disturbances[END_REF] to be incremental passivity from z to ŷ as presented in Definition 2. Assumption 4. There exists a pair of symmetric positive definite matrices P, P ∈ R n×n such that for some q, ℓ > 0, system (3) satisfies

P ∂f ∂x (x, u) + ∂f ∂x (x, u) ⊤ P ⪯ 0, (19)
P ∂f ∂x (x, u) + ∂f ∂x (x, u) ⊤ P -ℓC ⊤ C ⪯ -q P , (20)
for all x ∈ R nx , and all u ∈ U .

Inequality [START_REF] Pavlov | Incremental passivity and output regulation[END_REF] implies that the plant dynamics (3) is incrementally dissipative according to the notion introduced in [START_REF] Pavlov | Incremental passivity and output regulation[END_REF]. Additionally, ( 19) is a sufficient condition to satisfy the boundedness requirement in Assumption 2, as shown in the next subsection.

Remark 1. In the linear case, a necessary and sufficient condition for [START_REF] Pavlov | Incremental passivity and output regulation[END_REF] is that σ(A) ⊂ C -.

Now consider that the output injection gain

κ is selected as κ(x, y -C x) = ℓP -1 C ⊤ (y -C x). (21)
Then, combining [START_REF] Kellett | A compendium of comparison function results[END_REF] with inequality [START_REF] Pavlov | Incremental passivity and output regulation[END_REF], we can show that the dynamics in [START_REF] Marino | Adaptive stabilization of linear systems with outputs affected by unknown sinusoidal disturbances[END_REF] are incrementally passive (see Definition 2) from the input z to the output y with the constant metric P . In order to ease the presentation, we assume ℓ = 1 in the rest of the section.

In parallel, the inequality ( 20) is a differential detectability condition (with constant metric) as developed for instance in [START_REF] Tsinias | Observer design for nonlinear systems[END_REF][START_REF] Praly | On observers with state independent error Lyapunov function[END_REF], see also [START_REF] Bernard | Observer design for continuous-time dynamical systems[END_REF]Section 4]. Inequality [START_REF] Angeli | A Lyapunov approach to incremental stability properties[END_REF] combined with a LaSalle-type argument (or [START_REF] Praly | Observers to the aid of "strictification" of Lyapunov functions[END_REF]) is sufficient to show that observer [START_REF] Beale | Robust control system design with proportional integral observer[END_REF], with the output injection gain κ selected as in [START_REF] Kellett | A compendium of comparison function results[END_REF], is a convergent observer.

We stress that Assumption 4 provides a sufficient condition to develop the next stability analysis, but it is not necessary as shown by a counterexample in Section 5.

Assumption 5. The extended system

ξ = F (ξ, u) = f (x, u) Φη , ξ = x η (22)
ζ = Hξ = C Γ ⊤ ξ,
with state ξ and output ζ, is differentially detectable, namely there exists a symmetric positive definite matrix

Q = Q ⊤ > 0 and µ, q > 0 such that Q ∂F ∂ξ (ξ, u) + ∂F ∂ξ (ξ, u) ⊤ Q -2µH ⊤ H ⪯ -qI (23)
for all ξ ∈ R nx+nη and all u ∈ U .

Assumption 5 imposes a minimal observability condition on the extended dynamics [START_REF] Farza | Improved high gain observer design for a class of disturbed nonlinear systems[END_REF]. Clearly, such a condition is necessary to distinguish between the state trajectories and the output disturbances; the objective of the system [START_REF] Astolfi | Stubborn and dead-zone redesign for nonlinear observers and filters[END_REF] is to filter the effect of the disturbance d without losing useful information from y needed to recover the estimate of the full state x. We stress that the knowledge of the matrix Q is needed only for analysis purposes but not for design.

To better understand the implications of Assumption 5, it is convenient to particularize it in the linear case scenario, i.e., for systems of the form (4). Indeed, in the linear case, Assumption 5 implies that the unstable modes of A and Φ are observable from y. Precisely, based on the results in [START_REF] Davison | New results on the controllability and observability of general composite systems[END_REF] and standard Hautus (PBH) test, and the fact that the pair (Φ, Γ ⊤ ) is observable, it can be deduced that the extended system [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF]

system is detectable if rank   A -sI 0 0 Φ -sI C Γ ⊤   = n x + n η , ∀s ∈ C + . ( 24
)
Note that equation ( 24) reduces to the following disjoint spectrum condition σ(A) ∩ σ(Φ) = ∅, which is similar to the non-resonance condition of other internal model based designs [START_REF] Francis | The internal model principle for linear multivariable regulators[END_REF]. As a last remark, it is worth noticing that different notions of non-resonance condition (although very similar from a conceptual point of view and completely equivalent in the linear case) have been proposed in [START_REF] Wang | Pre-processing nonlinear output regulation with non-vanishing measurements[END_REF]Assumption 5], [START_REF] Marconi | Non-resonance conditions for uniform observability in the problem of nonlinear output regulation[END_REF]Proposition 3] or [START_REF] Kawano | PBH tests for nonlinear systems[END_REF].

Checking Assumptions 4 and 5

Notice that Assumptions 4 and 5 require solving a set of matrix inequalities that depend on the state x and input u. That is, they require solving an infinite set of LMIs. Nonetheless, we can obtain a finite set of LMIs through convex relaxation, see e.g. [START_REF] Boyd | Linear matrix inequalities in system and control theory[END_REF]. Specifically, let A := {A 1 , . . . , A N } be a family of matrices in R nx such that ∂f ∂x (x, u) ∈ ConvexHull(A) for all (x, u). Then, at each (x, u) we have that

∂f ∂x (x, u) = N i=1 ϱ i (x, u)A i , for some ϱ i (x, u) such that N i=1 ϱ i (x, u) = 1 for all (x, u). With this in mind, inequality (19) reads as N i=1 ϱ i (x, u) P A i + A ⊤ i P ⪯ 0.
Consequently, any symmetric positive definite matrix P that is a solution of

P A i + A ⊤ i P ⪯ 0, i = 1, . . . , N
is also a solution of [START_REF] Pavlov | Incremental passivity and output regulation[END_REF]. Then, [START_REF] Angeli | A Lyapunov approach to incremental stability properties[END_REF] and Assumption 5 can be analyzed similarly.

Stability Analysis

The main result of this section is formalized as follows.

Theorem 2. Let Assumptions 1, 4 and 5 hold, and the observer feedback term being [START_REF] Kellett | A compendium of comparison function results[END_REF]. Then, Assumptions 2 and 3 are satisfied.

