Tuan Anh Tran

Carine Jauberthie

Françoise Le Gall

Louise Travé-Massuyès

Interval Kalman filter enhanced by positive definite upper bounds

Keywords: Uncertain linear systems, Kalman filter, Interval analysis, Gaussian noise, Covariance matrices

A method based on the interval Kalman filter for discrete uncertain linear systems is presented. The system under consideration is subject to bounded parameter uncertainties not only in the state and observation matrices, but also in the covariance matrices of Gaussian noises. The gain matrix provided by the filter is optimized to give a minimal upper bound on the state estimation error covariance for all admissible uncertainties. The state estimation is then determined by using interval analysis in order to enclose the set of all possible solutions with respect to the classical Kalman filtering structure.

INTRODUCTION

Extending the Kalman filter technique for uncertain linear systems is an active research topic that attracts increasing interest. For instance, a joint Zonotopic and Gaussian Kalman filter has been proposed in [START_REF] Combastel | Merging Kalman filtering and zonotopic state bounding for robust fault detection under noisy environment[END_REF] for discrete-time linear time-varying systems excited by Gaussian noises and bounded disturbances represented by zonotopes. The method minimizes a multi-objective optimization criterion which represents the compromise between the size of the zonotopic part and the covariance of the Gaussian distribution. Regarding to parameter uncertainties, several results have been derived on the design of optimal robust Kalman filters for discrete timevarying systems subject to norm-bounded uncertainties, for instance [START_REF] Zhe | Finite-horizon robust Kalman filtering for uncertain discrete time-varying systems with uncertain-covariance white noises[END_REF], [START_REF] Mohamed | Robust finitehorizon Kalman filtering for uncertain discrete-time systems[END_REF]. The optimality of these methods is based on finding an upper bound on the estimation error covariance for any acceptable modeling uncertainties and then to minimize the proposed upper bound. However, this approach may be conservative when there exist too many uncertainties in the system. Another possibility is to consider the model matrices (including the covariance matrices of Gaussian noises) as intervals containing all admissible values of parameters. A drawback of this approach is the singularity problem which may appear in interval matrix inversion. In [START_REF] Chen | Interval Kalman filtering[END_REF], the classical Kalman filter [START_REF] Kalman | A new approach to linear filtering and prediction problems[END_REF]) has been extended to this type of uncertain systems. The authors propose to bypass the singularity problem by using the upper bound of the interval matrix to be inverted. This method leads to a solution that is not guaranteed, i.e. the solution set may not include all the classical Kalman filter solutions consistent with the bounded uncertainties represented in the system. In [START_REF] Xiong | Fault detection using interval Kalman filtering enhanced by constraint propagation[END_REF], an improved interval Kalman filter (iIKF) has been proposed that solves the interval matrix inversion problem with the set inversion algorithm SIVIA (Set Inversion Via Interval Analysis) and constraint satisfaction problems (CSP) (see [START_REF] Jaulin | Applied Interval Analysis, with Examples in Parameter and State Estimation, Robust Control and Robotics[END_REF]). Nevertheless, this algorithm demands high computational time if there exist large uncertainties affecting the considered system (see [START_REF] Tran | Two stochastic filters and their interval extensions[END_REF]). Motivated by the above observations, this paper proposes a new interval Kalman filtering algorithm with two main goals: minimizing an upper bound for the estimation error covariance and enclosing the set of possible solutions of the filtering problem for interval linear systems. This algorithm, called Minimum Upper Bound of Variance Interval Kalman Filter (UBIKF), achieves a reasonable computational time. It also achieves the goal to enclose all the estimates consistent with the parameter uncertainties with a much less conservative manner than [START_REF] Xiong | Fault detection using interval Kalman filtering enhanced by constraint propagation[END_REF].

This paper is organized as follows. The problem formulation is described in Section 2. Section 3 reviews the main notions of positive semidefinite matrix and interval analysis which are necessary for the development of the new algorithm. Then the new interval Kalman filter is derived in Section 4, followed by a numerical example in Section 5. Conclusions and future works are presented in Section 6.

PROBLEM FORMULATION

Considering the following class of uncertain linear discretetime stochastic systems:

x k+1 = A k x k + w k , y k = C k x k + v k , (1)
where x k ∈ R nx is the state vector, y k ∈ R ny is the measurement vector and w k , v k are white Gaussian noise sequences with zero mean and covariance matrices Q and R. The structure of the classical Kalman filter for system (1) is represented as follows [START_REF] Kalman | A new approach to linear filtering and prediction problems[END_REF]):

x+ k = x- k + K k (y k -C k x- k ), (2)
where x+ k is the posterior state estimate, xk = A k x+ k-1 is the prediction of state a priori and K k is the gain matrix that minimizes the mean square error

E[(x + k -x k )(x + k -x k ) T ].
We consider the case where the matrices A k , C k and the covariance matrices Q, R are assumed bounded. Interval analysis is one of the tools to represent bounded uncertainties.

Interval analysis was developed by [START_REF] Moore | Interval analysis[END_REF] and is useful to deal with bounded uncertainties. Most of the notions of interval analysis can be found in [START_REF] Jaulin | Applied Interval Analysis, with Examples in Parameter and State Estimation, Robust Control and Robotics[END_REF] in which an interval [x] is defined as a closed and connected subset of R:

[x] = {x ∈ R | x ≤ x ≤ x} , (3
) where x and x are respectively the lower and upper bound. The center of [x] is defined by mid([x]) = (x + x)/2 and its radius is rad([x]) = (x -x)/2. The set of all intervals in R is noted as IR. An interval vector (or matrix) is a vector (or matrix) whose elements are considered as intervals. In this paper, an interval vector and an interval matrix are denoted as Since it is impossible to solve directly the Kalman filtering problem due to parameter uncertainties, our goal is to obtain an upper bound P + k such that:

E (x + k -x k )(x + k -x k ) T P + k , (4)
for the set of all models with parameters bounded by the above interval matrices. The envelope enclosing the set of state estimates corresponding to the gain K is then computed. This idea is similar to [START_REF] Xiong | Fault detection using interval Kalman filtering enhanced by constraint propagation[END_REF] in which the envelopes of the optimal gains and the state estimates given by the Kalman filtering procedure are determined by using interval analysis. In contrast, the proposed algorithm determines a gain matrix K which minimizes the trace of the upper bound on the error covariance instead of finding a set of gain in [START_REF] Xiong | Fault detection using interval Kalman filtering enhanced by constraint propagation[END_REF]. Since the gain matrix in our filter is punctual, it allows to reduce the conservatism and the computational time of interval operations.

PRELIMINARIES

This section introduces the notations used throughout the paper, including some definitions and properties of positive semidefinite matrices and some basic concepts of interval analysis.

Symmetric positive semidefinite matrices

Definition 1. A symmetric n × n real matrix M is positive semidefinite, denoted M 0, if for all non-zero ndimensional column vector z, the scalar z T M z is nonnegative.

All the diagonal elements of a positive semi-definite matrix are real and non-negative. A special case of positive semidefinite matrices is given in the following remark.

Remark 1. For any real matrix M , the product M M T is a positive semidefinite matrix.

Remark 1 comes from the fact that for any non-zero vector z of appropriate size, the expression z

T M M T z = M T z 2 2
is non-negative, where V 2 is the Euclidean norm of the real vector V defined as the square root of the sum of squares of its elements. Definition 2. Given two symmetric real matrices M, N with the same size, M is said greater or equal to N , written as M N , if M -N is positive semidefinite. Proposition 1. Given two non null matrices M, N with the same size and an arbitrary real number β > 0, the following inequality holds:

β -1 M M T + βN N T M N T + N M T .
(5)

Proof. Developing (M -βN )(M -βN ) T and using Remark 1, we obtain inequality (5). Remark 2. A positive value of β can be computed to minimize the trace of the left-hand side of inequality ( 5).

Differentiating the trace of the left-hand side of inequality (5) gives:

∂tr β -1 M M T + βN N T ∂β = - 1 β 2 tr M M T + tr N N T , (6)
∂ 2 tr β -1 M M T + βN N T ∂β 2 = 2 β 3 tr M M T 0, (7)
where tr(M) is the trace of the square matrix M. The positiveness of the second derivative guarantees the existence of the global minimum. Setting the first derivative to zero, we obtain:

β = tr(M M T )/tr(N N T ). ( 8
)
Proposition 2. Consider two n × n matrices M 1 , M 2 and M 1 M 2 , the following inequality holds for any m × n matrix N :

N M 1 N T N M 2 N T . (9)
Proof. The definition of a positive semidefinite matrix is used to show that N (M 1 -M 2 )N T 0. For any non-zero m-dimensional vector z, we have:

z T N (M 1 -M 2 )N T z = x T (M 1 -M 2 )x ≥ 0, since M 1 -M 2 0, where x = N T z ∈ R n . Therefore, N (M 1 -M 2 )N T 0.
An operation used in this paper is the square root of a positive semidefinite matrix M . Property 1. A symmetric positive semidefinite matrix M is orthogonally diagonalizable, i.e. there exist an orthogonal matrix V and a diagonal matrix D such that M = V DV T . Thus, the square root of M is The following property is useful to describe a quantity in terms of its nominal value and a bounded uncertainty: Property 3. (see [START_REF] Moore | Introduction to Interval Analysis[END_REF]). Given a real value x belonging to an interval [x], there exists a real value α ∈ [-1, 1] such that x = mid([x]) + α.rad ([x]) .

M 1/2 = V D 1/2 V T . Property 2. If M N 0 then M 1/2 N 1/2 0.
Using Property 3 for matrices, the following proposition is obtained: Proposition 3. Given an m × n real matrix M belonging to an interval matrix [M ], there exist mn real values α ij ∈ [-1, 1] with i ∈ {1, . . . , m}, j ∈ {1, . . . , n} such that:

M = mid([M ]) + m i=1 n j=1 α ij rad([M ij ]) , 1 (10)
where rad([M ij ]) is an m × n matrix whose ij-th element contains the radius of the ij-th element interval value of [M ] and the other elements are zero.

In the case of symmetric matrices, the following representation should be considered: Proposition 4. Given an n × n real symmetric matrix M belonging to a symmetric interval matrix [M ], there exist

n(n + 1)/2 real values α ij ∈ [-1, 1] such that: M =mid([M ]) + diag (rad ([M ])) diag α ii + n-1 i=1 n j=i+1 α ij rad [M i,j ] , (11)
where diag (rad ([M ])) is a diagonal matrix containing the radius of diagonal elements of [M ], rad [M i,j ] is a symmetric matrix whose ij-and ji-th elements are the radius of [M ij ] and [M ji ] and the other elements are zero. matrix diag α ii is diagonal and

α ij ∈ [-1, 1], for all 1 ≤ i ≤ j ≤ n.
Classical operations for intervals, interval vectors, interval matrices (e.g. +,-,×, ÷,. . . ) are extensions of the same operations for reals, real vectors, real matrices. For instance, the following results will be applied in the next sections of this paper:

[x] + [y] = x + y, x + y , [x] -[y] = x -y, x -y , [x] × [y] =
M 2 =   1 + β -1 mid,ii + n-1 i=1 n j=i+1 β -1 mid,ij   (mid ([M ])) 2 +   1 + β mid,ii + n-1 i=1 n j=i+1 β -1 ii,ij   (diag(rad([M ]))) 2 + n-1 i=1 n j=i+1 1 + β mid,ij + β ii,ij + n-1 k=i n l=j+1 β kl,ij rad([M i,j ]) 2 ,
where :

β mid,ii = tr {(mid([M ])) 2 } /tr{(diag(rad([M ]))) 2 }, β mid,ij = tr{(mid([M ])) 2 }/tr{[rad([M i,j ])] 2 }, β ii,ij = tr{(diag(rad([M ]))) 2 }/tr{[rad([M i,j ])] 2 }, β kl,ij = tr{[rad([M k,l ])] 2 }/tr{[rad([M i,j ])] 2 }.
The matrix M is computed using Property 1 for M 2 .

Proof. The representation of a symmetric matrix

M ∈ [M ] in Proposition 4 is used to compute M 2 = M M T : M 2 = (mid ([M ])) 2 + diag(rad([M ])) diag α ii 2 diag (rad([M ])) + {mid ([M ])} diag α ii diag(rad([M ])) + diag(rad([M ]))diag α ii {mid ([M ])} + n-1 i=1 n j=i+1 {mid ([M ])} α ij rad([M i,j ]) + n-1 i=1 n j=i+1 α ij rad([M i,j ]) {mid ([M ])} + n-1 i=1 n j=i+1 α ij rad([M i,j ]) diag α ii diag(rad([M ])) + n-1 i=1 n j=i+1 diag(rad([M ]))diag α ii α ij rad([M i,j ]) + n-1 i=1 n j=i+1 n-1 k=1 n l=k+1 {α ij [rad([M i,j ])]}{α kl [rad([M k,l ])]}.
The upper bound M 2 of the matrix M 2 is obtained by using Proposition 1, Remark 2, Proposition 2 and noting that 0 diag α ii 2 I and 0 ≤ (α ij ) 2 ≤ 1 for all -1 ≤ α ij ≤ 1. Property 2 gives M M , for all symmetric positive semidefinite matrix M belonging to [M ].

