Daniel Sajan

Rajesh George
email: george@csi.tu-darmstadt.de

K G K Komban

P Warrier

V P N Radhakrishnan

C P G Nampoori

Vallabhan

Sajan ; George

K G K ; Warrier

Sajan D George

Rajesh Komban

P Radhakrishnan

V P N Nampoori

C P G Vallabhan

Thermal characterisation of nanocrystalline porous CePO4 ceramics

Keywords: ceramics, nanostructured ceramics, phonons, thermal transport ceramics, nanostructured ceramics, phonons Photoacoustic, nanocrystalline ceramics, sol-gel method, thermal diffusivity

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Introduction

Research towards a better understanding of the physical properties of heterogeneous solids has both scientific and technological importance [1][2]. In recent years, rare earth phosphates are becoming increasingly important for several applications which include ceramic materials, catalysts, photoluminescence materials, dielectric substrates, metal surface treatment agents, optical materials, etc [3][4][5][6]. Other potential applications of these materials include host matrices for immobilization of actinide radio nuclear wastes, solid protonic conductors and weak fiber matrix interfaces in ceramic-ceramic composites or thermal protection coatings [7][8][9][10]. Of late, synthesis of nanocrystalline rare earth phosphates has been attempted via various methods such as sol-gel method, surfactant directed hydrothermal approach, etc [11][12].

Cerium phosphate is an important class of rare earth phosphate ceramic, which has immense use in high temperature applications due to its high melting point. Apart from its potential as storage material for nuclear waste and poison to automobile catalyst, recently, it has been used as proton conduction membrane in hydrogen fuel cells [13]. However, the chemical and physical properties of these ceramics depend greatly on the processing technique and conditions. Although various aspects of nanocrystalline ceramics are already investigated, measurement of thermal parameters of sol-gel prepared nanocrystalline CePO 4 is not investigated in detail.

Since the proposal of R-G theory of photoacoustic effect in solids [14], photothermal methods have been used for determining several material properties in different states of matter where conventional spectroscopic methods fail [15][16][17]. Amongst various photothermal techniques, photoacoustic method (PA) has become more popular due to it simplistic approach. The PA effect directly looks into the heat generated in the sample due to nonradiative de-excitation processes following an optical excitation of the sample with modulated or pulsed light [17]. Data on the thermal properties are vital for any material employed in devices with thermal dissipation. Thermal diffusivity is a unique and important thermophysical parameter, which measures the rate of diffusion of heat in a material.

Physically, the inverse of thermal diffusivity is a measure of the time required to establish thermal equilibrium in a system in which transient temperature change has occurred [18][19][20][21].

As a result, thermal diffusivity is directly related to micro structure and thereby the processing conditions of ceramics.

In our previous work, we reported thermal characterization of porous LaPO 4 ceramics using photothermal technique [21]. In this paper, work is focused on the evaluation of thermal diffusivity of nanocrystalline CePO 4 prepared via sol-gel process and the influence of sintering temperature on thermal diffusivity. Sintering is the process whereby interparticle pores in a granular material are eliminated by atomic diffusion driven by capillary forces. The process of sintering can profoundly influence the porosity of nanocrystalline ceramics and thereby the heat transport through materials. Porous ceramics are receiving considerable attention due to its applications in variety of areas such as thermal and noise isolation, liquid and gas infiltration, catalyst products, electronics and biomedical applications [22][23][24]. Hence, a detailed investigation of the influence of sintering on porosity and hence on thermal diffusivity is of great relevance, especially from the device fabrication point of view.

Sample Preparation

Cerium nitrate, 99.9% pure (M/s Indian Rare Earths Ltd., India) and orthophosphoric acid (AR grade, 88%, Qualigens Fine Chemicals, India) are used as starting materials. A 0.03 M (9.239 g) solution of Ce(NO 3 ) 3 .6H 2 O is prepared in de-ionized water. Orthophoshporic acid solution (2,084g, 1.191 mL) is added drop-wise to the cerium nitrate solution under constant stirring. An off-white precipitate of cerium phosphate was obtained. The precipitate is filtered and washed free of nitrates and phosphates, and is further re-dispersed in de-ionized water. Electrostatic stabilization is achieved by the addition of 20% nitric acid, with the pH maintained in the range of 1.8 -2. The suspension is stirred continuously for about 4 hours.

The colloidal sol thus obtained is further subjected to ultrasonication for 15 minutes to obtain stable cerium phosphate precursor sol. Gelation of the sol is carried out by exposing to ammonia atmosphere for a period of two days. Cerium phosphate gel thus obtained is dried at 80 0 C and calcined at 400 0 C at the rate 10 0 C/min under air atmosphere to yield Cerium Phosphate gel powder. The relative density of cerium phosphate sintered at different temperatures has been measured using Archimedes displacement method. The porosity of specimens having relative density is evaluated knowing that [25], and is reported in Table I.

The phase identification is carried out by XRD (Philips PW 1710) on the calcined powders using Cu Kα radiation (Figure 1). The gel calcined at 400 0 C shows the transformation of rhabdophane to hexagonal form. The major hexagonal cerium phoshpate is identified with JCPDS file no 4-362 and the presence of rhabdophane (JCPDS file no 35-0614) in the XRD pattern also shows the incomplete transformation of hexagonal phase.

Sample calcined at 800 0 C shows broad peaks of monazite type monoclinic cerium phosphate (JCPDS file no 32-0199) which became sharp above this temperature. The crystallite size calculated using X ray line broadening method using Scherrer equation , where is crystalline size, is wavelength of radiation (1.54 for Cu Kα radiation), is the corrected peak width at half maximum intensity, and θ is the angle of diffraction at the peak position, is shown in

Theoretical background

According to one dimensional model of Rosencwaig and Gersho [14], the pressure variation at the front surface of an optically thick where is the optical absorption length) sample irradiated with a chopped beam of monochromatic radiation depends on the thermal diffusivity of the sample. The theoretical expression for may be written as

In the above expression

Where and

is the ratio between effusivities of backing material ( and the sample ( ) and and ( 6)

Here and indicates the length and thermal conductivity of the corresponding part of the geometry shown in Figure 4 where the subscripts and refers to gas, sample and backing respectively. is the ambient pressure (temperature), is the specific heat ratio of air.

is the absorbed power of optical radiation. The effusivity of the gas can be neglected in comparison to the effusivity of sample, since their ratio is always less than 1%. The term X depends on the modulation frequency through the product , which can be written as

where characteristic frequency is given by (8) When thermal diffusion length is greater than sample thickness, thermal properties of the backing material determine the PA signal. But in the thermally thick regime, PA signal is independent of the thermal properties of backing material. For a given sample thickness, there is a transition from thermally thin regime to thermally thick regime with variation in chopping frequency. Such a transition is exhibited in the amplitude spectrum of PA signal, which is evinced by the change in slope and by knowing the transition frequency and thickness of the specimen; thermal diffusivity can be evaluated using equation (8).

Results and Discussions

Initially, the experimental setup used for the investigation is calibrated by evaluating the thermal diffusivity of GaAs and Al. The measured values of thermal diffusivity (0.260 ± 0.03 cm 2 s -1 and 0.978 ± 0.03 cm 2 s -1 respectively) agree well with earlier reported values [18,27]. Figure 5 shows the amplitude spectrum of PA signal for sample sintered at resistance, etc and thereby affect the propagation of thermal energy carriers [28][29][30][31][32]. The appearance of porosity amounts to a new phase in ceramic materials prepared through the gel route. In the case of porous ceramics, heat transport could be primarily due to two processes, phonon and radiative transport [33]. The former relates to the movement of atoms in the crystalline lattice, whereas the latter is due to electron transitions between energy levels in these atoms. If we neglect the relation between energy levels and spacing between the atoms, then both mechanisms can be considered independently. Such an assumption seems to be reasonable if the amplitude of atomic oscillations is considerably smaller than the lattice constant [22]. In general, the radiative contribution arises from photon emission caused by electron transport. However, in the case of dielectrics at room temperature, the density of electrons is negligibly small and thermal energy is carried mainly by phonons. The influence of pores on thermal conductivity value has already been investigated [31][32][33]. For cases in which the damping wavelength of thermal waves is greater than or of the order of typical grain size (L), "thermal disturbance" is not affected by the heterogeneity of sample and the measurements then yield the "global thermal diffusivity" [START_REF] Litoysky | Trends in Heat, Mass and Momentum Transfer[END_REF], which is the case for all samples in the present study. However, the pores in the lattice can act as scattering centres for phonons and hence affect the phonon mean free path and consequently their thermal As reported earlier [28][29][30], density variations and hence porosity variations caused by sintering temperature of samples under investigation can affect the thermal conductivity value. By considering porous ceramics as a two-phase network in which pores are randomly embedded in a solid matrix, the thermal diffusivity of the specimen is given by the expression,

, where is the thermal conductivity of the specimen having porosity . and are the density and specific heat capacity of the sample with zero porosity [START_REF] Kerrisk | [END_REF].

If the thermal conductivity of the porous material is modified by the same ratio as that of the ratio between actual density and maximum density, then thermal conductivity is given by Leob equation [36],

. However, such a methodology to evaluate thermal conductivity yields thermal diffusivity value that is independent of porosity, which contradicts our experimental observation. All the studies done on samples prepared via sol-gel route show a dependence of thermal diffusivity value on porosity. A similar influence of porosity on heat transport of ceramics has already been investigated [37]. In order to incorporate the influence of porosity on the propagation of thermal waves and hence on the measured thermal diffusivity values, Sanchez-Lavega et.al [38] samples under investigation on porosity indicates that the samples are below their percolation threshold. Percolation threshold is the limit above which a specific physical property (e.g, thermal diffusivity, hardness, stiffness etc) is insensitive to further changes in relative density.

