Marie Amélie
email: marieamelie.morlais@free.fr

Utility maximization in a jump market model

Keywords: Utility maximization, Backward Stochastic Differential Equations (BSDE) with jumps, stochastic exponential, BMO martingale MSC classification (2000): 91B28, 91B16, 60H10

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Utility Maximization in a jump market model

Introduction and motivation

The aim of this paper is to solve the utility maximization problem in a discontinuous framework: in this setting, the price process of stocks is assumed to be a local martingale driven by an independent one dimensional brownian motion and a Poisson point process.

This problem is a classical one in the financial literature. But, as opposed to most of the papers dealing with the same problem (among them, we cite [START_REF] Biagini | On the super replication price of unbounded claims[END_REF] or [START_REF] Schachermayer | Utility maximization in incomplete markets[END_REF]), we cannot rely on duality results, since we do not impose the convexity of the constraint set. The idea consists rather in adapting a dynamic method based on the dynamic programming principle: in the paper [START_REF] Hu | Utility maximization in incomplete markets[END_REF] dealing with the same topic, the authors already used this method to derive a specific BSDE. But, contrary to the present paper, they work in a brownian filtration and hence, results for quadratic BSDEs are available. Therefore, our main contribution is to establish new existence results for solution of the BSDE obtained in this discontinuous setting: the essential difficulty is to handle both the presence of jumps and the presence of a quadratic term. To this end and especially to handle the quadratic growth of the driver of the BSDE, we first apply the method introduced in the Brownian setting in [START_REF] Kobylanski | Backward stochastic differential equations and partial differential equations with quadratic growth[END_REF]. The model is very analogous to [START_REF] Becherer | Bounded solutions to Backward SDE's with jumps for utility optimization and indifference hedging[END_REF] but, contrary to this last reference, we assume here that the price process has jumps and that there exists additional constraints on the portfolio.

The rest of the paper is structured as follows: in Section 2, we describe the market model by giving some preliminary remarks, specifying all the notations and natural

The model and preliminaries

We consider a probability space (Ω, F, P) equipped with two independent stochastic processes:

. A standard (one dimensional) brownian motion: W =(W t ) t∈[0,T ] .

. A real-valued Poisson point process p defined on [0, T ] × R \ {0}. Referring to chapter 2 in [START_REF] Ikeda | Stochastic differential equations and diffusion processes[END_REF], we denote by N p (ds, dx) the associated counting measure, such that its compensator is Np (ds, dx) = n(dx)ds, and the Levy measure n(dx) (also denoted by n) is positive and satisfies n({0}) = 0 and n((

1 ∧ |x|) 2 ) := R\{0} (1 ∧ |x|) 2 n(dx) < ∞ . (1)
These two processes are considered on [0, T ]: T is called the horizon or maturity time in the financial context and, in all the sequel, it is fixed and deterministic. For technical reasons, n is assumed to be finite and, in all the paper, we denote by F the filtration generated by the two processes W and N p (and completed by N , consisting in all the P-null sets). Using the same notations as in [START_REF] Ikeda | Stochastic differential equations and diffusion processes[END_REF], we denote by Ñp (ds, dx) ( Ñp (ds, dx) := N p (ds, dx) -Np (ds, dx)) the compensated measure, which is a martingale random measure: in particular, for any predictable and locally square integrable process F , the stochastic integral F • Ñp := F s (x) Ñp (ds, dx) is a locally square integrable martingale. We denote by Z • W (resp. U • Ñp ) the stochastic integral of Z w.r.t. W (resp. the stochastic integral of U w.r.t. Ñp ). Furthermore, the filtration F has the predictable representation property: i.e., for any local martingale K of F, there exists two predictable processes Z and U such that

∀ t, K t = K 0 + Z • W t + U • Ñp t .
(In Section 2.2, we provide a definition of the hilbertian spaces, in which these stochastic integrals are considered).

Description of the model

The financial market consists in one risk-free asset (assumed to have zero interest rate) and one single risky asset, whose price process is denoted by S.

In particular, the stock price process is a one dimensional local martingale satisfying

dS s = S s-b s ds + σ s dW s + R * β s (x) Ñp (ds, dx) . (2)
All processes b, σ and β are assumed to be bounded and predictable and β satisfies: β > -1. This last condition implies that the stochastic exponential E(β • Ñp ) is positive, P-a.s.: hence, the process S is itself almost surely positive. The boundedness of β, σ and θ ensures both existence and uniqueness results for the SDE [START_REF] Barles | Backward stochastic differential equations and integral-partial differential equations[END_REF]. Then, provided that: σ = 0, we can define θ by: θ s = σ -1 s b s (P-a.s. and for all s). The process θ, also called market price of risk process, is supposed to be bounded and, under this assumption, the measure P θ with density

dP θ dP = E T (- . 0 θ s dW s ),
is a risk-neutral measure: this means that, under P θ , the price process S is a local martingale.

In what follows, we introduce the usual notions of trading strategies and self financing portfolio, assuming that all trading strategies are constrained to take their values in a closed set denoted by C. In a first step and to make easier the proofs, this set C is supposed to be compact (this assumption is relaxed in the last section). Due to the presence of constraints in this model with finite horizon T , not any F T -measurable random variable B is attainable by using contrained strategies. In that context, we adress the problem of characterizing dynamically the value process associated to the exponential utility maximization problem (in the sequel, we denote by U α the exponential utility function with parameter α, which is defined on R by:

U α (•) = -exp(-α•)). Definition 1 A predictable R-valued process π is a self-financing trading strategy, if it takes its values in a constraint set C and if the process X π,t,x such that ∀ s ∈ [t, T ], X π,t,x s := x + s t π s dS s S s- , (3)
is in the space H 2 of semimartingales (see chapter 4, [START_REF] Protter | Stochastic integration and differential equations[END_REF]). Such a process X π := X π,t,x stands for the wealth of an agent having strategy π and wealth x at time t. Now, as soon as the constraint set C is compact, the set consisting of all constrained strategies satisfies an additional integrability property.

Lemma 1 Under the assumption of compactness of the constraint set C, all trading strategies π := (π s ) s∈[t,T ] as introduced in Definition 1 satisfy {exp(-αX π τ ), τ F-stopping time } is a uniformly integrable family. (4)

We denote by A t the admissibility set where the subscript t indicates that we start the wealth dynamics at time t: more precisely, this set consists in all the strategies whose restriction to the interval [0, t] is equal to zero and which satisfy both Definition 1 and the condition (4). In Section 4.3, we provide a justification of this last technical condition, which has already been used in [START_REF] Hu | Utility maximization in incomplete markets[END_REF] in a Brownian setting. The usual and much more restrictive admissibility condition consists in assuming that the wealth process X π is bounded from below (uniformly over all strategies π).

Before proving this result, we introduce another notion which can also be found in [START_REF] Dellacherie | Théorie des martingales[END_REF]: a martingale M is said to be in the class of BMO martingales if there exists a constant c, c > 0, such that, for all F-stopping time τ , esssup

Ω E Fτ ( M T -M τ ) ≤ c 2 and |∆M τ | 2 ≤ c 2 .
(In the continuous case, the BMO property follows from the first condition, whereas, in the discontinuous setting, we need to ensure the boundedness of the jumps of M ). The following result, referred as Kazamaki's criterion and also stated in [START_REF] Kazamaki | A sufficient condition for the uniform integrability of exponential martingales[END_REF], relates the martingale property of a stochastic exponential to a BMO condition.

Lemma 2 (Kazamaki's criterion) Let δ be such that: 0 < δ < ∞ and M a BMO martingale satisfying: ∆M t ≥ -1 + δ, P-a.s. and for all t, then E(M ) is a true martingale.

Proof of Lemma 1

We first rely on the expression (3) of X π and on the definition of θ to claim

dX π t = π t σ t θ t dt + π t σ t dW t + R\{0} π t β t Ñp (dt, dx).
Applying a generalized Itô's formula to U = e -αX π (one reference is Theorem 5.1, Chapter 2 in [START_REF] Ikeda | Stochastic differential equations and diffusion processes[END_REF]), we get

dU t = U t -απ t σ t dW t + R\{0} (e -απtβt -1) Ñp (dt, dx) +U t -απ t σ t θ t dt + α 2 2 |π t σ t | 2 dt + R\{0} (e -απtβt -1 + απ t β t )n(dx)ds).
Using the definition of the stochastic exponential E(M ) of M , U has the multiplicative form

U t = U 0 E t ( . 0 -απ s σ s dW s + . 0 R\{0} (e -απsβs -1)) Ñp (ds, dx))e Āπ t , (5)
where the process Āπ is such that

Āπ t = t 0 -απ s σ s θ s + α 2 2 |π s σ s | 2 + R\{0} (e -απsβs -1 + απ s β s )n(dx) ds.
Hence, Āπ is a bounded process, thanks to the boundedness of the parameters of the SDE (2), the finiteness of the measure n and the compactness of the constraint set C). Besides, thanks to Lemma 2, the stochastic exponential appearing in ( 5) is a true martingale and the desired property (4) results from this decomposition.

Preliminaries

In the sequel, we denote by S ∞ (R) the set of all adapted processes Y with càdlàg paths (càdlàg stands for right continuous with left limits) such that: esssup

t,ω |Y t (ω)| < ∞, L 2 (W ) denotes the set of all predictable processes Z such that E T 0 |Z s | 2 ds < ∞.
and L 2 ( Ñp ) denotes the set of all P ⊗ B(R \ {0})-measurable processes U such that

E [0,T ]×R * |U s (x)| 2 n(dx)ds < ∞.
P stands for the σ-field of all predictable sets of [0, T ]× Ω and B(R \{0}) the Borel field of R \ {0}. We introduce L 0 (n) (denoted by L 0 (n, R, R \ {0}) in [START_REF] Becherer | Bounded solutions to Backward SDE's with jumps for utility optimization and indifference hedging[END_REF]) as being the set of all the functions u, which map R in R\{0}. This set is equipped with the topology of convergence in measure.