MAIN RESULTS

Like the classical Kalman filter, the filter proposed in this paper can be designed in two steps: prediction and correction.

Prediction step

In the prediction step, the state estimate from the previous time step and the transition model are used to predict the state at the current time step. Thanks to natural interval extension, this step is performed as follows:

[x k ] -= [A k ] [x k-1 ] + , (12)
where the sign (-) means the result of the prediction step and (+) represents the estimations. The error covariance matrix can also be predicted using interval analysis:

[P k ] -= [A k ] P + k-1 [A k ] T + [Q] . (13)
Equation ( 13) provides an interval matrix enclosing all possible covariance matrices given the interval matrices [A k ], [Q] and the estimated covariance matrix P + k-1 of the previous step. An upper bound P - k of [P k ] -can be determined by applying Proposition 5, i.e. P -

k [P k ] -.

Correction step

The state estimate at time step k is computed by the Equation ( 2

), given x- k ∈ [x k ] -and C k ∈ [C k ].
In order to reduce the effect of the dependency problem [START_REF] Jaulin | Applied Interval Analysis, with Examples in Parameter and State Estimation, Robust Control and Robotics[END_REF]), the Equation ( 2) is rearranged as follows:

[x k ] + = (I -K k [C k ]) [x k ] -+ K k y k . ( 14
)
The box [x k ] + encloses all possible values of x+ k . The gain matrix K k is determined as follows. The expression of the error covariance matrix after the correction step, for any

C k ∈ [C k ], R ∈ [R]
, is:

P + k = (I -K k C k ) P - k (I -K k C k ) T + K k RK T k .
(15) An upper bound of P + k can be obtained by using Proposition 3 for the matrix C k , then developing Equation ( 15).

The terms (I

-K k mid([C k ]))P - k (α ij rad([M ij ]) )
T can be rewritten as follows:

(I -K k mid ([C k ])) P - k 1/2 α ij rad M ij P - k 1/2 T . Similarly, (α ij rad([M ij ]) )P - k (α mn [rad([M mn ])])
T are also divided into two parts:

α ij rad([M ij ]) P - k 1/2 α mn [rad([M mn ])] P - k 1/2 T ,
for all i, m ∈ {1, . . . , n y } and j, n ∈ {1, . . . , n x }. Proposition 1 is then applied to these pairs of matrices with β = 1 to obtain the following inequality:

P + k (n x n y + 1) (I -K k mid([C k ])) P - k (I -K k mid([C k ])) T + (n x n y + 1) K k ny i=1 nx j=1 rad([C ij k ]) P - k rad([C ij k ]) T K T k + K k RK T k = P + k .
(16) where R [R] is determined with Proposition 5.

Having the expression of P +

k as a function of K k , we look for K k , the value that minimizes the trace of P + k . The first and second derivatives of tr P + k with respect to K k are:

dtr P + k dK k = -2P - k,ny mid([C k ]) T + 2K k mid([C k ])P - k,ny mid([C k ]) T + 2K k   ny i=1 nx j=1 rad([C ij k ])P - k rad([C ij k ]) T   + 2K k R, d 2 tr P + k dK 2 k = 2mid([C k ])P - k,ny mid([C k ]) T + 2 ny i=1 nx j=1 rad([C ij k ]) P - k rad([C ij k ]) T + 2R,
where P - k,ny = (n x n y + 1)P - k . The second derivative is always definite positive that guarantees the existence of a minimum value for tr P + k and K k is obtained from the first derivative:

K k = P - k,ny mid([C k ]) T S -1 k , (17)
where:

S k = mid([C k ])P - k,ny mid([C k ]) T + + ny i=1 nx j=1 rad([C ij k ]) P - k,ny rad([C ij k ]) T + R. ( 18
)
The expression of the covariance matrix bound P + k is obtained by Equation ( 16) using K k given in Equation ( 17):

P + k = (I -K k mid([C k ])) P - k,ny . (19
) In summary, the UBIKF allows to find a gain matrix K k , a covariance upper bound P + k and an interval state estimate [x k ] + at each iteration such that for any real matrices

A k , C k , Q, R belonging to the interval matrices [A k ], [C k ], [Q], [R], we have: x+ k ∈ [x + k ] P + k P + k , (20)
where x+ k and P + k are the state estimate and the covariance matrix given by the Kalman filtering procedure for the linear system (A k , C k , Q, R) and the gain matrix K k . The proposed filter has the similar objective as [START_REF] Xiong | Fault detection using interval Kalman filtering enhanced by constraint propagation[END_REF], i.e. to bound the set of all state estimates. Nevertheless, it should provide less conservative results since the gain matrix is punctual and so the conservatism of interval analysis is reduced. This is illustrated in the case study of Section 5.

NUMERICAL EXAMPLE

The filter proposed in this paper is applied to a state estimation problem of a typical proton exchange membrane fuel cell system (PEMFC). The main components of the system are shown in Figure 1. The fuel cell converts the chemical energy into electricity by consuming oxygen and hydrogen provided by air and hydrogen supply system. The linear model used in this paper can be found in [START_REF] Rotondo | Robust fault diagnosis of proton exchange membrane fuel cells using a takagi-sugeno interval observer approach[END_REF]. It is derived from the nonlinear model presented in [START_REF] Pukrushpan | Control-Oriented Modeling and Analysis for Automotive Fuel Cell Systems[END_REF] under the following assumptions: the stack temperature T st is constant; the temperature and humidity of the inlet reactant flows are perfectly controlled; the anode and cathode volumes of multiple fuel cells are lumped as a single stack anode and cathode volumes; all the reactant behave as ideal gases. The stack current I st is considered as the system input. A state space representation of order 9 is presented in [START_REF] Rotondo | Robust fault diagnosis of proton exchange membrane fuel cells using a takagi-sugeno interval observer approach[END_REF] with the following state variables: mass of oxygen (m O2 ), mass of hydrogen (m H2 ), mass of nitrogen (m N2 ), air mass in the supply manifold (m sm ), air pressure in the supply manifold (p sm ), air pressure in the return manifold (p rm ), compressor speed (ω cp ), mass of water in the anode (m w,an ) and in the cathode (m w,ca ). An Euler discretization has been performed with the sample time T s = 0.04s to obtain the following discrete model:

x k+1 = A k x k + B k I st,k + D k + w k , y k = C k x k + v k , (21)
where w k and v k are zero-mean Gaussian process noise and measurement error vectors. The measured outputs of the system are compressor speed (ω cp ), mass of oxygen (m O2 ), mass of hydrogen (m H2 ), mass of nitrogen (m N2 ), air pressure in the supply manifold (p sm ), air pressure in the return manifold (p rm ), pressure in the anode (p an ), pressure in the cathode (p ca ). The covariance matrices Q and R of process noises and measurement errors are given by: Q = diag(10 -12 ; 10 -12 ; 10 -12 ; 10 -12 ; 10 2 ; 10 2 ; 10; 10 -12 ; 10 -12 ), R = diag(10; 10 -10 ; 10 -10 ; 10 -10 ; 10 2 ; 10 2 ; 10 2 ).

A test with time-varying parameters has been designed. The elements a 1i (first line), a 2i (second line), and a 7i (seventh line) of the matrix A k oscillate with sinusoidal law having the magnitude of 5% of their nominal values. These variation laws are assumed unknown and must be considered as uncertainties by the filters. The interval matrix [A k ] containing all admissible parameters is chosen as follows:

mid([A k ]) = A 0 k , rad([A k ]) = 0.05 A 0 k , (22)
where A 0 k is the nominal value of A k given in [START_REF] Rotondo | Robust fault diagnosis of proton exchange membrane fuel cells using a takagi-sugeno interval observer approach[END_REF].

Moreover, the compressor speed sensor, mass of hydrogen sensor, mass of oxygen sensor are assumed to provide 2% of uncertainty. Therefore, the interval matrix [C k ] is defined as:

mid

([C k ]) = C 0 k , rad([C k ]) = 0.02 C 0 k , (23)
where C 0 k is the nominal value of C k . The covariance matrix of process noise is considered to be certain, i.e. [Q] = Q, while the measurement error covariance matrix is assumed to be uncertain with 50% uncertainty.

mid

([R]) = R , rad([R]) = 0.5|R|. ( 24
)
Besides, this numerical example is also tested with the improved interval Kalman filter (iIKF) presented in [START_REF] Xiong | Fault detection using interval Kalman filtering enhanced by constraint propagation[END_REF]. The results of this algorithm are compared with those of the UBIKF in Figure 2 and in Table 1, including the root mean square error (RMSE) for each state, the percentage of time steps when the confidence interval contains the real state O, and the computational time (for 7500 time steps). Both of these two filtering algorithms have the same goal to provide the envelopes of the set of all possible state estimates of the filtering problem in a linear system with bounded uncertainties. The 99.7% confidence intervals 2 given by these two filters (Figure 2) enclose the real states at every time step (O = 100%). The confidence intervals of the proposed filter are tighter than those given by iIKF. Moreover, the gain in computational time of the new filter compared to iIKF is about 3.8 (Table 1). These advantages come from the fact that the UBIKF deals with the real matrices while the iIKF applies the constraint satisfaction problem and the set inversion techniques for interval matrices. Therefore, the UBIKF allows to obtain less conservative result with reasonable computational time. iIKF 5.4.10 -5 100 1.5.10 -5 100 156 100 511 s UBIKF 4.8.10 -5 100 1.4.10 -5 100 177 100 133 s

The initial interval Kalman filter in [START_REF] Chen | Interval Kalman filtering[END_REF] has also been applied to this example. The results given by this algorithm diverge due to the large uncertainties in the model.

CONCLUSION

This paper presents an efficient method to find a positive definite upper bound of an interval matrix, that is applied to propose an interval filter for uncertain linear systems. This approach allows to bound the set of all possible state estimations given by the Kalman filtering structure for any 2 The 99.7% confidence intervals of a multivariate Gaussian distribution N (µ, Σ) are µ -3 diag(Σ), µ + 3 diag(Σ) . For UBIKF, the 99.7% confidence intervals are computed using the box [x k ] + and the upper bound of the covariance matrix. admissible parameter uncertainties. It has been applied to a model-based state estimation of a PEMFC and has been compared with the improved interval Kalman filter [START_REF] Xiong | Fault detection using interval Kalman filtering enhanced by constraint propagation[END_REF]. The new filter provides convergent state confidence intervals with reasonable width and small computational time.

Further work will target complexity analysis of the proposed filter and the other interval Kalman filters. It will give a theoretical result about the computational time of the considered filtering algorithms.

  [x] and [M ]. The set of n-dimensional interval vectors (or m × n interval matrices) is denoted as IR n (or IR m×n ). Given [M ] ∈ IR m×n , the two functions mid([M ]) and rad([M ]) provide two m × n real matrices containing the centers and radius of elements of [M ]. The matrices A k , C k , Q and R of the system (1) are represented by interval matrices, denoted [A k ], [C k ], [Q] and [R], containing all possible values of each parameter.

3. 2

 2 Interval analysis Definition 3. An interval matrix [M ] ∈ IR n×n is said to be symmetric if the real matrices mid([M ]) and rad([M ]) are symmetric.

  min(xy, xy, xy, xy), max(xy, xy, xy, xy) . For the interval matrix multiplication, the ij-th interval [C ij ] of the product [C] = [A][B] encloses all possible values of the ij-th element of real matrices C = AB, for any A ∈ [A], B ∈ [B]. The interval matrix obtained [C] may contain real matrices D that are not the result of the multiplication of any real matrices A ∈ [A] and B ∈ [B]. This conservatism is generated by the multi-occurrence problem detailed in Moore et al. [2009]. Proposition 5. Given a symmetric interval matrix [M ] ∈ IR n×n , there exists a symmetric positive semidefinite 1 The expression is a developed form of the Hadamard product. matrix M that bounds all symmetric positive semidefinite matrices M ∈ [M ], i.e. M -M 0. We denote M [M ]. The expression of M 2 is:

Fig. 1 .

 1 Fig. 1. Fuel cell system

Fig. 2 .