Proof. First, recall the following identity

g(1) -g(0) = 1 0 ∂g ∂s (s)
ds valid for any C 1 function g : R → R. Hence, by denoting g(s) := f (x + (s -1)x, u), where x := x -x, we get

f (x, u)-f (x-x, u) = 1 0 ∂f ∂x (x+(s-1)x, u)ds x (25)
for all x, x ∈ R nx and u ∈ R nu . Similar derivations will be used all throughout the proof. We start by showing that Assumption 2 holds. To this end, consider the Lyapunov function V = x ⊤ P x with P given by [START_REF] Pavlov | Incremental passivity and output regulation[END_REF]. Differentiating along solutions and using [START_REF] Zemouche | Circle criterion-based H∞ observer design for lipschitz and monotonic nonlinear systemsenhanced lmi conditions and constructive discussions[END_REF] we obtain

V = 2x ⊤ P f (x, u) = 2x ⊤ P 1 0 ∂f ∂x (sx, u)ds x ≤ x ⊤ 1 0 P ∂f ∂x (sx, u) + ∂f ∂x (sx, u) ⊤ P ds x ≤ 0 ,
where the last inequality comes from [START_REF] Pavlov | Incremental passivity and output regulation[END_REF]. We conclude that the set X 0 = {x ⊤ P x ≤ c}, c = sup x∈X0 |x ⊤ P x| is forward invariant. Next, consider any two solution pairs (x, η) and (x ′ , η ′ ) to system [START_REF] Pyrkin | Adaptive controller for linear system with input delay and output disturbance[END_REF] with the output injection gain [START_REF] Kellett | A compendium of comparison function results[END_REF], denote

e := x η = x -x′ η -η ′ , δ := d -d ′ ,
and define ξ = x η . The e-dynamics evolves according to

ẋ = ψ(x, x, u) -LC x -LΓ ⊤ η + Lδ η = (Φ -ΓΓ ⊤ )η -ΓC x + Γδ (26)
with

L = P -1 C ⊤ and ψ(x, x, u) := f (x, u) -f (x -x, u), or, in compact notation, ė = Θ( ξ, e, u) -KHe + Kδ (27)
with

H := C Γ ⊤ , K := L Γ , Θ( ξ, e, u) := F ( ξ, u) -F ( ξ -e, u), F (ξ, u) := f (x, u) Φη .
Consider the Lyapunov function

V = 1 2 x⊤ P x + 1 2 η⊤ η. (28)
The derivative of the first term V x = 1 2 x⊤ P x along the solutions to [START_REF] Moreno | Observer design for nonlinear systems: A dissipative approach[END_REF] gives

Vx = x⊤ P ψ(x, x, u) -LC x -LΓ ⊤ η + Lδ .
Using Assumption 4, we have the following property

x⊤ P ψ(x, x, u) ≤ 0. (29)
Indeed,

2x ⊤ P [f (x, u) -f (x -x, u)] = 2x ⊤ P 1 0 ∂f ∂x (x + (s -1)x, u)ds x = x⊤ 1 0 P ∂f ∂x ((x ′ , s), u) + ∂f ⊤ ∂x ((x ′ , s), u)P ds x,
where in the second step we used the identity [START_REF] Zemouche | Circle criterion-based H∞ observer design for lipschitz and monotonic nonlinear systemsenhanced lmi conditions and constructive discussions[END_REF] and in the third step we used the compact notation (x ′ , s) for x′ = x+(s-1)x. Thus, (29) follows from [START_REF] Pavlov | Incremental passivity and output regulation[END_REF]. Then, since in view of [START_REF] Kellett | A compendium of comparison function results[END_REF],

P L = C ⊤ , the derivative of V x satisfies Vx ≤ -x ⊤ C ⊤ (C x + Γ ⊤ η -δ). (30)
Next, we compute the derivative of the second term

V η = 1 2
η⊤ η. Recalling the skew-symmetric properties of the matrix Φ, we have

Vη = η⊤ [(Φη -ΓΓ ⊤ )η -ΓC x + Γδ] = -η ⊤ Γ(Γ ⊤ η + C x -δ). (31)
Combining inequalities [START_REF] Praly | On observers with state independent error Lyapunov function[END_REF] and ( 31), we finally obtain

V = Vx + Vη ≤ -(η ⊤ Γ + x⊤ C ⊤ )(Γ ⊤ η + C x -δ) ≤ -|He| 2 + |He| |δ| . (32)
The next steps consists on "strictifying" the Lyapunov function V in order to derive the incremental input-tostate stability condition. To derive a strict Lyapunov function, we follow the observer-based approach proposed in [START_REF] Praly | Observers to the aid of "strictification" of Lyapunov functions[END_REF]. Specifically, consider the Lyapunov function U (e) = e ⊤ Qe together with Assumption 5 and (23). Then, the derivative of U (e) along the solutions of [START_REF] Marino | Adaptive notch filters are local adaptive observers[END_REF] gives

U = 2e ⊤ Q(Θ( ξ, e, u) -KHe + Kδ) = 2e ⊤ Q(Θ( ξ, e, u) -µQ -1 He) + 2e ⊤ Q(µQ -1 He -KHe + Kδ).
Recalling the definition of Θ, the first term becomes

2e ⊤ Q F ( ξ, u) -F ( ξ -e, u) -µQ -1 He = 2e ⊤ 1 0 Q ∂F ∂ξ ( ξ + (s -1)e, u)dse -2µe ⊤ H ⊤ He = e ⊤ 1 0 Q ∂F ∂ξ (( ξ′ , s), u) + ∂F ∂ξ (( ξ′ , s), u) ⊤ Q -2µH ⊤ H ds e,
where in the second step we used similar derivations as those in [START_REF] Zemouche | Circle criterion-based H∞ observer design for lipschitz and monotonic nonlinear systemsenhanced lmi conditions and constructive discussions[END_REF], and in the third step we used the compact notation ( ξ′ , s) for ξ′ = ξ + (s -1)e. As a consequence, using [START_REF] Wang | High-gain observers with limited gain power for systems with observability canonical form[END_REF] we obtain 2e ⊤ Q(Θ( ξ, e, u) -µQ -1 He) ≤ -q|e| 2 for all ( ξ, e). Moreover, using the Young's inequality, the derivative of U is estimated as

U ≤ -q|e| 2 + 2|He|(µ + |QK|)|e| + 2|e||QK| |δ|. (33)
Finally, let ν > 0 be a positive parameter to be fixed and define the Lyapunov function W = νV + U . Combining [START_REF] Davison | New results on the controllability and observability of general composite systems[END_REF] with [START_REF] Marconi | Non-resonance conditions for uniform observability in the problem of nonlinear output regulation[END_REF], the derivative of W is computed as

Ẇ ≤ -|e| |He| q -µ -|QK| ⋆ ν |e| |He| + |e| ν|H| + 2|QK| |δ|
Hence, selecting ν such that qν > (µ + |QK|) 2 , the previous shows the existence of ε, ρ v > 0 such that

Ẇ ≤ -ε|e| 2 + ρ v |δ| 2 .
Hence, W is a (uniform in the trajectories of the plant x) global exponential incremental ISS Lyapunov function [START_REF] Angeli | A Lyapunov approach to incremental stability properties[END_REF], concluding the proof of the theorem.