L 2 (n) (resp. L ∞ (n))
is the subset of all functions in L 0 (n) such that: E(

T 0 |u(x)| 2 n(dx)) < ∞ (resp. u takes bounded values).
A solution of a BSDE with jumps, which is given by its terminal condition B and its generator f , is a triple of processes (Y , Z, U ) defined on

S ∞ (R) × L 2 (W ) × L 2 ( Ñp ) such that: T 0 |f (s, Y s , Z s , U s )|ds < ∞, P-a.s.
and satisfying

Y t = B+ T t f (s, Y s-, Z s , U s )ds- T t Z s dW s - T t R * U s (x) Ñp (ds, dx). ( 6
)
Throughout this paper, we study a specific BSDE with jumps whose terminal condition coincide with the contingent claim B, which is a bounded F Tmeasurable random variable, and whose generator f is independent of y (this is the case in the financial application). Besides and since we do not work on a brownian filtration, the processes Z and U have to be predictable, for any solution of the BSDE [START_REF] Delbaen | The mathematics of arbitrage[END_REF] .

A solution of such BSDEs with jumps is usually defined on S 2 ×L 2 (W )× L 2 ( Ñp ), S 2 being equipped with the following norm:

|Y | S 2 = E( sup t∈[0,T ] |Y t | 2 ) 1 2
(our main references for studies on BSDEs with jumps, are [START_REF] Barles | Backward stochastic differential equations and integral-partial differential equations[END_REF] and [START_REF] Royer | Backward stochastic differential equations with jumps and related non-linear expectations[END_REF]). But, in this paper, the results of the aforementionned papers cannot be applied, since the generator of the BSDE we are interested in, does not satisfy the usual conditions (it is not Lipschitz w.r.t. its variable z).

3 The quadratic BSDE with jumps

Main assumptions

In all the sequel (except in the proof of a priori estimates), we use the explicit form of the generator

f (s, z, u) = inf π∈C α 2 |πσ s -(z + θ s α )| 2 + |u -πβ s | α -θ s z - |θ s | 2 2α , (7)
where the processes β, θ and σ are defined in Section 2.1. This expression of the generator will be justified in Section 4.3. We introduce the notation | • | α as being the convex functional such that

∀ u ∈ (L 2 ∩ L ∞ )(n), |u| α = R\{0} exp(αu(x)) -αu(x) -1 α n(dx), = R\{0} g α (u(x))n(dx),
with the real function g α defined by: g α (y) = exp(αy)-αy-1 α . As in [START_REF] Becherer | Bounded solutions to Backward SDE's with jumps for utility optimization and indifference hedging[END_REF], we assume the finiteness of the measure n. In all that paper, B is a bounded F T -measurable random variable and we use these two standing assumptions on the generator f (H 1 ). The first assumption denoted by (H 1 ) consists in specifying both a lower and an upper bound

∀ z, u ∈ R × (L 2 ∩ L ∞ )(n) -θ s z -|θs| 2 2α ≤ f (s, z, u) ≤ α 2 |z| 2 + |u| α , P-a.s
. and for all s.

(H 2 ). The second assumption, referred as (H 2 ), consists in two estimates: the first one deals with the increments of the generator f w.r.t. z

∃ C > 0, κ ∈ BM O(W ), ∀ z, z ′ ∈ R, ∀u ∈ L 2 (n(dx)), |f (s, z, u) -f (s, z ′ , u)| ≤ C(κ s + |z| + |z ′ |)|z -z ′ |
The second estimate deals with the increments w.r.t.

u ∀z ∈ R, ∀ u, u ′ ∈ (L 2 ∩ L ∞ )(n(dx)), f (s, z, u) -f (s, z, u ′ ) ≤ R\{0} γ s (u, u ′ )(u(x) -u ′ (x))n(dx),
with the following expression for γ s (u, u ′ ) for all s

γ s (u, u ′ ) = sup π∈C 1 0 g ′ α (λ(u -πβ s ) + (1 -λ)(u ′ -πβ s )(x))dλ 1 u≥u ′ + inf π∈C 1 0 g ′ α (λ(u -πβ s ) + (1 -λ)(u ′ -πβ s )(x)dλ 1 u<u ′ ,
and for any fixed s, ω, this last expression makes sense as soon as u and u ′ are in (L2 ∩ L ∞ )(n(dx)). Now, for any arbitrary predictable processes U , U ′ taking their values in L 2 ∩ L ∞ (n)), we can define a process 2 γ by

γs = γ s (U s , U ′ s ), (8)
and its predictability results from its expression in terms of both the predictable processes U , U ′ and β.

Comments on these assumptions • The expression [START_REF] Dellacherie | Théorie des martingales[END_REF] of the generator f clearly satisfies the assumption (H 1 ). As in [START_REF] Briand | BSDE with quadratic growth and unbounded terminal value[END_REF], where the authors work in a brownian setting, this growth condition is useful to establish precise a priori estimates for any solution of the BSDE in S ∞ (R)×L 2 (W )×L 2 ( Ñp ). These estimates are given in lemma 3 in the next paragraph and, analogously to [START_REF] Hu | Utility maximization in incomplete markets[END_REF], their existence is the starting point of the proof of the main existence result.

To check the first estimate in (H 2 ) we define, analogously as in [START_REF] Hu | Utility maximization in incomplete markets[END_REF] and for any z, z

′ ∈ R and u ′ ∈ (L 2 ∩ L ∞ )(n(dx)),      λ s (z, z ′ ) := f (s,z,u)-f (s,z ′ ,u) z-z ′ , if z -z ′ = 0, λ s (z, z ′ ) := 0, otherwise. It entails: λ s (z, z ′ ) ≤ C κ s + |z| + |z ′ |
, and hence, the generator f

given by ( 7) satisfies this first assumption with C := α 2 and with κ depending only on θ and on the two reals α and sup π∈C |π|.

• To obtain the second estimate in (H 2 ), we rely on

f (s, z, u)-f (s, z, u ′ ) ≤ sup π∈C R * g α ((u -πβ s )(x)) -g α ((u ′ -πβ s )(x)n(dx) ≤ sup π∈C R * 1 0 g ′ α λ(u -πβ s ) + (1 -λ)(u ′ -πβ s ) dλ (u -u ′ )(x)n(dx) ,
and we use that:

(u -u ′ )(x) = (u -u ′ )(x)1 u≥u ′ -(-(u -u ′ )(x))1 u<u ′ ,
and also: -sup

π (-A π ) = inf π A π , with A π such that A π := 1 0 g ′ α λ(u -πβ s ) + (1 -λ)(u ′ -πβ s ) dλ ,
to claim that the expression given in (H 2 ) on page 7 can be taken for γ. To obtain the BMO property of the process given by ( 8), we use the compactness of C and we assume that both processes U and U ′ take their values

L 2 ∩ L ∞ (n(dx)) with |U s | L ∞ (n) , |U ′ s | L ∞ (n) ≤ K, to argue that ∃ δ K , CK > 0, s.t. -1 + δ K ≤ γ s (U s , U ′ s )
≤ CK , which entails, in particular, that this process is in BMO( Ñp ). This BMO property is essential in the proof of the uniqueness result to justify the use of Girsanov's theorem.

Theoretical results

We now state the uniqueness and existence results for solutions of the BSDE (6) with parameters (f , B).

Theorem 1 (Uniqueness) f given by ( 7) and the terminal condition B being bounded, the BSDE of the form [START_REF] Delbaen | The mathematics of arbitrage[END_REF]

with parameters (f , B) has at most one solution (Y , Z, U ) in S ∞ (R) × L 2 (W ) × L 2 ( Ñp ).
Theorem 2 (Existence) f given by ( 7) and the terminal condition B being bounded, there exists at least one solution of the BSDE [START_REF] Delbaen | The mathematics of arbitrage[END_REF]

in S ∞ (R)× L 2 (W ) × L 2 ( Ñp ).

A priori estimates

In this part, we establish a priori estimates for any solution of BSDE [START_REF] Delbaen | The mathematics of arbitrage[END_REF] with a bounded terminal condition B and with a generator g, which may be different from f but satisfies (H 1 ).

Lemma 3 For any BSDE of the form ( 6) with a generator g satisfying (H 1 ) and a bounded terminal condition B, there exists three constants C 1 , C 2 and C 3 depending only on |B| ∞ , |θ| S ∞ (R) and α, and such that, for any solution

(Y , Z, U ) in S ∞ (R) × L 2 (W ) × L 2 ( Ñp ) and for any F-stopping time τ , τ taking its values in [0, T ], (i) P-a.s. and for all t, t ∈ [0, T ], C 1 ≤ Y t ≤ C 2 , (ii) E Fτ ( T τ |Z s | 2 ds + T τ R * |U s (x)| 2 n(dx)ds) ≤ C 3 .
Corollary 1 Under the same assumptions than in lemma 3 on the parameters g and B and for any solution (Y ,

Z, U ) in S ∞ (R) × L 2 (W ) × L 2 ( Ñp ) of the BSDE (6),
• there exists a predictable version Ũ of U such that:

Ũ ≡ U (in L 2 ( Ñp )).
Noting U instead of Ũ , this process satisfies

(i)(a) |U s | L ∞ (n) ≤ 2|Y | S ∞ (R) , and (i)(b) ∀ s, |U s | 2 L 2 (n) ≤ 4n(R\{0})|Y | 2 S ∞ (R) .
• This equivalence result is satisfied

∃ C > 0 1 C E [0,T ]×R\{0} |U s (x)| 2 n(dx)ds ≤ E T 0 |U s | α ds ≤ CE [0,T ]×R\{0} |U s (x)| 2 n(dx)ds, (9)
with a constant C depending only on α and |Y | S ∞ (R) .

Proof of the corollary To justify the first assertion (i)(a), we write

|∆Y t | = |Y t -Y t-| = R\{0} |U t (x)|N p ({t}, dx), (10)
and referring to the same notations as in chapter 2, [START_REF] Ikeda | Stochastic differential equations and diffusion processes[END_REF], we have

R\{0} |U t (x)|N p ({t}, dx) = |U t (p(t))|1 (t∈Dp) , (11)
(the random set D p stands for the jump times of the Poisson point process). Defining Ũ by Ũt

(x) = U t (x)1 |Ut(x)|≤2|Y | S ∞ (R) ,
we obtain a predictable process, and it results from ( 10) and ( 11) that:

Ũt (p(t))1 (t∈Dp) = U t (p(t))1 (t∈Dp ) . As a consequence E T 0 R\{0} |( Ũ -U )(s, x)| 2 Np (ds, dx) = E   s∈Dp, s≤T |U (s, p(s)) -Ũ (s, p(s))| 2   = 0,
from which we deduce: U = Ũ (in L 2 ( Ñp )). We can now assume that, for any solution (Y, Z, U ), U satisfies:

|U t | L ∞ (n) ≤ 2|Y | S ∞ (R)
, P-a.s. and for all t.

