Nyassoke Titi

Gaston Clément

Kamdem Jules

Louis Aimé Fono

Fishery Management in a Regime Switching Environment: Utility Based Approach

Keywords: regime switching, floods, crra utility, logistic growth, mean-reverting prices, viscosity solution, Howard's algorithm

In this paper we study the problem of optimal fishing for regime switching, which may be regarded as sequential optimal problem with changes of regimes.

The growth dynamics of a given fish species is described by the differential stochastic logistic model in which we take into account two states: prior or during floods and after. The resulting dynamic programming principle leads to a system of variational inequalities, by means of viscosity solutions approach, we prove the existence and uniqueness of the value functions. Then numerical approximation is used to answer the question: what is the optimal fishing effort for a sustainable fishery?

Introduction

The simplest population model commonly used in fisheries is the logistic growth model extended to include catch:

dB dt = rB(1 - B K ) -C (1)
where B is the biomass of the stock, r is the intrinsic rate of growth, K(the carrying capacity) is the biomass the stock would tend toward if unfished, and C is the catch rate.

The catch C is constant in quota management. Normally the catch is assumed to be proportional to fishing effort and to stock size, which results in the model proposed by [START_REF] Schaefer | Some aspects of the dynamics of populations important to the management of the commercial marine fisheries[END_REF] :

dB dt = rB(1 - B K ) -qEB (2)
where E is the fishing effort and q is a parameter describing the efficiency of the fishing gear However, the environment is subject to significant random fluctuations that affect the population per capita natural growth rate. The effect of these fluctuations can be approximated by a white noise σ (t), where (t) is a standard white noise and σ > 0 measures the strength of environmental fluctuations see [START_REF] Brites | Fisheries management in random environments: Comparison of harvesting policies for the logistic model[END_REF]. Therefore, the above ODE Eq. ( 2) must be updated to the stochastic differential equation (SDE) which can be written in the standard format:

dB(t) = rB(t)(1 - B(t) K )dt -qE(t)B(t)dt + σB(t)dW (t) (3)
where W (t) is a standard Wiener process. We will assume that r -qE > σ 2 /2, otherwise the population will rendered extinct (see [START_REF] Braumann | Stochastic differential equation models of fisheries in an uncertain world: extinction probabilities, optimal fishing effort, and parameter estimation[END_REF] ).

Environmentally driven long-term changes in fish populations, which can play a major role in determining how such populations respond to fishing pressure, are rapidly being recognized as a critical problem in fisheries science ((4)).

The life cycle of African fish species of river is closely related to the seasons -reproduction almost always occurring just prior to, or during, floods ((5), ( 6), [START_REF] Svensson | Fresh water fishes from the Gambia river (British West Africa): results of the Swedish expedition 1931[END_REF], [START_REF] Carey | Breeding seasons and quantitative data on gonads and ova for certain fish species[END_REF]).

Floods appear to be essential for the completion of their reproductive cycle for most species: the absence of floods due to the drought in the Sahel has caused a decline in fish reproduction in the central Niger Delta, the Senegal River and Lake Chad (Stauch, personal communication).

There is some evidence that flood intensity acts in favor of reproduction, as it has been observed that the structured age class related to the high floods in the Kafu were more varied (( 9)).

In our study we consider only two seasons: the dry season with intensive fishing and reduced reproduction, the flood period with reduced fishing and intensive reproduction.

A regime switching model provides an alternate approach to capturing nonconstant drift and volatility terms for the stochastic process followed by the biomass of fish. Therefore, the above SDE Eq. ( 3) must be updated to the another stochastic differential equation that captures the regime swithing :

dB(t) = r α(t) B(t)(1 - B(t) K
)dt -qE α(t) (t)B(t)dt + σ α(t) B(t)dW (t) [START_REF] Maccall | Fishery-management and stock-rebuilding prospects under conditions of low-frequency environmental variability and species interactions[END_REF] where α(t) refer to regimes and there are 2 regimes, i.e, α(t) ∈ {1, 2}

Certainly, there is some evidence that uncertainty in price parameters leads to changes in the optimal policy ((10)), and the number of studies that include uncertainty in both the biological stock dynamics and the price dynamics is steadily increasing. The models in [START_REF] Kvamsdal | Harvesting in a fishery with stochastic growth and a mean-reverting price[END_REF] have considered stochastic meanreverting prices and when compared with the typical geometric Brownian motion model, a mean-reverting price better reflect basic, microeconomic ideas about supply behavior (see [START_REF] Insley | A real options approach to the valuation of a forestry investment[END_REF]).

Let the instantaneous profit from the harvest of the stock biomass π(B t , h t ) be given as:

π(B t , h t ) = P t h t -c(B t , h t ) (5)
where, h t denotes the volume of harvest, B t the stock of the resource, c(B t , h t ) is the cost function, both at time t and P t the mean-reverting (actual or spot) price of the harvest at the time of decision making. This can be modeled by the following process:

dP t = θ(p 0 -p1 h -P t )dt + σ P dW P (t) (6)
The parameters are positve constants, θ is the reversion speed, p0 is a maximum price, p1 is the slope of the inverse demand curve and σ P is the volatility of the spot price (see [START_REF] Kvamsdal | Harvesting in a fishery with stochastic growth and a mean-reverting price[END_REF]).

Many works set the problem, in the infinite horizon time, as follows:

max ht +∞ 0 e -βt π(B t , h t )dt (7)
Previous work finds, almost without exception, that all fishers are risk-averse (( 13), [START_REF] Dupont | Price uncertainty, expectations formation and fishers' location choices[END_REF], and ( 15)). Under an expected-utility theory (EUT) specification of choice under uncertainty, we assume a constant relative risk-aversion (CRRA) utility function defined as U (x) = x 1-γ 1 -γ where x signifies the lottery prize and γ is the CRRA coefficient to be estimated: with γ = 0 denoting risk neutrality, γ > 0 indicating risk aversion, and γ < 0 denoting risk loving (( 16)).

In recent years, further emphasis has been put on developing models for optimal management of these stochastic natural resources ((17); ( 18); ( 19)). Although the number of studies in bioeconomic modeling that include the stochastic dynamics are increasing, they are still not adequate.

The time horizon also plays a crucial role in optimal policies and the usual infinite horizon framework problem requires the existence of linear growth conditions on drift part of the logistic process for our solution to hold, raising the question of whether another solution may exist or not. In this paper, we consider the finite time horizon T with utility on both profit and remaining biomass.

The outline of this paper is as follows. In Section II we formulate a stochastic optimal control problem. Section III the optimal strategies to the utility maximization problem are derived. In Section IV we present examples to illustrate the results. Finally, in section V we end with some summarizing comments.

2 Mathematical model

Stochastic logistic growth model

Throughout this paper we let (Ω, F, {F t } t≥0 , P) be a complete probability space with a filtration {F t } t≥0 satisfying the usual conditions (i.e. it is increasing and right continuous while F 0 contains all P-null sets). Let W (t) and W P (t), t ≥ 0, be scalar independant Brownian motions defined on this probability space.

After specifying a stochastic model for biological growth that we use in this paper. The valuation of the biomass can be described in terms of the following variables:

t is time, t ∈ [0; T ] and T is finite-horizon of time. α(t) is a right-continuous-time Markov chain, F t -adapted with finite state space S = {1; 2} and generator Q = (q ij ) ∈ R 2 × R 2 such that q ij ≥ 0 for i = j and 2
j=1 q ij = 0. We assume that the Markov chain α(.) is independent of the Brownian motions W P (.) and W (.) B(t) stock of fish biomass at time t. Whith initial condition B(0) > 0 r α(t) intrinsic rate of growth in regime α(t) E α(t) is the fishing effort which depends on the current regime α(t) σ α(t) is volatility in regime α(t). In any time r α(t) -qE > σ 2 α(t) /2

We set an SDE under regime switching of the form:

dB(t) = f (t, B(t), α(t))dt + g(t, B(t), α(t))dW (t) (8)
on t ≥ 0 with initial value B(0

) = b ∈]0; K[, where
f : R + × R + × S → R and g : R + × R + × S → R (9)
This equation can be regarded as the result of the following 2 equations:

dB(t) = f (t, B(t), i)dt + g(t, B(t), i)dW (t); i = 1, 2 (10)
switching from one to the other according to the movement of the Markov chain.

Recalling that

f (t, B(t), i) = r i B(t)(1 - B(t) K ) -qE i (t)B(t) and g(t, B(t), i) = σ i B(t) (11)
B(t) is an unknown stochastic process, that is, the solution to Eq.( 8) satisfying

the initial condition B(0) = b such that 0 < b < K.
B t,i = K exp[(r i -qE -(1/2)σ 2 i )t + σ i W t ] (K/B 0 ) + r i t 0 exp[(r i -qE -(1/2)σ 2 i )s + σ i W s ]ds i = 1, 2. (12)

The Mean-reverting spot price 78

The version of the Ornstein-Uhlenbeck (OU) process we employ here is described by

dP t = θ(p 0 -p1 h -P t )dt + σ P dW P (t) (13)
where the parameters are positve constants, θ is the reversion speed, p0 is a maximum price, p1 is the slope of the inverse demand curve and σ p is the volatility of the spot price. Note that the mean (or long-term) price p0 -p1 h may depend upon the harvest level. W P (t) is standardized Brownian motion as before. Its solution for an initial condition P (0) = p is

P t = pe -θt + (p 0 -p1 h)(1 -e -θt ) + σ t 0 e -θ(t-s) dW s (14)
One of the most convenient properties is that

P t ∼ N p0 -p1 h + (p + p0 -p1 h)e -θt , σ 2 2θ (1 -e -2θt ) (15)

The optimization problem

The cost of harvest per unit time is assumed to depend on effort and to have a quadratic form given by

c(B t , E t ) = (c 1 + c 2 E(t))E(t) (16)
where c 1 , c 2 > 0 are constants. The quadratic cost structure incorporates the case where the fishermen need to use less efficient vessels and fishing technologies or pay higher overtime wages to implement an extraordinary high effort (see [START_REF] Clark | Mathematical Bioeconomics: The Optimal Management Resources[END_REF], ( 21)).

By substituting the values in equation 5, the profit function can be rearranged as:

π(B t , P t , E) = (qB t P t -c 1 -c 2 E(t))E(t) (17)
where, c 1 and c 2 are positive parameters.

For a time t in the horizon [0, T ], we define a performance criterion for each i ∈ S as:

V i (t, b t , p t ) =E bt,pt,i T t e -β(s-t) π(B s , P s , E s , i) 1-γ 1 -γ ds + e -β(T -t) V i (B T ) (18) =E bt,pt,i T t e -β(s-t) l(s, B s , P s , E s , i)ds + e -β(T -t) m(T, B T ) (19)
We will start the optimization at time t = 0. Let b 0 = b, p 0 = p with b, p ∈ ]0; +∞], we have

V i (0, b, p) =E b,p,i T 0 e -βs π(B s , P s , E s , i) 1-γ 1 -γ ds + e -βT V i (B T ) (20) =E b,p,i T 0 e -βs l(s, B s , P s , E s , i)ds + e -βT m(T, B T ) (21)
Here E b,p,i is the conditional expectation given B(0) = b, P (0) = p and α(0) = i under P, where T is the finite time horizon β > 0 is a discount factor.