Above percolation threshold, particle-particle contact is insufficient to fully transmit the physical forces [23].

The calculated value of empirical constant (=1.98) > 1 suggests that the effect of porosity on heat conduction in all the samples under study is not a mere density effect (air holes in the bulk specimen). A value of suggests that thermal diffusivity value becomes equal to and thus thermal diffusivity value is independent of porosity. A value of suggests that porous ceramics can exhibit higher thermal diffusivity value compared to samples with zero porosity, which contradicts our experimental observation as well as the scattering mechanisms in porous ceramics. Evaluation of thermal conductivity using the relation thus affects the propagation of phonons through the specimen, apart from porosity (or density). In the case of larger grain sized nanocrystals, majority of heat is carried away by well defined quanta of propagating and polarized vibrational modes (i.e phonons). Hence the actual value of thermal parameters is determined by the mean free path of phonons between the scattering events. Mean free path of phonons are essentially determined by various scattering mechanisms in ceramics such as phonon-phonon scattering, phonon-electron scattering, phonon-grain boundary scattering, phonon-defect scattering, etc [30]. In the case of samples investigated here, phonon-electron scattering can be neglected as the densities of free electrons are negligibly small. However, grain boundaries affect the propagation of phonons and hence their thermal diffusivity value. If the grain boundary has a significant thickness, phonons will be scattered a few times inside the grain boundary phase and the thermal conductivity of grain boundary phase plays a significant role in determining the effective thermal parameters. If the grain boundaries are thin enough, phonons may scatter once at the grain boundary and then the phonon scattering sites in sample grains will control the effective thermal parameters of sintered samples [39]. SEM analysis of our samples sintered at 1300 0 C (Figure 7) and 1500 0 C (Figure 8) shows that grain size increases from 1 µm to 2 µm for samples sintered at 1300 0 C and 1500 0 C respectively. As the sintering temperature increases, grain size increases resulting in reduction in grain boundaries and a consequent increase in phonon mean free path and hence thermal diffusivity value. A recent investigation shows that for dense samples, the phonon mean free path is smaller than the grain size at all processing temperatures [40]. Hence, in addition to porosity, the defects in grains also contribute to the variation in thermal diffusivity [41] and this explains the value of n >1. An investigation on porous ceramic alumina shows that the solid area fraction (SAF) in the direction of heat flow also affects the measured thermal diffusivity value [23]. Enhancement in solid area fraction with increase in sintering temperature results in cohesive thermal transport with reduced scattering due to defects and interfaces. Thermal contact resistance at the interface is related to macroscopic thermal conductivity ( and to the grain conductivity through the relation , where is the grain size [42]. Increase in sintering temperature results in enhanced contribution from grain conductivity due to more ordered structure and reduction in interface thermal resistance.

Conclusions:

In this article, we measured the thermal diffusivity value of CePO

Figure 2 .

 2 The crystallite size ranges from 10 to 40 nm and it increases with sintering temperature. The calcined powder was compacted at a pressure of 200 MPa to pellets of size 11 mm and thickness around 500 µm and sintered in the range of 800 to 1700 0 C, at a heating rate of 10 0 C/min and soaked for 3hours. BET surface area and pore size distribution of the gel after calcination at 400 and 800 °C were adsorption after degassing the powders at 200 °C for 4h using Micromeritics Gemini 2375 V5.01 surface area analyzer. The pore size distribution was obtained by Barrett-Joiner-Halenda (BJH) method from the desorption curve of the isotherm using the Micromeritics StarDriver version 2.03 software. Experimental A schematic view of experimental set up is shown in Figure 3. Optical radiation from an Argon ion laser at 488 nm (Liconix 5300) is used as the source of excitation, and it is intensity modulated using a mechanical chopper (Stanford Research Systems SR 540) before it strikes the sample surface. Detection of pressure fluctuations (photoacoustic signal) in the cell cavity is done using a sensitive microphone (Knowles BT 1754). The amplitude of the photoacoustic signal is measured using a dual phase lock-in amplifier (Stanford Research Systems SR 830). The laser power used for the present studies is 50 mW with a stability of ± 0.5%. A homemade PA cell is employed for the present studies. The present cell is made using an acrylic polymer (Perspex) and the volume of the cell is designed such that PA amplitude is maximum but still large enough to avoid thermoviscous damping of acoustic signals at the cell wall. The PA cell is fabricated on the cell material with 5cm diameter and 1cmthickness by drilling bore of diameter 3mm, along its thickness direction, through its center. Another fine bore of diameter 1.5 mm pierced at the middle of the main chamber and directed perpendicular to it serves as the acoustic coupler between the main chamber and the microphone. At a distance of 8mm from the main chamber the microphone is fixed to the orifice of side tube. In this case, the thermal diffusion length in air is smaller than acoustic cell diameter and thus one dimensional model of R-G theory can be used. More details about the set up can be found elsewhere[26].

  800 0 C. All other samples show similar behavior (not shown here). The thermal diffusivity values of cerium phosphate sintered at different temperatures was evaluated from the amplitude spectrum of PA signal and is given in TableI. From the values, it is obvious that sintering temperature affects porosity of the specimen and thereby influences the thermal diffusivity value. Thermal transport properties of polycrystalline dielectric materials such as investigated here are directly influenced by materials composition, structure and arrangement of phases. In the case of heterogenous materials such as ceramics, local variation in thermal parameters can occur because of the presence of different solid phases, pores, boundary

  diffusivity value. Results of pore size distribution and N 2 adsorption characteristics of cerium phosphate gel after calcination at 400 0 C and 800 0 C are given in Figures6 (a) and 6 (b) respectively. The presence of micropores (<20 Å) with extended mesoporosity (20-500 Å) is seen in calcined gels. On heating to 800 °C volume of smaller pores up to around 100 Å decreases considerably and correspondingly decrease in surface area from 96 to 49 m 2 /g is observed when the gel is heated at 400 0 C and 800 0 C respectively. The BET isotherms with characteristic loop clearly indicate a type IV, i.e., mainly mesoporous textures, although at relatively low pressure (P/P0) it shows microporous character. Compared with 800 o C calcined sample, the volume of adsorption of N 2 is much higher in the case of 400 o C all relative pressure. Generally, as the pore sizes are much smaller than the damping wavelength of thermal waves, the measured values in this study are the effective thermal parameters of specimen.

Figure 1 .

 1 Figure 1. M -monazite, H -hexagonal, R-rhabdophane,

Figure 2 .Figure 3 .Figure 4 .Figure 5 .Figure 6 Figure 7 .Figure 8 .

 2345678 Figure 2. Plot of crystalline size versus temperature

Figure 1

 1 Figure 1.

Table 1 .

 1 thermal resistance. The present investigation also shows that, by measuring thermophysical parameters, the simple and elegant PA technique can throw light into structural variations of ceramics caused by variations in processing conditions. Such tunability in thermal properties with processing conditions and evaluation of thermal diffusivity of these materials could find applications in the microelectronic and optolectronic industry. Thermal diffusivity of CePO 4 ceramics

	4 prepared by the sol-gel

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Vu Thanh
email: thanh@ceremade.dauphine.fr

Nam Ceremade

Optimal control of direction of reflection for jump diffusions

Keywords: Optimal control, Dynamic programming, Skorokhod problem, discontinuous viscosity solutions MSC Classification (2000): 93E20, 60G99, 49L25, 91B28

We extend the optimal control of direction of reflection problem introduced in Bouchard [4] to the jump diffusion case. In a Brownian diffusion framework with jumps, the controlled process is defined as the solution of a stochastic differential equation (SDE) reflected at the boundary of a domain along oblique directions of reflection which are controlled by a predictable process which may have jumps. We also provide a version of the weak dynamic programming principle of Bouchard and Touzi [5] adapted to our context and which is sufficient to provide a viscosity characterization of the associated value function without requiring the usual heavy measurable selection arguments nor the a-priori continuity of the value function.

Introduction

The aim of this paper is to study a class of optimal control problem for reflected processes on the boundary of a bounded domain O, whose direction of reflection can be controlled. When the direction of reflection γ is not controlled, the existence of a solution to reflected SDEs was studied in the case where the domain O is a half space by El Karoui and Marchan [START_REF] Chaleyat-Maurel | Reflexion discontinue et systeme stochastiques[END_REF], and, Ikeda and Watanabe [START_REF] Ikeda | Stochastic differential equations and diffusion processes[END_REF]. Tanaka [START_REF] Tanaka | Stochastic differential equations with reflecting boundary condition in convex region[END_REF] considered the case of convex sets. More general domains have been discussed by Dupuis and Ishii [START_REF] Dupuis | SDEs with oblique reflection on nonsmooth domains[END_REF], where they proved the strong existence and uniqueness of solutions in two cases. In the first case, the direction of reflection γ at each point of the boundary is single valued and varies smoothly, even if the domain O may be non smooth. In the second case, the domain O is the intersection of a finite number of domains with relatively smooth boundaries. Motivated by applications in financial mathematics, Bouchard [START_REF] Bouchard | Optimal reflection of diffusions and barrier options pricing under contraints[END_REF] then proved the existence of a solution to a class of reflected SDEs, in which the oblique direction of reflection is controlled. This result is restricted to Brownian SDEs and to the case where the control is a deterministic combination of an Itô process and a continuous process with bounded variation. In this paper, we extend Bouchard's result to the case of jump diffusion and allow the control to have discontinuous paths.