 2 Fig. 2. Real state and 99.7% confidence intervals (upper bound (UB) and lower bound (LB)) given by iIKF and the UBIKF for mass of oxygen m O2 and mass of hydrogen m H2 .

Table 1 :

 1 Results of the iIKF and the proposed UBIKF

	m O 2 RMSE O(%) RMSE O(%) RMSE O(%) m H 2 ω cp	Time


Thomas Ondet
email: thomasondet@hotmail.com

Béatrice Muscatelli-Groux

Cédric Coulouarn

Sacha Robert

Thomas Gicquel

Aude Bodin

Vincent Lagente

Jean-Alexis Grimaud

The release of pro-inflammatory cytokines is mediated via MAP-kinases rather than inflammasome signaling pathway in keratinocytes

Keywords: Keratinocytes, Toll like receptors, Inflammation, MAP kinase

Toll like receptors (TLRs) are expressed in the skin and airway epithelial tissues, which are the most important sites of host-pathogen interactions. TLRs recognize the three-dimensional structure of pathogen associated molecules and are thus useful markers of the innate immune response. Here, we investigated the role of lipopolysaccharides (LPS) and monosodium urate (MSU) crystals in the activation of the TLR and Nod Like Receptors (NLR) pathways in human keratinocytes. Analysis of the inflammasome compounds revealed that NLRP3 and TLR4, both of which are components of

Accepted Article

inflammasome complexes involved in the activation of IL-1β, were not expressed in keratinocytes.

Transcriptomic analysis showed that combination of both MSU and LPS priming do not elicit significant result compare to MSU treatment that induced the expression of TLR2, IL-6 and IL-8/CXCL8 in the keratinocyte cell line HaCaT. Furthermore, MSU promoted the phosphorylation of Erk1/2 and MAPK14/p38α MAP kinases. We concluded that MSU stimulates a pro-inflammatory response in keratinocytes via MAP Kinase pathway to induce production of IL-8/CXCL8 and TLR2.

Introduction

The skin is the first line of defense against invading pathogens [START_REF] Miller | Toll-like receptors in skin[END_REF] . Its epithelial tissue structure is mainly composed of keratinocytes, which form the largest surface of the body in contact with the outside. Keratinocytes play a prominent role in primary immunity. Moreover, the cross-talk between keratinocytes and fibroblasts is important for the inflammatory process of wound healing and control extracellular matrix homeostasis [START_REF] Grimaud | Matrix receptors to cytokines: from concept to control of tissue fibrosis dynamics[END_REF] .

The inflammatory process is associated with the activation of cell signaling pathways leading to the release of various mediators including cytokines and chemokines, which in turn leads to tissue remodeling, wound healing and skin repair. One intracellular signaling pathway associated with the inflammatory process is the NLRP3 inflammasome, which is characterized by the activation and the secretion of IL-1β [START_REF] Martinon | Gout-associated uric acid crystals activate the NALP3 inflammasome[END_REF][START_REF] Martinon | The inflammasome: a molecular platform triggering activation of inflammatory caspases and processing of proIL-beta[END_REF] .

Accepted Article

Toll-like receptors (TLR) are pattern recognition receptors (PRR) directed against highly conserved molecular patterns called pathogen-associated molecular patterns (PAMPs). Ten functional TLRs targeting specific structures (lipids, proteins, nucleic acids or crystals) have been identified in humans [START_REF] Miller | Toll-like receptors in skin[END_REF][START_REF] Kawai | The role of pattern-recognition receptors in innate immunity: update on Toll-like receptors[END_REF] . In monocytes, TLRs activate several signaling pathways such as MAP kinases, leading to the production of sever IL-1β, IL-6 and IL-8/CXCL-8. Keratinocytes express these cytokines constitutively in very low amounts in resting conditions and their production is increased upon stimulation [START_REF] Kawai | The role of pattern-recognition receptors in innate immunity: update on Toll-like receptors[END_REF] . TLR4 constitutes the main receptor of LPS [START_REF] Hoshino | Cutting edge: Toll-like receptor 4 (TLR4)-deficient mice are hyporesponsive to lipopolysaccharide: evidence for TLR4 as the Lps gene product[END_REF] . Several studies have found that TLR4 is expressed on keratinocytes [START_REF] Panzer | TLR2 and TLR4 expression in atopic dermatitis, contact dermatitis and psoriasis[END_REF] whereas others have not [START_REF] Baker | Normal keratinocytes express Toll-like receptors (TLRs) 1, 2 and 5: modulation of TLR expression in chronic plaque psoriasis[END_REF] . Others still have found that TLR4 expression depends on the stage of keratinocyte differentiation [START_REF] Pivarcsi | Differentiation-regulated expression of Toll-like receptors 2 and 4 in HaCaT keratinocytes[END_REF][START_REF] Köllisch | Various members of the Toll-like receptor family contribute to the innate immune response of human epidermal keratinocytes[END_REF] , which is correlated with the presence or absence of the inflammasome in keratinocytes.

Several inflammasome pathways have been described, including the canonical pathway which consists of three main effectors, namely NLRP3, the pro-caspase-1 and the apoptosis speck-like protein containing a CARD (ASC) adapter, the later bridging the interactions between NLRP3 and the pro-caspase-1. Assembly of the NLRP3 inflammasome leads to the cleavage of cytosolic pro-IL-1β into the mature pro-inflammatory cytokine IL-1β by activated caspase-1 [START_REF] Martinon | The inflammasome: a molecular platform triggering activation of inflammatory caspases and processing of proIL-beta[END_REF] . Monosodium uric (MSU) crystals activate the inflammasome via the ASC adapter protein, leading to the activation of caspase-1 by the complex, which cleaves the immature form of pro-IL-1β into an active form [START_REF] Martinon | Gout-associated uric acid crystals activate the NALP3 inflammasome[END_REF] .

We previously reported that the combination of MSU crystals with LPS activates the NLRP3 inflammasome in human macrophages and results in the production of IL-1β [START_REF] Gicquel | IL-1β production is dependent on the activation of purinergic receptors and NLRP3 pathway in human macrophages[END_REF] . In this study, we investigated the mechanisms of production of pro-inflammatory cytokines from keratinocytes activated with MSU, notably the involvement of NLR-inflammasome, TLR and MAP kinase pathways.

Accepted Article

Results

Synthesis of pro-inflammatory cytokines from keratinocytes stimulated with MSU and LPS

MSU treatment significantly increased the abundance of IL-1β, IL-6 and IL-8/CXCL8 mRNA in both HEK and HaCaT cells (Figure 1). Up-regulation of the pro-IL-1β gene suggests that keratinocytes may produce activated IL-1β through the inflammasome signaling pathway. To investigate the involvement of the inflammasome in keratinocytes, cells were stimulated by LPS, as TLR4 agonist, in association with MSU. Nevertheless, LPS priming before MSU stimulation or LPS alone did not resulted in IL-1β expression (data not shown), suggesting that keratinocytes are unable to produce IL-1β despite of the increase of mRNA transcript.

In contrast, MSU treatment significantly induced the production of IL-6 and IL-8/CXCL8 proteins in keratinocytes whereas LPS alone did not (Figure 2). Moreover, LPS priming before MSU treatment did not further boost IL-6 and IL-8/CXCL8 expression. The glucocorticoid dexamethasone significantly impaired the production of both IL-6 and IL-8/CXCL8 in MSU and LPS plus MSU-stimulated cells.

Investigation of TLR and NLR inflammasome pathways in keratinocytes

In order to identify signaling pathways involved, we further examined the mRNA expression of various Nod-like (NLRP1, NLRP2, NLRP3, NLRP6, NLRC4, AIM2) and toll-like receptors (TLR 1 to 10) in keratinocytes (Figure 3). We used THP-1 macrophages as a comparative positive control because these cells are described to express high levels of various TLR and NLR inflammasomes [START_REF] Martinon | The inflammasome: a molecular platform triggering activation of inflammatory caspases and processing of proIL-beta[END_REF][START_REF] Zarember | Tissue Expression of Human Toll-Like Receptors and Differential Regulation of Toll-Like Receptor mRNAs in Leukocytes in Response to Microbes, Their Products, and Cytokines[END_REF] .

Accepted Article

The expression of TLR2 was moderately expressed in both HEK and HaCaT but was significantly upregulated after MSU treatment. Others TLRs were moderately or not at all expressed in both HEK and HaCaT. For example, TLR4 is not expressed in primary keratinocytes and barely detectable in the HaCaT cell line.

We did not detect NLRP3 mRNA in keratinocytes suggesting that the NLRP3-inflammasome pathway is not involved in the activation of keratinocytes. NRLP6, NLRC4 and AIM2 mRNA were barely detectable whereas NLRP1 and NLRP2 mRNA were present at higher levels (Figure 3).

Involvement of the MAP kinase pathway in MSU stimulation

To investigate the signaling pathways associated with MSU activation, we used a proteome profiler array to analyze the status of 43 kinase phosphorylation sites in keratinocytes stimulated with MSU and in unstimulated control cells. MSU significantly induced the phosphorylation of ERK1/2 and MAPK14/p38α, both of which are involved in the MAP kinase pathway. In contrast, the phosphorylation of WNK1 kinase was significantly down-regulated by MSU) (Figure 4a).

Then, we also analyzed the time course of MAPK14/p38α phosphorylation in MSU-stimulated keratinocytes by western blotting, using unstimulated keratinocytes as a control. MAPK14/p38α phosphorylation appears to be maximum after 45 minutes of MSU treatment, whereas total MAPK14/p38α showed no variation (Figure 4b).

Accepted Article

Involvement of the MAP kinase pathway in the production of IL-8

To confirm the involvement MAPK14/p38α and ERK1/2 in HaCaT, we used selective inhibitors of these kinases (Figure 5). U0126 is a potent and selective inhibitor of MEK1 and MEK2 which blocks intracellularly the phosphorylation and activation of ERK [START_REF] Favata | Identification of a novel inhibitor of mitogen-activated protein kinase kinase[END_REF] and SB239063 is a MAPK14/p38α

inhibitor selective for the p38 α and β isoforms [START_REF] Underwood | SB 239063, a potent p38 MAP kinase inhibitor, reduces inflammatory cytokine production, airways eosinophil infiltration, and persistence[END_REF] . Both inhibitors (U0126 and SB239063) significantly reduced the production of IL-8/CXCL8 by HaCaT Cells stimulated with MSU (Figure 5).

Gene expression profiling of keratinocytes stimulated with MSU

To investigate the mecanisms involved in the inflammatory process in keratinocytes by MSU, we analyzed the gene expression profile of HaCaT cells incubated for 12 h with MSU at 1mg/ml (Figure 6). 12h was determined as an optimal time point for gene expression analysis by qPCR analysis (data not shown). We selected 12h rather than 24h to obtain a moderate inflammation response. 24h of MSU stimulation corresponds to the mRNA maximun expression of cytokines expression (IL-6 & CXL8/IL8) but could decrease other proteins of interests.

Gene expression analysis resulted in the modulation of 128 genes (P< 0.005, fold change > 2), including 91 up-regulated and 37 down-regulated genes.

Gene ontology analysis revealed that various family of genes up-regulated by MSU treatment were involved in inflammation, including cytokine of the CCL family (5 genes) and chemokines of the CXCL family (7 genes). Consequently of pro inflammatory conditions, keratinocytes response leads to keratinocytes differentiation by producing S100 proteins family (S100A8, S100A9, S100A12) and small proline-rich protein involved in cross-bridging proteins for the keratinocytes differentiation (SPRR1A, SPRR2A, SPRR2C, SPRR2D, SPRR2D, SPRR2E, SPRR2F) (Error! Reference source not found.).

Accepted Article

The second family of genes of interest is involved in response to MSU includes innate response genes as defensin (defensin, beta 4A) and gene associated to Fas-or tumor necrosis factor type alpha-induced apoptosis immediate early response genes (IER3). Of note, IERS and some other proteins including dual specificity phosphatase (DUSP6) are known to be involved in phosphorylation for ERK1/2 proteins [START_REF] Garcia | IEX-1: A new ERK substrate involved in both ERK survival activity and ERK activation[END_REF][START_REF] Oliveira | NRAS mutation causes a human autoimmune lymphoproliferative syndrome[END_REF] (Error! Reference source not found.).

Discussion

Keratinocytes are the main cellular constituents of the skin epidermis and are an important line of defense against pathogens. When activated, they participate of the inflammatory process through the increased production of several cytokines and/or chemokines [START_REF] Ozawa | Localization of IL-8 and complement components in lesional skin of psoriasis vulgaris and pustulosis palmaris et plantaris[END_REF] . In this study, we investigated how different signaling pathways related to innate immunity are involved in this production. We used the activation of HEK and HaCaT keratinocyte cells by MSU as an experimental model. Indeed, various studies showed that MSU is important for the induction of pro-inflammatory responses and monocyte recruitment, leading to the secretion of large amounts of IL-6 and IL-8/CXCL8 proteins [24-26].