We say that the control process E(t) is admissible if the following tree conditions are satisfied:

1. the SDE (4) for the state process B(t) has a unique strong solution;

2. the SDE (6) for the state process P (t) has a unique strong solution;

3. E b,p,i

T 0 |e -βt π(B t , P t , E t , i) 1-γ 1 -γ |dt + |e -βT V (B T )| < ∞.
Write A for the set of admissible controls. The number of tools, gears, hours, vessels and manpower is finite and limited, so we require the set A in which the controls take values to be bounded. The stochastic control problem is to find an optimal control E * ∈ A i such that:

v i (b, p) = sup E∈Ai V i (b, p) (22)
3 Main results

The Hamilton-Jacobi-Bellman equations associated with this problem is a variational inequality involving, at least heuristically, a nonlinear second order parabolic differential equations :

∂v i ∂t (t, b t , p t ) + sup E∈Ai -βv i (t, b t , p t ) + π(B t , P t , E t ) 1-γ 1 -γ + Lv i (t, b t , p t ) = 0, (23)
v i (T, b t , p t ) = κ γ B 1-γ T 1 -γ f or i ∈ {0; 1}; κ > 0 ( 24
)
where L is an operator defined by:

Lv i (t, B, P ) = θ(p 0 -p1 qEB -P ) ∂v i ∂p (t, B, P ) + f (B, E i ) ∂v i ∂b (t, B, P ) + 1 2 σ 2 P ∂ 2 v i ∂p 2 (t, B, P ) + 1 2 g 2 (B, E) ∂ 2 v i ∂b 2 (t, B, P ) + q ij (v j (t, B, P ) -v i (t, B, P )) ( 25
)
As it is well-known, there is not in general a smooth solution of the equation [START_REF] Liu | Optimal stopping of switching diffusions with state dependent switching rates[END_REF] hence we find the solution in the viscosity sense, as introduced by ( 22), in subsection 3.2. Recall that E * f ree is the optimal solution of equation 23. The optimal harvest rule E * (t) can be described as follows

E * (t) =          0 if E * free (t) < 0 E * free (t) if 0 ≤ E * free (t) ≤ E max E max if E * free (t) > E max (26)
In addition to these, we know that the fishery is valueless if the stock goes extinct and therefore add the condition V i (0, P t ) = 0, which must hold for all P t and i. 100

On the regularity of value functions

In this section, we study the growth and continuity properties of the value functions.

We shall make the following assumptions: there exist ρ > 0 such that for all

s, t ∈ [0; T ], b, b ∈ R + , p, p ∈ R + and E ∈ A |l(t, b, p, E) -l(s, b , p , E)| + |m(b, p) -m(b , p )| ≤ ρ [|t -s| + |b -b | + |p -p |] (27)
and the global linear growth conditions:

|l(t, b, p, E)| + |m(b, p)| ≤ ρ [1 + |b| + |p|] (28)
Lemma 3.1. Let ( 27) and (28) hold. For any k ∈ [0; 2] there exists

C = C(k; K; T ) > 0 such that for all h, t ∈ [0; T ], b, p, b t , p t ∈ R + : E|B t,bt h | k ≤ C(1 + |b t | k ); E|P t,pt h | k ≤ C(1 + |p t | k ) (29) E|B t,bt h -b t | k ≤ C(1 + |b t | k )h k 2 ; E|P t,pt h -p t | k ≤ C(1 + |p t | k )h k 2 (30) E|B t,bt h -B t,b t h | k ≤ C|b t -b t | 2 ; E|P t,pt h -P t,p t h | k ≤ C|p t -p t | 2 (31) E sup 0≤s≤h |B t,bt h | k ≤ C(1 + |b t | k )h k 2 ; E sup 0≤s≤h |P t,pt h | k ≤ C(1 + |p t | k )h k 2 (32) Proof 3.1. See Appendix B 102 Proposition 3.1.
For any i ∈ S, the value function denoted by v i (s, b, p) satisfies a linear growth condition and is also Lipschitz in (b, p) uniformly in t.

There exists a constant C > 0, such that

0 ≤ v i (s, b s , p s ) ≤ C(1 + |b s | + |p s |), ∀(s, b s , p s ) ∈ [0; T ] × R + × R + (33) |v i (s, b s , p s ) -v i (s, b s , p s )| ≤ C(|b s -b s | + |p s -p s |), ∀s ∈ [0; T ], b s , b s ∈ R + , p s , p s ∈ R + (34) Proof 3.2.
We first show that v is Lipschitz in (b, p), uniformly in t and its linear growth condition.

v i (s, b s , p s ) = sup E∈A E T s e -β(u-s) l(i, u, B s,bs u , P s,ps u , E u )du + e -β(T -s) m(B s,bs T , P s,ps T ) (35
|v i (s, b s , p s ) -v i (s, b s , p s )| ≤ sup E∈Ai E T s e -β(u-s) l(i, u, B s,bs u , P s,ps u , E u )-l(i, u, B s,b s u , P s,p s u , E u ) du + e -β(T -s) m(B s,bs T , P s,ps T ) -m(B s,b s T , P s,p s T ) ≤ sup E∈Ai E T s l(i, u, B s,bs u , P s,ps u , E u ) -l(i, u, B s,b s u , P s,p s u , E u ) du + m(B s,bs T , P s,ps T ) -m(B s,b s T , P s,p s T ) ≤ sup E∈Ai E T s |B s,bs u -B s,b s u |+|P s,ps u -P s,p s u | du+ |B s,bs T -B s,b s T |+|P s,ps T -P s,p s T | ≤ sup E∈Ai T s E |B s,bs u -B s,b s u |+E|P s,ps u -P s,p s u | du+ E|B s,bs T -B s,b s T |+E|P s,ps T -P s,p s T | |v i (s, b s , p s ) -v i (s, b s , p s )| ≤ C |b s -b s | + |p s -p s | (36)
2. from linear growth condition (28) on l ; m and from estimate (3.1), with k=1,

|v i (s, b s , p s )| ≤ sup E∈Ai E T s l(i, u, B s,bs u , P s,ps u , E u ) du + m(B s,bs T , P s,ps T ) ( 37
)
|v i (s, b s , p s )| ≤ ρ sup E∈Ai E T s 1 + |B s,bs u | + |P s,ps u | du + 1 + |B s,bs T | + |P T | (38) |v i (s, b s , p s )| ≤ ρ sup E∈Ai T s 1 + E|B s,bs u | + E|P s,ps u | du + 1 + E|B s,bs T | + E|P s,ps T | ( 39
)
|v i (s, b s , p s )| ≤ C (1 + |b s | + |p s |) (40)
Proposition 3.2. Under assumptions (49) and (28

) the value function v ∈ C 0 ([0; T ] × R + × R + ). More precisely, there exists a constant C > 0 such that for all t, s ∈ [0; T ], b t , b s ∈ R + , p t , p s ∈ R + , |v i (t, b t , p t ) -v i (s, b s , p s )| ≤ C (1 + |b t | + |p t |)|s -t| 1 2 + |b t -b s | + |p t -p s | (41)
Proof 3.3. Let 0 ≤ t < s ≤ T . To prove continuity property in time t, we use the dynamic programming principle.

v i (t, b, p) = sup E∈Ai E T t e -β(u-t) l(i, u, B t,bt u , P t,pt u , E u )du + e -β(T -t) m(B t,bt T , P t,pt T ) (42) = sup E∈Ai E s t e -β(u-t) l(i, u, B t,bt u , P t,pt u , E u )du + e -β(s-t) v(s, B t,bt s , P t,bt s , i) (43) = sup E∈Ai E s-t o e -βu l(i, t + u, B t,bt t+u , P t,pt t+u , E t+u )du + e -β(s-t) v(s, B t,bt s-t , P t,pt s-t , i) (44) 0 ≤ v i (t, b t , p t ) -v i (s, b s , p s ) = sup E∈Ai E s-t 0 e -β(u) l(i, u, B t,bt u , P t,pt u , E u )du + e -β(s-t) v(s, B t,bt s-t , P t,bt s-t , i) -v(s, b s , p s , i) + e -β(s-t) -1 v(s, b s , p s , i) (45)
Applying linear growth condition [START_REF] Barles | Convergence of numerical schemes for degenerate parabolic equations arising in finance theory[END_REF] on l, noting that 0 ≤ 1-e -β(s-t) ≤ β(s-t) and v satisfies (3.1) , we deduce that:

|v i (t, b t , p t ) -v i (s, b s , p s )| ≤ sup E∈Ai E s-t 0 l(i, u, B t,bt u , P t,pt u , E u ) du+ e -β(s-t) v(s, B t,bt s-t , P t,bt s-t , i)-v(s, b s , p s , i) + e -β(s-t) -1 v(s, b s , p s , i) ≤ (s-t) 1 2 s-t 0 sup E∈Ai E l(i, u, B t,bt u , p t,pt u , E u ) 2 du 1 2 + sup E∈Ai E e -β(s-t) v(s, B t,bt s-t , P t,pt s-t , i) -v(s, b s , p s , i) + sup E∈Ai E e -β(s-t) -1 v(s, b s , p s , i) ≤ (s-t) 1 2 s-t 0 ρ 2 sup E∈Ai 1 + E|B t,bt u | + E|p t,pt u | 2 du 1 2 + sup E∈Ai E v(s, B t,bt s-t , P t,pt s-t , i) -v(s, b s , p s , i) + β|s -t| sup E∈A E |v(s, b s , p s , i)| ≤ |s-t| 1 2 s-t 0 ρ sup E∈Ai 1 + E|B t,bt u | + E|P t,pt u | du + sup E∈Ai E v(s, B t,bt s-t , P t,pt s-t , i)-v(s, b s , p s , i) + β|s -t| sup E∈Ai E v(s, b s , p s , i) ≤ C |s -t| 1 2 s-t 0 (1 + E|B t,bt u | + E|P t,pt u |)du + (|b t -b s | + |p t -p s |) + β(1 + |b s | + |p s |)|s -t| ≤ C (1 + |b t | + |p t |)|s -t| 1 2 + |b t -b s | + |p t -p s | (46)

Existence of viscosity solution

In this section we will first define what we mean by viscosity solutions. Then 104 we will prove that the value function is a viscosity solution.

From the optimization problem (23), we derive the Bellman equations as follows:

∂v i ∂t (t, B, P ) + sup E∈Ai -βv i (t, B, P ) + π 1-γ 1 -γ + θ(p 0 -p1 qEB -P ) ∂v i ∂p (t, B, P ) + r i B 1 - B K -qEB ∂v i ∂b (t, B, P ) + 1 2 σ 2 P ∂ 2 v i ∂p 2 (t, B, P ) + 1 2 σ 2 B 2 ∂ 2 v i ∂b 2 (t, B, P ) + q ij (v j (t, B, P ) -v i (t, B, P )) = 0 (47)
The corresponding Hamiltonian has the following form:

H i, s, B, P, u i , ∂u i ∂s , ∂u i ∂b , ∂u i ∂p , ∂ 2 u i ∂b 2 , ∂ 2 u i ∂p 2 = ∂u i ∂s (s, B, P )+ sup E∈Ai -βu i (s, B, P )+ π(B s , P s , E s ) 1-γ 1 -γ +Lu i (s, B, P ) = 0 (48)
We have the following systems:

     H i, s, B, P, u i , ∂u i ∂s , ∂u i ∂b , ∂u i ∂p , ∂ 2 u i ∂b 2 , ∂ 2 u i ∂p 2 = 0 f or (i, s, B, P ) ∈ S × [0; T i ] × R + × R + u i (T, B, ) = κ γ B 1-γ T 1-γ f or i, j ∈ {0; 1} κ > 0. ( 49
)
we recall that

π(B s , P s , E s ) = (qB s P s -c 1 -c 2 E(s))E(s) (50)
In order to study the possibility of existence and uniqueness of a solution of 106 (49), we use a notion of viscosity solution introduced by ( 22).