As a first step, we start with an associated deterministic Skhorokhod problem:

ϕ(t) = ψ(t) + ∫ t 0 γ(ϕ(s), ε(s))1 ϕ(s)∈∂O dη(s), ϕ(t) ∈ O, (1.1)
where η is a non decreasing function and γ is controlled by a control process ε taking values in a given compact set E of R l . Bouchard [START_REF] Bouchard | Optimal reflection of diffusions and barrier options pricing under contraints[END_REF] proved the strong existence of a solution for such problems in the family of continuous functions when ε is a continuous function with bounded variation. Extending this result, we consider the Skhorokhod problem in the family of càdlàg functions with finite number of points of discontinuity. The difficulty comes from the way the solution map is defined at the jump times.

In this paper, we will investigate on a particular class of solutions, which is parameterized through the choice of a projection operator π. If the value ϕ(s-) + ∆ψ(s) is out of the closure of the domain at a jump time s, we simply project this value on the boundary ∂O of the domain along the direction γ.

The value after the jump of ϕ is chosen as π(ϕ(s-) + ∆ψ(s), ε(s)), where the projection π along the oblique direction γ satisfies y = π(y, e) -l(y, e)γ(π(y, e), e), for all y / ∈ Ō and e ∈ E, for some suitable positive function l. This leads to ϕ(s) = (ϕ(s-) + ∆ψ(s)) + γ(ϕ(s), ε(s))∆η(s), with ∆η(s) = l(ϕ(s-) + ∆ψ(s), ε(s)). When the direction of reflection is not oblique and the domain O is convex, the function π is just the usual projection operator and l(y) coincides with the distance to the closure of the domain Ō.

We next consider the stochastic case. Namely, we prove the existence of an unique pair formed by a reflected process X ε and a non decreasing process L ε satisfying { X(r) = x + ∫ r t F (X(s-))dZ s + ∫ r t γ(X(s), ε(s))1 X(s)∈∂O dL(s), X(r) ∈ Ō, for all r ∈ [t, T ]

(1.2)

where Z is the sum of a drift term, a Brownian stochastic integral and an adapted compound Poisson process, and the control process ε belongs to the class E of E-valued càdlàg predictable processes with bounded variation and finite activity. As in the deterministic case, we only study a particular class of solutions, which is parameterized by π. This means that whenever X is not in the domain O because of a jump, it is projected on the boundary ∂O along the direction γ and the value after the jump is also chosen as π (X(s-) + F (X(s-))∆Z s , ε(s)).

In section 3, we then introduce an optimal control problem, which extends the framework of [START_REF] Bouchard | Optimal reflection of diffusions and barrier options pricing under contraints[END_REF] to the jump diffusion case,

v(t, x) = sup ε∈E J(t, x; ε) (1.3)
where the cost function

J(t, x; ε) is defined as E [ β ε t,x (T )g(X ε t,x (T )) + ∫ T t β ε t,x (s)f (X ε t,x (s))ds
] with

β ε t,x (s) = e - ∫ s t ρ(X ε t,x (r-))dL ε t,x (r)
, f, g, ρ are some given functions, and the subscript t, x means that the solution of (1.2) is considered from time t with the initial condition x. As usual, the technical key for deriving the associated PDEs is the dynamic programming principle (DPP). The formal statement of the DPP may be written as follows, for τ in the set T (t, T ) of stopping times taking values in [t, T ], v(t, x) = ṽ(t, x), (1.4) where ṽ(t, x)

:= sup ε∈E E [ β ε t,x (τ )v(τ, X ε t,x (τ )) + ∫ τ t β ε t,x (s)f (X ε t,x (s))ds ] ,
see [START_REF] Bouchard | Optimal reflection of diffusions and barrier options pricing under contraints[END_REF], Fleming and Soner [START_REF] Fleming | Controlled Markov Processes and Viscosity Solutions[END_REF] and Lions [START_REF] Lions | Optimal Control of Diffusion Processes and Hamiltion-Jacobi-Bellman Equations[END_REF]. Bouchard and Touzi [START_REF] Bouchard | Weak Dynamic programming principle for viscosity solutions[END_REF] recently discussed a weaker version of the classical DPP (1.4), which is sufficient to provide a viscosity characterization of the associated value function, without requiring the usual heavy measurable selection argument nor the a priori continuity on the associated value function. In this paper, we apply their result to our context:

v(t, x) ≤ sup ε∈E E [ β ε t,x (τ )[v * , g](τ, X ε t,x (τ )) + ∫ τ t β ε t,x (s)f (X ε t,x (s))ds ] , (1.5)
and, for every upper semi-continuous function

φ such that φ ≤ v * , v(t, x) ≥ sup ε∈E E [ β ε t,x (τ )[φ, g](τ, X ε t,x (τ )) + ∫ τ t β ε t,x (s)f (X ε t,x (s))ds ] , (1.6)
where v * (resp. v * ) is the upper (resp. lower) semi-continuous envelope of v, and [w, g](s, x) := w(s, x)1 s<T + g(x)1 s=T for any map w define on [0, T ] × Ō. This allows us to provide a PDE characterization of the value function v in the viscosity sense. We finally extend the comparison principle of Bouchard [START_REF] Bouchard | Optimal reflection of diffusions and barrier options pricing under contraints[END_REF] to our context.

Following are some notations that will be used through out this paper.

Notations. For T > 0 and a Borel set

K of R d , D f ([0, T ], K) is the set of càdlàg functions from [0, T ]
into K with a finite number of discontinuous points, and

BV f ([0, T ], K) is the subset of elements in D f ([0, T ], K) with bounded variation. For ε ∈ BV f ([0, T ], K), we set |ε| := ∑ i≤n |ε i |,
where |ε i | is the total variation of ε i . We denote by N ε [t,T ] the number of jump times of ε on the interval [t, T ]. In the space R d , we denote by ⟨•, •⟩ natural scalar product and by ∥ • ∥ the associated norm. Any element of R d is viewed as a column vector. For x ∈ R d , we denote by B(x, r) the open ball of radius r > 0 and center x. M d is the set of square matrices of dimension n, Trace [M ] is the trace of M ∈ M d and M * is its transposition. For a set K ⊂ R d , we note K c its complement and ∂K its boundary. Given a smooth map φ on [0, T ] × R d , we denote by ∂ t φ its partial derivatives with respect to its first variable, and by Dφ and D 2 φ the partial gradient and Hessian matrix with respect to its second variable. If nothing else is specified, identities involving random variables have to be taken in the a.s. sense.

The SDEs with controlled oblique reflection

The deterministic problem.

For sake of simplicity, we first explain how we construct a class of solutions for the deterministic Skorokhod problem (SP), given an open domain O ⊂ R d and a continuous deterministic map ψ:

ϕ(t) = ψ(t) + ∫ t 0 γ(ϕ(s), ε(s))1 ϕ(s)∈∂O dη(s) , ϕ(t) ∈ Ō ∀ t ≤ T .
(SP)

In the case where the direction of reflection γ is not controlled, i.e. γ is a smooth function from R d to R d satisfying |γ| = 1 which does not depend on ε, Dupuis and Isshi [START_REF] Dupuis | SDEs with oblique reflection on nonsmooth domains[END_REF] proved the strong existence of a solution to the SP when O is a bounded open set and there exists r ∈ (0, 1) such that

∪ 0≤λ≤r B(x -λγ(x), λr) ⊂ O c , ∀x ∈ ∂O. (2.1)
In the case of controlled directions of reflection, Bouchard [START_REF] Bouchard | Optimal reflection of diffusions and barrier options pricing under contraints[END_REF] showed that the existence holds whenever the condition (2. In all this paper, E denotes a given compact subset of R l for some l ≥ 1.

In order to extend this result to the case where ψ is a deterministic càdlàg function with finite number of points of discontinuity, we focus on the definition of the solution value at the jump times. At each jump time s, the value after the jump of ϕ is chosen as π(ϕ(s-) + ∆ψ(s), ε(s)), where π is the image of a projection operator on the boundary ∂O along the direction γ satisfying following conditions: G2. This means that the value of ϕ just after the jump at time s is defined as

ϕ(s) = (ϕ(s-) + ∆ψ(s)) + γ(ϕ(s), ε(s))∆η(s),
where ∆η(s) = l(ϕ(s-) + ∆ψ(s), ε(s)), or equivalently ∆ϕ(s) = ∆ψ(s) + γ(ϕ(s), ε(s))∆η(s).

In view of the existence result of Bouchard [START_REF] Bouchard | Optimal reflection of diffusions and barrier options pricing under contraints[END_REF], we already know that the existence of a solution to (SP) is guaranteed between the jump times and that the uniqueness between the jump times holds if (ψ, ε)

∈ BV f ([0, T ], R d ) × BV f ([0, T ], R l ).
By pasting together the solutions at the jumps times according to the above rule, we clearly obtain an existence on the whole time interval [0, T ] when ψ and ε have only a finite number of discontinuous points.

Lemma 2.1 Assume that G1 and G2 hold and fix ψ, ε

∈ D f ([0, T ], R d ).
Then, there exists a solution (ϕ, η) to (SP) associated to (π, l), i.e. there exists (ϕ, η)

∈ D f ([0, T ], R d ) × D f ([0, T ], R) such that (i) ϕ(t) = ψ(t) + ∫ t 0 γ(ϕ(s), ε(s))1 ϕ(s)∈∂O dη(s), (ii) ϕ(t) ∈ Ō f or t ∈ [0, T ],
(iii) η is a non decreasing f unction,

(iv) ϕ(s) = π(ϕ(s-) + ∆ψ(s), ε(s)) and ∆η(s) = l(ϕ(s-) + ∆ψ(s), ε(s)) f or s ∈ [0, T ].
Moreover, the uniqueness holds if

(ψ, ε) ∈ BV f ([0, T ], R d ) × BV f ([0, T ], R l ).