Numerical Simulations

5.1. Linear case Consider a linear system of the form (4) with

A =   0 1.5 0 -1.5 0 0 1 0 -1   , C = 1 0 0 0 1 1 , B =   0 0 0  
where the measured output is corrupted with an additive disturbance generated by the exosystem [START_REF] Shafai | LTR design of discrete-time proportional-integral observers[END_REF] with ω 1 = 0, ω 2 = 1 and ω 3 = 2π. Assume that the states of the system are estimated through a linear observer of the form [START_REF] Cecilia | A masking protocol for private communication and attack detection in nonlinear observers[END_REF] with the gain L = 0.994 0.093 0 0.704 0.094 1.534 . In the simulations we considered the following set of initial conditions: x(0) = (1, 1, 1), x(0) = 0 and w(0) = (1, 0.2, 0, 0.2, 0) and v is taken as a bounded realization of white noise of variance 0.01 in both outputs.

Naturally, the presence of the output disturbance prevents the convergence of the state-estimation error x to zero. This fact can be seen in Fig. 1. The objective is to robustify the observer in front of the presented output disturbance. Note that the pair (A, C) is detectable, i.e., [START_REF] Angeli | A Lyapunov approach to incremental stability properties[END_REF] is satisfied. Moreover, the matrix A is negative semidefinite. As a consequence, there exists a matrix P such that [START_REF] Pavlov | Incremental passivity and output regulation[END_REF] holds. Moreover, the unstable modes of A are observable and are located at λ = ±1.5j, which are disjointed from the unstable modes of Φ, that are located at {±6.28j, ±1j, 0}, thus, Assumption 5 is also satisfied.

Therefore, according to Theorem 1 and Theorem 2, if the output-estimation error ỹ is filtered through an internal model filter [START_REF] Astolfi | Stubborn and dead-zone redesign for nonlinear observers and filters[END_REF], the state-estimation error converges asymptotically to zero. Fig. 1 presents the stateestimation error of an observer filtered with the internal model filter [START_REF] Astolfi | Stubborn and dead-zone redesign for nonlinear observers and filters[END_REF]. The filter completely rejects the modeled part of the output disturbance.

Nonlinear case

Consider the ball-and-beam system studied in [START_REF] Wu | Robust trajectory tracking for incrementally passive nonlinear systems[END_REF] ẋ

= f (x, u) :=        x 2 x 1 x 2 4 -g sin x 3 J b /(M R 2 ) + 1 x 4 - x 1 x 2 x 4 + gx 2 cos x 3 x 2 1 + J /M + J b /M        +     0 0 0 1     u y = Cx + d := 1 0 0 0 0 0 1 0 x + d (34)
where x = (x 1 , . . . , x 4 ) ∈ R 4 is the state vector, y = (y 1 , y 2 ) ∈ R 2 is the measured output, d is an output disturbance, and the input u ∈ R is designed as The system also satisfies [START_REF] Angeli | A Lyapunov approach to incremental stability properties[END_REF] with the same matrix P for all ℓ > 0. Therefore, an observer of the form ( 5) can be implemented with a feedback term as in [START_REF] Kellett | A compendium of comparison function results[END_REF]. Now, assume that the output additive disturbance d is generated by the exosystem [START_REF] Shafai | LTR design of discrete-time proportional-integral observers[END_REF] with ω k = 20 √ k for k = 1, 2, 3, initial condition w(0) = (5, 0, 5, 0, 5, 0), and v is a bounded realization of a white noise of variance 0.0005 and 0.00001 in the outputs y 1 and y 2 , respectively. Notice that the ball and beam system and the considered disturbance model evolve in two different time-scales. Therefore, the detectability condition in Assumption 5 is satisfied.

u = 2x 1 x 2 x 4 + gx 1 cos x 3 x 2 1 + J /M + J b /M + 24(J b /R 2 + M )x 1 M g + 50(J b /R 2 + M )x 2 M g -35x 3 -10x 4 + sin t.
Due to the presence of the output disturbance d the observer does not converge to the true value. This fact can be seen in Figure 2. The objective is to redesign the proposed observer to obtain the bound in (8). Since the system satisfies Assumptions 4-5, according to Theorems 1-2 adding an internal model filter to the observer as in [START_REF] Marino | Adaptive stabilization of linear systems with outputs affected by unknown sinusoidal disturbances[END_REF] obtains the desired bound. Indeed, the evolution of the estimation error of the filter-observer architecture in ( 12) is depicted in Figure 2. As it can be seen, the estimation error converges to a bound of lower magnitude. That is, the filter rejects the periodic disturbance and ensures the bound in [START_REF] Astolfi | On the use of low-pass filters in high-gain observers[END_REF], which validates the result.

Non-passive example

The objective of this example is to show that the passivity condition in Assumption 4 is only a sufficient (but not necessary) condition to ensure the δISS bound in Assumption 3. Precisely, consider the system in [START_REF] Fan | Observer design for systems with multivariable monotone nonlinearities[END_REF] ẋ = Ax + Gϑ(Hx), y = Cx + d where x = (x 1 , x 2 , x 3 ) ∈ R 3 , y ∈ R, A =

The observer gains are L = , K = [ 1.92 0.61 ]. The objective is to robustify the observer in front of a disturbance of the form (6), with ω 1 = 10π and ω 2 = 20π, w 1 (0) = w 3 (0) = 1 and v = 0. In the presence of such disturbance, the observer estimation not only converges to an oscillatory trajectory, but presents a significant bias with respect to the true value, see Fig. 3. This behaviour is a consequence of introducing the noise through the nonlinear feedback term [START_REF] Boyd | Linear matrix inequalities in system and control theory[END_REF]. Moreover, even though the approach proposed in [START_REF] Fan | Observer design for systems with multivariable monotone nonlinearities[END_REF] can be implemented for the original system, it leads to infeasible linear matrix inequalities for the extended system of the form [START_REF] Farza | Improved high gain observer design for a class of disturbed nonlinear systems[END_REF]. Notice that the system does not satisfy the incremental dissipativity assumption in Assumption 4, as some states drift to infinity. Nonetheless, in the considered numerical simulation, the filter-observer architecture satisfies the stability condition in Assumption 3. Thus, according to Theorem 1, the internal model filter exactly rejects the modeled part of the disturbance, as depicted in Fig. 3.