108

Let denote the set of measurable functions on [0; T ] × R + × R + with polynomial growth of degree q ≥ 0 as,

C q ([0; T ] × R + × R + ) = {φ : [0; T ] × R + × R + , measurable | ∃C > 0, |φ(t, b, p)| ≤ C(1 + |b| q + |p| q )}. ( 51
) Definition 3.1. We say that u i ∈ C 0 ([0; T ] × R + × R + ) is called i. a viscosity subsolution of (49) if for any i ∈ S, u i (T, b, p) ≤ κ γ b 1-γ T 1-γ , for all b ∈ R + , p ∈ R + and for all functions φ ∈ C 1,2,2 ([0; T ] × R + × R + ) ∩ C 2 ([0; T ]×R + ×R + ) and ( t, b, p) such that u i -φ attains its local maximum at ( t, b, p), H i, t, b, p, φ( t, b, p), ∂φ( t, b, p) ∂s , ∂φ( t, b, p) ∂b , ∂φ( t, b, p) ∂p , ∂ 2 φ( t, b, p) ∂b 2 , ∂ 2 φ( t, b, p) ∂p 2 ≥ 0 (52)
ii. a viscosity supersolution of (49) if for any i ∈ S, Proof 3.4. We establish the viscosity super-and sub-solution properties, respectively in the following two steps.

u i (T, b, p) ≥ κ γ b 1-γ T 1-γ , for all b ∈ R + , p ∈ R + and if for all functions φ ∈ C 1,2,2 ([0; T ] × R + × R + ) ∩ C 2 ([0; T ] × R + × R + )
Step 1. v i (t, b t , p t ), i = 1; 2 is a viscosity super-solution of (47).

We already know that v ∈ C 0 ([0; T ] × R + × R + ). We first note that

v i (T, b, p) = κ γ B 1-γ T 1-γ so, the boundary condition at time t = T is clearly satisfied. Let (s, b s , p s ) ∈ [0; T ] × R + × R + , i ∈ S and φ ∈ C 1,2,2 ([0; T ] × R + × R + ) ∩ C 2 ([0; T ] × R + × R + )
such that v i (., ., .) -φ(., ., .) has a local minimum at (s, b s , p s ). Let N(b s , p s ) a neighborhood of (s, b s , p s ) where v i (., ., .) -φ(., ., .) take its minimum, we introduce a new test-function ψ as follows:

ψ(., ., ., j) =      φ(., ., .) + [v i (s, b s , p s ) -φ(s, b s , p s )], if j = i v i (., ., .), if j = i. ( 54
)
This helps us to suppose without any loss of generality that this minimum is equal to 0.

Let τ α be the first jump time of α(t) = α(t)

Recalling that (B bs s , P ps s ) = (b s , p s ) and using Equations ( 54) and (56), we have

E bs,ps,i e -βθs∧h ψ(θ s ∧ h, B θs∧h , P θs∧h , α(θ s ∧ h)) ≥ + v i (s,
r i b s 1 - b s K -qEb s ∂ψ(s, b s , p s , i) ∂b -θ(p 0 -p1 qEb s -p s ) ∂ψ(s, b s , p s , i) ∂p - 1 2 σ 2 b 2 s ∂ 2 ψ(s, b s , p s , i) ∂b 2 - 1 2 σ 2 P ∂ 2 ψ(s, b s , p s , i) ∂p 2 -q ij [v j (s, b s , p s ) -v i (s, b s , p s )] -l(i, s, b s , p s , E s ) ≥ 0 (60)
This shows that the value function v i (t, b t , p t ), i = 1; 2 satisfies the viscos-118 ity super-solution property (53).

Step 2. v i (t, b t , p t ), i = 1; 2 is a viscosity sub-solution of (47).

We argue by contradiction. Assume that there exist an i 0 ∈ S, a point

(s, b s , p s ) ∈ [0; T ] × R * + × R * + and a testing function φ i0 ∈ C 1,2,2 ([0; T ] × R * + × R * + ) ∩ C 2 ([0; T ] × R * + × R * +
) such that v i0 (., ., .) -φ i0 (., ., .) has a local maximum at (s, b s , p s ) in a bounded neighborhood N(b s , p s ), v i0 (s, b s , p s ) = φ i0 (s, b s , p s ), and

min - ∂φ i0 (s, b s , p s ) ∂t + inf E∈Ai 0 βv i0 (s, b s , p s )- r i0 b s 1 - b s K -qEb s ∂φ i0 (s, b s , p s ) ∂b -θ(p 0 -p1 qEb s -p s ) ∂φ i0 (s, b s , p s ) ∂p - 1 2 σ 2 b 2 s ∂ 2 φ i0 (s, b s , p s ) ∂b 2 - 1 2 σ 2 P ∂ 2 φ i0 (s, b s , p s ) ∂p 2 -q i0j [v j (s, b s , p s )-v i0 (s, b s , p s )] -l( i0 , s, b s , p s , E s ) ; v i0 (T, b s , p s ) -κ γ B 1-γ T 1 -γ > 0 i 0 = j (61)
By the continuity of all functions involved in (61) (v i0 , φ i0 , φ i0 , q ij , l, ...), 120 there exist a δ > 0 and an open ball and

B δ (b s , p s ) ⊂ N(b s , p s ) such that - ∂φ i0 (t, b t , p t ) ∂t + inf E∈Ai 0 βv i0 (t, b t , p t )- r i0 b t 1 - b t K -qEb t ∂φ i0 (t, b t , p t ) ∂b -θ(p 0 -p1 qEb t -p t ) ∂φ i0 (t, b t , p t ) ∂p - 1 2 σ 2 b 2 s ∂ 2 φ i0 (t, b t , p t ) ∂b 2 - 1 2 σ 2 P ∂ 2 φ i0 (t, b t , p t ) ∂p 2 -q i0j [v j (t, b t , p t )-v i0 (t, b t , p t )] -l( i0 , t, b t , p t , E t ) > δ i 0 = j; (t, b t , p t ) ∈ B δ (b s , p s ) (62) and v i0 (T, b t , p t ) -κ γ B 1-γ T 1 -γ > δ (t, b t , p t ) ∈ B δ (b s , p s ) (63) Let θ δ = min{t ≥ s : (t,
v i0 (T, b t , p t ) -κ γ B 1-γ T 1 -γ > δ (t, b t , p t ) ∈ B δ (b s , p s ) ( 65
)
As previously, we can replace φ i0 by a new test-function ψ defined as follows:

ψ(., ., ., j) = For details see [START_REF] Liu | Optimal stopping of switching diffusions with state dependent switching rates[END_REF]. It follows that

     φ i0 (., ., .), if j = i 0 v i0 (., ., .), if j = i 0 . (66)
+ e -βτ [κ γ B 1-γ T 1 -γ ]1 {τ <θ} + δE bs,ps,i0 (θ∧τ )
v i0 (s, b s , p s ) ≥ sup τ ∈[s;T ];E∈A E bs,ps,i0 + (θ∧τ ) s e -βt l( i0 , t, B t , P t , E t ) dt + e -βθ v i0 (θ, B θ , P θ , α(θ))1 {τ ≥θ} + e -βτ [κ γ B 1-γ T 1 -γ ]1 {τ <θ} + C 0 δ 1 -E bs,ps,i0 e -βτα (71)
which is a contradiction to the DP principle since E bs,ps,i0 e -βτα < 1.

Therefore the value function v i (t, b t , p t ), i = 1; 2 is a viscosity sub-solution of the system (2.8).

This completes the proof of Theorem 3.1

Comparison principle: uniqueness of the viscosity solution

In this section, we prove a comparison result from which we obtain the uniqueness of the solution of the coupled system of partial differential equations.

In proving comparison results for viscosity solutions, the notion of parabolic we define the parabolic superjet:

P 2,+ v(t, b, p, i) = (c, q, M ) ∈ R × R 2 × S 2 : v(s, b , p , i) ≤ v(t, b, p, i) + c(s -t) + q.((b -b), (p -p)) + 1 2 ((b -b), (p -p)).M ((b -b), (p -p)) + o(|((b -b), (p -p))| 2 ) as (s; b , p ) → (t; b, p) (72)
and its closure:

P2,+ v(t, b, p, i) = (c, q, M ) = lim n→∞ (c n , q n , M n ) with (c n , q n , M n ) ∈ P 2,+ v(t n , b n , p n , i) and lim n→∞ (t n , b n , p n , v(t n , b n , p n , i)) = (t, b, p, v(t, b, p, i)) (73)
Similarly, we define the parabolic subjet P2,v(t, b, p, i) = -P2,+ (-v)(t, b, p, i)

and its closure P2,-v(t, b, p, i) = -P2,+ (-v)(t, b, p, i)
It is proved in [START_REF] Linos | Optimal control of diffustion processes and hamilton-jacobibellman equations part i: the dynamic programming principle and application[END_REF] that

P 2,+(-) v(t, b, p, i) = φ ∂t (t, b, p, i), D (b,p) φ(t, b, p, i), D 2 (b,p) φ(t, b, p, i)
and v -φ has a global maximum (minimum) at (t, b, p, i)

The previous notions lead to new definition of viscosity solutions.

Definition 3.3. u i ∈ C 0 ([0; T ] × R * + × R * + ) satisfying the polynomial growth condition is a viscosity solution of (49) if (1) for any test-function φ ∈ C 1,2,2 ([0; T ] × R * + × R * + ) if (t, b, p
) is a local maximum point of u i (., ., .) -φ(., ., .) and if (c, q, L 1 ) ∈ P2,+ u(t, b, p, i)

with c = ∂φ(t, b, p)/∂t; q = D (b,p) φ(t, b, p) and L 1 ≤ D 2 (b,p) φ(t, b, p), then H i, s, b, p, u i , ∂u i ∂s , ∂u i ∂b , ∂u i ∂p , ∂ 2 u i ∂b 2 , ∂ 2 u i ∂p 2 ≤ 0 (75)
in this case u is a viscosity subsolution,

(2) for any test-function

φ ∈ C 1,2,2 ([0; T ] × R * + × R * + ) if (t, b, p
) is a local minimum point of u i (., ., .) -φ(., ., .) and if (c, q, L 2 ) ∈ P2,u(t, b, p, i) with c = ∂φ(t, b, p)/∂t; q = D (b,p) φ(t, b, p) and L 2 ≥ D 2 (b,p) φ(t, b, p), then

H i, s, b, p, u i , ∂u i ∂s , ∂u i ∂b , ∂u i ∂p , ∂ 2 u i ∂b 2 , ∂ 2 u i ∂p 2 ≥ 0 (76)
in this case u is a viscosity supersolution, It is proved in [START_REF] Barles | Second-order elliptic integro-differential equations: viscosity solutions' theory revisited[END_REF] that this new definition and the previous one are equivalent. We refer the reader to the mentioned paper for a proof. The last definition is particular suitable for the discussion of a maximum principle which is the backbone of the uniqueness problem for the viscosity solutions theory.