Remark 2.1 (i)

When the domain O is convex and the reflection is not oblique, i.e. γ(x, e) ≡ n(x) for all x ∈ ∂O with n(x) standing for the inward normal vector at the point x ∈ ∂O, then the usual projection operator together with the distance function to the boundary ∂O satisfies assumption G2.

(ii) Clearly, the Lipschitz continuity conditions of π and l are not necessary for proving the existence of a solution to (SP). They will be used later to provide some technical estimates on the controlled processes, see Proposition 3.1 below.

SDEs with oblique reflection.

We now consider the stochastic version of (SP). Let W be a standard n-dimensional Brownian motion and µ be a Poisson random measure on R d , which are defined on a complete probability space (Ω, F, P), such that W and µ are independent. We denote by F := {F t } 0≤t≤T the P-complete filtration generated by (W, µ). We suppose that µ admits a deterministic (P, F)-intensity kernel μ(dz)ds which satisfies

∫ T 0 ∫ R d μ(dz)ds < ∞ . (2.
3)

The aim of this section is to study the existence and uniqueness of a solution (X, L) to the class of reflected SDEs with controlled oblique reflection γ:

X(t) = x + ∫ t 0 F s (X(s-))dZ s + ∫ t 0 γ(X(s), ε(s))1 X(s)∈∂O dL(s), (2.4)
where (F s ) s≤T is a predictable process with values in the set of Lipchitz functions from R d to M d such that F (0) is essentially bounded, and Z is a R d -valued càdlàg Lévy process defined as

Z t = ∫ t 0 b s ds + ∫ t 0 σ s dW s + ∫ t 0 ∫ R d β s (z)µ(dz, ds), (2.5) where (t, z) ∈ [0, T ] × R d → (b t , σ t , β t (z)) is a deterministic bounded map with values in R d × M d × R d .
As already mentioned, an existence of solutions was proved for Itô processes, i.e. β = 0, and the controls ε with continuous path and essentially bounded variation in [START_REF] Bouchard | Optimal reflection of diffusions and barrier options pricing under contraints[END_REF]. In this paper, we extend this result to the case of jump diffusion and to the case where the controls ε can have discontinuous paths with a.s. finite activity. As in the deterministic case, we only consider a particular class of solutions which is parameterized by the projection operator π. Namely, X is projected on the boundary ∂O through the projection operator π whenever it is out of the domain because of a jump. The value after the jump of X is chosen as π(X(s-) + F s (X(s-))∆Z s , ε(s)).

In order to state rigorously the main result of this section, we first need to introduce some additional notations and definitions. For any Borel set K, we denote by D F ([0, T ], K) the set of K-valued adapted càdlàg semimartingales with finite activity and BV F ([0, T ], K) the set of processes in D F ([0, T ], K) with a.s. bounded variation on [0, T ]. We set E := BF F ([0, T ], E) for case of our notations.

Definition 2.1 Given x ∈ O and ε ∈ E, we say that (X, L) ∈ D F ([0, T ], R d )×D F ([0, T ], R
) is a solution of the reflected SDEs with direction of reflection γ, projection operator (π, l)

and initial condition x, if      X(t) = x + ∫ t 0 F s (X(s-))dZ s + ∫ t 0 γ(X(s), ε(s))1 X(s)∈∂O dL(s), X(s) ∈ Ō ∀ s ≤ T, L is a non decreasing process, X(s) = π(X(s-) + F s (X(s-))∆Z s , ε(s)) , ∆L(s) = l(X(s-) + F s (X(s-))∆Z s , ε(s)) ∀ s ≤ T .
(2.6)

We now state the main result of this section.

Theorem 2.1 Fix x ∈ O and ε ∈ E, and assume that G1 and G2 hold. Then, there exists an unique solution (X, L)

∈ D F ([0, T ], R d ) × D F ([0, T ], R) of the reflected SDEs (2.6
) with oblique direction of reflection γ, projection operator (π, l) and initial condition x.

Proof. Let {T k } k≥1 be the jump times of (Z, ε). Assume that (2.6) admits a solution on [T 0 , T k-1 ] for k ≥ 1, with T 0 = 0. It then follows from Theorem 2.2 in [START_REF] Bouchard | Optimal reflection of diffusions and barrier options pricing under contraints[END_REF] that there exists an unique solution (X, L) of (2.6) on [T k-1 , T k ). We set

X(T k ) = π(X(T k -) + F T k (X(T k -))∆Z T k , ε(T k )) so that X(T k ) = X(T k -) + F T k (X(T k -))∆Z T k + γ(X(T k ), ε(T k ))1 X(T k )∈∂O ∆L(T k ), with ∆L(T k ) = l(X(T k -) + F T k (X(T k -))∆Z T k , ε(T k )). Since N (Z,ε) [0,T ] < ∞ P-a.
s, an induction leads to an existence result on [0, T ]. Uniqueness follows from the uniqueness of the solution on each interval [T k-1 , T k ), see Theorem 2.2 in [START_REF] Bouchard | Optimal reflection of diffusions and barrier options pricing under contraints[END_REF].

2
3 The optimal control problem 3.1 Definitions.

We now introduce the optimal control problem which extends the one considered in [START_REF] Bouchard | Optimal reflection of diffusions and barrier options pricing under contraints[END_REF]. The set of control processes ζ := (α, ε) is defined as A × E, where A is the set of predictable processes taking values in a given compact subset A of R m , for some m ≥ 1. The family of controlled processes (X α,ε t,x , L α,ε t,x ) is defined as follows. Let b, σ and χ be continuous maps on Ō × A and O × A × R d with values in R d , M d and R d respectively. We assume that they are Lipchitz continuous with respect to their first variable, uniformly in the others, and that χ is bounded with respect to its last component. It then follows from Theorem 2.1 that, for (t,

x) ∈ [0, T ] × O and (α, ε) ∈ A × E, there exists an unique pair (X α,ε t,x , L α,ε t,x ) ∈ D F ([0, T ], R d ) × D F ([0, T ], R) which satisfies X(r) = x + ∫ r t b(X(s), α(s))ds + ∫ r t σ(X(s), α(s))dW s + ∫ r t ∫ R d χ(X(s-), α(s), z)µ(dz, ds) + ∫ r t γ(X(s), ε(s))1 X(s)∈∂O dL(s), ∀t ≤ r ≤ T,(3.1) X(s) ∈ Ō ∀s ∈ [t, T ], L is non decreasing, (3.2) (X(s), ∆L(s)) = ∫ R d (π, l) (X(s-) + χ(X(s-), α(s), z), ε(s)) µ(dz, {s}), for s ∈ [t, T ]. (3.3)
Let ρ, f, g be bounded Borel measurable real valued maps on Ō × E, Ō × A and Ō, respectively. We assume that g is Lipchitz continuous and ρ ≥ 0. The functions ρ and f are also assumed to be Lipchitz continuous in their first variable, uniformly in their second one. We then define the cost function

J(t, x; ζ) := E [ β ζ t,x (T )g(X ζ t,x (T )) + ∫ T t β ζ t,x (s)f (X ζ t,x (s), α(s))ds ] , ( 3.4)
where

β ζ t,x (s) := e - ∫ s t ρ(X ζ t,x (r-),ε(r))dL ζ t,x (r) , for ζ = (α, ε) ∈ A × E.
The aim of the controller is to maximize J(t, x; ζ) over the set A t × E t of controls in A × E which are independent on F t , compare with [START_REF] Bouchard | Weak Dynamic programming principle for viscosity solutions[END_REF]. The associated value function is then defined as

v(t, x) := sup ζ∈At×Et J(t, x; ζ).

Dynamic programming.

In order to provide a PDE characterization of the value function v, we shall appeal as usual to the dynamic programming principle. The classical DPP (1.4) relates the time-t value function v(t, •) to the later time-τ value v(τ, •), for any stopping time τ ∈ T (t, T ). Recently Bouchard and Touzi [START_REF] Bouchard | Weak Dynamic programming principle for viscosity solutions[END_REF] provided a weaker version of the DPP, which is sufficient to provide a viscosity characterization of v. This version allows us to avoid the technical difficulties related to the use of non-trivial measurable selection arguments or the a-priori continuity of v. From now, for t ≤ T , we denote by T t (τ 1 , τ 2 ) the set of elements in T (τ 1 , τ 2 ) which are independent on F t . The weak version of the DPP reads as follows:

Theorem 3.1 Fix (t, x) ∈ [0, T ] × Ō and τ ∈ T t (t, T ), then v(t, x) ≤ sup ζ∈At×Et E [ β ζ t,x (τ )[v * , g](τ, X ζ t,x (τ )) + ∫ τ t β ζ t,x (s)f (X ζ t,x (s))ds ] , (3.5)
and

v(t, x) ≥ sup ζ∈At×Et E [ β ζ t,x (τ )[φ, g](τ, X ζ t,x (τ )) + ∫ τ t β ζ t,x (s)f (X ζ t,x (s))ds ] , (3.6)
for any upper semi continuous function φ such that v ≥ φ.

Arguing as in [START_REF] Bouchard | Weak Dynamic programming principle for viscosity solutions[END_REF], the result follows once J(•; ζ) is proved to be lower semicontinuous for all ζ ∈ A × E.

In our setting, one can actually prove the continuity of the above map.