Conclusion

This work proposes an observer redesign method to reduce the effect of output disturbances of the form [START_REF] Shafai | LTR design of discrete-time proportional-integral observers[END_REF]. The main idea is to connect in cascade the already designed observer with a filter that includes an internal model of the modeled part of the disturbance. It has been established that, if the filter-observer architecture satisfies a particular stability condition, then the state estimation error is ISS with respect to the unstructured term of the disturbance and the structured terms are exactly eliminated. Moreover, this work shows that, for any incremental dissipative system that satisfies a particular differential detectability property, it is possible to design a filter that guarantees stability of the overall observer. Numerical simulations confirm that the incremental dissipativity property is a sufficient but not necessary condition for stability.

Future works will focus on relaxing this dissipativity condition and using Riemannian metrics (see, e.g., [START_REF] Sanfelice | Convergence of Nonlinear Observers on R n With a Riemannian Metric (Part I)[END_REF] and [START_REF] Bernard | Observer design for continuous-time dynamical systems[END_REF]Section 4]). Additionally, future works will explore the practical application of the proposal in multi-agent distributed problems [START_REF] Chopra | Passivity-based control of multi-agent systems[END_REF] and masking protocols [START_REF] Cecilia | A masking protocol for private communication and attack detection in nonlinear observers[END_REF]

Figure 1 :

 1 Figure 1: Evolution of the state-estimation error without (blue) and with (red) internal model filter.

1 ,

 1 H = [ 0 1 0 0 0 1 ] and ϑ(x) =

Figure 2 :

 2 Figure 2: Evolution of the state-estimation error without (blue) and with (red) internal model filter.

Figure 3 : 2 .

 32 Figure 3: Evolution of the state-estimation error without (blue, red, yellow) and with (purple, green, ciano) internal model filter.

Note that, in this case, the overall goal could be simply solved by taking L = 0 as output injection gain so that to obtain an exponentially stable observer not affected by any output disturbance.

⋆ This research was partially supported by the French Grant ANR ALLIGATOR (ANR-22-CE48-0009-01), by the Spanish Ministry of Universities funded by the European Union -NextGen-erationEU (2022UPC-MSC-93823).

This work is part of the Project MAFALDA (PID2021-126001OB-C31) funded by MCIN/ AEI /10.13039/501100011033 and by "ERDF A way of making Europe". This work is part of the project MASHED (TED2021-129927B-I00), funded by MCIN/ AEI/10.13039/501100011033 and by the European Union Next GenerationEU/PRTR.


Pavel V Gapeev
email: p.v.gapeev@lse.ac.uk

Monique Jeanblanc
email: monique.jeanblanc@univ-evry.fr

On the construction of conditional probability densities in the Brownian and compound Poisson filtrations *

Keywords: Mathematics Subject Classification 2010: Primary 60G44, 60J65, 60G40. Secondary 60G35, 60H10, 91G40 Conditional probability density process, Brownian motion, compound Poisson process, Jacod's equivalence hypothesis

, it is possible to construct an associated random time on some extended probability space admitting such a given supermartingale as conditional survival process and we shall check that this construction (with a particular choice of supermartingale) implies that Jacod's equivalence hypothesis, that is, the existence of a family of strictly positive conditional probability densities for the random times with respect to the reference filtration, is satisfied. We use the components of the multiplicative decomposition of the constructed supermartingales to provide explicit expressions for the conditional probability densities of the random times on the Brownian and compound Poisson filtrations.

Introduction

In the models of quickest change-point (disorder) detection, one usually starts with the probability space enhanced with a random time and another source of randomness such as a Brownian motion or a compound Poisson process. The quickest detection problems seek to determine stopping times at which the alarms should be sounded to indicate changes in the probabilistic characteristics of continuously observable stochastic processes. These detection times of alarms are sought to be as close as possible to the unknown and unobservable random times of change in either the drift rate of the observable Brownian motion or the intensity and the jump distribution of the observable compound Poisson process. In order to solve the problems, the appropriate stochastic differential equations are derived for the (supermartingale) survival conditional probability processes of the random times or the equivalent (submartingale) posterior probability process on the given Brownian or compound Poisson reference filtrations. These processes play the role of sufficient statistics in the appropriate quickest detection problems with rewards containing a linear combination of the false alarm probabilities and the expected linear delay penalties. More precisely, the optimal detection times in these problems represent the first hitting times by either the survival conditional probability or the posterior probability processes of certain boundaries which are found as solutions of the associated free-boundary problems for ordinary or partial (integro-)differential (see, e.g. Shiryaev [START_REF] Shiryaev | Stochastic Disorder Problems[END_REF] and the references therein).

In this paper, we proceed in a more natural opposite direction and present a construction of supermartingales valued in the interval [0, 1] as solutions of certain stochastic differential equations on the given Brownian or compound Poisson reference filtrations. Then, we apply the results of Jeanblanc and Song [START_REF] Jeanblanc | Default times with given survival probability and their Fmartingale decomposition formula[END_REF] to construct the associated random times on the appropriate extended probability spaces. These properties lead to the satisfaction of Jacod's equivalence hypothesis, that is, to the existence of strictly positive conditional densities for the random times with respect to the reference filtrations. Such assumptions are usually satisfied in the classical models of credit risk theory in which the random default times have given strictly positive conditional densities with respect to the reference filtrations reflecting the information observable from the associated models of financial markets (see, e.g. Aksamit and Jeanblanc [START_REF] Aksamit | Enlargement of Filtration with Finance in View[END_REF] for further discussions on Jacod's hypothesis and credit risk models). We provide a multiplicative decomposition for the constructed supermartingale and use the resulting components to derive the families of the conditional probability densities of the random times on the Brownian and compound Poisson filtrations.

The paper is organised as follows. We present a general framework for the model in Section 2. Then, we construct the appropriate supermartingales in the case of a reference filtration generated by a Brownian in Section 3, and in the case of a reference filtration generated by a compound Poisson process in Section 4.

The framework

We give a model for constructions of supermartingales valued in [0, 1] with respect to a reference filtration.

For this purpose, we work on a filtered probability space (Ω, G, F, P), where F is a given (reference) filtration. We call an F-conditional density any family of strictly positive F-martingales

(p(u) = (p t (u)) t≥0 ; ∀u ≥ 0), parameterised by u ∈ [0, ∞), such that p is O(F) ⊗ B([0, ∞))- measurable and ∞ 0 p t (u) du = 1, ∀t ≥ 0 . (2.1)
Actually, there are very few explicit examples of such densities in the literature (see, e.g. [2, Chapter 4], [START_REF] El Karoui | Conditional default probability and density[END_REF] and [START_REF] Jeanblanc | Explicit model of default time with given survival probability[END_REF]). In this paper, we will show how to construct such families (p(u); ∀u ≥ 0) in two cases: when F is a Brownian filtration, and when F is a compound Poisson filtration. In both cases, we start with a nonnegative bounded F-adapted process λ = (λ t ) t≥0 satisfying

t 0 λ s ds < ∞, ∀t ≥ 0 , ∞ 0 λ s ds = ∞, (P-a.s.) , (2.2)
and we construct a supermartingale G = (G t ) t≥0 valued in (0, 1] such that G 0 = 1 and the process (G t exp( t 0 λ s ds)) t≥0 is a (strictly) positive F-local martingale (see, e.g. [START_REF] Gapeev | Constructing random times with given survival process and applications to valuation of credit derivatives[END_REF]). In this case, using the work of [START_REF] Jeanblanc | Default times with given survival probability and their Fmartingale decomposition formula[END_REF], on the extended measurable space (Ω × [0, ∞), G ⊗ B([0, ∞))), one can construct a positive random variable τ (in fact, τ (ω ′ , ω ′′ ) = ω ′′ ) and a probability measure