Before state next lemma, we first introduce the inf and sup-convolution operations we are going to use.

R α [U ](z, r) = sup |Z-z|≤1 U (Z) -r • (Z -z) - |Z -z| 2α (77) R α [V ](z, r) = inf |Z-z|≤1 V (Z) + r • (Z -z) + |Z -z| 2α (78) R α [U ](z, r) is called the modified sup-convolution and R α [V ](z, r) the modified inf-convolution. Notice that R α [V ](z, r) = -R α [-U ](z, r) Lemma 3.2. (nonlocal Jensen-Ishii's lemma (25))
For any i ∈ S, let u i (., ., .) and v i (., ., .) be, respectively, a usc and lsc function

defined on [0; T ]×R + ×R + and φ ∈ C 1,2,2 ([0; T ]×R 2 + ×R 2 + )∩C 2 ([0; T ]×R 2 + ×R 2 + ) if ( t, ( b1 , p1 ), ( b2 , p2 )) ∈ [0; T ] × R 2 + × R 2 + is a zero global maximum point of u i (t, b, p)-v i (t, b , p )-φ(t, (b, p), (b , p )) and if c-d := D t φ( t, ( b1 , p1 ), ( b2 , p2 )),
q := D (b,p) φ( t, ( b1 , p1 ), ( b2 , p2 )), r := -D (b ,p ) φ( t, ( b1 , p1 ), ( b2 , p2 )) , then for any K > 0, there exists α(K) > 0 such that, for any 0 < α < α(K), we have: there exist sequences t k → t, (b k , p k ) → ( b1 , p1 ), (b k , p k ) → ( b2 , p2 ), q k → q, r k → r, matrices M k , N k and a sequence of functions φ k , converging to the function φ α := R α [φ](((b, p), (b , p )), (q, r)) uniformly in R 2 + × R 2 + and in C 2 (B(( t, ( b1 , p1 ), ( b2 , p2 )), K)), such that

u i (t k , (b k , p k )) → u i ( t, ( b1 , p1 )), v i (t k , (b k , p k )) → v i ( t, ( b2 , p2 )) ( 79
) (t k , (b k , p k ), (b k , p k ))
is a global maximum of u i (., (., .))-v i (., (., .))-φ(., (., .), (., .))

(c k , q k , M k ) ∈ P2,+ u i (t k , (b k , p k )) (80)
(-d k , r k , N k ) ∈ P2,-v i (t k , (b k , p k )) (81) - 1 α   I 0 0 I   ≤   M k 0 0 -N k   ≤ D 2 (b,p),(b ,p ) φ(t k , (b k , p k ), (b k , p k )) (82) Here c k -d k = ∇ t φ(t k , (b k , p k ), (b k , p k )), q k = ∇ (b,p) φ(t k , (b k , p k ), (b k , p k )), r k = ∇ (b ,p ) φ(t k , (b k , p k ), (b k , p k ))
and φ α ( t, ( b1 , p1 ), ( b2 , p2 )) = φ( t, ( b1 , p1 ), ( b2 , p2 )), 162 ∇φ α ( t, ( b1 , p1 ), ( b2 , p2 )) = ∇φ( t, ( b1 , p1 ), ( b2 , p2 ))

Now we can state our comparison result.

u i (t, b, p) ≤ v i (t, b, p) f or all (t, b, p, i) ∈ [0; T ] × R + × R + × S (83)
Proof 3.5. For , , δ, λ > 0, we define the auxiliary functions φ :

(0; T ] × R 2 + × R 2 + → R and Ξ: [0; T ] × R 2 + × R 2 + × S by φ(t, (b, p), (b , p )
) = t + 1 2 |(b, p)-(b , p )| 2 +δe λ(T -t) (|(b, p)| 2 +|(b , p )| 2 ) (84)
and

Ξ(t, (b, p), (b , p ), i) = v i (t, b, p) -u i (t, b , p ) -φ(t, (b, p), (b , p )) (85)
By using the linear growth of v i and u i , we have for each i ∈ S lim

|(b,p)|+|(b ,p )|→∞ Ξ(t, (b, p), (b , p ), i) = -∞ (86)
Then, since v i and u i are uniformly continuous with respect to (t, b, p) on each compact subset of [0; T ]×R + ×R + ×S and that S is a finite set, Ξ attains its global maximum at some finite point belonging to a compact

K ⊂ [0; T ] × R 2 + × R 2 + × S say, t δ , (b 1δ , p 1δ ), (b 2δ , p 2δ ), α δ . Observing that 2Ξ t δ , (b 1δ , p 1δ ), (b 2δ , p 2δ ), α δ ≥ Ξ t δ , (b 1δ , p 1δ ), (b 2δ , p 2δ ), α δ + Ξ t δ , (b 1δ , p 1δ ), (b 2δ , p 2δ
), α δ and using the uniform continuity of v i and u i on K we have

1 |(b 1δ , p 1δ ) -(b 2δ , p 2δ )| 2 ≤ v i t δ , (b 1δ , p 1δ ) -v i tδ , (b 2δ , p 2δ ) +u i t δ , (b 1δ , p 1δ ) -u i t δ , (b 2δ , p 2δ ) ≤ 2C|(b 1δ , p 1δ ) -(b 2δ , p 2δ )| (87) Thus, |(b 1δ , p 1δ ) -(b 2δ , p 2δ )| ≤ 2C ( 88
)
where C is a positive constant independent of , , δ, λ. From the inequality, 166 2Ξ T, (0, 0), (0, 0),

α δ ≤ 2Ξ t δ , (b 1δ , p 1δ ), (b 2δ , p 2δ ), α δ (89)
and the linear growth for v i and u i , we have:

δ |(b 1δ , p 1δ ) | 2 + |(b 2δ , p 2δ | 2 ≤ e -λ(T -t δ ) v i t δ , b 1δ , p 1δ -v i T, 0, 0 + u i T, 0, 0 -u i t δ , b 2δ , p 2δ ≤ e -λ(T -t δ ) C 2 1 + |(b 1δ , p 1δ )| + |(b 2δ , p 2δ )| (90) It follows that δ |(b 1δ , p 1δ ) | 2 + |(b 2δ , p 2δ | 2 (1 + |(b 1δ , p 1δ )| + |(b 2δ , p 2δ )| ≤ C 2 (91)
Consequently, there exists C δ > 0 such that

|(b 1δ , p 1δ )| + |(b 2δ , p 2δ )| ≤ C δ (92)
This inequality implies that for any fixed δ ∈ (0, 1), the sets {(b 1δ , p 1δ ), > 0}

and {(b 2δ , p 2δ ), > 0} are bounded by C δ independent of . It follows from inequalities (90) and (92) that, possibly if necessary along a subsequence, named again t δ , (b 1δ , p 1δ ), (b 2δ , p 2δ ), α δ that there exist (b 1δ0 , p 1δ0 ) ∈ R 2 + , t δ 0 ∈ (0, T ] and α δ 0 ∈ S such that: lim

↓0 (b 1δ , p 1δ ) = (b 1δ0 , p 1δ0 ) = lim ↓0 (b 1δ , p 1δ ), lim ↓0 t δ = t δ0 , lim ↓0 α δ = α δ0 .
If t δ = T then writing that Ξ t, (b, p), (b, p), α δ ≤ Ξ T, (b 1δ , p 1δ ), (b 2δ , p 2δ ), α δ , we have

u i (t, b, p) -v i (t, b, p) - t -2δe λ(T -t) (|(b, p)| 2 ) ≤ u i (T, (b 1δ , p 1δ )) -v i (T, (b 2δ , p 2δ )) - T - 1 2 |(b 1δ , p 1δ ) -(b 2δ , p 2δ )| 2 -δ(|(b 1δ , p 1δ )| 2 + |(b 2δ , p 2δ )| 2 ) ≤ u i (T, (b 1δ , p 1δ )) -v i (T, (b 2δ , p 2δ )) = [u i (T, (b 1δ , p 1δ )) -v i (T, (b 1δ , p 1δ ))] + [v i (T, (b 1δ , p 1δ )) -v i (T, (b 2δ , p 2δ ))] ≤ C 1 |(b 1δ , p 1δ ) -(b 2δ , p 2δ )| (93)
where the last inequality follows from the uniform continuity of v i and by as-

sumption that u i (T, (b 1δ , p 1δ )) = κ γ b 1-γ T 1-γ = v i (T, (b 1δ , p 1δ ))
Sending , , δ ↓ 0 and using estimate (88), we have: u i (t, b, p) ≤ v i (t, b, p) . Assume now that t δ < T . Applying Lemma 3.2 with u i , v i and φ(t, (b, p), (b , p )) at the point (t δ , (b 1δ , p 1δ ), (b 2δ , p 2δ ), α δ ) ∈ (0; T ) × R 2 + × R 2 + × S, for any ζ ∈ (0, 1) there are d ∈ R, M δ , N δ ∈ S 2 such that:

d - t 2 -λδe λ(T -t) (|(b δ , p δ )| 2 + |(b δ , p δ )| 2 ), 1 ((b δ , p δ ) -(b δ , p δ )) + 2δe λ(T -t) (b δ , p δ ), M δ + 2δe λ(T -t) I ∈ P2,+ v(t, b, p, i) d, 1 ((b δ , p δ )-(b δ , p δ ))-2δe λ(T -t) (b δ , p δ ), N δ -2δe λ(T -t) I ∈ P2,-v(t, b, p, i) (94
  I 0 0 I   ≤   M δ 0 0 -N δ   ≤ 2   I -I -I I   (96) It follows that (b δ , p δ )M δ   b δ p δ   -(b δ , p δ )N δ   b δ p δ   = ((b δ , p δ ), (b δ , p δ ))   M δ 0 0 -N δ             b δ p δ     b δ p δ           ≤ ((b δ , p δ ), (b δ , p δ ))   2   I -I -I I               b δ p δ     b δ p δ           ≤ 2 |(b δ , p δ ) -(b δ , p δ )| 2 (97)
Furthermore, the definition of the viscosity subsolution u i and supersolution v i implies that

min βu i0 (t δ , b δ , p δ ) -d - t 2 δ -λδe λ(T -t δ ) (|(b δ , p δ )| 2 + |(b δ , p δ )| 2 ) + inf E∈Ai 0 -r i0 b δ 1 - b δ K -qEb δ 1 (b δ -b δ ) + 2δe λ(T -t) b δ -θ(p 0 -p 1 qEb δ -p s ) 1 (p δ -p δ )+2δe λ(T -t) p δ - 1 2 (σb δ ; σ P )(M δ +2δe λ(T -t) I)   σb δ σ p   -q i0j [u j (t δ , b δ , p δ ) -u i0 (t δ , b δ , p δ )] -l( i0 , t δ , b δ , p δ , E t δ ) ; u i0 (T, , b δ , p δ ) -κ γ B 1-γ T 1 -γ ≤ 0 i 0 = j (98)
and

168 min βv i0 (t δ , b δ , p δ )-d+ inf E∈Ai 0 -r i0 b δ 1- b δ K -qEb δ 1 (b δ -b δ )+2δe λ(T -t) b δ -θ(p 0 -p 1 qEb δ -p s ) 1 (p δ -p δ )+2δe λ(T -t) p δ - 1 2 (σb δ ; σ P )(N δ -2δe λ(T -t) I)   σb δ σ p   -q i0j [v j (t δ , b δ , p δ ) -v i0 (t δ , b δ , p δ )] -l( i0 , t δ , b δ , p δ , E t δ ) ; v i0 (T, , b δ , p δ ) -κ γ B 1-γ T 1 -γ ≥ 0 i 0 = j (99)
Let us define operators A E (x, v, φ, X, Z) and B E (x, v).