To justify the estimate in L 2 (n), we rely both on Cauchy Schwarz inequality and on the finiteness of the Levy measure. As soon as U takes its values in L 2 ∩ L ∞ (n), the equivalence result (9) holds. This last result is useful to prove Lemma 5 also referred as "monotone stability" result and inspired by the work of Kobylanski in the Brownian setting in [START_REF] Kobylanski | Backward stochastic differential equations and partial differential equations with quadratic growth[END_REF].

Proof of Lemma 3 To establish (i), we assume the existence of a solution of the BSDE (6) with parameters (g, B) and we apply Itô's formula to

e αY e αYt -e αB = T t αe αYs g(s, Z s , U s ) - α 2 |Z s | 2 -|U s | α ds - T t αe αYs Z s dW s - T t R *
e αYs (e αUs(x) -1) Ñp (dx, ds).

However, to be more rigorous, we should apply the standard procedure of localization, i.e we should first introduce the sequence of stopping times

τ m = inf 0≤t≤T { t 0 e 2αYs |Z s | 2 ds ≥ m} ∪ { t 0 R\{0} e 2αYs |e αUs(x) -1| 2 n(dx)ds ≥ m} ,
and then take the conditional expectation w.r.t. F t , before passing to the limit as m goes to ∞. Hence and without loss of generality, we just take the conditional expectation w.r.t. F t in Itô formula applied to e αY and use the upper bound in (H 1 ) to obtain

e αYt -E Ft (e αB ) ≤ E Ft ( T t αe αYs (g(s, Z s , U s ) - α 2 |Z s | 2 -|U s | α ) ≤0 ds).
Since: E Ft (e αB ) ≤ e α|B|∞ , the right-hand side in (i) holds true with C 2 = |B| ∞ . To obtain the left-hand side in (i), we introduce A such that

A := (g(s, Z s ) + (Z s θ s + |θs| 2 2α
)): such a process A is non negative, thanks to (H 1 ). Hence, we have

Y t = Y T + T t g(s, Z s , U s )ds - T t Z s dW s - T t R * U s (x) Ñp (ds, dx) = Y T + (A T -A t ) - T t (Z s θ s + |θ s | 2 2α )ds - T t Z s dW s - T t R * U s (x) Ñp (ds, dx).
We now introduce the density of a new probability measure P θ

dP θ = E T (- . 0 θ s dW s )dP.
The BMO property of -θ • W entails that the continuous stochastic exponential E(-. 0 θ s dW s ) is a true martingale. Hence, P θ is an equivalent probability measure under which the process W θ := W + . 0 θ u du, is a standard Brownian motion. Finally, we get

Y t = Y T + (A T -A t ) - T t |θ s | 2 2α ds - T t Z s dW θ s - T t R * U s (x) Ñp (ds, dx).
(12) Under P θ , the two last terms are local martingales bounded from below. Using a standard localization procedure, we obtain the existence of (τ n ) such that the two local martingales stopped at time τ n are F-martingales. Taking the conditional expectation w.r.t. F t and under P θ , we get

Y t∧τn ≥ E θ Y T ∧τn - T t |θ s | 2 2α ds|F t ,
E θ standing for the expectation under P θ . Applying now the bounded convergence theorem and relying on the boundedness assumption on θ (given by (H 1 )), we can let n tend to +∞ in [START_REF] Ikeda | Stochastic differential equations and diffusion processes[END_REF] to claim

Y t = E θ Y t |F t ≥ -|B| ∞ - |θ| 2 S ∞ T 2α , P θ -a.s.
This holds true P-a.s., because of the equivalence of P θ and P and finally, the assertion (i) is satisfied with C 1 such that

C 1 = -|B| ∞ - |θ| 2 S ∞ T 2α .
To prove assertion (ii), we apply Itô's formula to (Y -C 2 ) 2 (C 2 := |B| ∞ is the upper bound in (i)). Since Z • W and U • Ñp are true martingales (as square integrable stochastic integrals), we next integrate this formula between an arbitrary stopping time τ and T and take the conditional expectation w.r.t. F τ to get

E Fτ ((Y τ -C 2 ) 2 -(Y T -C 2 ) 2 ) = E Fτ T τ 2(Y s -C 2 )g(s, Z s , U s )ds -E Fτ T τ |Z s | 2 ds + T τ R * |U s (x)| 2 n(dx)ds .
To give an upper bound of Y s -C 2 )g(s, Z s , U s , we first rely on and

     0 ≥ (Y -C 2 ) ≥ -2|B| ∞ + |θ| 2 S ∞ T 2α , g(s, Z s , U s ) ≥ -θ s Z s -|θs| 2 2α ,
to deduce the existence of a constant C depending only on θ and α such that

(Y s -C 2 )g(s, Z s , U s ) ≤ C |θ s Z s | + |θ s | 2 2α . ( 13
)
We now split the discussion into two cases:

• if θ ≡ 0, then g is bounded by 0 from below and it implies

E Fτ ((Y τ -C 2 ) 2 -(Y T -C 2 ) 2 ) ≤ -E Fτ T τ |Z s | 2 ds + T τ R * |U s (x)| 2 n(dx)ds ,
which leads to

E Fτ T τ |Z s | 2 ds + T τ R * |U s | 2 n(dx)ds ≤ |Y T -C 2 | 2 ≤ 4|Y | S ∞ .
• else if θ = 0, we use: ab ≤ 1 2 (a 2 + b 2 ) in [START_REF] Kobylanski | Backward stochastic differential equations and partial differential equations with quadratic growth[END_REF] to argue the existence of a bounded process

C 1 (•) such that 2(Y s -C 2 )g(s, Z s , U s ) ≤ C 1 (s) + 1 2 |Z s | 2 .
From straightforward computations, we get

1 2 E Fτ T τ |Z s | 2 ds + T τ R * |U s | 2 n(dx)ds ≤ E Fτ T τ C 1 (s)ds + (Y T -C 2 ) 2 .
and hence, (ii) follows.

Uniqueness

Proof of Theorem 1

The idea consists in using a linearization procedure and in justifying, as in [START_REF] Hu | Utility maximization in incomplete markets[END_REF], the application of Girsanov's theorem. We first consider (Y 1 , Z 1 , U 1 ) and (Y 2 , Z 2 , U 2 ) two solutions of the BSDE with jumps given by (f , B) and C any positive constant such that:

|Y i | S ∞ (R) ≤ C (thanks to (i) in Lemma 3, we can set C := |C 1 | + |C 2 |). Then, we introduce Ŷ , Ẑ and Û Ŷ = Y 1 -Y 2 , Ẑ = Z 1 -Z 2 , Û = U 1 -U 2 .
τ being an arbitrary F-stopping time, we apply Itô's formula to Ŷ between t ∧ τ and τ ,

Ŷt∧τ -Ŷτ = τ t∧τ (f (s, Z 1 s , U 1 s ) -f (s, Z 2 s , U 2 s ))ds - τ t∧τ Ẑs dW s - τ t∧τ R *
Ûs Ñp (ds, dx).

Since the generator does not satisfy the usual conditions: it is not Lipschitz neither in z nor in u), we rely on assumption (H 2 ), i.e. on the estimates of the increments of the generator f We first introduce λ

:= λ(Z 1 , Z 2 )      λ s (Z 1 s , Z 2 s ) = f (s,Z 1 s ,U 1 s )-f (s,Z 2 s ,U 1 s ) Z 1 s -Z 2 s , if Ẑ = 0, λ s (Z 1 s , Z 2 s ) = 0, else.
Thanks to (H 2 ), we get

f (s, Z 1 s , U 1 s ) -f (s, Z 2 s , U 2 s ) = f (s, Z 1 s , U 1 s ) -f (s, Z 2 s , U 1 s ) + f (s, Z 2 s , U 1 s ) -f (s, Z 2 s , U 2 s ) ≤ λ s (Z 1 s , Z 2 s ) Ẑs + R * γ s (U 1 s , U 2 s ) Ûs (x)n(dx),
where the process γ = (γ s (U 1 s , U 2 s )) is defined analagously as in ( 8) and for all s by γs = sup

π∈C 1 0 g ′ α (λ(U 1 s -πβ s ) + (1 -λ)(U 2 s -πβ s )(x))dλ 1 u≥u ′ + inf π∈C 1 0 g ′ α (λ(U 1 s -πβ s ) + (1 -λ)(U 2 s -πβ s )(x)dλ 1 u<u ′ .
As in subsection 3.1, γ is explicitely expressed in terms of both the two predictable processes U i (i = 1, 2) and the predictable process β and hence, it is itself predictable. Thanks to Corollary 1, we also obtain that for i :=1, 2,

|U i s | L ∞ (n) ≤ 2|Y i | S ∞ (R) ≤ 2C, (
C is given at the beginning of this proof), which implies

∃ δ > 0, C > 0, s.t. -1 + δ ≤ γ s (U 1 s , U 2
s ) ≤ C, P-a.s. and for all s.

Since the process λ satisfies

|λ s (Z 1 s , Z 2 s )| ≤ C(κ s + |Z 1 s | + |Z 2 s |),
the BMO properties of λ • W and γ • Ñp result from the assumption (H 2 ) and also from the BMO property of . 0 Z i dW s , which holds true for i = 1, 2 (we refer to (ii) in Lemma 3). Defining M 1 and M 2 such that

M 1 = λ • W and M 2 = γ • Ñp ,
and setting: dQ = E(M 1 + M 2 )dP, it results from Kazamaki's criterion that E(M 1 + M 2 ) is a true martingale and hence, Q is an equivalent probability measure. Using Girsanov's theorem, we set:

W λ := W -W, λ • W and Ñ γ := Ñp -Ñp , γ • Ñp and we introduce ∀ t, M t = t 0 Ẑs dW λ s + t 0 R * Ûs (x) Ñ γ (ds, dx),
which is a local martingale under Q. We conclude by relying on a standard localization procedure: there exists a sequence (τ n ) of stopping times converging by increasing to T and such that: M t∧τ n is a martingale. Hence, under the measure Q, Ŷ•∧τ n is a submartingale, which yields

Ŷt∧τ n ≤ E Q Ŷτ n |F t .
Using the bounded convergence theorem, we finally get: Ŷt ≤ 0, Q-a.s. and P-a.s., because of the equivalence of P and Q. Thanks to the symmetry of this problem, we can conclude: Ŷ = 0.

Existence

To prove the existence for a solution of the BSDE (6) with parameters (f , B) and with f given by ( 7), we proceed in three main steps and, for more convenience, we provide here a description of each step:

• The first step consists in the construction of an approximating sequence (f n ) of f such that each generator is lipschitz.