Q such that Q(τ > t | F t ) = G t , and Q Ft = P Ft , ∀t ≥ 0 .
In particular, if W is an (F, P)-standard Brownian motion, it is also an (F, Q)-standard Brownian motion. We recall that Jacod's equivalence hypothesis holds, if there exists a family of F-conditional densities (p(u); ∀u ≥ 0) such that

Q(τ > u | F t ) = ∞ u p t (v) dv, ∀t ≥ 0, ∀u ≥ 0 .
The family of processes (p(u); ∀u ≥ 0) is then called the conditional density.

Remark 2.1 We refer the reader to the seminal papers of Jacod [START_REF] Jacod | Grossissement initial, hypothèse (H ′ ) et théorème de Girsanov[END_REF], Grorud and Pontier [START_REF] Grorud | Insider trading in a continuous time market model[END_REF], and Amendinger [START_REF] Amendinger | Martingale representation theorems for initially enlarged filtrations[END_REF], and to the book [START_REF] Aksamit | Enlargement of Filtration with Finance in View[END_REF] for more details on Jacod's hypothesis. Note that the knowledge of the conditional density allows to give the decomposition of any martingale in the reference filtration as a semimartingale in the initial (and progressive) enlargement with τ (see, e.g. [START_REF] Aksamit | Enlargement of Filtration with Finance in View[END_REF], [START_REF] Jeanblanc | Martingale representation property in progressively enlarged filtrations[END_REF], [START_REF] Fontana | The strong predictable representation property in initially enlarged filtrations under the density hypothesis[END_REF], [START_REF] Bielecki | Special semimartingales and shrinkage of filtration[END_REF] and [START_REF] Gapeev | Projections of martingales in enlargements of Brownian filtrations under Jacod's equivalence hypothesis[END_REF]- [START_REF] Gapeev | Projections of martingales in enlargements of filtrations under Jacod's equivalence hypothesis for marked point processes[END_REF]).

The case of a Brownian filtration

In this section, we consider the case in which F is the filtration generated by a standard Brownian motion W . We use the notation of the framework described in Section 2.

Supermartingales valued in [0, 1]

Proposition 3.1 Let us consider the stochastic differential equation

dG t = -λ t G t dt -G t (1 -G t ) ρ t dW t , G 0 ∈ (0, 1] , (3.1)
where λ is a nonnegative F-adapted process such that the condition of (2.2) holds and ρ an F-adapted process satisfying

t 0 ρ 2 s ds < ∞, (P-a.s.), ∀t ≥ 0 . (3.2)
Then, the equation in (3.1) admits a (pathwise) unique (continuous

) solution G = (G t ) t≥0 which is valued in [0, 1]. Proof: Let us set T 1 = inf{t > 0 | G t = 1} and T 0 = inf{t > 0 | G t = 0}.
Obviously, the equality G t = 0 holds, ∀t ≥ T 0 , and we also have G t < 1 on {0 < t < T 1 }. Then, applying Itô's formula (see, e.g. [27, Chapter VI, Theorem 1.2]) to the change of variables Φ t = (1 -G t )/G t , ∀t ≥ 0, we obtain that the stochastic differential equation of (3.1) is equivalent to the one

dΦ t = λ t (1 + Φ t ) + Φ 2 t 1 + Φ t ρ 2 t dt + Φ t ρ t dW t , Φ 0 ∈ [0, ∞) . (3.3)
Assuming that the solution Φ to the equation in (3.3) exists, it is non-negative. Moreover, since the coefficients of the stochastic differential equation in (3.3) are Lipschitz continuous on [0, ∞) and of a linear growth, it follows from the result of [24, Chapter IV, Theorem 4.8] that the equation in (3.3) admits a pathwise unique (strong) solution process Φ = (Φ t ) t≥0 , which does not explode at any t ≥ 0. In this case, because the process G starts at some G 0 ∈ (0, 1], we may conclude from the structure of the coefficients of the equation in (3.3) for the process Φ = (1 -G)/G that T 0 = ∞ (P-a.s.). Moreover, by means of the comparison results for pathwise solutions of stochastic differential equations in [25, Theorem 3.2] and [9, Theorem 1], we see that the inequality

Φ 0 exp t 0 ρ s dW s - 1 2 t 0 ρ 2 s ds ≤ Φ t , ∀t ≥ 0 , (3.4)
holds. Hence, due to the assumption in (3.2), we see from (3.4) that the process Φ does not touch 0 after the time 0, when being started at Φ 0 ≥ 0. This fact implies that the process G = 1/(1 + Φ) does not touch 1 after the time 0, so that T 1 = ∞ (P-a.s.). □ Remark 3.2 Note that the equation in (3.1) has the same structure as the appropriate stochastic differential equations for the posterior probability processes Π = (Π t ) t≥0 of the occurrence of the random change-point (disorder) time defined by Π t = 1 -G t , ∀t ≥ 0, in the quickest change-point detection problems for Wiener and more general diffusion processes studied in [28, Chapter IV, Section 4] (see also [29, Chapter IV, Section 4], and [26, Chapter VI, Section 22], as well as [START_REF] Gapeev | The Wiener disorder problem with finite horizon[END_REF]- [START_REF] Gapeev | Bayesian quickest detection problems for some diffusion processes[END_REF]). It is shown in the sources mentioned above that the optimal stopping times of alarms in the quickest detection problems are given by the first times at which the processes Π hit boundaries which are determined as solutions to the associated free-boundary problems for ordinary or partial differential operators. In the case of observable Wiener processes, the optimal hitting boundaries for the processes Π are constant on the allowed infinite observation intervals, but they are time-dependent when the allowed observation time intervals are finite.

In the case of observable more general diffusion processes, the optimal hitting boundaries for the processes Π depend on the running values of the observation processes.