LPS has been largely reported to be a potent activator for the production of IL-1β from macrophages

21

.

In the present study, we found that LPS neither stimulates nor potentiates the production of IL-1β from keratinocytes. However, it is still unclear whether keratinocytes synthesize IL-1β. Some studies have detected IL-1β protein in these cells [START_REF] Feldmeyer | The inflammasome mediates UVB-induced activation and secretion of interleukin-1beta by keratinocytes[END_REF][START_REF] Dombrowski | Cytosolic DNA triggers inflammasome activation in keratinocytes in psoriatic lesions[END_REF] whereas others found that although keratinocytes express IL-1β, they do not produce its active form [START_REF] Mizutani | Human keratinocytes produce but do not process prointerleukin-1 (IL-1) beta. Different strategies of IL-1 production and processing in monocytes and keratinocytes[END_REF] . Given that the production of IL-1β depends on the NLRP3-inflammasome pathway, we further explored the involvement of this signaling pathway in keratinocytes stimulated with MSU. Interestingly, by investigating the expression of various NLRinflammasomes, we observed that NLRP3 was not expressed in keratinocytes, whereas NLRP1, NLRP2, NLRP6, NLRC4 and AIM2, were moderately expressed.

Accepted Article

To investigate why IL-1 β protein is not produced in keratinocytes and to gain insight into the role of the innate immune response and LPS, we analyzed the mRNA levels of the 10 functionally active TLRs in humans in the HaCaT cell line and HEK human keratinocyte progenitors stimulated with MSU.

TLR1, TLR2, TLR3 and TLR5 were expressed in both HaCaT and HEK cells. TLR4 and TLR10 were specifically expressed in HaCaT cells whereas TLR6 was specifically expressed in HEK cells. The remaining TLR transcripts were not detected in either cell types. These results suggest that TLR2 can form heterodimers with TLR1 to recognize triacyl lipopeptide, and to a lesser extent with TLR6 to recognize diacyl lipopteptide [START_REF] Takeda | Recognition of lipopeptides by Toll-like receptors[END_REF] , in keratinocytes. These heterodimers can recognize many lipopeptides from the cell wall of bacteria and homodimers of TLR2 can recognize bacterial peptidoglycan and some heat shock proteins [START_REF] Akira | Pathogen recognition and innate immunity[END_REF] . We found that TLR7, TLR8, and TLR9 are not expressed in keratinocytes, suggesting that these cells are not able to recognize single stranded RNA or CpG from bacteria or viruses [START_REF] Köllisch | Various members of the Toll-like receptor family contribute to the innate immune response of human epidermal keratinocytes[END_REF] . The function of TLR10 is still unclear. This receptor was only expressed in HaCaT cells and was significantly down-regulated by MSU stimulation. Nevertheless, our results confirm new observations linking the up-regulation of TLR2 to the anti-inflammatory properties of TLR10 [START_REF] Oosting | Human TLR10 is an anti-inflammatory pattern-recognition receptor[END_REF] . Our finding that keratinocytes express TLR3 confirms previous results

showing that these cells are able to sense genetic material from necrotic cells, including UVdamaged keratinocytes [START_REF] Borkowski | Activation of TLR3 in keratinocytes increases expression of genes involved in formation of the epidermis, lipid accumulation, and epidermal organelles[END_REF] . These results are consistent with previous in vivo observations showing that TLR2 is up-regulated in response to pro-inflammatory stimuli in psoriatic skin [START_REF] Panzer | TLR2 and TLR4 expression in atopic dermatitis, contact dermatitis and psoriasis[END_REF][START_REF] Begon | Expression, subcellular localization and cytokinic modulation of Toll-like receptors (TLRs) in normal human keratinocytes: TLR2 up-regulation in psoriatic skin[END_REF] . Recent studies imply a role for TLRs in wound healing and the maintenance of skin barrier homeostasis, and demonstrate that TLR2 stimulates the formation of tight junctions [START_REF] Kuo | Activation of epidermal toll-like receptor 2 enhances tight junction function: implications for atopic dermatitis and skin barrier repair[END_REF] . TLR3 and TLR4 may also promote wound healing [START_REF] Chen | Toll-like receptor 4 has an essential role in early skin wound healing[END_REF][START_REF] Suga | TLR4, rather than TLR2, regulates wound healing through TGF-β and CCL5 expression[END_REF][START_REF] Borkowski | Toll-Like Receptor 3 Activation Is Required for Normal Skin Barrier Repair Following UV Damage[END_REF] . These mechanisms remain poorly understood but may involve the hyperproliferation of keratinocytes associated with the up-regulation TLR2 in psoriatic skin [START_REF] Begon | Expression, subcellular localization and cytokinic modulation of Toll-like receptors (TLRs) in normal human keratinocytes: TLR2 up-regulation in psoriatic skin[END_REF] .

Accepted Article

Thus, we observed that the main components of the NLRP3-inflammasome and TLR4, are not expressed in primary keratinocytes and are moderately expressed in HaCaT cells. These data are consistent with the absence of the production of IL-1β from keratinocytes in our study and are not consistent with the proposal of the involvement of NLRP3-caspase 1 pathway in keratinocytes [START_REF] Feldmeyer | The inflammasome mediates UVB-induced activation and secretion of interleukin-1beta by keratinocytes[END_REF][START_REF] Watanabe | Activation of the IL-Accepted Article 1beta-processing inflammasome is involved in contact hypersensitivity[END_REF] .

We also clearly showed that following MSU stimulation, HEK and HaCaT cells release IL-6 and IL-8/CXCL8. Both proteins are strongly up-regulated in skin during pro-inflammatory disorders such as psoriasis, and lead to leucocyte migration and keratinocyte proliferation and differentiation in the upper level of the epidermis [START_REF] Gillitzer | Upper Keratinocytes of Psoriatic Skin Lesions Express High Levels of NAP-1/IL-8 mRNA In Situ[END_REF][START_REF] Nickoloff | Cellular localization of interleukin-8 and its inducer, tumor necrosis factor-alpha in psoriasis[END_REF] . Interestingly, the gene expression profiling showed that several cytokines were also up-regulated by MSU. A similar cytokine expression profile is observed in skin lesions, such as psoriatic skin, to which both Th2 and Th1 cells are frequently recruited [START_REF] Meyer | IL-32 is expressed by human primary keratinocytes and modulates keratinocyte apoptosis in atopic dermatitis[END_REF][START_REF] Suga | The role of IL-32 in cutaneous T-cell lymphoma[END_REF] .

It remains unclear how MSU is recognized by the cell and leads to cytokine release [START_REF] Ghaemi-Oskouie | The role of uric acid as an endogenous danger signal in immunity and inflammation[END_REF] . One study [START_REF] Liu-Bryan | Innate immunity conferred by Tolllike receptors 2 and 4 and myeloid differentiation factor 88 expression is pivotal to monosodium urate monohydrate crystal-induced inflammation[END_REF] suggests that TLRs are able to recognize the crystal structure of MSU. Based on the Proteome Profiler TM Array results, we found that MSU selectively leads to the phosphorylation of two MAPkinase proteins: MAPK14/p38α and ERK1/2. This observation suggests that these two kinases are involved in the recognition of the MSU and their activation leads to the up-regulation of TLR2 as occurs in several skin disorders for the recognition of microbial pathogens [START_REF] Kawai | The role of pattern-recognition receptors in innate immunity: update on Toll-like receptors[END_REF][START_REF] Yu | Serum amyloid A Induces IL-1β secretion from keratinocytes via the NLRP3 Inflammasome[END_REF] . This mechanism is similar to that reported for Poly(I:C), a TLR3 agonist that stimulates IL-8/CXCL8 production in keratinocytes [START_REF] Köllisch | Various members of the Toll-like receptor family contribute to the innate immune response of human epidermal keratinocytes[END_REF][START_REF] Borkowski | UVB Radiation Illuminates the Role of TLR3 in the Epidermis[END_REF] . We then used selective inhibitors to confirm the involvement of MAPK14/p38α and ERK1/2 in IL-8/CXCL8 expression. These inhibitors impaired IL-8/CXCL8 expression in response to MSU, suggesting the involvement of ERK1/2 and MAPK14/p38α in this process as described previously in monocytes [START_REF] Liu | Extracellular signalregulated kinase 1/extracellular signal-regulated kinase 2 mitogen-activated protein kinase signaling and activation of activator protein 1 and nuclear factor kappaB transcription factors play central roles in interleukin-8 expression[END_REF] . The inflammasome-independent pro-inflammatory NF-kB signaling pathways in keratinocyte proliferation and differentiation has been largely demonstrated [START_REF] Begon | Expression, subcellular localization and cytokinic modulation of Toll-like receptors (TLRs) in normal human keratinocytes: TLR2 up-regulation in psoriatic skin[END_REF][START_REF] Takao | Expression of NF-κB in epidermis and the relationship between NF-κB activation and inhibition of keratinocyte growth[END_REF][START_REF] Wullaert | NF-κB in the regulation of epithelial homeostasis and inflammation[END_REF] .

Accepted Article

Here we have demonstrated that alternative pathway involving the MAPkinase ERK1/2 or MAPK14/p38α are also involved in keratinocyte inflammatory response. Inhibition of MAPkinase ERK1/2 or MAPK14/p38α did not completely abolish IL-8 synthesis, which confirms the involvement of additional signaling pathway including NF-ƘB signaling. This data are in correlation with genes profile expression results, several genes coding for proteins involved in the ERK1/2 phosphorylation pathway including DUSP6 and IERS3 [START_REF] Garcia | IEX-1: A new ERK substrate involved in both ERK survival activity and ERK activation[END_REF][START_REF] Oliveira | NRAS mutation causes a human autoimmune lymphoproliferative syndrome[END_REF] are upregulated upon MSU stimulation.

In conclusion, we showed that MSU may induce the production of IL-6 and IL-8/CXCL8 proteins and up-regulates TLR2 in keratinocytes. However, we failed to detect IL-1β protein in keratinocytes, although MSU stimulated the production of IL-1β mRNA. MSU also up-regulated the phosphorylation of ERK1/2 and MAPK14/p38α proteins in the MAP Kinase pathway, which in turn stimulated the production of IL-6 and IL-8/CXCL8. Our results are consistent with in vivo observations in psoriasis [START_REF] Begon | Expression, subcellular localization and cytokinic modulation of Toll-like receptors (TLRs) in normal human keratinocytes: TLR2 up-regulation in psoriatic skin[END_REF] , this disease is associated to keratinocytes hyper proliferative activity, keratinocytes differentiation, cytokines expression, and TLR2 up-regulation.

Material and methods

Cell culture

Normal human keratinocytes (HEK) were obtained from Cellntec (Bern, Switzerland) and grown in CnT-57 PCT Epidermal Keratinocyte Medium supplemented with bovine pituitary extract (Cellntec, Bern, Switzerland). HaCaT cells were obtained from Cell Lines Service (Eppelheim, Germany) [START_REF] Boukamp | Normal keratinization in a spontaneously immortalized aneuploid human keratinocyte cell line[END_REF] , and were grown in DMEM (PAN biotech, Aidenbach, Germany) supplemented with 10% FBS (Life Technology, Grand Island, NY, USA), L-glutamine and non-essential amino acids. Cells and supernatant were harvested in 24-well plates after stimulation for 24 h with either lipopolysaccharide (LPS) tlrl-3pelps LPS from E. coli O111:B4 (Invivogen, Toulouse, France) at 1µg/ml

Accepted Article

or MSU (MonoSodium Urate) at 1mg/ml in endotoxin free conditions. MSU was obtained as described previously [START_REF] Martinon | Gout-associated uric acid crystals activate the NALP3 inflammasome[END_REF] and optimal concentration was determined at 1000µg/ml without cytotoxicity. Cells were also stimulated with dexamethasone at 10µM in serum-free conditions (Sigma, Saint-Quentin Fallavier, France). Signaling pathway inhibitors were tested in the HaCaT cell line. HaCaT were incubated for 2 h with either the MEK inhibitor U0126 monoethanolate (3μM, 5μM) or the MAPK14/p38α inhibitor SB239063 (20μM, 10μM), and were then incubated for 24 h with MSU at 1mg/ml [START_REF] Victoni | Roflumilast N-oxide prevents cytokine secretion induced by cigarette smoke combined with LPS through JAK/STAT and ERK1/2 inhibition in airway epithelial cells[END_REF] . THP-1 cells were purchased from American Type Culture Collection (Manasass, VA, USA.) and maintained in RPMI 1640 medium supplemented with 10% FCS, 50 UI/mL penicillin, 50 µg/mL streptomycin, 2mM L-glutamine and 1mM sodium pyruvate at 37°C and 5% CO 2 in a humidified incubator. THP-1 cells were differentiated by 10ng/ml of Phorbol-myristate-acetate (PMA) for 3 days, followed by 24 h without PMA in RPMI 1640 medium supplemented with 2% FCS.