Proposition 3.1 Fix (t 0 , x 0 ) ∈ [0, T ] × Ō and ζ ∈ A × E. Then, we have lim (t,x)→(t0,x0) J(t, x; ζ) = J(t 0 , x 0 ; ζ). ( 3.7)
Proposition 3.1 will be proved later in the Subsection 3.3. Before providing the proof of the DPP, we verify the consistency with deterministic initial data assumption, see Assumption A4 in [START_REF] Bouchard | Weak Dynamic programming principle for viscosity solutions[END_REF].

Lemma 3.1 (i) Fix (t, x) ∈ [0, T ] × Ō, (ζ, θ) ∈ A t × E t × T t (t, T ). For P-a.e ω ∈ Ω, there exists ζω ∈ A θ(ω) × E θ(ω) such that E [ β ζ t,x (T )g(X ζ t,x (T )) + ∫ T t β ζ t,x (s)f (X ζ t,x (s), α(s))ds|F θ ] (ω) = β ζ t,x (θ(ω))J(θ(ω), X ζ t,x (θ)(ω); ζω ) + ∫ θ(ω) t β ζ t,x (s)(ω)f (X ζ t,x (s)(ω), α(s)(ω))ds. (ii) For t ≤ s ≤ T, θ ∈ T t (t, s), ζ ∈ A s × E s and ζ := ζ1 [t,θ) + ζ1 [θ,T ] , we have, for P-a.e. ω ∈ Ω, E [ β ζ t,x (T )g(X ζ t,x (T )) + ∫ T t β ζ t,x (s)f (X ζ t,x (s), α(s))ds|F θ ] (ω) = β ζ t,x (θ(ω))J(θ(ω), X ζ t,x (θ)(ω); ζ) + ∫ θ(ω) t β ζ t,x (s)(ω)f (X ζ t,x (s)(ω), α(s)(ω))ds.
Proof. In this proof, we consider the space (Ω, F, P) as being the product space

C([0, T ], R d ) × S([0, T ], R d ), where S([0, T ], R d ) := {(t i , z i ) i≥1 : t i ↑ T, z i ∈ R d },
equipped with the product measure P induced by the Wiener measure and the Poisson random measure µ. We denote by ω or ω a generic point. We also define the stopping operator ω r • := ω r∧• and the translation operator T r (ω) := ω •+r -ω r . We then obtain from direct computations that, for ζ = (α, ε)

∈ A t × E t : E [ β ζ t,x (T )g(X ζ t,x (T )) + ∫ T t β ζ t,x (s)f (X ζ t,x (s), α(s))ds|F θ ] (ω) = β ζ t,x (θ) (ω) ∫ [ β ζ(ω θ(ω) +T θ(ω) (ω)) θ(ω),X ζ t,x (θ)(ω) (T ) g ( X ζ(ω θ(ω) +T θ(ω) (ω)) θ(ω),X ζ t,x (θ)(ω) (T ) ) + ∫ T θ(ω) β ζ(ω θ(ω) +T θ(ω) (ω)) θ(ω),X ζ t,x (θ)(ω) (s)f ( X ζ(ω θ(ω) +T θ(ω) (ω)) θ(ω),X ζ t,x (θ)(ω) (s), α(ω θ(ω) + T θ(ω) (ω))(s) ) ds ] dP(T θ(ω) (ω)) + ∫ θ(ω) t β ζ t,x (s)(ω)f (X ζ t,x (s)(ω), α(s)(ω))ds = β ζ t,x (θ)(ω)J(θ(ω), X ζ t,x (θ)(ω); ζω ) + ∫ θ(ω) t β ζ t,x (s)(ω)f (X ζ t,x (s)(ω), α(s)(ω))ds, where ω ∈ Ω → ζω (ω) := ζ(ω θ(ω) + T θ(ω) (ω)) ∈ A θ(ω) × E θ(ω) .
This leads proves (i). The assertion (ii) is proved similarly by using the fact that θ(ω) ∈ [t, s] for P-a.e. Note that {U (s, y; r) : (s, y) ∈ [0, T ] × Ō, 0 < r < r(s, y)} is a Vitali covering of [0, T ] × Ō. It then follows from the Vitali's covering Theorem that there exists a countable sequence

ω ∈ Ω. 2
{t i , x i } i∈N so that [0, T ] × Ō ⊂ ∪ i∈N U (t i , x i ; r i ) with r i := r(t i , x i ). We can then extract a partition {B i } i∈N of [0, T ] × Ō and a sequence {t i , x i } i≥1 satisfying (t i , x i ) ∈ B i for each i ∈ N, such that (t, x) ∈ B i implies t ≤ t i , and
J(•; ζ i ) -φ ≥ ε on B i , for some ζ i ∈ A ti × E ti . ( 3.9)
We now fix ζ ∈ E t × A t and τ ∈ T t (t, T ) and set

ζ := ζ1 [t,τ ) + 1 [τ,T ] ∑ n≥1 ζ n 1 {(τ,X ζ t,x (τ ))∈Bn} . Note that ζ ∈ A t × E t since τ , (ζ n ) n≥1 and X ζ t,
x are independent of F t . It then follows from (ii) of Lemma 3.1 and (3.9) that

J(t, x, ζ) -E[g(X ζ t,x (T ))1 {τ =T } ] -E[ ∫ τ t β ζ t,x (s)f (X ζ t,x (s))ds] ≥ E [ β ζ t,x (τ )J(τ, X ζ t,x (τ ); ζ)1 {τ <T } ] ≥ E [ β ζ t,x (τ )φ(τ, X ζ t,x (τ ))1 {τ <T } ] -ϵ. This implies that v(t, x) ≥ E [ β ζ t,x (τ )[φ, g](τ, X ζ t,x (τ )) + ∫ τ t β ζ t,x (s)f (X ζ t,x (s))ds
] . In this section, we prove the continuity of the cost function in the (t, x)-variable as follows.

1. We first show that the map J(•; α, ε) is continuous if µ and ε are such that

N µ [t,T ] ≤ m P -a.s and ε ∈ E b k , for some m, k ≥ 1,
where E b k is defined as the set of ε ∈ E such that |ε| ≤ k and the number of jump times of ε is a.s smaller than k, and

N µ [t,T ] := µ(R d , [t, T ]
). This result is proved as a consequence of the estimates of X and β in Lemma 3.2 presented below together with the Lipchitz continuity conditions on f, g and ρ.

Lemma 3.2 Fix k, m ∈ N. Assume that G1 and G2 hold, N µ [t,T ] ≤ m P-a.e and ε ∈ E b k . Then, there exist a constant M > 0 and a function λ so that, for all t ≤ t ′ ≤ T and x, x ′ ∈ Ō, we have

E [ sup t ′ ≤s≤T |X α,ε t,x (s) -X α,ε t ′ ,x ′ (s)| 4 ] ≤ M |x -x ′ | 4 + λ(|t -t ′ |), (3.10) E [ sup t ′ ≤s≤T | ln β α,ε t,x (s) -ln β α,ε t ′ ,x ′ (s)| ] ≤ M |x -x ′ | + λ(|t -t ′ |), (3.11)
where lim a→0 λ(a) = 0.

Proof. In order to prove the result, we use a similar argument as in Proposition 3.1 [START_REF] Bouchard | Optimal reflection of diffusions and barrier options pricing under contraints[END_REF] on the time intervals where (Z, ε) is continuous. We focus on the differences which come from the points of discontinuity of (Z, ε). From now, we denote (X, L) := (X α,ε t,x , L α,ε t,x ) and (X ′ , L ′ ) := (X α,ε t ′ ,x ′ , L α,ε t ′ ,x ′ ). We only prove the first assertion, the second one follows from the same line of arguments. Let {T i } i≥1 be the sequence of jump times on [t ′ , T ] of (Z, ε) and T 0 := t ′ . By the same argument as in Proposition 3.1 of [START_REF] Bouchard | Optimal reflection of diffusions and barrier options pricing under contraints[END_REF] we obtain that

E[ sup r∈[Ti,Ti+1) |X(r) -X ′ (r)| 4 ] ≤ C 1 E[|X(T i ) -X ′ (T i )| 4 ], for some C 1 > 0.
(3.12)

It follows from (3.3) that E[|X(T i+1 ) -X ′ (T i+1 )| 4 ] ≤ E[| ∫ R d [π(χ(X(T i+1 -), α(T i+1 ), z), ε(T i+1 )) -π(χ(X ′ (T i+1 -), α(T i+1 ), z), ε(T i+1 ))]µ(dz, {T i+1 })| 4 ]
This, together with (2.3) and the Lipschitz continuity assumption on χ and π, implies that

E[|X(T i+1 ) -X ′ (T i+1 )| 4 ] ≤ C 2 E[|X(T i+1 -) -X ′ (T i+1 -)| 4 ], for some C 2 > 0 . (3.13)
Using the previous inequality and (3.12), we deduce that there exists C > 0 s.t

E[ sup r∈[Ti,Ti+1] |X(r) -X ′ (r)| 4 ] ≤ CE[|X(T i ) -X ′ (T i )| 4 ].
Applying an induction argument, it implies to

E[ sup r∈[Ti,Ti+1] |X(r) -X ′ (r)| 4 ] ≤ C i E[|X(t ′ ) -X ′ (t ′ )| 4 ]. Since N µ [t,T ] ≤ m and N ε [t,T ] ≤ k, we have E[ sup t ′ ≤s≤T |X(s) -X ′ (s)| 4 ] ≤ C m+k E[|X(t ′ ) -x ′ | 4 ], for some C > 1. ( 3
|X(r) -x ′ | 4 ] ≤ C1 E[|X( Ti ) -x ′ | 4 + | Ti+1 -Ti |]. (3.15)
Recalling (3.3) and the Lipschitz continuity assumption on π, we deduce that there exists C2 > 0 so that

E[|X( Ti+1 ) -x ′ | 4 ] = E [ | ∫ R d [π(X( Ti+1 -) + χ(X( Ti+1 -), α( Ti+1 ), z), ε( Ti+1 )) -x]µ(dz, { Ti+1 })| 4 ] = E [ | ∫ R d [π(X( Ti+1 -) + χ(X( Ti+1 -), α( Ti+1 ), z), ε( Ti+1 )) -π(x, ε( Ti+1 ))]µ(dz, { Ti+1 })| 4 ] ≤ C2 E [ |X( Ti+1 -) -x ′ | 4 + ∫ R d |χ(X( Ti+1 -), α( Ti+1 ), z)| 4 µ(dz, { Ti+1 })
] .