Remark 3.3 This study is easily extended to the case in which F is still a Brownian filtration, but the process G = (G t ) t≥0 satisfies the more general that in (3.1) stochastic differential equation

dG t = -G t dΛ t -G t (1 -G t ) ρ t dW t , G 0 ∈ (0, 1] , (3.5)
where Λ = (Λ t ) t≥0 is a positive continuous increasing process started at Λ 0 = 0. This is particularly the case when τ is honest, since, in that case, the dynamics of the process

G
= M Y -A Y ,
where M Y is a continuous (uniformly integrable) martingale, so that the process G := Y ∧ 1 is a supermartingale valued in (0, 1]. Then, an application of Tanaka's formula for semi-martingales [22, Formula 4.1.15], by virtue of x ∧ y = x -(x -y) + , leads to

dG t = 1 1 {Yt≤1} dY t - 1 2 dL 1 t (Y ), G 0 ∈ (0, 1] .
Note that any positive supermartingale admits a multiplicative decomposition Y t = N t e -Λt , ∀t ≥ 0. Hence, there exists an F-adapted process ϑ = (ϑ t ) t≥0 satisfying the integrability condition t 0 ϑ 2 s ds < ∞, (P-a.s.), ∀t ≥ 0 , and such that

dY t = -Y t dΛ t -Y t ϑ t dW t , Y 0 ∈ (0, ∞) .
Therefore, we see that

dG t = -G t 1 1 {Yt≤1} dΛ t + 1 2 dℓ t -1 1 {Yt≤1} G t (1 -G t ) ρ t dW t , G 0 ∈ (0, 1] ,
where we have

dℓ t = 1 G t dL 1 t (Y ) and ρ t = ϑ t 1 -G t , ∀t ≥ 0 .
continuous and ρ is square integrable.

The multiplicative decomposition for the process G

As we recalled in Section 2 above, the process G = (G t ) t≥0 defined in (3.1) when started at G 0 = 1 admits a multiplicative decomposition G t = e -Λt N t , ∀t ≥ 0, where N = (N t ) t≥0 started at N 0 = 1 is a continuous local martingale and

Λ t = t 0 λ s ds, ∀t ≥ 0 . (3.6)
In this case, the equality N t = G t e Λt holds, ∀t ≥ 0, and, according to Itô's formula, the process N follows the stochastic differential equation

dN t = -e Λt G t (1 -G t ) ρ t dW t = -N t (1 -G t ) ρ t dW t , N 0 = 1 , (3.7)
and the explicit expression

N t = exp - t 0 (1 -G s ) ρ s dW s - 1 2 t 0 (1 -G s ) 2 ρ 2 s ds , ∀t ≥ 0 . (3.8)
For further use, we define the processes L = (L t ) t≥0 and K = (K t ) t≥0 by the explicit expressions

L t = exp - t 0 ρ s dW s - 1 2 t 0 ρ 2 s ds + t 0 G s ρ 2 s ds , ∀t ≥ 0 , (3.9)
and

K t = exp t 0 G s ρ s dW s - 1 2 t 0 G 2 s ρ 2 s ds , ∀t ≥ 0 , (3.10)
which admit the stochastic differential equation

dL t = -L t ρ t (dW t -G t ρ t dt), L 0 = 1 , (3.11)
and

dK t = K t G t ρ t dW t , K 0 = 1 . (3.12)
By virtue of the integration-by-parts formula, it is straightforward to see from (3.8) and (3.9)-(3.10) that the equality K t L t = N t holds, ∀t ≥ 0.

Strictly positive martingales M u decreasing w.r.t. u

From [START_REF] Jeanblanc | Default times with given survival probability and their Fmartingale decomposition formula[END_REF], we deduce that any solution Ψ u = (Ψ u t ) t≥u of the problem

dΨ u t = -Ψ u t (e -Λt /(1 -G t ))dN t , ∀u ≤ t < ∞ , Ψ u u = x , (3.13)
for any x ∈ [0, 1] and u ≥ 0 fixed, is a martingale, increasing with respect to u ∈ [0, ∞) and is valued in [0, 1]. Here, the components in the equation of (3.13) are given by (3.6) and (3.7).

For x = 1-G u fixed, one can construct on an extended probability space (see above), a random time τ and a probability measure Q such that Q and P coincide on the filtration F = (F t ) t≥0 and Q(τ > u | F t ) = Ψ u t , ∀t ≥ 0, ∀u ≥ 0 fixed (see [START_REF] Jeanblanc | Default times with given survival probability and their Fmartingale decomposition formula[END_REF]Section 3] and [START_REF] Li | Random times and multiplicative systems[END_REF]). Note that

dΨ u t = Ψ u t G t ρ t dW t , ∀t ≥ u , (3.14)
so that, for any u ≥ 0 fixed, we have

Ψ u t = Ψ u u exp t u G s ρ s dW s - 1 2 t u G 2 s ρ 2 s dW s , ∀t ≥ u , (3.15)
or

Ψ u t = Ψ u u K t K u = K t 1 -G u K u , ∀t ≥ u . (3.16)
Then, by integration-by-parts formula, after some easy computations, we get

d 1 -G u K u = λ u K u G u du , (3.17)
and

Ψ u t = K t u 0 λ s K s G s ds, ∀t ≥ u . (3.18)
It follows that Ψ u is differentiable w.r.t. u and, for any t ≥ 0 fixed, we have

d du Ψ u t = K t λ u K u G u = K t λ u K u e -Λu K u L u = K t λ u e -Λu L u , ∀t ≥ u . (3.19)
Setting M u t = 1 -Ψ u t , ∀t ≥ u, ∀u ≥ 0 fixed, we find

dM u t = (1 -M u t )(e -Λt /(1 -G t ))dN t , ∀u ≤ t < ∞ M u u = G u , (3.20)
where the process G = (G t ) t≥0 is given by (3.5) and the process N = (N t ) t≥0 admits the stochastic differential of (3.7). Then, we can show that the following assertion holds.

Proposition 3.4 For any u ≥ 0 fixed, the following expression holds

M u t = N t e -Λt + K t t u N s K s λ s e -Λs ds , ∀t ≥ u , (3.21)
where the components are given by the formulas in (3.6) and (3.8)-(3.10).

Proof: For any u ≥ 0 fixed, after some algebraic computations, we get that the expressions

dM u t = G t (1 -G t ) ρ t dW t -(M u t -N t e -Λt ) G t ρ t dW t = -G t (-1 + G t + M u t -G t ) ρ t dW t = (1 -M u t ) e -Λt 1 -G t dN t , ∀t ≥ u , (3.22)
hold.

□

For any 0 ≤ t < u, in order to preserve the martingale property of M u , we define

M u t = E[M u u | F t ] = E[G u | F t ] = E[N u e -Λu | F t ].
Furthermore, if λ is deterministic and N a true martingale, we have M u t = N t e -Λu , ∀t < u .