RT-qPCR

Total RNA was extracted with the RNeasy Plus Mini Kit Qiagen and quantified with the nanodrop sytem (Wilmington, DE, USA). RT-PCR was performed with 1µg of RNA that had been reverse transcribed with the High-Capacity cDNA Reverse Transcription Kit (Life Technology, Grand Island, NY, USA). The SYBR Green PCR Master Mix from Life Technology (Grand Island, NY, USA) was used for all reactions. The expression of the gene of interest was normalized to that of the housekeeping gene GAPDH.

Gene expression profiling

Transcriptomic analysis was performed as described previously [START_REF] Coulouarn | Hepatocyte-stellate cell cross-talk in the liver engenders a permissive inflammatory microenvironment that drives progression in hepatocellular carcinoma[END_REF] . The cells were incubated with MSU (1mg/ml) for 12 h. RNA was extracted from cells at 80% confluence (n=3) with the QIAzol Lysis Reagent RNeasy kit (Qiagen) and 200ng of total RNA was transcribed and labeled in vitro using the Agilent Low-Input QuickAmp Labeling kit (Agilent Technologies) in the presence of Cy3-CTP

Accepted Article

according to the manufacturer's protocol. After purification using the RNeasy mini kit (Qiagen), the yield and specific activity of cRNA were determined using a NanoDrop Spectrophotometer (yield, 6.2±0.85μg cRNA; specific activity, 13.46±1.59 pmol Cy3/μg cRNA).

An equal amount (600ng) of Cy3-labeled cRNA was fragmented and subsequently hybridized for 18 h onto human SurePrint G3 8x60K pangenomic microarrays (Agilent Technologies), which were then washed and scanned according to the manufacturer's instructions (Agilent Technologies). Gene expression data were further processed using Feature Extraction (version 10.7) and GeneSpring (version 11.0) software (Agilent Technologies). Filtration of array data resulted in the selection of 24,198 non-flag positive and significant gene features. Inter-array normalization was performed using the 75th percentile signal value. The results of MSU-treated and control samples were compared with a t-test (p>0.05 and fold change FC>2). Modular and Singular Enrichment Analysis was performed from GeneCodis analysis [START_REF] Tabas-Madrid | GeneCodis3: a non-redundant and modular enrichment analysis tool for functional genomics[END_REF][START_REF] Nogales-Cadenas | GeneCodis: interpreting gene lists through enrichment analysis and integration of diverse biological information[END_REF][START_REF] Carmona-Saez | GENECODIS: a webbased tool for finding significant concurrent annotations in gene lists[END_REF] Microarray data are publicly available from the gene expression omnibus (GEO) database (www.ncbi.nlm.nih.gov/geo; GSE65931).

ELISA

Cytokines were quantified by ELISA with the Human DuoSet from R&D systems (Minneapolis, MN), for, IL-6 (DY206) and IL-8 (DY208) according to the manufacturer's protocol.

Western blotting

Cells were collected and washed with cold PBS. Phospho-p38α (#9211) antibody and total-p38α (#9212) antibody were purchased from Cell Signaling Technology (Beverly, MA, USA). Cells were harvested in lysis buffer (ref: 895561, R&D Systems, Minneapolis, USA). The Micro BCA Protein Assay

Figures legendsFigure 1 :

 1 Figures legends

Figure 2 :

 2 Figure 2: Effects of dexamethasone on the production of IL-6 and IL-8/CXCL8 from HaCat cells treated with LPS and MSU and both LPS+MSU.HaCaT cells were treated for 24 h with LPS (1µg/ml), MSU (1mg/ml) or with LPS for 24 h followed by MSU and dexamethasone (10µM) for another 24 h. Cells were cultured in serum-free conditions. The mean ± STDV of three experiments is shown. IL-6 and IL-8/CXCL8 were measured by ELISA, * p<0.05 compared to control, α p<0.05 compared to MSU.

Figure 3 :

 3 Figure 3: Comparison of TLR and NLR gene expression in HEK and HaCat keratinocytes and THP-1 macrophages. a) The amount of mRNA detected by qPCR is indicated as follows: (-) no detectable expression or Ct value above 32, (+) normal expression between 28 and 32 Ct, and (++) under 28 Ct. RT-PCR for TLRs and NLRs in HaCaT and HEK, THP-1 was used as a positive control. b) HEK and HaCaT were stimulated for 24 h with MSU at 1mg/ml. The abundance of each TLR mRNA was quantified by qPCR and compared with that in control cells. Only detectable mRNA is shown on graphs. The mean ± STDV of three experiments is shown. * p<0.05 compared to control.

Figure 4 :

 4 Figure 4: Effect of MSU on the phosphorylation status of various kinases involved in key signaling pathways in HaCaT and kinetics of MAPK14/p38α phosphorylation. a) Cells were incubated for 45 min in serum-free medium with MSU (which corresponds to the time at which MAPK14/p38α phosphorylation is maximal). Human phosphoprotein arrays were then probed with cell extracts.The relative signal intensity is expressed as a percentage of the unstimulated control. b) Cells were incubated at different time points and activation was stopped in lysis buffer. MAPK14/p38α phosphorylation after MSU treatment was compared to that in control conditions and normalized to total MAPK14/p38α.The mean ± STDV of three experiments is shown. * p<0.05 compared to control.

Figure 5 :

 5 Figure 5: Inhibitors of MEK1/2 (U0126 monoethanolate) and MAPK14/p38α (SB 239063) significantly impair IL-8/CXCL8 production in response to MSU in HaCaT cells. Cells were incubated in serum-free conditions for 2 h in the presence of the MEK1/2 inhibitor U0126 or the MAPK14/p38α inhibitor SB 239063 and then incubated without MSU (Vehicle 1) or 24 h with MSU at 1mg/ml (vehicle 2). IL-8/CXCL8 was quantified by ELISA. The mean ± STDV of three experiments is shown. * p<0.05 compared to control. α p<0.05 compared to treatment.

Figure 6 :

 6 Figure6: Gene expression profiling of keratinocytes (HaCaT) incubated for 12 h with MSU (1mg/ml). a) Hierarchical clustering analysis of genes differentially expressed between MSU and control (CTRL) samples. Each column represents a gene. b) Gene ontology analysis classifies the differentially expressed genes according to their function. The categories containing the largest numbers of up-regulated genes are protein binding, receptor, chemokine and cytokine activity. Upregulated genes included TLR2 (included in protein binding and receptor activity categories), IL-8/CXCL8 (chemokine activity) and IL-1β (cytokine and growth factor activity). The pie chart shows the main families of up-regulated genes.

Table 1 : Deposition of Microarray Data

 1

				Genes up-regulated by MSU in keratinocytes
	GeneSymbol	FCAbsolute	p-value	EntrezGeneID RefSeqAccession GeneName
	CCL2	13.43511	0.008603666	6347	NM_002982	chemokine (C-C motif) ligand 2
	STEAP1	2.6475887	0.0032068007 26872	NM_012449	six transmembrane epithelial antigen of the prostate 1
	ABCA12	2.4267173	0.0023810326 26154	NM_173076	ATP-binding cassette, sub-family A (ABC1), member 12
	ANKK1	3.1858988	0.0027284503 255239	NM_178510	ankyrin repeat and kinase domain containing 1
	BCL3	2.0136309	0.013269338	602	NM_005178	B-cell CLL/lymphoma 3
	BIRC3	5.319158	0.017337652	330	NM_001165	baculoviral IAP repeat containing 3
	CCL17	4.020964	0.011919154	6361	NM_002987	chemokine (C-C motif) ligand 17
	CD47	2.023733	5.98002E-4	961	NM_198793	CD47 molecule
	CD83	2.6760464	0.04108729	9308	NM_004233	CD83 molecule
	CH25H	2.6119664	0.004160655	9023	NM_003956	cholesterol 25-hydroxylase
						chemokine (C-X-C motif) ligand 1 (melanoma growth stimulating
	CXCL1	3.6461887	0.0015183772 2919	NM_001511	activity, alpha)
						chemokine (C-X-C motif) ligand 1 (melanoma growth stimulating
	CXCL1	4.473043	2.130726E-4	2919	NM_001511	activity, alpha)
	CXCL2	4.432413	0.002278508	2920	NM_002089	chemokine (C-X-C motif) ligand 2
	CXCL5	8.463041	0.010513073	6374	NM_002994	chemokine (C-X-C motif) ligand 5
	CXCL6	6.2538757	0.0051556756 6372	NM_002993	chemokine (C-X-C motif) ligand 6 (granulocyte chemotactic protein 2)
	CYB5R2	2.0837421	0.023170486	51700	NM_016229	cytochrome b5 reductase 2
	DEFB4A	2.678952	0.039558947	1673	NM_004942	defensin, beta 4A
	DUSP5	2.2878127	0.012084142	1847	NM_004419	dual specificity phosphatase 5
	DUSP6	2.4933317	0.0012751031 1848	NM_001946	dual specificity phosphatase 6
	FOSL1	2.4588754	0.0061446475 8061	NM_005438	FOS-like antigen 1
	GFPT2	8.238503	0.0051870644 9945	NM_005110	glutamine-fructose-6-phosphate transaminase 2
	GOLGA7B	2.2784798	0.00750335	401647	NM_001010917	golgin A7 family, member B
	HAS3	2.03934	0.01052052	3038	NM_005329	hyaluronan synthase 3

Acknowledgments

The authors thank Ahmad Sharanek and Alex Edelman & Associates for correcting the manuscript and Tatiana Victoni for helpful advice.

Accepted Article

Kit was purchased from Pierce (Rockford, IL, USA). Acrylamide gel NuPAGE® Novex® 4-12% Bis-Tris Midi Protein Gels (Life technology, Grand Island, NY, USA) and Nitrocellulose Trans-Blot® Turbo™ Mini membranes (Biorad, Hercules, CA, USA) were used.

Human Phosphoprotein Array

Cells were harvested at 10 7 cells/ml after 45 min incubation with MSU. Protein was quantified with the BCA Protein Assay Kit from Pierce (Rockford, IL, USA). A total of 700µg of whole cell lysate was applied to the phosphoprotein array according to the manufacturer's instructions (Proteome Profiler Human Phosphokinase Array kit, R&D Systems, Minneapolis, USA).

Statistical analysis

Experiments were performed in triplicate. Results are presented as the mean ± STDV. Comparisons between the experimental and control conditions were performed with a Mann Whitney test.

For transcriptomic analysis, comparisons were made with an unpaired t-test. The cut off fold change was 2 with a p value of 0.05.

Conflict of Interest

The authors state no conflict of interest.

Accepted Article


Caroline Hillairet
email: caroline.hillairet@ensae.fr

Ying Jiao
email: ying.jiao@univ-lyon1.fr

Anthony Réveillac
email: anthony.reveillac@insa-toulouse.fr

Pricing formulae for derivatives in insurance using the Malliavin calculus *

Pricing formulae for derivatives in insurance using the

Malliavin calculus *

Introduction

Risk analysis in the context of insurance or reinsurance is often based on the study of properties of a so-called cumulative loss process L := (L t ) t∈[0,T ] over a period of time [0, T ] where T > 0 denotes the maturity of a contract. Usually, L takes the form

L t := Nt i=1 X i , t ∈ [0, T ],
where N := (N t ) t∈[0,T ] is a counting process, and the random variables (X i ) i∈N * represent the amount of the claims. A typical contract in reinsurance is the Stop-Loss contract that offers protection against an increase in either (or both) severity and frequency of a company's loss experience. More precisely, Stop-loss contracts provide to its buyer (another insurance company) the protection against losses which are larger than a given level K and its payoff function is given by a "call" function. In some cases, there is also an upper limit given by some real number M , which specifies the maximal reimbursement amount. Thus the payoff of such a contract is given by

Φ(L T ) =      0, if L T < K; L T -K, if K ≤ L T < M ; M -K, if L T ≥ M.
(1.1)

In full generality the risk carried out by the claims is neither hedgeable nor related to a financial market, hence the premium of the Stop-Loss is equal to E[Φ(L T )] which immediately re-writes as

E[Φ(L T )] = E L T 1 {L T ∈[K,M ]} -KP [L T ∈ [K, M ]] + (M -K)P [L T ≥ M ] . (1.2)
There is a large number of papers describing how to approximate the compound distribution function of the cumulated loss L T , and to compute the Stop-Loss premium. The aggregate claims distribution function can in some cases be calculated recursively, using, for example, the Panjer recursion formula, see Panjer [START_REF] Harry | Recursive evaluation of a family of compound distributions[END_REF] and Gerber [START_REF] Hans | On the numerical evaluation of the distribution of aggregate claims and its stop-loss premiums[END_REF]. Various approximations of Stop-Loss reinsurance premiums are described in the literature, some of them assuming a specific dependence structure.