The previous inequality together with (3.15) leads to

E[ sup r∈[ Ti, Ti+1] |X(r)-x ′ | 4 ] ≤ CE [ |X( Ti ) -x ′ | 4 + | Ti+1 -Ti | + ∫ R d |χ(X( Ti+1 -), α( Ti+1 ), z)| 4 µ(dz, { Ti+1 }) ] ,
for some C > 0.

Then,

E[ sup t≤r≤t ′ |X(r) -x ′ | 4 ] ≤ C m+k |x -x ′ | 4 + C|t ′ -t| + CE[ ∫ t ′ t ∫ R d |χ(X(s-), α(s), z)| 4 μ(dz)ds],
for some C > 1.

Since χ is bounded with respect to z, we then conclude that

E[ sup t≤r≤t ′ |X(r) -x ′ | 4 ] ≤ M ′ |x -x ′ | 4 + λ(|t ′ -t|),
for some M ′ > 0 and a function λ satisfying lim a→0 λ(a) = 0. 2

2.

We now provide the proof of Proposition 3.1 in the general case. Fix (α, ε) ∈ A × E. We denote by {T m } m≥1 the sequence of jump times of µ on the interval ]t, T ] of (Z, ε), T 0 := t and

µ (m) (A, B) := µ(A, B ∩ [t, T m ]) for A ∈ B R d , B ∈ B [0,T ]
where B R d and B [0,T ] denote the Borel tribes of R d and [0, T ] respectively. Let (X

(m) t,x , L (m) t,x , β (m) t,x , J m (t, x; α, ε (m)
)) be defined as (X α,ε t,x , L α,ε t,x , β α,ε t,x , J(t, x; α, ε)) in (3.1), (3.1),(3.3) and (3.4) with µ (m) in place of µ and

ε (m) := ε1 [t,T ′ m ) + ε(T ′ m )1 [T ′ m ,T ] in place of ε,
where

T ′ m := sup{s ≥ t : N ε [t,s] ≤ m, |ε|(s) ≤ m} ∧ T m .
Since f, g and β are bounded on the domain O, there exists a constant M > 0 such that, for all (t, x) ∈ [0, T ] × R d , we have

|J m (t, x; α, ε (m) ) -J(t, x, y; α, ε)| ≤ E [ |β (m) (T )g(X (m) (T )) -β α,ε t,x (T )g(X α,ε t,x,y (T ))|1 A (m) c t ] +E [ | ∫ T t β (m) (u)f (X (m) (u)) -β α,ε t,x (u)f (X α,ε t,x (u))du|1 A (m) c t ] ≤ M P { A (m) c t } ≤ M P { A (m) c 0 } .
where

A (m) t := {ω ∈ Ω : N µ [t,T ] ≤ m, N ε [t,T ] ≤ m, |ε|(T ) ≤ m}. Since N µ [0,T ] ≤ ∞ P-a.
s and ε is a process with bounded variation and finite activity, then P{A (m) c 0 } converges to 0 when m goes to ∞. Hence, sup

(t,x)∈[0,T ]× Ō |J m (•; α, ε (m) ) -J(•; α, ε)| → 0, when m → ∞. (3.16)
It then follows the first part of proof that J m (•; α, ε (m) ) is continuous map. This leads to the required result (3.7). 2

The PDE characterization.

We are now ready to provide a PDE characterization for v. Note that the fact that the process X ζ t,x is projected through the projection operator π whenever it exists the domain O because of a jump implies that the associated Dynkin operator is given, for values (a, e) of the control process, by

L a,e φ(s, x) := ∂ t φ(s, x) + ⟨b(x, a), Dφ(s, x)⟩ + 1 2 Trace [ σ(x, a)σ * (x, a)D 2 φ(s, x) ] + ∫ R d [e -ρ(x,e)l a,e x (z) φ(s, π a,e x (z)) -φ(s, x)]μ(dz),
for smooth functions φ, where (π a,e x (z), l a,e x (z)) := (π, l)(x + χ(x, a, z), e).

Also note that the probability of having a jump at the time where the boundary is reached is 0. It follows that the reflection terms does not play the same role, from the PDE point of view, depending whenever the reflection operate at a point of continuity or at a jump time. In the first case, it corresponds to a Neumman type boundary condition, while, in the second case, it only appears in the Dynkin operator which drives the evolution of the value function in the domain as described above. This formally implies that v should be a solution of Bφ = 0, (3.17)

where

Bφ :=        min (a,e)∈A×E {-L a,e φ -f (•, a)} on [0, T ) × O min e∈E H e φ on [0, T ) × ∂O φ -g on {T } × Ō ,
and, for a smooth function φ on [0, T ] × Ō and (a, e) ∈ A × E, H e (x, y, p) := ρ(x, e)y -⟨γ(x, e), p⟩ and H e φ(t, x) := H e (x, φ(t, x), Dφ(t, x)).

Since v is not known to be smooth a-priori, we shall appeal as usual to the notion of viscosity solutions. Also note that the above operator B is not continuous, so that we have to consider its upper-and lowersemicontinuous envelopes to properly define the PDE. From now, given a function w on

[0, T ] × Ō, we set      w * (t, x) = lim inf (t ′ ,x ′ )→(t,x), (t ′ ,x ′ )∈[0,T )×O w(t ′ , x ′ ) w * (t, x) = lim sup (t ′ ,x ′ )→(t,x), (t ′ ,x ′ )∈[0,T )×O w(t ′ , x ′ ) , for (t, x) ∈ [0, T ] × Ō.
Definition 3.1 A lower semicontinuous (resp. upper semicontinuous ) function w on [0, T ] × Ō is a viscosity super-solution (resp. sub-solution )of (3.17) if, for any test function φ ∈ C 1,2 ([0, T ] × Ō) and (t 0 , x 0 ) ∈ [0, T ]× Ō that achieves a local minimum (resp. maximum) of w-φ so that (w-φ)(t 0 , x 0 ) = 0, we have B + φ ≥ 0 (resp. B -φ ≤ 0), where

B + φ :=        Bφ on [0, T ] × O min (a,e)∈A×E max {-L a,e φ -f (•, a), H e φ} on [0, T ) × ∂O φ -g on {T } × ∂O , B -φ :=          Bφ on [0, T ] × O min { min (a,e)∈A×E {-L a,e φ -f (•, a)}, min e∈E H e φ } on [0, T ) × ∂O min{φ -g, min e∈E H e φ} on {T } × ∂O ,
A local bounded function w is a discontinuous viscosity solution of (3.17) if w * (resp. w * ) is a supersolution (resp. sub-solution) of (3.17).

We can now state our main result.

Theorem 3.2 Assume that G1 and G2 hold. Then, v is a discontinuous viscosity solution of (3.17).

The proof of this result is reported in the subsequent sections.

3.4.1

The super-solution property.

Let φ ∈ C 1,2 ([0, T ] × Ō) and (t 0 , x 0 ) ∈ [0, T ] × Ō be such that min(strict) [0,T ]× Ō (v * -φ) = (v * -φ)(t 0 , x 0 ) = 0.
1. We first prove the required result in the case where (t 0 , x 0 ) ∈ [0, T ) × ∂O.

Then, arguing by contradiction, we suppose that min

(a,e)∈A×E max{-L a,e φ(t 0 , x 0 ) -f (x 0 , a), H e φ(t 0 , x 0 )} ≤ -2ϵ < 0. Define ϕ(t, x) := φ(t, x) -|t -t 0 | 2 -η|x -x 0 | 4 ,
so that, for η > 0 small enough, we have min

(a,e)∈A×E max{-L a,e ϕ(t 0 , x 0 ) -f (x 0 , a), H e ϕ(t 0 , x 0 )} ≤ -2ϵ, recall (2.
3). This implies that there exists (a 0 , e 0 ) ∈ A × E and δ > 0 such that t 0 + δ < T and

max{-L a0,e0 ϕ(t, x) -f (x, a 0 ), H e0 ϕ(t, x)} ≤ -ϵ on B ∩ [0, T ] × Ō, (3.18)
where B := [t 0 -δ, t 0 + δ) × B(x 0 , δ).

Let (t n , x n ) n be a sequence in [0, T ) × O converging to (t 0 , x 0 ) such that v(t n , x n ) → v * (t 0 , x 0 ) ,
and set (X n , L n , β n ) := (X a0,e0 tn,xn , L a0,e0 tn,xn , β a0,e0 tn,xn ). Obviously, we can assume that (t n , x n ) ∈ B. Let τ n and θ n be the first exit times of (s, X n (s)) s≥tn and (X n (s)) s≥tn from B and O, respectively. Using Itô's Lemma, we have

E [β n (τ n )ϕ (τ n , X n (τ n ))] = ϕ(t n , x n ) + E [∫ τn tn β n (s-)L a0,e0 ϕ(s, X n (s-))ds ] -E [∫ τn tn β n (s)H e0 ϕ(s, X n (s))dL c n (s) ] ,
where L c n denotes the continuous part of L n . It follows from (3.18) that

E[β n (τ n )ϕ (τ n , X n (τ n ))] ≥ ϕ(t n , x n ) -E [∫ τn tn β n (s)f (X n (s), a 0 )ds ] + E [∫ τn tn ϵβ n (s)dL c n (s)
] .