(3.23)

The set of conditional probability density processes p(u)

Let us note that, by virtue of (3.19), the family of the conditional probability density processes (p(u) = (p t (u)) t≥0 ; ∀u ≥ 0) with respect to Lebesgue measure is defined by

p t (u) = K t λ u e -Λu L u , ∀t ≥ u . (3.24)
We also deduce that

1 -M u t = Ψ u t = K t u 0 λ s e -Λs L s ds = u 0 p t (s) ds, ∀t ≥ u , (3.25)
and, for any u ≥ 0 fixed, using the martingale property of p(u), we get

p t (u) = E[K u λ u e -Λu L u | F t ] = E[λ u G u | F t ], ∀t ≤ u . (3.26)
Finally, we check that the property in (2.1) holds. For this, we note that the equalities

∞ 0 p t (s) ds = t 0 K t λ s e -Λs L s ds + ∞ t E[K s λ s e -Λs L s | F t ] ds = Ψ t t + ∞ t E[N s λ s e -Λs | F t ] ds = Ψ t t + E ∞ t λ s G s ds F t , ∀t ≥ 0 , (3.27)
hold and, from the stochastic differential equation satisfied by G in (3.1), we get

∞ t λ s G s ds = -G t - ∞ t G s (1 -G s ) ρ s dW s , ∀t ≥ 0 . (3.28)
If λ is a deterministic function, the process p(u) admits the stochastic differential

d t p t (u) = λ u e -Λu 1 1 {u<t} G t K t L u -1 1 {u≥t} (1 -G t ) K t L t ρ t dW t = p t (u) 1 1 {u<t} G t -1 1 {u≥t} (1 -G t ) ρ t dW t , p 0 (u) = 1 , (3.29)
so that the representation

p t (u) = λ u e -Λu exp t 0 φ s (u) dW s - 1 2 t 0 φ 2 s (u) ds , ∀t ≥ 0 , ∀u ≥ 0 , (3.30)
holds with

φ t (u) = 1 1 {u<t} G t -1 1 {u≥t} (1 -G t ) ρ t = 1 1 {u<t} -(1 -G t ) ρ t , ∀t ≥ 0 , ∀u ≥ 0 . (3.31)

The case of a compound Poisson filtration

In this section, we consider the case in which F is the filtration generated by a compound Poisson process. We use the notation of the framework described in Sections 2 and 3.

The conditional probability process

Suppose that there exists a compound Poisson process X = (X t ) t≥0 (or a pure jump Lévy process) with a finite intensity Lévy measure ν(dx) which is a positive σ -finite measure on B(R) satisfying the conditions

ν({0}) = 0, x 2 ∧ 1 ν(dx) < ∞ and |x| ν(dx) < ∞ . (4.1)
Assume that the process X generates the reference filtration F = (F t ) t≥0 which is complete under the probability measure P. Note that, in the compound Poisson case with ν(R) < ∞, the process X admits the representation

X t = N ′ t i=1 Ξ i , where N ′ = (N ′ t )
t≥0 is a Poisson process of intensity κ > 0 and (Ξ i ) i∈N is a sequence of independent identically distributed random variables with the distribution ν(dx)/κ , where N ′ and (Ξ i ) i∈N are independent under P. We denote by µ the jump measure of the process X defined by

µ (0, t] × A = 0<s≤t 1 1 {∆Xs∈A} , ∀t ≥ 0 , (4.2)
for any Borel set A ∈ B(R), where we set ∆X t = X t -X t-, ∀t ≥ 0 (see, e.g. [18, Chapter II, Section 4]).

Proposition 4.1 Let us consider the stochastic differential equation

dG t = -λ t G t-dt - R G t-(1 -G t-)(Υ(x) -1) 1 + (1 -G t-)(Υ(x) -1) µ(dt, dx) -dt ν(dx) , G 0 ∈ (0, 1] , (4.3)
where λ is a nonnegative F-adapted process such that the conditions of (2.2) hold, and Υ(x) is a continuous function such that the inequality Υ(x) > 1 holds, ∀x ∈ R, as well as the conditions Proof: Let us define

T 1 = inf{t > 0 | G t ≥ 1} and T 0 = inf{t > 0 | G t ≤ 0}
with the fact, G being a supermartingale the equality G t ≤ 0 holds, ∀t ≥ T 0 , as well as G t < 1 on {0 < t < T 1 }.

It is shown by means of Itô's formula (see, e.g. [START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF]Chapter I,Theorem 4.57]) that the process Φ = (Φ t ) t≥0 defined by Φ t = (1 -G t )/G t , ∀t ≥ 0, admits the stochastic differential

dΦ t = λ t (1 + Φ t-) dt (4.5) + Φ t- R Υ(x) -1 µ(dt, dx) - 1 + Φ t- 1 + Φ t-Υ(x) dt ν(dx) , Φ 0 ∈ [0, ∞).
Due to the condition Υ > 1, the solution Φ to the equation in (4.5) is non-negative. Since the coefficients of this stochastic differential equation in (4.5) are Lipschitz continuous on [0, ∞) and of a linear growth, it follows from the result of [18, Chapter III, Theorem 2.32] that the equation in (4.5) admits a pathwise unique (strong) solution process Φ = (Φ t ) t≥0 which does not explode at any t ≥ 0. In this case, because the process G starts at some G 0 ∈ (0, 1], we may conclude from the structure of the coefficients of the equation in (4.5) for the process Φ = (1 -G)/G that T 0 = ∞ (P-a.s.).

Let us now denote by (S n ) n∈N the sequence of jumps of the Poisson process N ′ and put S 0 = 0. Then, we see that, between any two jumps of N ′ , the process Φ follows the equation

dΦ t dt = λ t (1 + Φ t-) - R Φ t-(1 + Φ t-)(Υ(x) -1) 1 + Φ t-Υ(x) ν(dx), ∀t ∈ (S n-1 , S n ] , (4.6)
such that t ∈ [0, T 0 ∧ T 1 ], while, at any jump times of N ′ , we have

Φ Sn = Υ(∆X Sn ) -1 Φ S n-, ∀n ∈ N . (4.7)
Hence, due to the assumptions that Υ(x) > 1 holds, ∀x ∈ R, as well as the conditions in (4.4) are satisfied, we see from (4.6) and (4.7) that the process Φ does not touch 0 after the time 0, when being started at Φ 0 ≥ 0, so that T 1 = ∞ (P-a.s.). □ Remark 4.2 Note that the equation in (4.3) has the same structure as the appropriate stochastic differential equations for the posterior probability process Π = (Π t ) t≥0 of the occurrence of the random change-point (disorder) time defined by Π t = 1 -G t , ∀t ≥ 0, in the quickest change-point detection problems for a compound Poisson process studied in [START_REF] Gapeev | The disorder problem for compound Poisson processes with exponential jumps[END_REF], [START_REF] Dayanik | Compound Poisson disorder problem[END_REF], and [START_REF] Dayanik | Compound Poisson disorder problems with nonlinear detection delay penalty cost functions[END_REF]. It is shown in the sources mentioned above that the optimal stopping times of alarms in the quickest detection problem are given by the first times at which the processes Π hit boundaries which are determined as solutions to the associated free-boundary problems for integro-differential operators. In the case of observable compound Poisson processes, the optimal hitting boundaries for the processes Π are constant on the allowed infinite observation time intervals.