In analogy with the celebrated Black-Scholes formula, we aim in this paper to express the first term of the right-hand side of (1.2) in terms of a building block which represents the distribution function of the terminal loss L T . This feature is hidden in the Black-Scholes model since the terminal value of the stock has an explicit lognormal distribution. More specifically, we aim in computing E L T 1 {L T ∈[K,M ]} by using the building block

x → P [L T ∈ [K -x, M -x]].
Note that, on the credit derivative market, the payoff function (1.1) can also be related to Collateralized Debt Obligations (CDOs) where there are several tranches, and so several K and M levels, which are expressed in proportion of the underlying which is the loss of a given asset portfolio.

Stop-Loss contracts are the paradigm of reinsurance contracts, but we aim in dealing with more general payoffs whose valuation involves the computation of the quantity

E LT h (L T ) , (1.3)
where h : R + → R + is a Borelian map and where L is of the form LT := N T i=1 Xi , involving claims Xi which are related to the ones X i of the original loss L T . To be more precise, LT will be the effective loss covered by the reinsurance company whereas L T is the loss quantity that activates the contract. Typical examples will be given in Section 2.1. Once again, this is similar to the valuation of CDOs tranches where the recovery rate is often supposed to be a random variable of beta distribution with mean 40% whereas the realized rate, often revealed only after the formal bankruptcy, does not necessarily match with this value.

In this paper we provide an exact formula for (1.3) in terms of the building block x → E [h(L T + x)] (or of a related quantity for the more general situation (1.3), see (3.4) for a precise statement). This goal will be achieved by using one of the Malliavin calculus available for jump processes. Before turning to the exposition of the model, we emphasize that this methodology goes beyond the analysis of pricing and finds for instance application in the computation of the Expected Shortfall of contingent claims in the realm of risk measures. Indeed, the expected shortfall is a useful risk measure, that takes into account the size of the expected loss above the value at risk. Formally it is defined as

ES α (-L T ) = E [-L T |-L T > V @R α (-L T )] , α ∈ (0, 1).
As it is well-known, the expected shortfall coincides with Average Value at Risk (AV@R), that is

ES α (-L T ) = AV @R(-L T ) := 1 1 -α 1 α V @R s (-L T )ds, if and only if P[-L T ≤ q + -L T (t)] = t, t ∈ (0, 1)
, where q + -L T (t) denotes the quantile of level t of -L T (see Section 2.2.2 for a precise definition). However, already in the trivial example where the size claims X i are constant equal to 1, this property fails as L T = N T is a Poisson random variable which exhibits a discontinuous distribution function. However, our approach gives an alternative explicit computation of E[L T 1 {L T <β} ] and thus of ES α (-L T ) as

ES α (-L T ) = -E[L T 1 {L T <β} ] P(L T < β) , β := -V @R α (-L T ).
We conclude this section with some comments about the modeling of the claims X i and Xi .

In the classic Cramer-Lundberg model, the claims are independent and identically distributed (i.i.d.) and in addition independent of the counting process N which happens to be an inhomogeneous Poisson process. In this work we consider a doubly stochastic Poisson process N and we allow dependency between the size of the claims, their arrivals and the intensity of N . In particular we do not assume a Markovian setting. The impact of certain dependence structure on the Stop-Loss premium is studied in the reinsurance literature, such as in Albers [START_REF] Albers | Stop-loss premiums under dependence[END_REF], Denuit et al. [START_REF] Denuit | Does positive dependence between individual risks increase stop-loss premiums?[END_REF] or De Lourdes Centeno [START_REF] De | Dependent risks and excess of loss reinsurance[END_REF], but those works usually assume dependency between the successive claim sizes and the arrival intervals. Nevertheless, in the ruin theory literature, some contributions already propose explicit dependencies among inter-arrival times and the claim sizes, such as Albrecher and Boxma [START_REF] Albrecher | A ruin model with dependence between claim sizes and claim intervals[END_REF], Boudreault, Cossette, Landriault and Marceau [START_REF] Boudreault | On a risk model with dependence between interclaim arrivals and claim sizes[END_REF] and related works. A general framework of dependencies is proposed by Albrecher, Constantinescu, and Loisel [START_REF] Albrecher | Explicit ruin formulas for models with dependence among risks[END_REF] in which the dependence arises via mixing through a so-called frailty parameter. Recently, Albrecher et al. [START_REF] Albrecher | The single server queue with mixing dependencies[END_REF] extend duality results that relate survival and ruin probabilities in the insurance risk model to waiting time distributions in the 'corresponding' queueing model. The risk processes have a counterpart in workload models of queueing theory, and a similar mixing dependencies structure is considered in a queueing context. Besides, our framework extends the mixing approach of [START_REF] Albrecher | Explicit ruin formulas for models with dependence among risks[END_REF] and [START_REF] Albrecher | The single server queue with mixing dependencies[END_REF] by allowing nonexchangeable family of random variables for the claims amounts. In a similar way, in the credit risk modeling we can also suppose that the recovery rate depend on the underlying default intensity such as in Bakshi, Madan and Zhang [START_REF] Bakshi | Understanding the role of recovery in default risk models: empirical comparisons and implied recovery rates[END_REF].

We proceed as follows. We first make clear in Section 2 our model for the loss process and present the insurance contracts for which we will propose a pricing formula. The latter will be stated and proved as Theorem 3.5 in Section 3. Particular cases of this result to several types of contracts in insurance are also given in this section. Finally, explicit examples are presented in Section 4.

Model Setup

In this section, we describe the loss process and the associated reinsurance contracts we will study. Throughout this paper, T will denote a positive finite real number which represents the final horizon time.

The Loss process

We begin by introducing the loss process L := (L t ) t∈[0,T ] where the size of claims and their arrival times are correlated. Let (N t ) t∈[0,T ] be a Cox process (also called doubly stochastic Poisson process) with random intensity (λ t ) t∈[0,T ] , whose jump times, denoted by (τ i ) i∈N * , model the arrival times of the claims. We suppose that the claim size X i depends on both the cumulated intensity defined by Λ t := t 0 λ s ds and the claim arrival time τ i . Moreover, it will also depend on some random variable ε i where we suppose that (ε i ) i∈N * is a sequence of positive i.i.d. random variables independent of the Cox process N . More precisely, the loss is given by

L t := Nt i=1 X i e -κ(t-τ i ) , with X i := f (τ i , Λ τ i , ε i ), t ∈ [0, T ], (2.1)
where κ is the discounting factor and f : R 3 + → R + is a bounded deterministic function. We provide several examples as below.

Example 2.1.

1. In the classic ruin theory, the claim size is often supposed to be independent of the arrival and the intensity process. In this case, we have f (t, ℓ, x) = x.

2. In the second example, we suppose that the dependence of f on the exogenous factor ε is linear and the linear coefficient is a function of the cumulated intensity Λ rescaled by time, i.e., Λt t , which stands for some mean level of the intensity. For instance, let

f (t, ℓ, x) = ℓ t x.
In this example, if ε i follows an exponential distribution with parameter 1, then X i = f (τ i , Λ τ i , ε i ) follows an exponential distribution with parameter

τ i Λτ i conditionally to the vector (τ i , Λ τ i ).

Generalized loss process

We can also consider a more general case where the realized claim sizes (X i ) i∈N * are not exactly the ones that are computed to activate the reinsurance contract. More precisely, assume that in addition to the factors (ε i ) i∈N * , there exists a family of i.i.d. positive random variables (ϑ i ) i∈N * which may depend on the random variables ε i 's. Let g : R 4 + → R + be a deterministic bounded function. We can define a modified cumulative loss process as

Lt := Nt i=1 g(τ i , Λ τ i , ε i , ϑ i )e -κ(t-τ i ) , t ∈ [0, T ]. (2.2)
More precisely, although the insurance contract is triggered by the loss process L, the compensation amount can depend on some other exogenous factors (ϑ i ) i∈N * . This would mean for instance that the amounts ϑ i 's are much lower than the ε i 's. A typical example is given by the housing insurance market on the American East Coast. Indeed, this region is seasonally exposed to hurricanes of different magnitudes. Most of the damages impacts the houses of the insured who may as well buy contracts on other belongings such as cars which are much less valuable. After a hurricane episode, the re-insurance Stop-Loss contract will be activated on the basis of the total damages L T on the houses (which are represented by the claims ε i ) whereas the effective damages LT will also include all other insured belongings (which would be modeled by the ϑ i ). In the special case where the function g does not depend on the fourth variable, the general loss Lt reduces to the standard loss defined in (2.1). We give below some examples of the joint distribution

(ε i , ϑ i ).
Example 2.2.

1. The first natural case is that ε i and ϑ i are independent random variables. For example, each of them can follow an exponential distribution (or Erlang distribution) with different positive parameters θ 1 and θ 2 .

2. We can introduce dependence between ε i and ϑ i by using the mixing method in [START_REF] Albrecher | Explicit ruin formulas for models with dependence among risks[END_REF]. Let ε i and ϑ i follow Pareto marginal distributions respectively and a dependence structure according to a Clayton copula (according to Example 2.3 in [START_REF] Albrecher | Explicit ruin formulas for models with dependence among risks[END_REF], this can be achieved by mixing the two Pareto marginal distributions where the mixing parameter follows a Gamma distribution).

3. Case of explicit dependence : let ε i follow a Pareto distribution and ϑ i follow a Weibull distribution with form or scaling parameter depending of ε i .

Reinsurance contracts and related quantities

Generalized Stop-loss Contrats

We have seen in the introduction the Stop-Loss contract whose payoff is given by Φ(L T ) where Φ has been defined in (1.1) and corresponds to a call spread, that is, the difference of two call functions. Our approach allows us to go beyond the case of the Stop-Loss contract. Consider now a contract where the reinsurance company pays

Φ(L T , LT ) =      0, if L T ≤ K LT -K, if K ≤ L T ≤ M M -K, if L T ≥ M , (2.3)
with LT defined in (2.2) if the a priori loss L T excesses some amount K or belongs to some interval [K, M ]. Then the price of such a contract is :

E LT 1 {L T >K} -KP [L T ∈ [K, M ]] + (M -K)P [L T ≥ M ] . (2.4)

Expected Shortfall

The expected shortfall is a useful risk measure which takes into account the size of the expected loss above the value at risk. We recall the Expected Shortfall with level α as

ES α (-L T ) = E [-L T |-L T > V @R α (-L T )] , α ∈ (0, 1).
where the definition of V @R is

V @R α (X) = -q + X (α) = q - -X (1 -α) with q + X (t) = inf{x| P[X ≤ x] > t} = sup{x| P[X < x] ≤ t} q - X (t) = sup{x| P[X < x] < t} = inf{x| P[X ≤ x] ≥ t}. It is well known that ES α (X) is equal to AV @R(X) := 1 1-α 1 α V @R s (X)ds if and only if P[X ≤ q + X (t)] = t, t ∈ (0, 1)
, which is in particular satisfied if the distribution function of X is continuous (see e.g. [10, Relation (4.38)]). However, the latter property fails already in the case where the size claims X i are constant. Thus one can not rely on the above relation and has to compute directly the conditional expectation ES α (-L T ).

We will provide an alternative expression for the expected shortfall. We denote by

β := -V @R α (-L T ), then ES α (-L T ) = -E L T 1 {L T <β} P[L T < β] where β = q + -L T (α) = inf{x| P[L T > -x] > α.
} So once again the key term to compute turns out to be the expectation E L T 1 {L T <β} .

General payoffs

More generally, we are interested in computing quantities of the form

E LT h (L T ) ,
where h : R + → R + is a Borelian map with E[h(L T )] < ∞. Since in our model, the counting process is given by a Cox process with stochastic intensity, the building block becomes the following mapping by using the conditional expectation

x → E h(L T + x)|(λ t ) t∈[0,T ] .
Note that the examples of Section 2.2.1 (respectively of Section 2.2.2) are contained in this setting by choosing h :

= 1 [K,M ] for some -∞ ≤ K < M ≤ +∞ (respectively h := 1 [-∞,β] and LT = L T ).
Our approach calls for some stochastic analysis material that we present in the next section.