• In a second step and referring to already known results, we justify the existence of solutions to the BSDEs given by (f n , B) and we provide precise estimates.

• The last step consists in justifying a stability result for the solutions of these BSDEs (analogous to the one provided by [START_REF] Kobylanski | Backward stochastic differential equations and partial differential equations with quadratic growth[END_REF]) and deduce from it the existence of a limit, which solves the original BSDE. The proof being constructive, we need the explicit form (7) of the generator, satisfying in particular the assumptions (H 1 ) and (H 2 ) stated in section 3.1.

Step 1: Approximation by a truncation argument

Our aim is to construct an increasing sequence (f m ), such that each f m satisfies the assumption (H 1 ) with the same parameters as f . For all m, m ≥ M , with M given by M := 2(|C 1 | + |C 2 |) (C 1 and C 2 are the constants given in (i) by Lemma 3), we define

f m f m (s, z, u) = inf π∈C α 2 |πσ s -(z + θ s α )| 2 ρ m (z) + R * g α (u -πβ s )(x)ρ M (u)(x)n(dx) -θ s z -|θs| 2 2α .
The truncation function ρ m , assumed to be continuously differentiable, is such that:

• The sequence (ρ m ) is increasing with respect to m.

• ρ m (x) : = 1, if |x| ≤ m, 0 ≤ ρ m (x) ≤ 1, if m ≤ |x| ≤ m + 1, and ρ m (x) = 0, if |x| ≥ m + 1.
Now, we list and check the properties of f m , required first to establish existence and uniqueness results for the BSDEs given by (f m , B) and also, to ensure the passage to the limit, as m goes to ∞.

1. Each generator f m has the Lipschitz property: i.e, for any m ≥ M ,

∃ C(m) > 0, ∀ t, ∀ z, z ′ ∈ R, ∀ u, u ′ ∈ L 2 ∩ L ∞ (n), |f m (t, z, u) -f m (t, z ′ , u ′ )| ≤ C m |z -z ′ | + ( R * (u(x) -u ′ (x)) 2 n(dx)) 1 2
.

To handle the increments w.r.t z and u, we proceed by linearization analogously as in Section 3.1. The Lipschitz property follows from the boundedness of the increments, which itself follows from the truncation.

2. For each f m and using Theorem 2.5 in [START_REF] Royer | Backward stochastic differential equations with jumps and related non-linear expectations[END_REF], we prove that the monotonicity assumption (H comp ) and the associate condition (A γ ) holds true. Both these conditions result from the assumption (H 2 ) on the increments in u. In particular, for each f m , (H 2 ) is satisfied with: γ m := γ, for all m, and, for any

u in L 2 (n), ρ M (u) is in L 2 ∩ L ∞ (n).
Hence, a comparison result holds true for the BSDEs given by (f m , B).

We have

sup m |f m (s, 0, 0)| ≤ |θ s | 2 2α , (14)
which entails that sup m |f m (s, 0, 0)| is in L 1 (ds ⊗ dP).

4. The sequence (f m ) converges to f in the following sense

∀ s, s ∈ [0, T ], ∀ z ∈ R, ∀ u ∈ L 2 ∩ L ∞ (n) , f m (s, z, u) → f (s, z, u), P-a.s. as m → ∞.
This convergence results from the truncation of the functionals of z and u defining f m . Besides and thanks to the increasing property of (ρ m ) and the positiveness of the square functional involving z, (f m ) is itself increasing.

Step 2: Useful properties of this approximation

Referring to the results in [START_REF] Royer | Backward stochastic differential equations with jumps and related non-linear expectations[END_REF] or in [START_REF] Pardoux | Generalized discontinuous backward stochastic differential equations, Backward stochastic differential equations[END_REF], we obtain existence and uniqueness for a solution (Y m , Z m , U m ) in S 2 × L 2 (W ) × L 2 ( Ñp ) of the BSDEs given by (f m , B). To give estimates of Y m in S ∞ , which may depend on m, we state an auxiliary result.

Lemma 4 Let (Y n , Z n , U n ) be a solution in S 2 × L 2 (W ) × L 2 ( Ñp ) of a BSDE of type (Eq2) given by (g n , B), with a generator g n L n -Lipschitz and a terminal condition B bounded, we have

∃ K(L n , T ) > 0, ∀ t, |Y n t | 2 ≤ K(L n , T )E | B| 2 + ( T t |g n (s, 0, 0)|dC s ) 2 |F t . (15)
(the detailed proof, adapted from [START_REF] Briand | A converse comparison theorem for BSDEs and related properties of g-expectation[END_REF], is given in Appendix A). The solution being in S ∞ (R) × L 2 (W ) × L 2 ( Ñp ), we would like to get free from the dependence in m of the estimates of Y m in S ∞ (R): to this end, we justify that the estimates in Lemma 3 hold true for the solution (Y m , Z m , U m ) and for all m: in fact, each f m satisfies the assumption (H 1 ) with the same parameters as f and hence,

• (Y m ) is uniformly bound in S ∞ (R).
• (Z m ) and (U m ) are uniformly bounded in their respective Hilbert spaces, i.e. L 2 (W ) and L 2 ( Ñp ).

As in Corollary 1, we obtain an equivalence result for (U m ) m : more precisely, there exists a constant C independent of m and satisfying

1 C [0,T ]×R * |U m s (x)| 2 n(dx)ds ≤ T 0 |U m s | α ds ≤ C [0,T ]×R * |U m s (x)| 2 n(dx)ds.
We now justify the existence of processes Ỹ , Z and Ũ , which are the respective limits in a specific sense of (Y m ) m , (Z m ) m and (U m ) m . Using the comparison result for BSDE with jumps given by [START_REF] Royer | Backward stochastic differential equations with jumps and related non-linear expectations[END_REF] and justified in

Step 1, (Y m ) m is increasing. Hence, we can define Ỹ Ỹs = lim m ր (Y m s ), P-a.s. and for all s.

Since (Z m ) and (U m ) are uniformly bounded in their respective BMO spaces and, in particular, in L 2 (W ) and L 2 ( Ñp ), we can extract from both sequences converging subsequences in the weak sense. We denote by Z and Ũ their respective weak limits.

Step 3: Convergence of the approximation

In this last step, we prove the convergence of (Y m , Z m , U m ) to a solution of the BSDE (6) with parameters f and B. To this end, we justify the passage to the limit as m goes to ∞ in

Y m t = Y m T + T t f m (s, Z m s , U m s )ds - T t Z m s dW s - T t R * U m s Ñp (ds, dx). ( 16
)
To achieve this, the essential step is to establish a "monotone stability" result, which is an adaptation of Proposition 2.4 in [START_REF] Kobylanski | Backward stochastic differential equations and partial differential equations with quadratic growth[END_REF].

Lemma 5 Assuming that (1) There exists a sequence (f m ) m such that, for all s and for all converging sequences (z m ) m and (u m ) m taking their values in R and L 2 (n(dx)), with

(u m ) uniformly bounded in L ∞ (n), lim ր f m (s, z m , u m ) := f (s, z, u), as m → ∞.
(2) Each f m satisfies (H 1 ) with the same parameters as f .

(

) There exists (Y m , Z m , U m ) m in S ∞ (R) × L 2 (W ) × L 2 ( Ñp ) 3
, which are solutions of the BSDEs given by (f m , B).

Then, (Y m , Z m , U m ) converges to ( Ỹ , Z, Ũ ) in the following sense

E sup t∈[0,T ] |Y m t -Ỹt | + |Z m -Z| L 2 (W ) + |U m -Ũ | L 2 ( Ñp) → 0, ( 17
)
and ( Ỹ , Z, Ũ ) solves the BSDE [START_REF] Delbaen | The mathematics of arbitrage[END_REF] given by (f , B).

We first check that (f m ) constructed in Step 2 satisfies (1) in Lemma 5

|f m (s, z m , u m ) -f (s, z, u)| ≤ |(f m -f )(s, z m , u m )| =(I) + |f (s, z m , u m ) -f (s, z, u)| =(II)
.

Thanks to the continuity with respect to z and u of f , whose expression is ( 7), (II) converges to zero. Furthermore, the boundedness of both sequences (u m ) and (z m ) ensures that, for m large enough, f and f m coincide: hence, (I) is equal to zero, which leads to the conclusion.

Proof of Lemma 5

We relegate to Appendix B the tedious proof of the strong convergence of (Z m ) and (U m ) in their respective Hilbert spaces and, assuming this, we prove the existence of a solution of the BSDE [START_REF] Delbaen | The mathematics of arbitrage[END_REF] with parameters (f , B). To identify ( Ỹ , Z, Ũ ) as a solution of the BSDE (6), we have to prove

(i) t 0 Z m s dW s → t 0 Zs dW s , as m → ∞, (ii) t 0 R * U m s Ñp (dx, ds) → t 0 R * Ũs Ñp (dx, ds), as m → ∞, (iii) t 0 f m (s, Z m s , U m s )ds → t 0 f (s, Zs , Ũs )ds, as m → ∞.
Firstly, since Z and Ũ are the weak limits of (Z m ) and (U m ), assertions (i) and (ii) correspond to the strong convergence, respectively, in L 2 (W ) for (i) and, in L 2 ( Ñp (dx, ds)) for (ii). Now, to prove (iii), we need to justify that the convergence holds true in L 1 (ds ⊗ dP). This is argued as follows:

• on the one hand and from (i) and (ii), we have, eventually along a subsequence, the convergence in ds ⊗ dP-measure of (Z m ) and (U m ). From this remark and using assertion (1) in lemma 5, we deduce the convergence in ds ⊗ dP-measure of the sequence (f m (s, Z m s , U m s )) to f (s, Zs , Ũs ). • on the other hand, we prove the uniform integrability of (f m (s, Z m s , U m s )), along the subsequence where (Z m ) and (U m ) converge. Using now assumption (H 1 ) satisfied by each f m and Corollary 1 to argue that (|U m | α ) is uniformly bounded, this implies

∃ K 1 ∈ L 1 (ds ⊗ dP), |f m (s, Z m s , U m s )| ≤ K 1 s + α|Z m s -Zs | 2 + α| Zs | 2 + C,
with K 1 and C depending only on the parameters α and θ appearing in (H 1 ). The strong convergence in L 1 (ds ⊗ dP) of (|Z m -Z| 2 ) m implying its uniform integrability, the result follows. Using (i), (ii) and (iii), we obtain that ( Ỹ , Z, Ũ ) satisfies

Ỹt = B + T t f (s, Zs , Ũs )ds - T t Zs dW s - T t R * Ũs Ñp (dx, ds). ( 18
)
To prove the convergence given by [START_REF] Protter | Stochastic integration and differential equations[END_REF] in Lemma 5, we just take the supremum over t and then the expectation in Itô's formula applied to Ỹ -

Y m E sup t∈[0,T ] | Ỹt -Y m t | ≤ E T 0 |f (s, Zs , Ũs ) -f m (s, Z m s , U m s )|ds + E sup t∈[0,T ] | T t ( Zs -Z m s )dW s | + E sup t∈[0,T ] | T t ( Ũs -U m s ) Ñp (ds, dx)| .
Now, thanks to the convergence given by (iii) and Doob's inequality for the square integrable martingales, it follows that:

E sup t | Ỹt -Y m t | → 0.
4 Application to finance: the compact case

The optimization problem

The utility maximization problem considered is associated with the exponential utility function U α with real parameter α U α (.) :=exp(-α•) . This problem consists in maximizing the expected value of the exponential utility of the portfolio (i.e the wealth at time T minus a liability denoted by B). More precisely, we characterize in this section the expression of the value process V B which is defined at time t by

V B t (x) = sup π∈At E(U α (x + T t π s dS s S s- -B)|F t ). (19)
B stands for the contingent claim, which is assumed to be an F T -measurable random variable and x is a constant standing for the welath at time t.