A E (t, b, p, w, X, Y Z) = r i0 b 1 - b K -qEb X + θ(p 0 -p1 qEb -p s )Y + 1 2 wZw (100) B E (t, b, p, v) = q i0j [v j (t, b, p) -v i0 (t, b, p)] (101)
by Subtracting these last two inequalities and remarking that min(x; y)-min(z; t) ≤ 0 implies either x -z ≤ 0 or y -t ≤ 0, we divide our consideration into two 170 cases:

Case 1

β u i0 (t δ , b δ , p δ )-v i0 (t δ , b δ , p δ ) + t 2 δ +λδe λ(T -t δ ) (|(b δ , p δ )| 2 +|(b δ , p δ )| 2 ) ≤ sup E∈Ai 0 l( i0 , t δ , b δ , p δ , E t δ ) -l( i0 , t δ , b δ , p δ , E t δ ) + sup E∈Ai 0 A E t δ , b δ , p δ , (σb δ ; σ P ), 1 (b δ -b δ )+2δe λ(T -t δ ) b δ , 1 (p δ -p δ )+2δe λ(T -t δ ) p δ , M δ + 2δe λ(T -t δ ) I -A E t δ , b δ , p δ , (σb δ ; σ P ), 1 (b δ -b δ )+2δe λ(T -t) b δ , 1 (p δ -p δ )+2δe λ(T -t δ ) p δ , N δ -2δe λ(T -t δ ) I + sup E∈Ai 0 B E (t δ , b δ , p δ u) -B E (t δ , b δ , p δ v)] ≡ I 1 + I 2 + I 3 (102)
In view of condition ( 27) on l and from estimate (3.1), we have the classical estimates of I 1 and I 2 :

I 1 ≤C|(b δ , p δ ) -(b δ , p δ )| (103)
I 2 ≤C( 1 |(b δ , p δ ) -(b δ , p δ )| 2 + 2δe λ(T -t δ ) (1 + |(b δ , p δ )| 2 + |(b δ , p δ )| 2 ) (104)
Using the Lipschitz condition for u and v, we have

I 3 ≤ 2C|(b δ , p δ ) -(b δ , p δ )| (105)
Writing that Ξ(t, (b, p), (b, p), i) ≤ Ξ(t δ , (b δ , p δ ), (b δ , p δ ), i) for i ∈ S and using the inequality (102),

u i (t, b, p) -v i (t, b, p) - t -2δe λ(T -t) |(b, p)| 2 ≤ v i (t δ , b δ , p δ ) -u i (t δ , b δ , p δ ) - t δ -2δe λ(T -t) |(b δ , p δ )| 2 ≤ 1 β I 1 + I 2 + I 3 - βt 2 δ - λ β δe λ(T -t δ ) (|(b δ , p δ )| 2 + |(b δ , p δ )| 2 ) (106) this implies u i (t, b, p) -v i (t, b, p) - t -2δe λ(T -t) |(b, p)| 2 ≤ 1 β I 1 + I 2 + I 3 - λ β δe λ(T -t δ ) (|(b δ , p δ )| 2 + |(b δ , p δ )| 2 ) (107)
Sending ↓ 0, with the above estimates of (I 1 ) -(I 2 ) -(I 3 ), we obtain:

u i (t, b, p)-v i (t, b, p)- t -2δe λ(T -t) |(b, p)| 2 ≤ 2δ β e λ(T -t0) C(1+2|(b 0 , p 0 )| 2 )-λ|(b 0 , p 0 )| 2 (108)
Choose λ sufficiently large positive (λ ≥ 2C) and send , δ → 0 + to conclude that u i (t, b, p) ≤ v i (t, b, p)

Case 2 the second case occurs if

u i0 (T, , b δ , p δ ) -v i0 (T, , b δ , p δ ) ≤ 0 (109)
and

finally that u i (t, b, p) ≤ v i (t, b, p)
This completes the proof.

The following corollary follows from Theorems 3.1 and 3.2.

Corollary 3.1. The value function v is a uniques viscosity solution of (47) that has at most a linear growth.

Monotone Finite Difference and Simulation

The determination of the effort value requires numerical computations. Thus, instead of arbitrary parameters values, we have decided to use realistic values.

We found a quite complete set of parameter values in (2) and [START_REF] Kvamsdal | Harvesting in a fishery with stochastic growth and a mean-reverting price[END_REF]. The time horizon was set at T i = 5 years. The complete set of parameter values is listed in Table 1.

Sample Realisations of Price and Stock

We chose the maximum fishing effort value for these sample realisations. is the unique viscosity solution of the associated system of HJB equations. As an application, the optimal effort is deduced by using Howard's algorithm.

These methodologies can be applied to similar comparison studies and other fishery models. This will be the subject of a further paper.

Appendix B Proof of lemma 3.1

1. Let k ∈ [0; 2] and h = s -t.

According to Hölder inequality Eη k ≤ Eη 2 k/2 for ∀k ≥ 0, E|P

) 1 .

 1 Using elementary inequality | sup A -sup B| ≤ sup |A -B|, Lipschitz con-dition (27) on l ; m and from estimate (3.1), with k=1,

se

  -βt dt + e -βτ 1 {τ <θ} (69) Now the estimate of the term E bs,ps,i0 (θ∧τ ) s e -βt dt + e -βτ 1 {τ <θ} .There exist a positive constant C 0 such that E bs,ps,i0 (θ∧τ ) s e -βt dt + e -βτ 1 {τ <θ} ≥ C 0 1 -E bs,ps,i0 e -βτα (70)

3 . 2 .

 32 superjet and subjet defined by Crandall, Ishii and Lions [19] is particularly useful. Thus, we begin by Definition Given v ∈ C o ([0; T ]×R×R×S) and (t; b; p; i) ∈ [0; T )×R×R×S,

Definition 3 . 4 .

 34 For any usc (upper semi-continuous ) function U : R m → R and any lsc (lower semi-continuous ) function V : R m → R, we set

164 Theorem 3 . 2 .

 32 (comparison principle): If u i (t, b, p) and v i (t, b, p) are continuous in (t, b, p) and are, respectively, viscosity subsolution and supersolution of the HJB system (47) with at most linear growth then

2 ≤

 2 p),(b ,p ) φ(t δ , (b δ , p δ ), (b δ , p δ )) + ζ D 2 (b,p),(b ,p ) φ(t δ , (b δ , p δ ), (b δ , p δ )) + ζ(2 + 4δ e λ(T -t) ) 2δ+4ζδ 2 e λ(T -t) )e λ(T -t)

  The logical requirement is that B(t) must be positive. The resulting stochastic differential equation does not satisfy the standard assumptions for existence and uniqueness of solutions, The solution of this equation is.(For more details see Appendix A )

namely, linear growth and the Lipschitz condition. Nevertheless, for any positive initial condition, the solution exists and is unique under a hypothesis that both f and g satisfy the local Lipschitz condition and in any time r α(t) -qE > σ 2 α(t) /2.

  and (t, b, p) such that u i -φ attains its local minimum

	at (t, b, p),
	H i, t, b, p, φ(t, b, p),	∂φ(t, b, p) ∂s	,	∂φ(t, b, p) ∂b	,	∂φ(t, b, p) ∂p	,	∂ 2 φ(t, b, p) ∂b 2	,	∂ 2 φ(t, b, p) ∂p 2	≤ 0
										(53)
	iii. a viscosity solution of (49) if it is both a viscosity sub-and a supersolution
	of equation (49)
	Theorem 3.1. Under assumptions (27), the value function v is a viscosity
	solution of (47).

  bs,ps,i , i.e. τ α = min{t ≥ s : α(t) = i}. Then τ α > s, a.s. Let θ s ∈ (s, τ α ) be such that the state ) starts at (b s , p s ) and stays in N(b s , p s ) for s ≤ t ≤ θ s . Applying the generalized Itô's formula to the switching process e -βt ψ(t, B t , P t , α(t)), taking integral from t = s to t = θ s ∧ h, where h > 0 is a positive constant, and then taking expectation we have E bs,ps,i e -βθs∧h ψ(θ s ∧ h, B θs∧h , P θs∧h , α(θ s ∧ h)) = ψ(s, B s , P s , i)+E bs,ps,i

	(B bs,i t	, P ps,i t
							s	θs∧h	e -βt -βψ(t, B t , P t , α(t))+	∂ψ(t, B t , P t , α(t)) ∂t
	+ r i B t 1-	B t K	-qEB t	∂ψ(t, B t , P t , α(t)) ∂b	+θ(p 0 -p 1 qEB t -P t )	∂ψ(t, B t , P t , α(t)) ∂p
		+	1 2	σ 2 B 2 t	∂ 2 ψ(t, B t , P t , α(t)) ∂b 2	+	1 2	σ 2

P ∂ 2 ψ(t, B t , P t , α(t)) ∂p 2 + q α(t)j (ψ(t, B t , P t , j) -ψ(t, B t , P t , α(t))) dt ; α(t) = j (55) From hypothesis, for any t ∈ [s, θ s ∧ h] v i (t, B bs t , P ps t ) ≥ φ(t, B bs t , P ps t )+v i (s, b s , p s )-φ(s, b s , p s ) ≥ ψ(t, B bs t , P ps t , i)

  b s , p s ) + E bs,ps,i+ q ij (v j (t, B t , P t ) -v i (t, B t , P t )) dt (57) By Bellman's principle ψ(s, b s , p s , i) = v i (s, b s , p s ) = sup

									θs∧h
									E∈Ai	E bs,ps,i	s	e -βt l(i, t, B s,bs t	, P s,ps t	, E t )dt
					+ e -β(θs∧h) v i (θ s ∧ h, B s,bs θs∧h , P s,ps θs∧h )
								θs∧h
			≥ sup E∈Ai	E bs,ps,i	s	e -βt l(i, t, B s,bs t	, P s,ps t	, E t )dt
								+ e -β(θs∧h) ψ(θ s ∧ h, B s,bs θs∧h , P s,ps θs∧h , i)	(58)
	Setting τ = E(θ s ∧ h) combining (57) and (58) and multiplying both sides
	by 1/(τ -s) > 0 , we obtain
	sup E∈Ai	E bs,ps,i	1 τ -s
		-	1 2	σ 2 B 2			∂b 2	-	1 2	σ 2
							s	θs∧h	e -βt -βv i (t, B t , P t ) +	∂ψ(t, B t , P t , α(t)) ∂t
	+ r i B t 1-	B t K	-qEB t	∂ψ(t, B t , P t , α(t)) ∂b	+θ(p 0 -p 1 qEB t -P t )	∂ψ(t, B t , P t , α(t)) ∂p
		+	1 2	σ 2 B 2 t	∂ 2 ψ(t, B t , P t , α(t)) ∂b 2	+	1 2	σ 2 P	∂ 2 ψ(t, B t , P t , α(t)) ∂p 2