The pricing formulae using the Malliavin calculus

In this section, we establish our main pricing formulae by using the Malliavin calculus. To this end, we first make precise the Poisson space associated to the loss process. Then we provide basic tools for the Malliavin calculus.

Construction of the Poisson space

The counting process and intensity process

We recall that the loss process involves the Cox process (N t ) t∈[0,T ] with its intensity and jump times, and the family of random variables (ε i ) i∈N * . We begin by introducing a general counting process which will be useful for the construction of (N t ) t∈[0,T ] on a suitable space.

Let Ω 1 be the set of (finite or infinite) strictly increasing sequences in ]0, +∞[ . We define a continuous-time stochastic process C on the set Ω 1 as

∀ (t, ω 1 ) ∈ [0, +∞[×Ω 1 , C t (ω 1 ) := card([0, t] ∩ ω 1 ). Let F C = (F C
t ) be the filtration generated by the process C, namely F C t := σ(C s , s ≤ t). It is known that there exists a unique probability measure P 1 on (Ω 1 , F C ∞ ) under which the process C is a Poisson process of intensity 1, that is, for every (s, t) ∈ [0, +∞) 2 , with s < t, the random variable C t -C s is independent of F C s and Poisson distributed with parameter t-s.

We then consider a probability space (Ω 2 , A, P 2 ) on which is defined :

(i) a positive stochastic process (λ t ) t∈[0,T ] such that T 0 λ s ds < +∞, P 2 -a.s.. (ii) a collection of i.i.d. R 2 + -valued bounded random variables (ε i , ϑ i ) i∈N * and a R 2 + -random variable (ε, ϑ) independent from (ε i , ϑ i ) i∈N * , with (ε, ϑ) L = (ε 1 , ϑ 1
) (where L = stands for the equality of probability distributions). We set µ the law of the pair (ε, ϑ). Assumption 3.1. We assume that λ is independent of (ε i , ϑ i ) i∈N * , and of (ε, ϑ).

F λ = (F λ t
) t∈[0,T ] be the right-continuous complete filtration generated by the stochastic process λ. Moreover, we set

Λ t := t 0 λ s ds, t ∈ [0, T ]. (3.1)
Let F ε,ϑ be the σ-algebra generated by (ε i ) i∈N * and (ϑ i ) i∈N * . Note that only (ε i ) i∈N * and (ϑ i ) i∈N * will be involved in the loss process and ε and ϑ are just independent copies which play an auxiliary role. We denote by µ the probability law of the couple (ε i , ϑ i ).

Assumption 3.2. Throughout this paper, we assume that : Λ T < +∞, P 2a.s..

The doubly stochastic Poisson process

We now consider the product space (Ω := Ω 1 × Ω 2 , F := F C ∞ ⊗ A, P := P 1 ⊗ P 2 ). By abuse of notation, any random variable Y on Ω 1 can be considered as a random variable on Ω which sends ω = (ω 1 , ω 2 ) to Y (ω 1 ). Similarly, any random variable Z on Ω 2 can be considered as a random variable on Ω which sends ω = (ω 1 , ω 2 ) to Z(ω 2 ).

We define a counting process N := (N t ) t∈[0,T ] on Ω by using a time change as

N t (ω 1 , ω 2 ) := C Λt(ω 2 ) (ω 1 ) = C t 0 λs(ω 2 )ds (ω 1 ), t ∈ [0, T ], (ω 1 , ω 2 ) ∈ Ω.
Note that for any t, N t is F C ∞ ⊗ F λ T -measurable random variable. Moreover, for any fixed ω 2 in Ω 2 , N t ( . , ω 2 ) is an inhomogeneous Poisson process on Ω 1 with intensity t → λ t (ω 2 ) with respect to the filtration (F C Λt(ω 2 ) ) t∈[0,T ] which reads as 1 E e iu(Nt-Ns) F λ s = E exp (e iu -1)

t s λ r dr F λ s , 0 ≤ s < t ≤ T,
where E denotes the expectation with respect to the measure P. For a process (u t ) t∈[0,T ] such that :

     u t is F-measurable, t ∈ [0, T ], for a.e. ω 2 ∈ Ω 2 , (u t (•, ω 2 )) t∈[0,T ] is (F C Λt(ω 2 ) ) t∈[0,T ] -predictable, E T 0 |u t |dt < +∞, (3.2)
we denote by

T 0 u s dN s (ω 1 , ω 2 ) the Lebesgue-Stieltjes integral of u(ω 1 , ω 2 ) against the mea- sure N (ω 1 , ω 2 ).
For any i ∈ N, we let τ i be the i-th jump time of the process N , namely

∀ ω = (ω 1 , ω 2 ) ∈ Ω, τ i (ω) := inf{t > 0, N t = C Λt(ω 2 ) (ω 1 ) ≥ i},
with the convention τ 0 = 0.

The Malliavin integration by parts formula

We can now state the Malliavin integration by parts formula on the product space. For any t ∈ [0, T ], and ω 1 ∈ Ω 1 which is of finite length or has a limit greater than t, we define ω 1 ∪ {t} in Ω 1 as the increasing sequence whose underlying set is the union of ω 1 and t. The effect of this operator is to add a jump at time t to the Poisson process N . Finally, for ω := (ω 1 , ω 2 ) ∈ Ω, and t ∈ [0, T ], we set ω ∪ {t} := (ω 1 ∪ {t}, ω 2 ), provided that ω 1 ∪ {t} is well defined. The following lemma is a direct extension of the one presented for example in [START_REF] Picard | Formules de dualité sur l'espace de Poisson[END_REF]Corollaire 5] or [START_REF] Picard | On the existence of smooth densities for jump processes[END_REF] (see also [START_REF] Privault | Stochastic analysis in discrete and continuous settings with normal martingales[END_REF]). Lemma 3.3. Let u : Ω × [0, T ] → R be a stochastic process which enjoys (3.2), and F : Ω → R be a bounded F-measurable random variable. Then the stochastic process (ω, t) → F (ω ∪ {t}) is well-defined P ⊗ dt-a.e. and

E F T 0 u s dN s F λ T ∨ F ε,ϑ = E T 0 u t F (• ∪ {t})λ t dt F λ T ∨ F ε,ϑ . (3.3)

The main result

In this section we present our main result concerning the computation of the quantity

E LT h (L T ) , 1 By a slight abuse of notation, E • F λ T := E • F C 0 ⊗ F λ T and E • F λ T ∨ F ε,ϑ := E • F C 0 ⊗ (F λ T ∨ F ε,ϑ ) .
where h : R + → R + is a Borelian map with E[h(L T )] < ∞ and where L T and LT are respectively defined in (2.1) and (2.2). We set

ϕ h λ (x) := E h(L T + x)|F λ T , x ∈ R + . (3.4)
It might be surprising at first glance to consider the conditional expectation given λ in the building block. In fact, as the intensity λ of N is random, it can be compared to a Black-Scholes model with independent stochastic volatility. In that context the Black-Scholes formula would be written in terms of the conditional law of the terminal value of the stock given the volatility (which would simply be a lognormal distribution with variance given by the volatility). Recall that for the insurance contract presented in Section 2.2.1, h := 1 [K,M ] and thus ϕ h λ coincides with the conditional distribution function of L T .

Before turning to the statement and the proof of the main result, note that

LT = T 0 Ẑs dN s , (3.5)
with

Ẑs := +∞ i=1 g(s, Λ s , ε i , ϑ i )e -κ(T -s) 1 (τ i-1 ,τ i ] (s), s ∈ [0, T ]. (3.6)
Moreover on the set {∆ s N = 0}, one has

Ẑs = g(s, Λ s , ε 1+Ns , ϑ 1+Ns )e -κ(T -s) . (3.7)
As Λ is a continuous process, Ẑ satisfies Relation (3.2), provided that

E T 0 | Ẑt |dt < +∞.
We start our analysis with the following lemma.

Lemma 3.4. Under Assumptions 3.1 and 3.2, for any t ∈ [0, T ], it holds that

g(t, Λ t , ε 1+Nt , ϑ 1+Nt )e -κ(T -t) , L T (• ∪ {t}), λ t L = g(t, Λ t , ε, ϑ)e -κ(T -t) , L T + f (t, Λ t , ε)e -κ(T -t) , λ t .
Proof. We set

L t := Nt i=1 f (τ i , Λ τ i , ε i )e -κ(T -τ i ) , L + t := Nt i=1 f (τ i , Λ τ i , ε i+1 )e -κ(T -τ i ) , t ∈ [0, T ].
We first precise the value of L T (ω ∪ {t}) for a fixed element t ∈ (0, T ) and for ω := (ω 1 , ω 2 ) in Ω such that t ∈ ω 1 and ω 1 ∪ {t} is well defined (the set of such ω has probability 1). By definition, we have that

L T (ω ∪ {t}) = N T (ω∪{t}) i=1 f (τ i (ω ∪ {t}), Λ τ i (ω∪{t}) (ω 2 ), ε i (ω 2 ))e -κ(T -τ i (ω∪{t}))
Note that one has

∀ i ∈ N, τ i (ω ∪ {t}) =      τ i (ω), if i ≤ N t (ω), t, if i = N t (ω) + 1 τ i-1 (ω), if i > N t (ω) + 1.
Therefore we can write L T (ω ∪ {t}) as the sum of three terms as follows

L T (ω ∪ {t}) = Nt(ω) i=1 f (τ i (ω 1 ), Λ τ i (ω) (ω 2 ), ε i (ω 2 ))e -κ(T -τ i (ω 1 )) + f (t, Λ t (ω 2 ), ε 1+Nt(ω) (ω 2 ))e -κ(T -t) + N T (ω)+1 i=Nt(ω)+2 f (τ i-1 (ω 1 ), Λ τ i-1 (ω) (ω 2 ), ε i (ω 2 ))e -κ(T -τ i-1 (ω 1 )) . (3.8)
By definition, the first term in the sum is just L t (ω). Moreover, by a change of index we can write the third term as

N T (ω) i=Nt(ω)+1 f (τ i (ω), Λ τ i (ω) (ω 2 ), ε i+1 (ω 2 ))e -κ(T -τ i (ω)) = L + T (ω) -L + t (ω). (3.9)
Therefore by (3.8) the following equality holds almost surely

f (t, Λ t , ε 1+Nt )e -κ(T -t) = (L T ( . ∪ {t}) -L t ) -(L + T -L + t ). (3.10)
Moreover, from the decomposition formula (3.8) we also observe that ε 1+Nt is independent of

L t + L + T -L + t given F C ∞ ⊗ F λ T .
In addition, by Assumption 3.1 the conditional law of

ε 1+Nt given F C ∞ ⊗ F λ T identifies with the law of ε since F ε is independent of F λ T .
We now compute the characteristic functions of the two random vectors of interest. Let χ be the characteristic function of the random vector

g(t, Λ t , ε 1+Nt , ϑ 1+Nt )e -κ(T -t) , L T (• ∪ {t}), λ t . Let (u 1 , u 2 , u 3 ) ∈ R 3 . One has χ(u 1 , u 2 , u 3 ) := E e iu 1 g(t,Λt,ε 1+N t ,ϑ 1+N t )e -κ(T -t) +iu 2 L T ( . ∪{t})+iu 3 λt = E e iu 3 λt e iu 1 g(t,Λt,ε 1+N t ,ϑ 1+N t )e -κ(T -t) +iu 2( Lt+e -κ(T -t) (f (t,Λt,ε 1+N t )+L + T -L + t ) = E e iu 3 λt e iu 2 (Lt+L + T -L + t ) e iu 2 e -κ(T -t) f (t,Λt,ε 1+N t ) e iu 1 e -κ(T -t) g(t,Λt,ε 1+N t ,ϑ 1+N t ) .
Since ε 1+Nt and ϑ 1+Nt are independent of

L t + L + T -L + t given F C ∞ ⊗ F λ T , we obtain that χ(u 1 , u 2 , u 3 ) = E e iu 3 λt e iu 2 e -κ(T -t) f (t,Λt,ε) e iu 1 e -κ(T -t) g(t,Λt,ε,ϑ) E e iµ 2 (Lt+L + T -L + t ) F C ∞ ⊗ F λ T ,
where we also use the fact that the probability law of (ε 1+Nt , ϑ 1+Nt ) given F C ∞ ⊗ F λ T coincides with µ (which, we recall, is the probability law of (ε, ϑ)). Moreover, from (3.9) we observe that

L t + L + T -L + t has the same law as L T conditioned on F C ∞ ⊗ F λ T . Therefore, we obtain χ(u 1 , u 2 , u 3 ) = E[e iu 3 λt e iu 2 e -κ(T -t) f (t,Λt,ε) e iu 1 e -κ(T -t) g(t,Λt,ε,ϑ) e iu 2 L T ] = E[e iu 2 e -κ(T -t) f (t,Λt,ε) e iu 1 (e -κ(T -t) g(t,Λt,ε,ϑ)+L T ) e iu 3 λt ],
which shows that χ coincides with the characteristic function of the vector

g(t, Λ t , ε, ϑ)e -κ(T -t) , L T + f (t, Λ t , ε)e -κ(T -t) , λ t .
The lemma is thus proved.