Statement of the main result

Theorem 3 For any constant x, the expression of

V B t (x), as defined in (19), is V B t (x) = -exp(-α(x -Y t )), (20)
where Y t is the first component of the solution (Y, Z, U ) of the BSDE [START_REF] Delbaen | The mathematics of arbitrage[END_REF] given by the parameters (f , B), and whose generator f is

f (s, z, u) = inf π∈C α 2 |πσ s -(z + θ α )| 2 + |u -πβ s | α -θz - |θ| 2 2α .
Moreover, there exists an optimal strategy π * such that: π * ∈ A t , and satisfying

π * s ∈ argmin π∈C α 2 |πσ s -(Z s + θ s α )| 2 + |U s -πβ s | α . (21)

Proof of theorem 3

The dynamic method As in [START_REF] Hu | Utility maximization in incomplete markets[END_REF], the aim is to construct a family of processes (R π ) π∈At , (t being fixed), which is defined on [t, T ] and which satisfies

(i) R π t = R t is a fixed F t -measurable random variable , (ii) R π T = -exp(-α(X π T -B)), (iii)
R π is a supermartingale for each π in A t , and there exists π * , π * ∈ A t , such that R π * is a martingale.

20

For this, we look for a process R π such that

∀ s ∈ [t, T ], R π s = U α (X π s -Y s ).
For sake of clarity, we use the notation X π instead of X π,t,x and we assume that there exists a triple (Y , Z, U )) solving a BSDE with jumps of the form (6), with terminal condition B and with a generator f to be determined.

Referring to Theorem 5.1, Chapter 2 in [START_REF] Ikeda | Stochastic differential equations and diffusion processes[END_REF]), we first apply a generalized Itô's formula to R π , for any strategy π,

R π s -R π t = -α s t R π u (π u σ u -Z u )dW u + s t R π u - R * (exp(-α(π u β u -U u )) -1) Ñp (du, dx)) -α s t R π u (π u b u + f (u, Z u , U u ))du + α 2 2 s t R π u |π u σ u -Z u | 2 du + s t R π u R * (exp(-α(π u β u -U u )) -1 + α(π u β u -U u )) Np (du, dx). R π satisfies: dZ = Z -dM π + ZdA π , with A π such that dA π u := α 2 2 |πσ u -Z u | 2 -α(πb u + f (u, Z u , U u )) du + R * g α (U u (x))n(dx)du,
and with M π t,s := Es(M π ) Et(M π ) , where E(M π ) denotes the Doleans-Dade exponential of the local martingale M π

M π u = (-α(πσ -Z) • W ) u = M 1 u + (exp(-α(πβ -U )) -1) • Ñp ) u = M 2 u ,
with M 1 (resp. M 2 ) standing for the continuous part of M π (resp. the discontinuous part). It follows that R π has the multiplicative form.

∀ s ≥ t, R π s = R π t M π t,s exp(A π s -A π t ). (22)
Since: (exp(-α(πβ -U )) -1) ≥ -1, P-a.s., the Doleans-Dade exponential of M 2 is a positive local martingale and hence, a supermartingale. The supermartingale condition in (iii) holds true, provided, for all π, the process Ãπ := exp(A π ) is non decreasing: this entails

α 2 2 |πσ u -Z u | 2 -α(πb u + f (u, Z u , U u )) + R * (exp(-α(πβ u -U u )) -1 + α(πβ u -U u ))n(dx) ≥ 0.
This condition holds true, if we define f as follows

f (s, z, u) = inf π∈C α 2 |πσ s -(z + θ α )| 2 + |u -πβ s | α -θ s z - |θ s | 2 2α .
Provided: u ∈ L 2 ∩ L ∞ , this assumption ensures that |u -πβ| α is finite for any π ∈ A t , thanks to the boundedness of β and π (π taking its values in the compact set C) and for any z, z ∈ R, f (s, z, u) is almost surely finite.

Expression of the value function and optimal strategies

To prove the supermartingale property of R π for any strategy π (π ∈ A), we rely on the results obtained in Section 3 on the BSDE with parameters (f , B).

For this, we use the multiplicative form (22) of R π obtained in the previous paragraph.

Being a stochastic exponential of a martingale with jumps strictly larger than -1, E(M π ) is a positive local martingale for any π, and consequently, there exists a sequence of stopping times (τ n ) converging to T such that M π .∧τ n is a martingale. Since e A π is non decreasing and R t is non positive, we can claim, on the one hand, that R π .∧τ n satisfies

∀ s ≤ u, ∀A ∈ F s , E(R π u∧τ n 1 A ) ≤ E(R π s∧τ n 1 A ). (23)
On the other hand, since R π t := -e -αX π t e αYt = -e -αx e αYt , we use both the uniform integrability of (e -αX π τ ) (resulting from Lemma 1), where τ runs over the set of all stopping times and the boundedness of Y to obtain the uniform integrability of (R π .∧τ n ) n . Hence, the passage to the limit as n goes to ∞ in ( 23) is justified and it implies

∀ s, s ≤ u, ∀ A ∈ F s , E(R π u 1 A ) ≤ E(R π s 1 A ), which entails the supermartingale property of R π .
To complete the proof, we justify the expression of V B (x) by proving the optimality of any strategy π * satisfying (21). In fact, for this expression of π * , we have:

A π * ≡ 0 and hence, R π * = R π * t M π * is a true martingale (π * is in A t , thanks to Lemma 1). As a result, sup π∈At E(R π T ) = R π * t = V B t (x).
Using that (Y, Z, U ) is the unique solution of the BSDE given by (f , B), we obtain the expression (20) for the value function.

Characterization of optimal strategies

The following lemma answers positively to both problems of existence and measurability of a strategy π * satisfying (21).

Lemma 6 Let Z and U be two predictable processes taking their values in R and L 2 (n(dx)) and C a subset of R.

a. The process F defined as below is again predictable

∀s ∈ [0, T ] , F (s, Z s , U s ) = inf π∈C α 2 |πσ s -(Z s + θ α )| 2 + |U s -πβ s | α .
b. There exists a predictable version of π * which attains, for all s, ω, the minimum taken over C of the sum of the two convex functionals

α 2 |πσ s -(Z s + θ α )| 2 and |U s -πβ s | α .
Proof of lemma 6 To prove assertion a., we first introduce the sequence (F n ) n of predictable processes

F n := F n (s, Z s , U s ) := inf {π n ∈Q, dist(π n , C) ≤ 1 n } α 2 |π n σ s -(Z s + θ α )| 2 + |U s -π n β s | α .
F n is a predictable process: in fact, on the one hand, the infimum is taken over a countable subset of Q. On the other hand, the functionals of the predictable processes Z and U are continuous. Besides and thanks to the fact that: F (s, Z s , U s ) = lim n F n (s, Z s , U s ), F is itself predictable as a limit of such processes. Now, to justify assertion b., we argue the existence of a predictable selection by applying a measurable selection theorem (one reference is [START_REF] Aumann | Measurable utility and the measurable choice theorem Editions du Centre[END_REF]) to the set

G := (t, ω) s.t. {|f (t, ω, π, Z t (ω), U t (ω)) -inf π∈C f (t, ω, π, Z t (ω), U t (ω))| + (1 -1 C (π)) = 0}.
The presence of the last term in the left-hand side ensures that the predictable choice of π := (π(s, ω)) takes its values in C.

The non compact case

To solve the problem (19) and give the expression of the value function in the non compact case, we need to solve the same BSDE with parameters (f , B) and with f always given by [START_REF] Dellacherie | Théorie des martingales[END_REF]. We again check that: f : (z, u) → f (s, z, u) is finitely valued (P-a.s. and for all s): in fact, f is a convex continuous functional of z and u, which tends to ∞, as |z| and |u| L ∞ goes to ∞ and hence, the infimum is attained. But, as soon as C is no more compact, the linearization procedure used to prove the uniqueness result cannot be applied: in this case, the BMO controls obtained in section 3.4 for the processes κ and γ do not hold anymore. This does not exclude a priori that there could be a unique solution.

The method used as well as the justifications of the proof of the existence result being very similar as in the compact case, we just give here an outline of the procedure. The method consists once again in constructing a good approximation: for this, we introduce a sequence of BSDEs with parameters (f m , B) such that:

1. The sequence (f m ) m is monotonic and it satisfies f m (s, z, u) → f (s, z, u), P-a.s. and for all s, as m → ∞.

2. All the BSDEs given by (f m , B) satisfy (H 1 ) and (H 2 ): hence, thanks to Section 3, both the existence, uniqueness and comparison results hold for this sequence of BSDEs.

We then define (f m )

f m (s, z, u) = inf π∈C m α 2 |πσ s -(z + θ s α )| 2 + |u -πβ s | α -θ s z - |θ s | 2 2α . (24)
C m being defined as follows: C m := C ∩ [-m, m] corresponding to the intersection with the compact subset of R.