θs∧h s e -βt βv i (t, B t , P t ) -∂ψ(t, B t , P t , α(t)) ∂t -r i B t 1-B t K -qEB t ∂ψ(t, B t , P t , α(t)) ∂b -θ(p 0 -p 1 qEB t -P t ) ∂ψ(t, B t , P t , α(t)) ∂p t ∂ 2 ψ(t, B t , P t , α(t)) P ∂ 2 ψ(t, B t , P t , α(t)) ∂p 2 -q ij [v j (t, B t , P t ) -v i (t, B t , P t )] -l(i, t, B s,bs t , P s,ps t , E t ) dt ≥ 0 (59) letting τ ↓ s and using the dominated convergence theorem, it turns out that e -βs -∂ψ(s, b s , p s , i) ∂t + inf E∈Ai βv i (s, b s , p s )-

  i0j [v j (t, B t , P t )-v i0 (t, B t , P t )] -l( i0 , t, B t , P t , E t ) > δ i 0 = j; (t, B t , P t ) ∈ B δ (b s , p s ) (64)

	have
		βv i0 (t, B t , P t ) -	∂φ i0 (t, B t , P t ) ∂t
	-r i0 b t 1-	B t K	-qE i0 B t	∂φ i0 (t, B t , P t ) ∂b	-θ(p 0 -p 1 qEB t -P t )	∂φ i0 (t, B t , P t ) ∂p
	-	1 2	σ 2 b 2 s	∂ 2 φ i0 (t, B t , P t ) ∂b 2	-	1 2	σ 2 P	∂ 2 φ i0 (t, B t , P t ) ∂p 2	-q

B t , P t ) ∈ B δ (b s , p s )} be the first exit time of (t, B t , P t ) (= (t, B s,bs t , P s,ps t )) from B δ (b s , p s ). Let θ = θ δ ∧ τ α where τ α is

the first jump time of α(t) bs,ps,i0 . Then θ > 0, a.s.. For 0 ≤ t ≤ θ, we

  in which we used E bs,ps,i e -βθ∧τ ψ(θ∧τ, B θ∧τ , P θ∧τ , α(θ∧τ )) = E bs,ps,i e -βθs∧τ ψ(θ∧ τ, B θ∧τ , P θ∧τ , α(θ ∧ τ )-) due to continuity. Noting that the integrand in the RHS of (67) is continuous in t. Using (64), (65) and that v i0 (t, B t , P t ) ≤ φ i0 (t, B t , P t ) in (67). Also noting that α(t) = i 0 for 0 ≤ t ≤ θ, it follows v i0 (s, b s , p s ) ≥ E bs,ps,i0 e -βθ∧τ φ i0 (θ ∧ τ, B θ∧τ , P θ∧τ , α(θ ∧ τ ))

			+		s	(θ∧τ )	e -βt βv i0 (t, B t , P t ) -	∂φ i0 (t, B t , P t ) ∂t
	-r i B t 1-	B t K	-qE i B t	∂φ i0 (t, B t , P t ) ∂b	-θ(p 0 -p 1 qEB t -P t )	∂φ i0 (t, B t , P t ) ∂p
		-	1 2	σ 2 B 2 t	∂ 2 φ i0 (t, B t , P t ) ∂b 2	-	1 2	σ 2 P	∂ 2 φ i0 (t, B t , P t ) ∂p 2
				(θ∧τ )
	+	1 2	σ 2 B 2 t	∂ 2 ψ(t, B t , P t , α(t)) ∂b 2	+	1 2	σ 2 P	∂ 2 ψ(t, B t , P t , α(t)) ∂p 2

For any stopping time τ ∈ [s; T ]. Applying Itô's formula to the switching process e -βt ψ(t, B t , P t , α(t)), taking integral from t = s to t = (θ s ∧ τ )and then taking expectation yield E bs,ps,i e -βθ∧τ ψ(θ ∧ τ, B θ∧τ , P θ∧τ , α(θ

∧ τ )) = v i0 (s, b s , p s )+E bs,ps,i (θ∧τ )s e -βt -βψ(t, B t , P t , α(t))+ ∂ψ(t, B t , P t , α(t)) ∂t + r i B t 1-B t K -qE i B t ∂ψ(t,

B t , P t , α(t)) ∂b +θ(p 0 -p 1 qEB t -P t ) ∂ψ(t, B t , P t , α(t)) ∂p + q α(t)j [v j (t, B t , P t ) -ψ(t, B t , P t , α(t))] dt ; α(t) = j (67) -q i0j [v j (t, B t , P t ) -v i0 (t, B t , P t )] dt ; i 0 = j (68) i.e 122 v i0 (s, b s , p s ) ≥ E bs,ps,i0 e -βτ v i0 (τ, B τ , P τ , α(τ ))1 {τ <θ} +e -βθ v i0 (θ, B θ , P θ , α(θ))1 {τ ≥θ} + (θ∧τ ) s e -βt l( i0 , t, B t , P t , E t ) + δ dt ≥ E bs,ps,i0 e -βτ [κ γ B 1-γ T 1 -γ + δ])1 {τ <θ} + e -βθ v i0 (θ, B θ , P θ , α(θ))1 {τ ≥θ} + (θ∧τ ) s e -βt l( i0 , t, B t , P t , E t ) + δ dt ≥ E bs,ps,i0 + s e -βt l( i0 , t, B t , P t , E t ) dt+e -βθ v i0 (θ, B θ , P θ , α(θ))1 {τ ≥θ}

Table 1 :

 1 Quadratic cost parameter 0.10 × 10 -6 SF U -2 year -1 Numerical parameters

		Parameters Description		Values	Units
		r 1 ; r 2		Intrinsic growth rate	0.71; 0.68	year -1
		K		Carrying capacity		80.5 × 10 6	kg
		q		Catchability coefficient	3.30 × 10 -6 SF U -1 year -1
		E max		Maximum fishing effort	0.7r/q	SF U
		B o		Initial population size	0.5K		kg
		β		Discount factor		0.05		year -1
		p 0		Price per unit yield	1.59		kg -1
		c 1		Linear cost parameter	96 × 10 -6	SF U -1 year -1
		c 2
		θ		Mean-reversion speed	0.59
		p0		Price of the stock		1.211
		p1		Strength of demand	0.0001
		σ		Volatility of the stock	0.3
		σ p		Volatility of the price	0.3
		T		Time horizon		5		years
		γ		risk aversion coefficient	0.3
	188
								1.6
		1.7
	price regime 1	1.5 1.6					price regime 2	1.3 1.4 1.5
		1.4						1.2
		0.0	0.2	0.4	0.6	0.8	1.0	0.0	0.2	0.4	0.6	0.8	1.0
				time						time
	biomass regime 1	3.5e+07 4.5e+07					biomass regime 2	3.2e+07 3.6e+07 4.0e+07
		0.0	0.2	0.4	0.6	0.8	1.0	0.0	0.2	0.4	0.6	0.8	1.0
				time						time

Table 2 :

 2 Optimal Effort

222 36

  |P t,pt h | 2 ≤ 3 |p t | 2 + ≤ C 1 + |p t | 2 (B.7)Using (B.1) and elementary inequalities (a 1 + a 2 ) k ≤ 2 k-1 (|a 1 | k + |a 2 | k ) and ( |a 1 + a 2 | ≤ |a 1 | + |a 2 |) we deduce E|P t,pt h | k ≤ C 1 + |p t | k (B.8) C|b t -b t | 2 ; E|Pt,pt C|p t -p t | 2 (B.16)

	2. We have
	|P t,pt h -p t | 2 ≤ 2	0	h	f (u + t, P t,pt u , i)du	2	+	0	h	u , i)dw u g (u + t, P t,pt	2
												(B.10)
	t,pt h | k ≤ E|P t,pt h | 2 k/2 Similar arguments as above we deduce	(B.1)
	Given that		dP (t) = f (t, P (t))dt + g (t, P (t))dw(t) E|P t,pt 0 h -p t | 2 ≤ C 1 h 2 1 + E P t,pt u du	(B.2) (B.11)
	According to the elementary inequality | using (B.7) we deduce	M i=1 a i | 2 ≤ M	M i=1 |a i | 2 ,
					h E|P t,pt f (u + t, P t,pt u , i)du	2	+	h	u , i)dw u g (u + t, P t,pt	2
						0						0
												(B.3)
	Thus, using Ito-isometry and Fubini's theorem
	E|P t,pt
						t,p t u , i). Then
	E|B t,pt h -B	t,p t							0	h	f (u+t, P t,pt u , P u , i)du t,p t	2	+E	0	h	ḡ(u+t, P t,pt u , P u , i)dw u t,p t	2
	E|P t,pt h -P	t,p t					0	h	h f (u+t, P t,pt E P t,pt u u , P t,p t 2 du u , i)	2	du+E	(B.5) 0 h ḡ(u+t, P t,pt u , B u , i) t,p t	2	du
	Applying Gronwall's inequality we obtain E|P t,pt 0 h -P t,p t	0 h E P t,pt u	-P u t,p t	2	du	(B.14)
	Hence					E|P t,pt h | 2 ≤ C 1 e C1h 1 + |p t | 2	(B.6)
	i.e				E|P t,pt h	-P	t,p t
	E|P t,pt Similar arguments as above we deduce E|B t,bt h -B t,b t h 4. Using Doob's inequality for submartingale. We get -P t,p t h | 2 The same reasoning gives.... E sup 0≤s≤h |B t,bt h | k ≤ C(1 + |b t | k )h k 2 ; E sup 0≤s≤h |P t,pt h |	k ≤ C(1 + |p t | k )h	k 2
						E|B t,bt h | k ≤ C 1 + |b t | k	(B.9) (B.17)
							40

h | 2 ≤ 3 |p t | 2 + h 0 E f (u + t, P t,pt u , i) 2 du + h 0 E g (u + t, P t,pt u , i) 2 du (B.4)

f' and g' satisfying the growth condition by definition, thus there exists

C 1 ∈ R such that E|P t,bt h | 2 ≤ C 1 1 + |p t | 2 + h -p t | 2 ≤ C 1 + |p t | 2 h (B.12) Hence E|P t,pt h -p t | k ≤ C 1 + |p t | k h k/2 (B

.13) 3. Let us define the process P t,pt s -P t,p t s . Put f (u + t, P t,pt u , P t,p t u

, i) = f (u + t, P t,pt u , i) -f (u + t, P t,p t u , i) and ḡ(u + t, P t,pt u , P t,p t u , i) = g (u + t, P t,pt u , i) -g (u + t, P h | 2 ≤ 3 |p t -p t | 2 +E h | 2 ≤ 3 |p t -p t | 2 +E h | 2 ≤ C |p t -p t | 2 + h | 2 ≤ C|p t -p t | 2 (B.15) h | k ≤ h | k ≤

We treat an finite-horizon optimal fishery problem in switching diffusion models. Using the viscosity solution approach, we prove that the value function

Acknowledgements

We wish to thank here

The Numerical Approximation

In this section, we present a numerical solution. We consider the switching process α(t) where α(t) ∈ S = {1, 2} represents the season. In particular, α(t) = 1 stands for the flood period with reduced fishing and intensive reproduction. and α(t) = 2 the dry season with intensive fishing and reduced reproduction. For our problems we need to ensure that our discretization methods converge to the viscosity solution and determine the optimal effort. Using the basic results of ( 26) and ( 27), this ensures that our numerical solutions convergence to the viscosity solution. For this purpose, we use the fully implicit upwind scheme which is unconditionally monotone.