We now turn to the statement and the proof of the main result of this paper.

Theorem 3.5. Recall that (ε i , ϑ i ) i∈N * and (ε, ϑ) are i.i.d. with common law µ. Under the Assumptions 3.1 and 3.2, it holds that

E LT h (L T ) = T 0 e -κ(T -t) E g(t, Λ t , ε, ϑ) λ t ϕ h λ f (t, Λ t , ε)e -κ(T -t) dt = R 2 + T 0 e -κ(T -t) E g(t, Λ t , x, y) λ t ϕ h λ f (t, Λ t , x)e -κ(T -t) µ(dx, dy) dt, (3.11)
where LT is defined in (2.2) and the mapping ϕ h λ (x) := E h(L T + x)|F λ T is defined in (3.4). Proof. Assumptions 3.1 and 3.2 are in force. Using the relation (3.5) and the integration by parts formula on the Poisson space (3.3), it holds that

E LT h (L T ) = E E LT h (L T ) F ε,ϑ ∨ F λ T = E E h (L T ) T 0 Z t dN t F ε,ϑ ∨ F λ T = E T 0 Z t h (L T (• ∪ {t})) λ t dt
By Relation (3.7) and the fact that the set {∆ t N = 0} is negligeable, we obtain

E LT h (L T ) = E T 0 g(t, Λ t , ε 1+Nt , ϑ 1+Nt )e -κ(T -t) h (L T (• ∪ {t})) λ t dt = T 0 E g(Λ t , ε 1+Nt , ϑ 1+Nt )e -κ(T -t) h (L T (• ∪ {t})) λ t dt.
Finally, by Lemma 3.4, the above formula leads to -t) .

E L T 1 {L T ∈[K,M ]} = T 0 E g(t, Λ t , ε, ϑ)e -κ(T -t) h L T + f (t, Λ t , ε)e -κ(T -t) λ t dt. Since ε is independent of F λ T ∨ F ε , one has E h L T + f (Λ t , ε)e -κ(T -t) F λ T ∨ σ(ε) = ϕ h λ f (t, Λ t , ε)e -κ(T
Therefore

E LT h (L T ) = T 0 E g(t, Λ t , ε, ε)e -κ(T -t) λ t ϕ h λ f (t, Λ t , ε)e -κ(T -t) , f (t, Λ t , ε)e -κ(T -t) dt = R 2 + T 0 e -κ(T -t) E g(t, Λ t , x, y) λ t ϕ h λ f (t, Λ t , x)e -κ(T -t) dt µ(dx, dy),
as asserted by the theorem.

Remark 3.6.

1. Note that from Equality (3.11), it is clear that our approach only requires the knowledge of the conditional law of L T given λ (via the mapping ϕ λ ) and not the one of the pair (L T , LT ). This seems to be particularly useful for the numerical approximation of the aforementioned expectation.

2. The theorem above provides us the relation of the pricing formula with respect to the intensity process (λ t ) t≥0 of the counting process.

Relation (3.11) allows us to give a lower (respectively upper) bound on the price if h is assumed to be convex (respectively concave).

Corollary 3.7. Under the assumptions of Theorem 3.5, it holds that :

(i) if h is convex, then E LT h (L T ) ≥ R 2 + T 0 e -κ(T -t) E g(t, Λ t , x, y) λ t h E[L T F λ T ] + f (t, Λ t , x)e -κ(T -t) µ(dx, dy) dt. (i) if h is concave, then E LT h (L T ) ≤ R 2 + T 0 e -κ(T -t) E g(t, Λ t , x, y) λ t h E[L T F λ T ] + f (t, Λ t , x)e -κ(T -t) µ(dx, dy) dt.
Proof. We prove (i) as statement (ii) follows the same line. As h is assumed to be convex, Jensen's inequality implies that

ϕ h λ (x) ≥ h E[L T F λ T ] + x , x ∈ R + .
The result is then obtained by plugging this estimate in Relation (3.11).

Applications and examples

In this section, we provide some application examples of our main result, in particular for the (generalized) stop-loss contract. Such explicit computations will also be useful for the CDO tranches and expected shortfall risk measure.

Computation of the building block

We first focus on the building block ϕ h λ (defined in (3.4)) when h := 1 {[K,M ]} :

ϕ λ (x) := ϕ h λ (x) = P L T ∈ [K -x, M -x]|F λ T , x ∈ R +
which corresponds to the payoff of a stop-loss contract or a CDO tranche. Let F ε := σ(ε i , i ∈ N * ). For any i ∈ N * , we set X i := f (τ i , Λ τ i , ε i ) and x in R + , we have

P L T ∈ [K -x, M -x] F λ T ∨ F ε = P N T i=1 X i e κτ i ∈ [(K -x)e κT , (M -x)e κT ] F λ T ∨ F ε = +∞ k=1 E k i=1 X i e κτ i ∈ [(K -x)e κT , (M -x)e κT ] N T = k, F λ T ∨ F ε P[N T = k|F λ T ] = +∞ k=1 e -T 0 λsds S k P k i=1 X i e κt i ∈ [(K -x)e κT , (M -x)e κT ] F λ T ∨ F ε λ t 1 dt 1 • • • λ t k dt k = +∞ k=1 e -T 0 λsds S k R k + 1 { k i=1 x i e κt i ∈[(K-x)e κT ,(M -x)e κT ]} L |λ X (1:k) (dx 1 , . . . , dx k )λ t 1 dt 1 • • • λ t k dt k , (4.1)
where

S k := {0 < t 1 < • • • < t k ≤ T }, X (1:k) := (X 1 , . . . , X k ) and L |λ X (1:k) (dx 1 , . . . , dx k ) := P X (1:k) ∈ (dx 1 , . . . , dx k ) F λ T .
It just remains to compute the joint distribution of the claims X (1:k) in different situations.

In particular, we provide below an explicit example.

Model on ε i : We assume that (ε i ) i∈N * are i.i.d. random variables with Pareto distributions

P(α ε , β ε ) with (α ε , β ε ) ∈ (R * + ) 2 whose density ψ ε is defined as ψ ε (z) = β ε α βε ε z βε+1 1 {z≥αε} dz. Choosing f (t, ℓ, x) := ℓ t x, the conditional distribution L |λ X (1:k) (dx 1 , . . . , dx k ) in Relation (4.1) becomes L |λ X (1:k) (dx 1 , . . . , dx k ) = P Λ t 1 t 1 ε 1 , • • • , Λ t k t k ε k ∈ (dx 1 , . . . , dx k ) F λ T = k i=1 P Λ t i t i ε i ∈ dx i F λ T = k i=1   βε t i Λt i α ε βε z βε+1 i    1 z i ≥ t i Λ t i αε dz i .
The next step to compute the right-hand side of Relation (3.11) is to specify the joint law of (ε 1 , ϑ 1 ).

Model on (ε i , ϑ i ) : We assume that (ε i , ϑ i ) i∈N * are i.i.d. random vectors, with marginal distributions following Pareto distributions P(α ε , β ε ) and P(α ϑ , β ϑ ) (for a set of parameters α ε , β ε , α ϑ , β ϑ > 0) respectively. The dependence structure is modeled through a Clayton copula with parameter θ > 0. We recall that the Clayton copula is

C(u, v) := u -θ + v -θ -1 -1 θ
and the density c of the Clayton copula is given by

c(u, v) := (1 + θ)(uv) -1-θ (u -θ + v -θ -1) -1 θ -2 .
The joint distribution of (ε 1 , ϑ 1 ) is then given by

µ(dx, dy) = c (F ε (x), F ϑ (y)) ψ ε (x)ψ ϑ (y)dxdy, with F ε (z) = 1 -α βε ε z βε and F ϑ (z) = 1 - α β ϑ ϑ z β ϑ .
Joint law of (λ t , Λ t ) : The final step in the computation of Relation (3.11) is to make precise the joint law of (λ t , Λ t ). More precisely, we need to compute

E g(t, Λ t , x, y) λ t ϕ λ K -f (t, Λ t , x)e -κ(T -t) , M -f (t, Λ t , x)e -κ(T -t)
Assume the intensity process (λ t ) t∈[0,T ] is given by

λ t = λ 0 exp(2βW t )
where W is a Brownian motion, and β a constant (non null). Then the cumulative intensity is

Λ t = λ 0 t 0 exp(2βW s )ds.
By Borodin and Salminen [START_REF] Andrei | Handbook of Brownian motion-facts and formulae[END_REF] (page 169), the joint law of (Λ t , W t ) is given by In that case we omit the dependency on Λ for the mappings f and g (as Λ t = tλ 0 ). e -κ(T -t) g(t, x, y) ϕ λ 0 f (t, x)e -κ(T -t) µ(dx, dy)dt, (recall that µ := L (ε,ϑ) ).

P (Λ t ∈
If we assume furthermore that f (t, x) = g(t, x, y) = x and κ = 0, then the loss process L T corresponds to the cumulated loss of the classic Cramer-Lundberg model. In this context, a huge literature deals with the computation of the ruin probability and related quantities, such as the discounted penalty function at ruin (Gerber-Shiu function). Others papers are concerned with the pricing of Stop-Loss contract. The pricing relies on the computation of a term of the form M K ydF (y) with F being the cumulative distribution function of the loss process L T , and the discussion in the literature mainly focuses on the derivation of the compound distribution function F (usually calculated recursively, using Panjer recursion formula and numerical methods/approximations) cf. [START_REF] Harry | Recursive evaluation of a family of compound distributions[END_REF] and [START_REF] Hans | On the numerical evaluation of the distribution of aggregate claims and its stop-loss premiums[END_REF]. Our Malliavin approach provides another formula which reads as

E LT 1 L T ∈[K,M ] = λ 0 T R +
x (F (Mx) -F (Kx))µ(dx).

(4.3)

Note that our result indeed coincides with the one obtained in [START_REF] Hans | On the numerical evaluation of the distribution of aggregate claims and its stop-loss premiums[END_REF]. Indeed, if one translates in a general setting Formula [11, (6)] (as µ is constrained to have a finite support in N in [START_REF] Hans | On the numerical evaluation of the distribution of aggregate claims and its stop-loss premiums[END_REF]) the distribution F satisfies

4. 2 A

 2 Black-Scholes type formula for generalized Stop-Loss contracts in the Cramer-LundbergAs an illustration, we conclude our analysis by specifying our result in the classic Cramer-Lundberg model. More precisely, we assume that the Cox process is an homogeneous Poisson process with constant intensity λ 0 > 0 and set h := 1 [K,M ] , with K < M . The building block reduces to the distribution functionϕ λ 0 (x) := ϕ h λ 0 (x) = P [L T ∈ [Kx, Mx]] , x ∈ R + . (4.2)

Corollary 4 . 1 .

 41 Under the assumptions of Theorem 3.5, it holdsE LT 1 L T ∈[K,M ] = λ 0

  ydF (y) = λ 0 T R + xdF (yx)µ(dx), from which one deduces that M K ydF (y) = λ 0 T M K R + xµ(dx)dF (yx) = λ 0 T R + x M K dF (yx)µ(dx) = λ 0 T R + x(F (Kx) -F (Mx))µ(dx),which is exactly (4.3).

  dv, W t ∈ dz) =

						λ 0 |β| 2v	exp -	λ 0 (1 + e 2βz ) 2β 2 v	i β 2 t/2	λ 0 e βz β 2 v	1 {v>0} dvdz
	where the function
			i y (z) =	ze π √ πy π 2 4y	0	∞	exp -zch(x) -	x 2 4y	sh(x) sin(	πx 2y	)dx.
	=	R 2	g(t, v, x, y)ϕ λ K -	v t	xe -κ(T -t) , M -	v t	xe -κ(T -t) λ 2 0 |β| 2v	e 2βz exp -	λ 0 (1 + e 2βz ) 2β 2 v	i β 2 t/2	λ 0 e βz β 2 v	1 {v>0} dvdz.

The expectation term in the right-hand side of equation

(3.11)

is then

E g(t, Λ t , x, y) λ t ϕ λ Kf (t, Λ t , x)e -κ(T -t) , Mf (t, Λ t , x)e -κ(T -t)

The authors acknowledge Projet PEPS égalité (part of the European project INTEGER-WP4)