Since ρ ≥ 0, the function

β n (•) is non increasing. Hence E [ β n (τ n )ϕ (τ n , X n (τ n )) + ∫ τn tn β n (s)f (X n (s), a 0 )ds ] ≥ ϕ(t n , x n ) + E[ϵβ n (τ n )L c n (τ n )]. Since Ō is bounded, φ -ϕ ≥ ζ on ∂ c p B for some ζ > 0,
where

∂ c p B := [t 0 -δ, t 0 + δ] × (B(x 0 , δ) c ∩ Ō) ∪ {t 0 + δ} × Ō. This, together with the fact that β n (τ n ) = 1, L n (τ n ) = 0 on {τ n < θ n } , leads to E [ β n (τ n )φ (τ n , X n (τ n )) + ∫ τn tn β n (s)f (X n (s), a 0 )ds ] ≥ ϕ(t n , x n ) + E [ζ1 τn<θn + β n (τ n )(ζ + ϵL c n (τ n ))1 τn≥θn ] .
Let c > 0 be a positive constant satisfying |ρ| ≤ c, we have

E [ β n (τ n )φ (τ n , X n (τ n )) + ∫ τn tn β n (s)f (X n (s), a 0 )ds ] ≥ ϕ(t n , x n ) + E[ζ1 τn<θn + e -cLn(τn) (ζ + ϵL c n (τ n ))1 τn≥θn ] ≥ ϕ(t n , x n ) + min{ζ; E[e -cΣ s≤τn ∆Ln(s) ] inf k≥0 ( e -ck (ζ + ϵk) ) }.
It follows from Jensen's inequality, the Lipschitz property of l, (2.3) and the fact that l(•, e 0 ) = 0 on

O that ln E [ e -c ∑ s≤T ∆Ln(s) ] ≥ -cE   ∑ s≤T ∆L n (s)   ≥ -cE   ∑ s≤T l(X n (s-) + χ(X n (s-), a 0 , ∆Z(s)), e 0 )   ≥ -cE [ ∫ T 0 ∫ R d l(X n (s-) + χ(X n (s-), a 0 , z), e 0 )μ(dz)ds ] ≥ -cE [ ∫ T 0 ∫ R d [l(X n (s-) + χ(X n (s-), a 0 , z), e 0 ) -l(X n (s-), e 0 )]μ(dz)ds ] ≥ -c ′ ∫ T 0 sup Ō×E |χ| ∫ R d μ(dz)ds > -∞.
Then, there exists ϵ 0 > 0 so that

E [ β n (τ n )φ (τ n , X n (τ n )) + ∫ τn tn β n (s)f (X n (s), a 0 )ds ] ≥ ϕ(t n , x n ) + ϵ 0 .
For n large enough, this leads to a contradiction to the statement (3.6) of Theorem 3.1.

2.

The proof is similar for (t 0 , x 0 ) ∈ [0, T )×O. Indeed, by a similar localization as above, we can restrict to the case where X n does not escape the domain O, expect possible by a jump, i.e. L c n (τ n ) = 0. It follows that a contradiction can be obtained by exactly the same argument as step 1. by only assuming min (a,e)∈A×E (-L a,e φ(t 0 , x 0 ) -f (x 0 , a)) < 0. When (t 0 , x 0 ) ∈ {T } × Ō, it follows from Proposition 3.1 and the fact that A t × E t ⊃ A T × E T for all t ≤ T that v is lower-semicontinuous at the points of {T } × Ō. This leads clearly to v * ≥ g on {T } × Ō. 2

3.4.2

The sub-solution property.

Let (t 0 , x 0 ) ∈ [t, T ] × Ō and φ ∈ C 1,2 ([0, T ] × Ō) such that max(strict) [0,T ]× Ō (v * -φ) = (v * -φ)(t 0 , x 0 ) = 0.
1. We first consider the case where (t 0 , x 0 ) ∈ {T } × ∂O. We argue by contradiction and suppose that min{min e∈E H e φ(t 0 , x 0 ), (φ -g)(t 0 , x 0 )} =: 2ϵ > 0.

Since A is compact, after replacing φ by (t, x) → φ(t, x) + √ T -t + ι, with ι > 0 small, we can assume that lim where

B := [T -δ, T ) × B(x 0 , δ).
It follows from the fact that v * -φ achieves a strict local maximum at (t 0 , x 0 ) and the fact that the domain O is bounded that sup

∂ c p B (v * -φ) =: -η < 0, (3.20)
where

∂ c p B := [T -δ, T ] × (B(x 0 , δ) c ∩ Ō) ∪ {T } × B(x 0 , δ). Let (t n , x n ) n be a sequence in [0, T ) × O converging to (t 0 , x 0 ) such that v(t n , x n ) → v * (t 0 , x 0 ) .
Obviously, we can assume that (t n , x n ) ∈ B. Fix (ε, α) ∈ E tn × A tn and define

(X n , L n , β n ) := (X
E [ β n (τ n )φ (τ n , X n (τ n )) 1 {τn<T } ] + E[β n (T ) (g(X n (T )) + ϵ) 1 {τn=T } ] ≤ φ(t n , x n ) -E [∫ τn tn β n (s)f (X n (s), α s )ds ] -E [∫ τn tn ϵβ n (s)dL c n (s) ] ≤ φ(t n , x n ) -E [∫ τn tn β n (s)f (X n (s), α s )ds ] -E[ϵβ n (τ n )L c n (τ n )].
This, together with (3.20), implies that

E[β n (τ n )[v * , g] (τ n , X n (τ n ))] ≤ φ(t n , x n ) -E [∫ τn tn β n (s)f (X n (s), α s )ds ] -E[(ϵ ∧ η)β n (τ n )] -E[ϵβ n (τ n )L c n (τ n )].
Recalling that |ρ| ≤ c for some c > 0, we deduce that

E[β n (τ n )[v * , g] (τ n , X n (τ n ))] ≤ φ(t n , x n ) -E [∫ τn tn β n (s)f (X n (s), α s )ds ] -E[(ϵ ∧ η)1 τn≤θn + e -cLn(τn) (ϵL c n (τ n ) + ϵ ∧ η)1 τn>θn ].
Then,

E [ β n (τ n )[v * , g] (τ n , X n (τ n )) + ∫ τn tn β n (s)f (X n (s), α s )ds ] ≤ φ(t n , x n ) -min{ϵ, η, νE[e -cΣ s≤τn ∆L(s) ]},
where ν := inf k≥0 (e -ck (εk + ζ ∧ ϵ)). Note that the same argument as in the proof of section 3.4.1 shows that E[e -cΣ s≤τn ∆L(s) ] ≥ κ > 0, for some κ independent on n and ζ.

Using the fact that lim n→∞ (v -φ)(t n , x n ) = 0 and (2.3), we may then find η ′ > 0, which is independent on ε, α and n, such that

v(t n , x n ) -η ′ ≥ E [ β n (τ n )[v * , g] (τ n , X n (τ n )) + ∫ τn tn β n (s)f (X n (s), α s )ds ] ,
for n large enough. This leads to a contradiction to (3.5) in Theorem 3.1.

2.

The case where (t 0 , x 0 ) ∈ [0, T ) × ∂O can be treated similarly by similar argument as in previous step, see (3.19). We can indeed use a localization in order to assume that τ n ≤ T -ϵ for some ϵ > 0. Therefore, we do not need to compare the values of v * with g. The case (t 0 , x 0 ) ∈ [0, T ] × O is also treated similarly by using a localization argument as in step 2 of the proof in section 3.4.1. 2

The comparison theorem.

We now prove a comparison result for B + φ ≥ 0 and B -φ ≤ 0 on [0, T ] × Ō, which implies that v is continuous on [0, T ) × O, admits a continuous extension to [0, T ] × Ō, and is the unique viscosity super-solution (resp. sub-solution) of B + φ ≥ 0 (resp. B -φ ≤ 0) on [0, T ] × Ō. We first introduce an equivalent definition of viscosity solutions, which eliminates the appearance of test function in the integral associated to the measure of jumps. Let us denote by G a,e the operator from [0

, T ] × Ō × R × R d × M d to R parameterized by a smooth function φ as G a,e (x, q, p, M ; φ) := q + ⟨b(x, a), p⟩ + f (x, a) + 1 2 Trace [σ(x, a)σ * (x, a)M ] + ∫ R d [e -ρ(x,e)l a,e x (z) φ(π a,e x (z)) -φ(x)]μ(dz) so that L a,e φ(t, x) + f (x, a) = G a,e (x, ∂ t φ(t, x), Dφ(t, x), D 2 φ(t, x); φ(t, •)).
We also define F ± as the operator associated to B ± by the implicit relation:

F ± (x, ∂ t φ(t, x), Dφ(t, x), D 2 φ(t, x); φ(t, •)) = B ± φ(t,
) ∈ [0, T ] × Ō, (q 0 , p 0 , M 0 ) ∈ P- Ō w(t, x) (resp. P+ Ō w(t, x)) and φ ∈ C 1,2 ([0, T ] × Ō) so that (t 0 , x 0 ) is a maximum (resp. minimum) point of w -φ, w(t 0 , x 0 ) = φ(t 0 , x 0 ), and
q 0 = ∂ t φ(t 0 , x 0 ), p 0 = Dφ(t 0 , x 0 ), M 0 ≥ D 2 φ(t 0 , x 0 ) (resp. M 0 ≤ D 2 φ(t 0 , x 0 )),
we have F + (x 0 , q 0 , p 0 , M 0 ; w(t 0 , •)) ≥ 0 ( resp. F -(x 0 , q 0 , p 0 , M 0 ; w(t 0 , •)) ≤ 0).