To approximate the solution to [START_REF] Liu | Optimal stopping of switching diffusions with state dependent switching rates[END_REF] we discretize variables t, p and B with stepsizes ∆t, ∆p and ∆B respectively. The value of v i at a grid point (t n ; p k ; B l ) in the regime i is denoted by v n k,l (i). The derivatives of v i are approximated by

Discretizing equation ( 23)

and rearranging the terms, we obtain

In addition, we consider that:

• the optimization starts at time t = 0 and ends at time t = T < +∞

• the time interval is uniformly partitioned as 0 = t 0 < t 1 < ... < t N = T with t n+1 -t n = ∆t = T N , n = 0, 1, ..., N -1;

• the state variable of biomass takes values within the interval [0, 2K], which is uniformly partitioned as 0 = B 0 < B 1 < ... < B m = 2K with B l+1 -

• the state variable of prices takes values within the interval [0, p max ], which is uniformly partitioned as 0 = p 0 < p 1 < ... < p m = p max with

• we have boundary conditions, a terminal conditions

and a initial condition v i (0, b, p) = 0.

If we define the constants

We can rewrite this difference equation in a more manageable form:

Writing (119) in a appropriate matrix form, sup

E∈Ai

Step 0: start with an initial value for the control E 0 . Compute the solution v 0

Optimal effort

We applied the Howard's algorithm: we compute the optimal effort in both

Appendix A Stochastic logistic growth with harvesting

Recalling the Itô's chain rules for solving the SDE dX

To linearize (A.3), set B -1 t = y t so that


Justin Dzuche

Christian Deffo Tassak

Jules Sadefo
email: jsadefo@gmail.com

Kamdem Louis

Aimé Fono

Christian Deffo Tassak

Jules Sadefo Kamdem
email: jules.sadefo-kamdem@umontpellier.fr

Louis Aimé Fono

On the first moments and semi-moments of fuzzy variables based on a new measure and application for portfolio selection with fuzzy returns

Keywords: Fuzzy measure, Fuzzy variable, Expected value, Variance and Semi-Variance, Skewness, Kurtosis and Semi-Kurtosis, Optimal portfolios

On the first moments

Introduction

Uncertainty can be viewed as an aspect of randomness or ambiguity in real life phenomena. Thereby, one needs a measure on fuzzy events (fuzzy measure) to analyze questions and problems dealing with such uncertainty. After Zadeh's work [START_REF] Zadeh | Fuzzy set as a basis of theory of possibility[END_REF] introducing possibility and necessity measures, a vast literature appeared on fuzzy theory and fuzzy logic based on the aforementioned measures. Twenty years later, Liu [START_REF] Liu | Expected value of fuzzy variable and fuzzy expected value models[END_REF] introduced credibility measure as the arithmetic mean of the two first measures. This measure has been used to address other questions dealing with uncertainty (see Liu [START_REF] Liu | Expected value operator of random fuzzy variable and random fuzzy expected models[END_REF], Li et al. [START_REF] Li | Mean-variance-skewness model for portfolio selection with fuzzy returns[END_REF], Huang et al. [START_REF] Huang | Mean-semivariance models for fuzzy portfolio selection[END_REF], Sadefo et al. [START_REF] Sadefo | Moments and semimoments for fuzzy portfolio selection[END_REF] and Tassak et al. [START_REF] Tassak | Characterization of order dominances on fuzzy variables for portfolio selection with fuzzy returns[END_REF]). More recently, Yang and Iwamura [START_REF] Yang | Fuzzy Chance-Constrained Programming with Linear Combination of Possibility Measure and Necessity measure[END_REF] introduced a new measure, denoted m λ , as a convex linear combination of possibility and necessity measures (the weight λ of the possibility measure in the combination is a real parameter in the interval [0; 1]) and they determined some of its axioms. This new measure generalized the three previous ones and has been used in fuzzy chance-constrained programming (Yang and Kakuzo [START_REF] Yang | Fuzzy Chance-Constrained Programming with Linear Combination of Possibility Measure and Necessity measure[END_REF], Dai et al. [START_REF] Dai | Optimal strategies for carbon capture, utilization and storage based on an inexact m λ -measure fuzzy chance-constrained programming[END_REF]) as a more generalized approach compared with credibility constrained programming ones to find optimal strategies for carbon capture. We used it in our recent conference paper to determine Expected value and Variance of a fuzzy variable (Dzuche et al. [START_REF] Dzuché | Expected value and variance of a fuzzy variable based on a new fuzzy measure[END_REF]). This paper focuses on the recent measure for two major findings that might be useful. First, the weight λ can be considered as the the decision maker confidence on the fuzzy event degree of realization such that degree 1 means a complete confidence and degree 0 means no confidence. The m λ -measure can become either possibility measure,or necessity measure or credibility measure respectively for the decision marker who is self confident, unconfident and neutral (the weight is equal to 1 2 ). It is a comprise of the two first measures and this finding is similar to the idea of Hurwicz decision criteria in Microeconomics which is a compromise of maximax and maximin criteria. On the other hand, due to the fact that the new measure generalizes the three previous ones in technical point of view, many theoretical results already solved with one of the three first measures can be generalized with the new measure. In addition, some specific results will be outlined.

The modest contribution of this paper is to study, by means of m λ , fuzzy variables characteristics and to implement obtained results for portfolio optimization models with fuzzy returns. More specifically, in the continuation of our recent work (Dzuche et al. [START_REF] Dzuché | Expected value and variance of a fuzzy variable based on a new fuzzy measure[END_REF]), we propose some new axioms of m λ , and we introduce fuzzy variables characteristics w.r.t. to this measure. We determine four first moments (expected value, variance, skewness and kurtosis) and two first semi-moments (semi-variance and semi-kurtosis)of trapezoidal and triangular fuzzy variables. We establish some properties of these characteristics and the obtained theoretical results are applied in portfolio selection with fuzzy returns.

The paper is organized as follows. Section 2 recalls useful notions on fuzzy sets, on four previous usual fuzzy measures. We display some new axioms of the more recent one. In Section 3, we introduce and study fuzzy variables characteristics. More precisely, we focus on the determination of four first moments and two first semi-moments of trapezoidal and triangular fuzzy variables and we establish some properties of these characteristics. Section 4 displays an application of these characteristics in optimal portfolios selection where assets returns proposed by Huang [START_REF] Huang | Mean-semivariance models for fuzzy portfolio selection[END_REF] are described by triangular fuzzy variables. We also apply these characteristics to find optimal portfolios by using Tokyo stock exchange data provided by Hasuike et al. [START_REF] Hasuike | Portfolio selection problems with random fuzzy variable return[END_REF]. Appendix contains proofs of some results.

Preliminaries

Throughout this paper, X is a nonempty set, namely, the universal set. A fuzzy subset A of X is defined by its membership function:

µ A : X → [0, 1] such that, to each x ∈ X, is associated µ A (x) representing the membership grade of x to A. A is denoted by {(x, µ A (x)), x ∈ X}. If ∀x ∈ A, µ A (x) ∈ {0, 1}, then A becomes a crisp subset of X.
Let ξ be a mapping from X to R described by its membership function µ interpreted as: for any x ∈ R, µ(x) represents the degree that ξ takes value x. ξ is said to be a fuzzy variable if ξ is measurable. A fuzzy variable ξ is normal if ∃x 0 ∈ R, µ(x 0 ) = 1. For ξ and η be two fuzzy variables and α a real number, their sum ξ + η is a fuzzy variable defined by: ∀x ∈ R, (ξ + η)(x) = ξ(x) + η(x) and αξ is a fuzzy variable defined by: ∀x ∈ R, (αξ)(x) = αξ(x). A fuzzy number ξ is a fuzzy variable that satisfies:

∃a, b, c, d ∈ R with a ≤ b ≤ c ≤ d such that (i) µ is upper semi-continuous, (ii) ∀r ̸ ∈ [a, d], µ(r) = 0, (iii) µ is increasing on [a
: ∀x ∈ R, µ(x) =        x-a b-a if a ≤ x ≤ b 1, if b ≤ x ≤ c x-d c-d if c ≤ x ≤ d 0 elsewhere . If b = c, then ξ = (a, b, d) is a triangular fuzzy number.
Figure 1 displays a trapezoidal fuzzy variable and a triangular one. In the following, we recall classical fuzzy measures proposed in the literature to deal with imprecision. Let B ⊂ X and ξ : X → R be a fuzzy variable whose membership function µ, that means, each element x of X is associated to a real number ξ(x) by means of ξ with the membership grade µ(ξ(x)). We have the following classical and well-known measures: the Possibility measure defined by P os(B) = sup x∈B µ(ξ(x)) (Zadeh [START_REF] Zadeh | Fuzzy set as a basis of theory of possibility[END_REF]); the Necessity measure defined by N ec(B) = 1 -P os(B c ) = inf x∈B µ(ξ(x)) (Zadeh [START_REF] Zadeh | Fuzzy set as a basis of theory of possibility[END_REF]) and the Credibility measure defined by Cred(B) = 1 2 [P os(B) + N ec(B)] (Liu [START_REF] Liu | Expected value of fuzzy variable and fuzzy expected value models[END_REF]). Yang and Iwamura [START_REF] Yang | Fuzzy Chance-Constrained Programming with Linear Combination of Possibility Measure and Necessity measure[END_REF] introduced a measure, denoted m λ and defined by for λ ∈ [0, 1],

m λ (B) = λ sup x∈X µ B (x)) + (1 -λ) inf x∈X µ B (x)) (1)
or equivalently, we have:

m λ ({ξ ∈ B}) = λP os({ξ ∈ B}) + (1 -λ)N ec({ξ ∈ B}).
In particular, we have: For λ = 0 (resp. λ = 1, resp. λ = 1 2 ) m λ = N ec (resp. m λ = P os, resp. m λ = Cr). They established some axioms of m λ such as universality, subadditivity for λ ≥ 1 2 and monotonicity. Notice that if P(X) is the power set of X, then the triplet (X, P(X), m λ ) is called a λ-fuzzy space. In addition, we have the following two new axioms of m λ .

1.   λm λ (B) + (1 -λ)m λ (B c ) = λ or (1 -λ)m λ (B) + λm λ (B c ) = λ   (2) 2. m λ (∪ i∈I B i ) = sup i∈I m λ (B i ) if sup i∈I m λ (B i ) ≤ λ where (B i
) i∈I is a countable family of subsets of X and I ⊂ N.