Thanks to the results obtained in Section 3, there exists a sequence (Y m , Z m , U m ) of solution of these BSDEs. Furthermore, since (f m ) is decreasing w.r.t. m, it results from the comparison result, which is a direct byproduct of Theorem 1, that (Y m ) is decreasing (P-a.s. and for all s). Using now the results of Section 4, we get

V B m (x) := U α (x -Y m 0 ).
This corresponds to the optimization problem at time t := 0 and with C m as constraint set: (Y m 0 ) being a decreasing sequence, (V B m (x)) is increasing, which implies

lim m ր V B m (x) = lim m ր U α (x -Y m 0 ) = U α (x -Y 0 ).
The last step consists in identifying the limit (Y, Z, U ) as a solution of the BSDE with parameters (f , B) and then conclude

V B (x) = U α (x -Y 0 ).
For this, we rewrite the proof of a "monotone stability" result analogous to lemma 5. We just state the result, which is the key ingredient of the existence result. • Each f m satisfies (H 1 ) (with parameters independent of m).

Then, (Y m , Z m , U m ) converges to ( Ỹ , Z, Ũ ) in the following sense

E( sup t∈[0,T ] |Y m t -Ỹt |) + |Z m -Z| L 2 (W ) + |U m -Ũ | L 2 ( Ñp) → 0. (25)
(Y m ) is decreasing and the triple ( Ỹ , Z, Ũ ) is in S ∞ × L 2 (W ) × L 2 ( Ñp ) and solves the BSDE with parameters (f , B).

For the rest of the proof, we refer the reader to Paragraph 3.5.3 and, in particular, to Appendix B for the strong convergence given by (25). The proof of this last convergence is the key point: we refrain from doing this since the proof given in Appendix B can be rewritten identically (if we replace the increasing sequence of generators by a decreasing one) and, in particular, the estimates of lemma 3 hold true for the sequences (Y m ), (Z m ) and (U m ).

Conclusion

In that paper, we have solved the utility maximization problem with portfolio constraints in a discontinuous setting and by means of BSDEs. Our setting is very similar as the one in [START_REF] Becherer | Bounded solutions to Backward SDE's with jumps for utility optimization and indifference hedging[END_REF], but, contrary to the author of the aforementionned paper, the presence of constraints in the model entails that the driver has quadratic growth w.r.t. its variable z. Hence, our main achievement consists in obtaining existence and uniqueness results for such quadratic BSDEs with jumps. This theoretical study and the use of the dynamic principle allow to give the expression of the value function for any time t in terms of the solution of the BSDE and characterize optimal strategies for the problem.

Due to the restrictions, some questions remain for further investigations: in particular, the case when the Levy measure is infinite is unsolved (this assumption is already given in [START_REF] Becherer | Bounded solutions to Backward SDE's with jumps for utility optimization and indifference hedging[END_REF]). Here, we also restrict our study to the exponential utility function and we mention here that, in the power utility case, the utility maximization problem with non zero liability is an open problem.

6.1 Omitted proofs 6.2 Appendix A: proof of lemma 4

For convenience for the reader, we give a detailed outline of the proof which is adapted from Proposition 2.2 in [START_REF] Briand | A converse comparison theorem for BSDEs and related properties of g-expectation[END_REF] to the discontinuous setting. Contrary to lemma 3, where the first component of the solution is supposed to be in S ∞ , in this proposition, Y n is only assumed to be in S 2 . We first write Itô's formula for (e Γt |Y n t | 2 ), Γ being a non negative constant which is explicited during the proof.

d(e Γt |Y n t | 2 ) = Γe Γt |Y n t | 2 dt + e Γt 2Y n t dY n t + d Y n t , (26)
with

2Y n t dY n t + d Y n t := -2Y n t g n (t, Z n t , U n t )dt + |Z n t | 2 + R\{0} |U n t (x)| 2 n(dx) dt
This leads to

E T t e Γu |Z n u | 2 + |U n u | 2 L 2 (n) du |F t ≤ 2 E e ΓT |Y n T | 2 + 2 T t e Γu |Y n u ||g n (u, 0, 0)|du|F t . (30)
We consider again the integrated form of (26) between s and T and then, we take the supremum over s (s

∈

Appendix B: End of the proof of Lemma 5

To prove the essential result stated in Lemma 5, i.e. the strong convergence of (Z m ) and (U m ), we adapt the method already used in [START_REF] Kobylanski | Backward stochastic differential equations and partial differential equations with quadratic growth[END_REF]. To this end, we consider the function Φ K Φ K (x) = e 2Kx -2Kx -1 2K = g 2K (x) , which is twicely continuously differentiable and satisfies    Φ K (0) = 0, and Φ K ,

Φ ′′ K ≥ 0, Φ ′ K (x) ≥ 0, if x ≥ 0, Φ ′′ K -2KΦ ′ K = 2K.
These properties imply that, taking m, p such that m ≥ p ≥ M , with M given as in Step 1 in the proof of existence, we have: Φ with Ĉ given by Ĉs = where the limit, as m goes to ∞ (p being fixed), in the right-hand side of this inequality results from Lebesgue's convergence theorem: in fact, we have that (Y m s ) converges to Ỹs , P-a.s. and for all s, and: Ĉ, |Z p -Z| 2 and | Z| 2 are in L 1 (ds ⊗ dP). Before justifying the passage to the limit as p goes to ∞, we use that Z and Ũ are the respective weak limits of (Z m ) and (U m ) to obtain and hence, the desired convergence result follows: in fact, the limit in the right-hand side, as p goes to ∞, is equal to zero, thanks to the Lebesgue's convergence theorem.

Lemma 7

 7 We denote by (Y m , Z m , U m ) the solutions of BSDEs with parameters (f m , B) and we assume that (f m ) is such that:• (f m ) satisfies the monotone convergence lim m ց f m (s, Z m s , U m s ) := f (s,Zs , Ũs ) P-a.s. and for all s, with f continuous w.r.t. the variables z and u.

- 1 ) 2 L 2 0 e

 1220 We set: K = 4α and, transferring into the left-hand side the terms containing |Z m,p | 2 or |U m,p | 8α , we rewrite the equation (31)Eφ K (Y m,p 0 ( Ĉs + C + 2α(|Z p s -Zs | 2 + | Zs | 2 ))ds.We rely once again on the result given in Corollary 1 to claim the existence of a constantC (n) ds ≤ E T 8αY m,p s |U m,p s | 8α ds.This entails, taking the limit inf as m goes to ∞ and p being fixedEφ K ( Ỹ0 -Y p ( Ỹs-Y p s ) -1)( Ĉs + C + 2α(|Z p s -Zs | 2 + | Zs | 2 ))ds.

)

  8α( Ỹs -Y p s ) + 1) | Zs -Z p s | 2 ds ≤ lim inf m |U m,p s | 2 L 2 (n) ds,(since: e 8αY m,p s ≥ 1). Transferring now in the left-hand side of the last inequality of page 30 the unique term containing | Z -Z p | 2 , we getE ( Ỹs-Y p s ) -1)( Ĉs + C + 2α| Zs | 2 )ds

  [t, T ]) sup t≤s≤T e Γs |Y n s | 2 ≤ e ΓT |Y n T | 2We now apply the Burkholder-Davis-Gundy inequality to the supremum of the square integrable martingale K and the relation:Cab ≤ C 2 2 a 2 + 1 2 b 2 , to deduce the existence of a constant C such that Γs |Y n s | 2 |F t ≤ E e ΓT |Y n T | 2 + 2 Γu |Y n u ||g n (u, 0, 0)|du|F t Γs |Y n s | 2 |F t .From now, this constant C is generic and may vary from line to line. Combining this previous inequality with (30), we deduce Γu |Y n u ||g n (u, 0, 0)|du|F t . Γu |Y n u | 2 |F t + C 2 2 E Γs |Y n s | 2 |F t , the result follows.

					+ 2	t	T	e Γu |Y n u ||g n (u, 0, 0)|du + sup
	E sup
		+ C 2 2 E	t	T	e Γu |Z n u | 2 + |U n u | 2 L 2 (n) du|F t +	1 2	E sup
				E sup t≤s≤T	e Γs |Y n s | 2 +	t	T	e Γu |Z n u | 2 + |U n u | 2 L 2 (n) du|F t
					≤ CE e ΓT |Y n T | 2 +
	To obtain the desired relation, we check
	CE	t	T	e Γu |Y n u ||g n (u, 0, 0)|du|F t
	≤ 1 2 E sup	t	T	e	Γ 2 u |g n (u, 0, 0)|du	2 |F t .
	Noting that: |Y n t | 2 ≤ E sup

t≤s≤T |K T -K s |.

t≤s≤T e T t e t≤s≤T e T t e t≤u≤T e t≤s≤T e

  3 2α |θ s | 2 + |θs| 2 2α . Using the convexity of: z → |z| 2 , we obtain Z| 2 + | Zs | 2 .These estimates lead toF m, p ≤ Ĉs + 2α |Z m,p s | 2 + |Z p s -Z| 2 + | Zs | 2 + C.

	α 2 |Z m s | 2 ≤ α 2	1 3 |3Z m,p s | 2 + 1 3 |3(Z p s -Zs )| 2 + 1 3 |3 Zs | 2
		≤ 3α 2 (|Z m,p s | 2 + |Z p s -Zs | 2 + | Zs | 2 ),
	and, similarly,	α 4	|Z p s | 2 ≤	α 2	|Z p s -

Since the increments of the generator f are computed on the trajectories of predictable processes,[START_REF] El Karoui | Backward stochastic differential equations in finance[END_REF] is the expression of the process we are interested in, especially in the proof of the uniqueness result.

is a true martingale. We fix t (t ∈ [0, T ]) and we rewrite the formula (26) in the integrated form between s and T , for any s, t ≤ s ≤ T ,

We then rely on the Lipschitz property of the generator g n

with:

, and using that:

Combining both (28) and (29), setting: Γ = 4(L n )


Cedric Deffo Sikounmo
email: cedric.deffo-sikounmo@univ-smb.fr

Eric Benoit
email: eric.benoit@univ-smb.fr

Stéphane Perrin

Stephane Perrin
email: stephane.perrin@univ-smb.fr

Modeling situations in an intelligent connected furniture environment

Keywords: Indentication of situation, Ontology, Internet of things, Context, Furniture

Modeling of the situations in an intelligent connected furnitures environment

Introduction

The Internet of Thing is perceived as a pervasive presence around us of a variety of things or objects interacting between each other. The given things are physical objects or virtual ones. The physical objects are mainly connected objects.

Virtual objects have no existence in the concrete world but they are also able to propose services. The goal of our study is to recognize high level states of the rooms and more generally of the home i.e. normal state, abnormal state or alarm.