See [START_REF] Crandall | User's guide to viscosity solution of second order Partial differential equation[END_REF] for the standard notations P+ Ō and P-Ō .

Motivated by the comparison result of Proposition 3.4 in [START_REF] Bouchard | Optimal reflection of diffusions and barrier options pricing under contraints[END_REF], we add some assumptions: where ēx ∈ argmax{ρ(x, e) : e ∈ E}. Note that G3(iii) is weaker than the corresponding assumption (3.17) in [START_REF] Bouchard | Optimal reflection of diffusions and barrier options pricing under contraints[END_REF]. We now prove a comparison result:

G3. (i)

and the fact that g is continuous, it implies that Ũ (T, x η ) ≤ g(T, x η ) ≤ Ṽ (T, x η ). This leads to a contradiction of (c). where C(n, λ) goes to 0 when λ → 0 and then n → ∞. This, together with the fact that

( Ũ -Ṽ -2ηH) ≤ ( Ũ -Ṽ -2ηH)(t η , x η ) = m > 0,
and ρ, l ≥ 0, implies that

κ[ Ũ (t η n , x η n ) -Ṽ (t η n , y η n )) -2ηH(t η , x η )] ≤ -2η[ f (x η , a η n ) + L a η
n ,e η n H(t η , x η )] + C(n, λ).

Finally, using (c) and (3.23), we deduce by sending λ → 0 and then n → ∞ that κm ≤ 0, which is the required contradiction.

2.

The case where U (t η , x η ) < 0, up to a subsequence, is treated similarly. The difference comes from the test function which is chosen as follows Those variables are well defined, when the condition (iii) of assumption G3 holds. Then, if, up to a subsequence, x η n ∈ ∂O for all n, we have ρ(x η n , e) Ũ (t η n , x η n ) -⟨γ(x η n , e), p η n ⟩ = (ρ(x η , e) -ρ(x η , ēη )) Ũ (t η , x η ) + ρ(x η , ēη ) Ũ (t η , x η )(

Proof of Theorem 3. 1

 1 The inequality (3.5) is clearly a consequence of (i) in Lemma 3.1. So it remains to prove the inequality(3.6). Fix ϵ > 0. In view of definition of J and v, there exists a family {ζ(s, y)} (s,y)∈[0,T ]× Ō such thatJ(s, y; ζ(s, y)) ≥ v(s, y) -ϵ/3, for all (s, y) ∈ [0, T ] × Ō.Using Proposition 3.1 and the upper semi continuity of φ, we can choose a family {r(s, y)} (s,y)∈[0,T ]× Ō ⊂ (0, ∞) such that J(s, y; ζ(s, y)) -J(•; ζ(s, y)) ≤ ϵ/3 and φ -φ(s, y) ≤ ϵ/3 on U (s, y; r(s, y)), (3.8) where U (s, y; r) := [(s -r) ∨ 0, s] × B(y, r). Hence, J(•; ζ(s, y)) ≥ φ -ϵ on U (s, y; r(s, y)).

2 3. 3

 23 Proof of Proposition 3.1.

t→T∂

  t φ(t, x) = -∞. Then, there exists δ > 0 such that min{ min(a,e)∈A×E {-L a,e φ -f (•, a)}, min e∈E H e φ, φ -g} ≥ ϵ on B ∩ [T -δ, T ] × Ō,(3.19)

  There exists b > 0 such that B(x -bγ(x, e), b) ∩ O = ∅, f or all (x, e) ∈ ∂O × E. (3.21) (ii) There exists a C 2 ( Ō) function h such that ⟨γ(x, e), D h(x)⟩ ≥ 1, f or all x ∈ ∂O and e ∈ E. (3.22) (iii) For all x ∈ ∂O, we have inf e∈E ⟨γ(x, e), γ(x, ēx )⟩ > 0,

  Ψ η n (t, x, y) := Θ(t, x, y) -|x -x η | 4 -|t -t η | 2 -n 2 |x -y| 2 -b-1 η ρ(x η , ēη ) Ũ (t η , x η )⟨γ(x η , ẽη ), x -y⟩ ,where ēη = ēx η , bη and ẽη ∈ E satisfy min e∈E ⟨γ(x η , e), γ(x η , ēx )⟩ = ⟨γ(x η , ẽη ), γ(x η , ēx )⟩ = bη > 0.

  [START_REF] Barles | Second-Order Elliptic Integro-Differential Equations: Viscosity Solutions' Theory Revisited[END_REF] is imposed uniformly in the control variable:G1. O is a bounded open set, γ is a smooth function from R d × E to R d satisfying |γ| = 1,and there exists some r ∈ (0, 1) such that ∪

	B(x -λγ(x, e), λr) ⊂ O c , ∀(x, e) ∈ ∂O × E.	(2.2)
	0≤λ≤r

  It follows from 1.1 and 1.2 that (t η n , x η n , y η n ) ∈ [0, T ) × O 2 for all n, after possibly passing to a subsequence. Then, there exists (a η n , e η n ) n≥1 ⊂ A × E such that

	0 ≥ κ Ũ (t η n , x η n ) -b η 1,n -⟨b(x η n , a η n ), p η n ⟩ --∫ R d [ e -ρ(x η n ,e η n )l a η n ,e η n x η n (z) [ Ũ (t η n , π a η n ,e η n x η n (z)) -Ũ (t η 1 Trace [ σ(x η n , a η n )σ * (x η n , a η n )X η,λ n 2 ] n , x η n )] μ(dz) -f (x η n , a η n ) ]
	0 ≤ κ Ṽ (t η n , y η n ) + b η 1,n -⟨b(y η n , a η n ), q η n ⟩ --∫ R d [ e -ρ(y η n ,e η n )l a η n ,e η n y η n (z) [ Ṽ (t η n , π a η n ,e η n y η n (z)) -Ṽ (t η 1 Trace [ σ(y η n , a η n )σ * (y η n , a η n )Y η,λ n 2 ] n , y η n )] μ(dz) -f (y η n , a η ] n ).
	It follows from (b), (3.26) and the Lipchitz continuity of our coefficients that
	κ[ Ũ (t η n , x η n ) -Ṽ (t η n , y η ≤ ∫ R d [ e -ρ(x η ,e η n )l a η n ,e η n x η n	(z) [( Ũ -Ṽ -2ηH)(t η n , π	a η n ,e η n x η

1.3. n ) -2ηH(t η , x η )] + 2η[ f (x η , a η n ) + L a η n ,e η n H(t η , x η )] + C(n, λ) n (z)) -( Ũ -Ṽ -2ηH)(t η , y η )]

] μ(dz),

Theorem 3.3 Suppose that G3 holds. Let V (resp. U ) be a lower-semicontinuous (resp. uppersemicontinuous) locally bounded map on [0, T ] × Ō. Assume that V is a viscosity supersolution of B + φ ≥ 0 on [0, T ] × Ō and U is a viscosity subsolution of B -φ ≤ 0 on [0, T ] × Ō. Then, V ≥ U on [0, T ] × Ō.

Proof. Fix κ > 0 and set Ũ (t, x) := e κt U (t, x), Ṽ (t, x) := e κt V (t, x), f (t, x) := e κt f (t, x) and g(t, x) := e κt g(x). Here, κ is chosen such that f (•, a) -L a,e H ≥ 0, for all (a, e) ∈ A × E, (3.23) where H(t, x) := e -κt-h(x) .

We argue by contradiction, and therefore assume that sup

It follows from the fact that the domain O is bounded that Φ η := Ũ -Ṽ -2ηH achieves its maximum at (t η , x η ) on [0, T ] × Ō and satisfies Φ η (t η , x η ) =: m > 0, for η > 0 small enough .

(3.25)

1. We first study the case U (t η , x η ) ≥ 0, up to a subsequence. We define the function Ψ η n on [0, T ]× Ō2 as

We deduce that the term on the second line is bounded in n so that, up to a subsequence,

Sending n → ∞ in the previous inequality and using the maximum

This, together with (3.25), implies that

In view of Ishii's Lemma, see Theorem 8.3 in [START_REF] Crandall | User's guide to viscosity solution of second order Partial differential equation[END_REF], we deduce that, for each λ > 0, there are real coefficients b η 1,n , b η 2,n and symmetric matrices X η,λ n and Y η,λ n such that

where

with

and I stands for the identical matrix with dimension d × d.

1.1. We first suppose that, up to a subsequence,

Since |γ| = 1, this implies that 2⟨γ(x η n , e), y η n -

where λ n is independent on e and comes to 0 when n → ∞. This, together with (3.22), (b), (3.27) and the fact that ⟨γ(x η , e), γ(x η , e η )⟩ ≤ 1 since |γ| ≤ 1, implies that

when n is large enough. Using similar argument as above, if, up to a subsequence, y η n ∈ ∂O, we then have H e η (y η n , Ṽ (t η n , y η n ), q η n ) < -ηH(t η , x η ) < 0, for all e ∈ E, when n is large enough.

1.2.

We now suppose that, up to a subsequence, t η n = T , for all n ≥ 1. In view of step 1.1, we have Ũ (T, x η n ) ≤ g(T, x η n ) and Ṽ (T, y η n ) ≥ g(T, y η n ), for all n ≥ 1. Then, passing to the limit, recalling (a)