The multiplicative decomposition for the process G

In this setting, the process G = (G t ) t≥0 defined in (4.3) when started at G 0 = 1 also admits a multiplicative decomposition G t = e -Λt N t , ∀t ≥ 0, where N = (N t ) t≥0 started at N 0 = 1 is a piecewise-continuous local martingale and Λ = (Λ t ) t≥0 , given by (3.6) is continuous, hence it is predictable. In this case, the equality N t = G t e Λt holds, ∀t ≥ 0, and, according to Itô's formula, similar to the formulas in (3.7)-(3.8) above, the process N admits the stochastic differentials

dN t = -e Λt G t-(1 -G t-) R Υ(x) -1 1 + (1 -G t-)(Υ(x) -1)
µ(dt, dx) -dt ν(dx)

= -N t-(1 -G t-) R Υ(x) -1 1 + (1 -G t-)(Υ(x) -1)
µ(dt, dx) -dt ν(dx) , N 0 = 1 , (4.8)

and the explicit expression

N t = exp - t 0 R
ln 1 + (1 -G t-)(Υ(x) -1) µ(ds, dx) (4.9)

+ t 0 R (1 -G s-)(Υ(x) -1) 1 + (1 -G s-)(Υ(x) -1)
ds ν(dx) , ∀t ≥ 0 .

Similar to the formulas in (3.9)-(3.12) above, we define the processes L = (L t ) t≥0 and K = (K t ) t≥0 by the explicit expressions

- t 0 R G s-(Υ(x) -1) 1 + (1 -G s-)(Υ(x) -1)
ds ν(dx) , ∀t ≥ 0 , which admit the stochastic differentials

dL t = L t- R 1 Υ(x) -1 µ(dt, dx) - Υ(x) 1 + (1 -G t-)(Υ(x) -1)
dt ν(dx) , L 0 = 1 , (4.12) and

dK t = K t- R G t-(Υ(x) -1) 1 + (1 -G t-)(Υ(x) -1)
µ(dt, dx) -dt ν(dx) , K 0 = 1 . (4.13)

By virtue of the integration-by-parts formula, it is straightforward to see from (4.9) and (4.10)-(4.11) that the equality K t L t = N t holds, ∀t ≥ 0.

We also note that if λ is a deterministic function, the process p(u) admits the stochastic differential

dp t (u) = λ u e -Λu 1 1 {u<t} G t-K t-L u -1 1 {u≥t} (1 -G t-) K t-L t- × R Υ(x) -1 1 + (1 -G t-)(Υ(x) -1)
µ(dt, dx) -dt ν(dx)

= p t-(u) 1 1 {u<t} G t--1 1 {u≥t} (1 -G t-) (4.19) × R Υ(x) -1 1 + (1 -G t-)(Υ(x) -1)
µ(dt, dx) -dt ν(dx) , p 0 (u) = 1 , so that the representation

ξ t-(u, x) = 1 1 {u<t} G t--1 1 {u≥t} (1 -G t-) Υ(x) -1 1 + (1 -G t-)(Υ(x) -1) = 1 1 {u<t} -(1 -G t-) Υ(x) -1 1 + (1 -G t-)(Υ(x) -1)
, ∀t ≥ 0 , ∀u ≥ 0 . (4.21)

RΥ(x) -1 2 ν

 2 (dx) < ∞ and R |x| Υ(x) ν(dx) < ∞ (4.4) are satisfied. (The first condition in (4.4) is used for the equivalent change of probability measure (see, e.g. [18, Chapter III, Theorem 5.34].) Then, the equation in (4.3) admits a (pathwise) unique (piecewise-continuous) solution G = (G t ) t≥0 which is valued in [0, 1].

1 -

 1 G s-)(Υ(x) -1) ds ν(dx) , ∀t ≥ 0 , 1 -G s-)(Υ(x) -1) µ(ds, dx)(4.11)

  p t (u) = λ u e -Λu (4.20) × exp t 0 R ln 1 + ξ s-(u, x) µ(ds, dx) -t 0 R ξ s-(u, x) ds ν(dx) , ∀t ≥ 0 , ∀u ≥ 0 , holds with

Acknowledgments. This research benefited from the support of the 'Chaire Marchés en Mutation', French Banking Federation and ILB, Labex ANR 11-LABX-0019. This research was supported by a Small Grant from the Suntory and Toyota International Centres for Economics and Related Disciplines (STICERD) at the LSE. The authors thank Kostas Kardaras for a helpful discussion on the proof of Proposition 3.1 as well as Lin Yiqing and Yin Liming for a fruitful exchange on the problem and a different proof of Proposition 3.1.

, Labex ANR 11-LABX-0019

Strictly positive martingales M u decreasing w.r.t. u

In this setting, from [START_REF] Jeanblanc | Default times with given survival probability and their Fmartingale decomposition formula[END_REF], we also deduce that any solution Ψ u = (Ψ u t ) t≥u of the problem

for any x ∈ [0, 1] and u ≥ 0 fixed, is a martingale, increasing with respect to u ∈ [0, ∞) and is valued in [0, 1]. Here, the components in the equation of (3.13) are given by (3.6) and (4.8). For x = 1-G u fixed, one can construct on an extended probability space (see above), a random time τ and a probability measure Q such that Q and P coincide on the filtration [START_REF] Jeanblanc | Default times with given survival probability and their Fmartingale decomposition formula[END_REF]Section 3] and [START_REF] Li | Random times and multiplicative systems[END_REF]). Note that

so that, for any u ≥ 0 fixed, we have

ds ν(dx) , ∀t ≥ u , and thus, the expressions in (3.16)-(3.19) hold with K = (K t ) t≥0 and L = (L t ) t≥0 given by (4.10) and (4.11). Similarly, setting M u t = 1 -Ψ u t , ∀t ≥ u, ∀u ≥ 0 fixed, we find

where the process G = (G t ) t≥0 is given by (4.3) and the process N = (N t ) t≥0 admits the stochastic differential of (4.8). Then, we can show that the following assertion holds, which is proved by means of arguments similar to the ones used in Proposition 3.4 above.

Proposition 4.3 For any u ≥ 0 fixed, the same expression as in (3.21) holds, where the components are given by the formulas in (3.6) and (4.9)-(4.11).

The set of conditional probability density processes p(u)

Let us finally recall the family of the conditional probability density processes (p(u) = (p t (u)) t≥0 ; ∀u ≥ 0) which are also defined as in (3.24). It is shown by means of standard arguments that the expressions of (3.25)-(3.27) hold in this setting, while the expression in (3.28) takes the form

µ(ds, dx)-ds ν(dx) , ∀t ≥ 0 . (4.18)