In addition, we want to be able to obtain intermediate states like "someone is in the kitchen" or "night mode". In the specic case of home activity and state measurement, we consider that furniture units are especially suitable to provide low level information. This is particularly due to their presence around the house and because they are non-intrusive nature. Recognizing and identifying house states or other high level information can be done using several methods. In this paper, we propose to use an ontology-based approach to represent a context made of an aggregation of realized situations. The used ontologies can be supplied and updated directly by sensors. As suggested by [START_REF] Bettini | A survey of context modelling and reasoning techniques[END_REF], we apply inference rules on the graph that represents the concept ontology i.e. the ontology of physical entities. This thus provides high level information. However, inference rules are applied on the graph representing the ontology of realized situations. The latter allows the designer to take into consideration dynamic modication of the measured scene without adding new rules. However, if a situation has multiple instances, the problem turns out to distinguish each instance of the others. In this paper, we propose to attach rules on the situation. Using this approach, we show that multiple instances of context within a situation are distinguishable.

In the next section on this paper, we discuss about several approaches to model and to represent information of context. In the section 3, we present the ontology based representation of a furniture community. Section 4 presents the representation of situations in a furniture community. Section 5 explains the recognition process, illustrated with an example.

State of the art

In the literature, several studies were realized on the various situations that can be observed in an intelligent environment. Several studies do not consider the distinction between a situation and a context [START_REF] Attard | Ontology-based situation recognition for context-aware systems[END_REF] [16] [START_REF] Guermah | An ontology oriented architecture for context aware services adaptation[END_REF]. The information of context is generally dened as any information being used to characterize the situation of an entity. Such entity can be a person, a place or an object that is considered as relevant for the interaction between a user and an application, including the user and the application themselves [START_REF] Dey | Understanding and using context[END_REF]. In this paper, which relates to a community of intelligent connected furniture units, we distinguish three types of context information as proposed in [START_REF] Attard | Ontology-based situation recognition for context-aware systems[END_REF] : Raw context: Context whose related information is directly acquired through sensors. For example, the localization, the current time or any environmental parameters.

Interpreted context: Information of context derived from the raw contexts.

For example: Sat on a chair; day periods like the morning, the afternoon or the the evening; activities such as sleeping, working.

Live context: Information obtained by recurring combination of the raw contexts and the interpreted contexts which makes sense for a human. For example : working at the oce, looking at the TV within the living room.

Several approaches propose to model context information. Tao Gu et al. [START_REF] Gu | An ontology-based context model in intelligent environments[END_REF] classied these approaches into three categories: Application oriented approach: This approach produces models of context for specic applications. These models are generally proprietary models and don't have a formal basis. They don't support the knowledge sharing between the various systems [START_REF] Kindberg | A web-based nomadic computing system[END_REF].

Model oriented approach: This category of models usually uses conceptual modeling approaches to represent the context. we can quote studies in [START_REF] Harter | The anatomy of a context-aware application[END_REF] and [START_REF] Wu | Sensor fusion for context understanding[END_REF] which uses the entity-relation model, studies in [START_REF] Henricksen | Modeling context information in pervasive computing systems[END_REF] which uses at the same time the entity-relation model and UML diagrams. Finally, studies in [START_REF] Henricksen | Generating context management infrastructure from high-level context models[END_REF] reformulate this model with the extension of the object-Role Modeling (ORM) [START_REF] Henricksen | Generating context management infrastructure from high-level context models[END_REF]. Although this approach takes into account the temporal aspect of context information, it badly supports the generalization of knowledge and the reasoning. In [START_REF] Gonzalez | Context-based representation of intelligent behavior in training simulations[END_REF], Avelino J. Gonzalez et al oers an intelligent, contextbased behavior representation from the dierent perceptions of a submarine for training simulations. They use a hierarchical representation of contexts and they add rules to determine the best tactics to adopt depending on the situations that arise.

Ontology oriented approach: The context can be regarded as a kind of specic knowledge. It can be modeled like an ontology. Indeed, the use of ontologies allows not only the context modeling, but also the inference engine based reasoning on the collected data [START_REF] Guermah | An ontology oriented architecture for context aware services adaptation[END_REF].

We will use this latter approach in the following section in order to represent dierent information from contexts in a community of connected and intelligent furniture units.

3 Representation of a furnitures community

Presentation

A community of furniture units is a set of furniture pieces of various species like table, chair, bed, or mirror. These furniture units lead to a given environment.

An environment must be understood as any place within where furniture pieces may exist. It can be for example a house, the parts of a house or an oce. In this paper, we dene a community of intelligent connected furniture units as a set of furniture units, objects and humans interconnected within an environment.

In this community, the furniture pieces and the other objects can be intelligent ones i.e. be able to acquire, to handle and to exchange knowledge. In such environment, the furniture pieces, the objects and the humans are considered as the actors of the system. In the case of a house, several furniture units communities related to the dierent rooms can be identied. A house is generally divided into multiple rooms: living room, room, kitchen, shower, balcony, garden, hollow, garage, etc. A furniture unit can be dedicated to a given room or not, for example a bed or a chair. In the same way, a category of furniture unit can have several variations which are specic to a given room. Moreover, the ontology also includes objects that are not furniture units but that interact with them. For example, a TV interacts with a TV cabinet, so it must be considered into the ontology. In addition, it must be putted forward that human activities related to furniture are part of the most raised activities in a house. As the data acquisition requires sensors, the intelligent connected furniture units found naturally their place in the smart home. The chosen tool for the classication of the data given by sensors is an ontology based one. Such kind of tool already shows its popularity in the eld of the engineering of knowledge due to its portability, its evolutionary and its exibility [START_REF] Gu | An ontology-based context model in intelligent environments[END_REF].

Representation

Let us rst describe the list of concepts and relations used to build the ontology.

The gure 1 presents an ontology of a community of smart connected furniture

Concepts

Actor is the basic concept used to represent the objects, the furniture units and the living beings in a community of connected entities.

Service is concept representing the various features or capabilities which an actor can oer.

Human is the concept that represents the human beings in the community. Object represents the non-human actors. Furniture_unit represents a piece of furniture. A furniture piece is described by its style, its category, its functions and by the whole of the parts which compose it. The style is a set of aesthetic criteria and of materials, which helps to recognize the furniture period and its design. The category of a furniture unit refers to its species: tables, chairs, beds, mirrors. A function of a furniture unit leads to the various services it can oer to its user. The construction of a furniture requires various parts which can depend on its category, such as a mounting in the case of a chair or a table, a plate in the case of a table, a back and a seat in the case of a chair. As a result, the ontology of a furniture unit includes the following properties: Style, category, features and parts. This ontology aggregates the various concepts of an environment made of a community of intelligent connected furniture units, connected objects and beings humans. Into the following section, we introduce the representation of the various situations which rise from the interpretation of the contexts of the concept instances. [START_REF] Bettini | A survey of context modelling and reasoning techniques[END_REF] Representation of the situations in furniture community

Presentation

We dene a situation as follow. A situation is dene on a set of actors. Each actor produces sensor data, the raw context, and interprets them to produce its interpreted context. The situation aggregates the raw contexts and the interpreted contexts issued from these actors. Its then appear as an ontology. We distinguish the denition and the realization of situation: The denition is a concept ontology, and the realization is an instance ontology. Actually, the denition of a situation is also called a reference situation and leads to the concept of infon as proposed by Keith D. in [START_REF] Devlin | Situation theory and situation semantics[END_REF].

In this section we provide a representation of the various situations that can be found in an environment holding a community of connected and intelligent furniture units. Given an environment we seek to represent the various actors and the relations between them as described in section 3. After this classication stage, the resulting ontology is instantiated and initialized according to the facts of the actor instances. The facts of a furniture unit instance comes from the measured values acquired by its sensors. From this ontology instantiated, we try to recognize the various situations of the environment and the relations which exist between them. Several denitions were proposed in the literature.

In [START_REF] Ye | Situation identication techniques in pervasive computing: A review[END_REF], Ye et al. dene a situation as an external semantic interpretation of the data sensor.

The interpretation term explains the fact that the situations allot signicances to the measured values.

The external term reects the fact that interpretation is the applications point of view rather than the sensors one.

The term semantic reects the fact that an interpretation gives a signicance to a set of relations between objects. The concerned sets are themselves dened by data given by sensors.

They alternatively dene a situation by the collection of the relevant contexts, by discovering the signicant correlations between them, and by the assignment of labels with a descriptive name.

An ontology in the articial intelligence domain can be dened as the explicit specications of a conceptualization; who is a means of representation, of knowledge sharing and of knowledge reasoning [START_REF] Gruber | Ontology. In: Encyclopedia of Database Systems[END_REF]. Several papers gives an outline of the approaches of reasoning on ontology based contexts [START_REF] Bettini | A survey of context modelling and reasoning techniques[END_REF], [START_REF] Compton | A survey of the semantic specication of sensors[END_REF] Here, the situations are dened in term of rules. Example 2. N i=1 context(x i ,user) -→ IsInvolvedIn(situation,user), N > 1

The example 1 is similar to the example 2 in both cases a situation is represented by rules. Andrey Boytsov et al. [START_REF] Boytsov | Situation awareness meets ontologies: A context spaces case study[END_REF] bases work which precedes to dene a generator of situations. Moreover, they present two concepts of situations: a potential situation is an entity; A potential situation is a relation. Gu and Al.

[15] develop system SOCAM (Service Oriented Context Aware Middelware) or the concept of situation is similar to that of CONON.

Representation

Concepts In the proposed approach, we distinguish the denition of a situation from its realization. The realization of a situation is materialized by the existence of a sub-graph on individuals linked by the relations Has_part, Part_of and Connected_to. A situation is then dened by a set of rules on concepts.

Thus, the mathematical representation of a situation can be written as follows: let N , the set of the nodes, V the set of the relations and R the set of the rules then the situation S can be dened as: S = G(N, V, R). A graph representation of the situation work is given by the gure 3 and the relation "Close_to" is describing in [START_REF] Deo Sikounmo | States measurement in a context of intelligent connected furnitures[END_REF].

Relations

The generalization: a situation generalizes another situation, if the realization of the second one implies the realization of the rst one. For example the situation to listen to the music is a specialization of the situation entertainment [START_REF] Ye | Representing and manipulating situation hierarchies using situation lattices[END_REF].

The composition: a situation can result from the composition of other smaller situations; for example the situation barbecue breaks up into these two situations uses a barbecue with gas and uses food. In [START_REF] Mccowan | Automatic analysis of multimodal group actions in meetings[END_REF] McCowan et al., dene a situation of group (for example 'discussion' or 'presentation') like a composition of situations of individual users (for example 'to speak', 'to write').

of dependence: a situation depends on another situation, if an occurrence of rst is determined by an occurrence of the second. The dependence can be one limited time or unlimited, as proposed by [START_REF]Location and Context Awareness, 4th International Symposium, LoCA 2009[END_REF]. Sometimes, in the long run the dependence can be more useful while deducing from the high level situations. For example, a situation of going to work to perhaps better infer a situation to return to the house of work than of other situations dependent on short ranges.

temporal sequence a situation can occur before or after another situation, or interlace with another situation; for example, the situation taken of the pill must be realized after the situation dinner [START_REF]Workshops Proceedings of the 7th IEEE International Conference on Data Mining (ICDM 2007)[END_REF].

Several techniques for the situation discovery were already proposed. Juan et al. Our contribution in this area consists of a complete situation discovery process based on specications techniques. In addition, it provides all situation instances in our instantiated concept ontology.

In the literature, the studies on the situation recognition in an pervasive computing environment assume that several basic situations are known without worrying about the way they are obtained. Moreover, a separation between the concept ontology of the sensors and the situations ontology is noticed. In this paper, the representation we propose makes it possible to describe furniture units and the various entities found in their environment. Thus, starting from an instantiation ontology presented in section 3, the situation references as described in section 4 is used for the situation discovery process. This last process provides the desired realized situations. The complete process is detailed bellow. In this paper, the instance of a concept ontology is an ontology where each constitutive entity is associated with a set of facts. Each fact is a measured value of an entity property.

Situation recognition

The situation recognition technique proposed here assumes the existence of an instance of concept ontology and a set of reference situations. The recognition of situations is rst processed from the entities of the instance of our concept ontology and then performed from the situations deduced from this rst stage.

Recognition of situations starting from the context instances

Recognition of situations starting from the raw and interpreted context instances is the rst way to recognize the various situations of an environment. Knowing a instantiated ontology, the contexts of the various instances are collected. The interpreted contexts also called basic situations are deduced from this collected information.

Example 4. We want to know if a person sat on a chair. For that, the chair is instrumented with sensors located on his base and back. A person is sitting on a chair if he is in contact with the chair on its seat points. Moreover, there are several types of sitting attitudes which depend of the contact between the human and the back of the chair. The gure 2 gives a representation of a basic situation reference.

Recognition of situations starting from the basic situations High level situations are raised from the basic situations. They are combinations of the basic situations and/or of the entities of our instantiated ontology. The recognition of situations makes it possible to present the various potential situations of the system, without being certain of their realization.

Realization situations

Given a instance ontology and a set of potential recognized situations, a situation is said to be realized if there exists at least a subgraph of the instance ontology graph that respects at least one of the rules of our reference situation.

Proposition 2. A situation

S = G(N, V, R) is realized if: ∃G i (N i , V, r i ); r i is veried, with N i = {i /(i is_instance_of c) ∧ (c ∈ N )} and r i ∈ R. G i is also called instance of the reference situation S.
To discover situation in an environment of intelligent connected furniture our system must carry out two stages which are:

Step 1 Recognition of all potential situations based on a list of reference situations.

Step 2 Discovering the situations realized starting from the potential situations extracted in step 1.

Thus, this approach enables us to list all the situations of an intelligent connected furniture environment. Moreover, for each new entity, the same process is reiterated. That makes this approach dynamic in time. Algorithm 2: Discovered situations realized.

Application

In a rst time, let an ontology instance describing an environment made up of three humans and two chairs and a table as described in the gure 4. In a second time the same instance of concept ontology with new relations as shown in the gure 5 is given. We seek to recognize the various situations in which the actors of our environment are concerned. For that, we dene the following situations By applying the steps presented in section 5.2 we have the following results: case of the gure 4

Step 1 (discovery of all potential situations) result of algorithm 1:

{Close to situation, Sitting situation, Working situation}

Step 2 (discovered of realized situations) result of algorithm 2:

{Sitting situation(Human_1, Chair_2), Sitting situation(Human_2, Chair_1)} Step 1 (discovery of all potential situations) result of algorithm 1:

{Close to situation, Sitting situation, Working situation}

Step 2 (discovered of realized situations) result of algorithm 2:

{Sitting situation(Human_1, Chair_2), Sitting situation (Human_2, Chair_1),

Close to situation(Table_1, Chair_1), working situation(Human_1)}.

Conclusion and perspectives

In this paper we propose a generic representation of the situations of a community of intelligent connected furniture units. For that, we chose a representation of the physical entities by using ontologies. This representation enables us to emphasize the various parts of a furniture and another entities which can collaborate or interact with furniture units. From the description of community of furniture units, we seek to recognize the various states of our community. Thus, to recognize such states, reference situations are modeled by graphs made of nodes and relations and by rules. From this representation of situations references, we dene algorithms which make it possible to nd potentially realizable situations and actually realized situations. Given examples illustrate the use of the proposed algorithms in order to provide desired actually realized situations.

The future work for the improvement of the presented study will concern the discovery of new situations in an autonomous way and the association of the masses to the various elements of the graph of a situation. This last point of improvement will make possible to perform uncertainty based reasoning using theories like: the probability theory, the fuzzy sub-set theory or the evidence theory.

Fig. 1 .

 1 Fig. 1. Ontology of a community of Furniture

  Instance_i represents the individuals i.e. the entities resulting from the instantiation of the various concepts. Except for services, each instance leads to a physical entity. Relations Is_a: concept -→ concept is the subsomption relation (hierarchisation). It is a transitive relation. Type: individual -→ concept binds concrete entities to an abstract concept. Part_of: Actor -→ Actor and Has_part: Actor -→ Actor are the part-to-whole mereological relations between Actor concepts. Oered_by: Service -→ Actor and oers: Actor -→ Service are relations establishing a link between an actor and the various services it oers. Connected_to: Actor -→ Actor is a topological relation used to formalize the interaction between actors. Part_of: Actor individual -→ Actor individual and Has_part: Actor individual -→ Actor individual are the part-to-whole mereological relations between Actor instances. They are inferred from the Is_instance_of and the mereological relations between Actor concepts. Connected_to: Actor individual -→ Actor individual is a topological relation used to translate the interaction between actor individuals. The instances of this relation are given by the observation of the individuals, i.e. the physical entities.

Example 1 .

 1 (?u locatedIn Bedroom) & (Bedroom lightLevel LOW) & (Bedroom drapeStatus CLOSED) -→ (?u situation SLEEPING) Gu et Al. [15] develop the SOCAM system (Service Oriented Context Aware Middelware) where the concept of situation is similar to the CONON one. Anagnostopoulos et al. [2] proposes a technical situation awareness based on the combination of ontology and fuzzy logic where the situations are concepts and relations.

Example 3 .

 3 Sitting situation is the graph where: N = {Chair, Seat, Human}, V = {P art_of, Has_part, Is_instance_of, connected_to} and R = [r 1 = {(P art_of (Seat, Chair)∨Has_part(Chair, Seat))∧Connected_to(Human, Seat)}]. Let Sitting_situation denoting the result of this inference. It represents the fact that the Sitting situation is realized. A graph of the situation position seat is given in gure 2.One can note the various steps of the realization of a situation which will be detailed in the section 5.2. First of all, the step (a) in the gure 2 represents the graph of the situation which refers to sitting. The step (b) shows us a instantiation of this graph with dierent instances from the concepts Seat, Chair and Human. The step (c) of the gure 2 established the relations between the instances while basing itself on the observation; an example we can quote the study of the data sensors.

Fig. 2 .

 2 Fig. 2. Sitting situation: (a) graph of the reference situation (b) instantiation of two individuals of types Chair and Human (c) observation of the connected_to relation between individuals

Fig. 3 .

 3 Fig. 3. Graph of the reference situation labeled work

[ 26 ]

 26 class them in two categories which are: Specications based technics These approaches generally build a model of situation with an a priori expert knowledge, and reasons on it with measured values as inputs. As example of technique we can quote Space Logic programming spacial and temporal logic, ontology and Fuzzy logic Training based techniques The large variability of the specications and identications of the situations, that come from factors like time or the place or the individual,returns inconvenient the specications based techniques. Training techniques were largely applied to the training of complex associations between the situations and the measured values. In [26] Juan et al. characterizes this training based techniques.

  references: Working situation is a graph or: N = {Sitting_situation, Human, T able}, V = {P art_of, Is_instance_of, connected_to, Close_to} and R = [r 1 = {sitting_situation∧Connected_to(Human, T able)∧Close_to(Chair, T able)}, r 2 = {Connected_to(Human, T able)}]. Sitting situation is a graph or: N = {Chair, Seat, Human}, V = {P art_of, Is_instance_of, connected_to} and R = [r 1 = {(P art_of (Seat, Chair)∨Has_part(Chair, Seat))∧Connected_to(Human, Seat)}]. Close to situation is a graph or:N = {Chair, T able}, V = {Close_to} and R = [r 1 = {(Close_to(Seat, Chair)}].

Fig. 4 .Fig. 5 .

 45 Fig. 4. Example of instance of an ontology representing an environment made up of chairs and humans

  Example 5. The recognition of a working situation is performs throw the recognition of the basic situations like a person sitting on a chair, a computer on a table, a running computer, a person in contact with the table and the chair near the table. The gure 3 gives a situation reference. In a general way a situation is known as potentially realizable or recognizable in an instance of our concept ontology O i if for any item of the set of the whole nodes situation N there exists an instance in O i . Proposition 1. S = G(N, V, R) is a potential situation (a situation denition) if: ∀c ∈ N, ∃i; i instance_of c and i ∈ O i .

  O i an ontology instance Input: LRS a list of the reference situations Output: LP S a list of the potential situations foreach s ∈ LRS do if ( ∀c ∈ N (s), ∃i; i instance_of c and i ∈ O i ) then : O i an ontology instance Input: LP S a list of the potential situations Output: LS a list of the situations carried

	LP S ← LP S + s
	end
	end
	return LP S;
	Algorithm 1: Discovered potential situations

Input:

Inputforeach s ∈ LP S do if ( ∃r ∈ R(s); r is veried ) then LS ← LS + s end end return LS;

Acknowledgments

This research is supported by the Universite Savoie Mont Blanc, the Association des Pays de Savoie and Miliboo Corporation.