When λ = 1 2 , the Axiom defined by (2) becomes the duality axiom of credibility measure defined by Liu [START_REF] Liu | Expected value of fuzzy variable and fuzzy expected value models[END_REF] as Cr(B) + Cr(B c ) = 1.

Let us end this Section by evaluating the measure of events ξ ≥ t and ξ ≤ t for t ∈ R when ξ is a trapezoidal fuzzy variable.

Example 1. m λ {ξ ≥ t} = λ sup r≥t µ(r) + (1 -λ)(1 -sup r<t µ(r)) and m λ {ξ ≤ t} = λ sup r≤t µ(r) + (1 -λ)(1 -sup r>t µ(r)).
We obtain:

m λ {ξ ≥ t} =                  0 if d ≤ t, λ(d -t) d -c if c ≤ t ≤ d, λ if b ≤ t < c, λ(t -a) + b -t b -a if a ≤ t < b, 1 if t < a m λ {ξ ≤ t} =                  1 if d ≤ t, λ(d -t) + t -c d -c if c ≤ t ≤ d, λ if b ≤ t < c, λ(t -a) b -a if a ≤ t < b, 0 if t < a.
Throughout this paper, ξ is a normal fuzzy variable.

In the following Section, we introduce four main characteristics of a fuzzy variable by means of m λ -measure. We compute them for trapezoidal and triangular fuzzy variables and determine some of their properties.

Characteristics of a fuzzy variable with respect

to the m λ -measure

In the following Subsection, we introduce and study the expected value and the variance of a fuzzy variable with respect to m λ -measure. Results of our recent conference paper (Dzuché et al. [START_REF] Dzuché | Expected value and variance of a fuzzy variable based on a new fuzzy measure[END_REF]) are a part of the findings of the Subsection.

Expected value and Variance

Definition 1. Let ξ be a fuzzy variable and λ ∈ [0, 1]. The expected value of ξ is defined by:

E λ [ξ] = ∫ 0 -∞ [m λ {ξ ≥ r} -1]dr + ∫ +∞ 0 m λ {ξ ≥ r}dr (3)
with the condition that, at least one of the two integrals is finite.

Remark 1. 1) When λ = 1 2 , we obtain the expected value defined by Liu [START_REF] Liu | Expected value of fuzzy variable and fuzzy expected value models[END_REF] by means of the credibility measure as follows:

E 1 2 [ξ] = ∫ +∞ 0 Cr{ξ ≥ r}dr - ∫ 0 -∞ Cr{ξ ≤ r}dr
provided that, at least one of the two integrals is finite. 2) Liu [START_REF] Liu | Expected value of fuzzy variable and fuzzy expected value models[END_REF] introduced for a normalized fuzzy variable, the upper expected value of ξ and the lower expected value of ξ respectively denoted E[ξ] and E[ξ] and defined by:

E[ξ] = ∫ +∞ 0 P os{ξ ≥ r}dr - ∫ 0 -∞ N ec{ξ ≤ r}]dr (4)
and

E[ξ] = ∫ +∞ 0 N ec{ξ ≥ r}dr - ∫ 0 -∞ P os{ξ ≤ r}]dr. ( 5
)
We have:

E λ [ξ] = λE[ξ] + (1 -λ)E[ξ] (6)
We now compute the expected values of trapezoidal and triangular fuzzy variables.

Example 2. -The expected value of a trapezoidal fuzzy variable ξ = (a, b, c, d)

based on the m λ -measure is defined by:

E λ [ξ] = (1 -λ) a + b 2 + λ c + d 2 .
-We deduce that the expected value of a triangular fuzzy variable ξ = (a, b, c) is given by:

E λ [ξ] = (1 -λ) a 2 + λ c 2 + b 2 . -If λ = 1 2
, we obtain expected values of Liu [START_REF] Liu | Expected value of fuzzy variable and fuzzy expected value models[END_REF] respectively for trapezoidal and triangular fuzzy variables, that is,

E 1 2 [ξ] = a+b+c+d 4 and E 1 2 [ξ] = a+2b+c 4 .
The following result establishes that the expected value of a trapezoidal fuzzy variable is increasing with respect to λ.

Proposition 1. Let ξ = (a, b, c, d) be a trapezoidal fuzzy variable, λ 1 , λ 2 ∈ [0, 1]. If λ 1 ≤ λ 2 then E λ 1 [ξ] ≤ E λ 2 [ξ].
Proof : Let ξ = (a, b, c, d) be a trapezoidal fuzzy variable and let us set:

f (λ) = (1 -λ) a+b 2 + λ c+d 2 . We have: f ′ (λ) = c-a 2 + d-b 2 ≥ 0 since c ≥ a and d ≥ b. So f = E λ [ξ] is increasing with respect to λ.
We end this Subsection by introducing the Variance of a fuzzy variable and we compute it in some cases.

Definition 2. Let ξ be a fuzzy variable such that E

[ξ] = e λ and λ ∈ [0, 1] .
The variance of ξ is defined by:

V λ [ξ] = E[(ξ -e λ ) 2 ] = ∫ +∞ 0 m λ {(ξ -e λ ) 2 ≥ r}dr. ( 7
)
In the following Example, we compute the variance of a trapezoidal fuzzy variable and a triangular one.

Example 3.

1. ξ = (a, b, c, d) is a trapezoidal fuzzy variable with expected value e λ .

We set:

α = b -a, γ = c -b, β = d -c, t = β(e λ -a) -α(d -e λ )
β -α and without loss of generality, we assume that: b < e λ < c, γ < α < β, c -e λ < e λ -a < d -e λ and c -e λ < t < e λ -a. We obtain:

V λ [ξ] = λ(c -e λ ) 2 + λ α [(e λ -a)(t 2 -(c -e λ ) 2 ) -2 3 (t 3 -(c -e λ ) 3 )] + λ β [(d -e λ )((e λ -a) 2 -t 2 ) -2 3 ((e λ -a) 3 -t 3 )] + λ β [ 1 3 (d -e λ ) 3 ) + 2 3 (e λ -a) 3 -(e λ -a) 2 (d -e λ )]. 2. ξ = (a, b, c) is a triangular fuzzy variable with expected value e λ . We set: α = b -a, β = c -b, t = β(e λ -a) -α(c -e λ )
β -α and without loss of generality, we assume that: b > e λ and α > β. We obtain:

V λ [ξ] = (b -e λ ) 2 -1-λ α [(b -e λ ) 2 (e λ -a) + 2 3 (b -e λ ) 3 ] + λ β [(c -e λ )(t 2 -(b -e λ ) 2 ) -2 3 (t 3 -(b -e λ ) 3 )] + λ α [(e λ -a)((c -e λ ) 2 - t 2 ) -2 3 ((c -e λ ) 3 -t 3 )] + λ α [ 1 3 (e λ -a) 3 + 2 3 (c -e λ ) 3 -(c -e λ ) 2 (e λ -a)].
The following result establishes useful properties (linearity and homogeneity) of the expected value and the variance respectively with respect to the m λ -measure.

Proposition 2. Let ξ and η be two fuzzy variables such that

E λ [ξ] < ∞ and E λ [η] < ∞, µ a nonnegative real number, ν a real number and λ ∈ [0, 1].

Skewness and Kurtosis

1.

{ E λ [ξ + η] = E λ [ξ] + E λ [η] E λ [µξ] = µE λ [ξ] . 2. V λ [µξ + ν] = µ 2 V λ [ξ].
Proof: 1) The proof of the linearity is given in the Section Appendix. 2) Since αξ + β is a fuzzy variable; on the other hand, by Definition 2 and the first Proposition 2 , we have:

V λ [µξ + ν] = E λ [(µξ + ν -E λ [µξ + ν]) 2 ] = E λ [(µξ + ν -µE λ [ξ] -ν) 2 ]. Thus, we have: V λ [µξ + ν] = E λ [(µξ -µE λ [ξ]) 2 ] = E λ [µ 2 (ξ -E λ [ξ]) 2 ]
. Finally, by the first result of Proposition 2, we have:

V λ [µξ + ν] = µ 2 E λ [(ξ -E λ [ξ]) 2 ] = µ 2 V λ [ξ].
Throughout this paper, we introduce new concepts and establish new results.

In the following Subsection, we introduce and study the Skewness and Kurtosis of a fuzzy variable with respect to the m λ -measure.

Skewness and Kurtosis

Definition 3. Let ξ be a fuzzy variable such that E λ [ξ] = e λ and λ ∈ [0, 1] . 1)
The skewness of ξ is defined by:

SK λ [ξ] = E λ [(ξ-e λ ) 3 ] = ∫ 0 -∞ [m λ {(ξ-e λ ) 3 ≤ r}-1]dr+ ∫ +∞ 0 m λ {(ξ-e λ ) 3 ≥ r}dr (8)
2) The kurtosis of ξ is defined by:

K λ [ξ] = E λ [(ξ -e λ ) 4 ] = ∫ +∞ 0 m λ {(ξ -e λ ) 4 ≥ r}dr. ( 9
)
We now compute skewness and kurtosis of a trapezoidal fuzzy variable and a triangular one with expected value e λ .

Example 4.

1. For a trapezoidal fuzzy variable ξ = (a, b, c, d). The Skewness SK λ [ξ] of ξ is given by:

SK λ [ξ] = 1 4(b -a) [(1-λ)(b-e λ ) 4 -λ(a-e λ ) 4 ]+ λ 4(d -c) [(d-e λ ) 4 -(c-e λ ) 4 ].

For a triangular fuzzy variable ξ = (a, b, c).

The Skewness SK λ [ξ] of ξ is given by:

SK λ [ξ] = 1 4(b -a) [(1-λ)(b-e λ ) 4 -λ(a-e λ ) 4 ]+ λ 4(c -b) [(c-e λ ) 4 -(b-e λ ) 4 ]. If λ = 1 2
, we obtain Li et al.'s result on Skewness ( [START_REF] Li | Mean-variance-skewness model for portfolio selection with fuzzy returns[END_REF]).

In the following Example, we compute kurtosis of a trapezoidal fuzzy variable and a triangular one.

Example 5.

1. ξ = (a, b, c, d) is a trapezoidal fuzzy variable with expected value e λ .

We set:

α = b -a, γ = c -b, β = d -c, t = β(e λ -a) -α(d -e λ )
β -α and without loss of generality, we assume that: b < e λ < c, γ < α < β, c -e λ < e λ -a < d -e λ and c -e λ < t < e λ -a. We obtain:

K λ [ξ] = λ(c -e λ ) 4 + λ α [(e λ -a)(t 4 -(c -e λ ) 4 ) -4 5 (t 5 -(c -e λ ) 5 )] + λ β [(d -e λ )((e λ -a) 4 -t 4 ) -4 5 ((e λ -a) 5 -t 5 )] + λ β [ 1 5 (d -e λ ) 5 ) + 4 5 (e λ -a) 5 -(e λ -a) 4 (d -e λ )].
2. ξ = (a, b, c) is a triangular fuzzy variable with expected value e λ .

We set:

α = b -a, β = c -b, t = β(e λ -a) -α(c -e λ )
β -α and without loss of generality, we assume that: b > e λ and α > β. We obtain:

K λ [ξ] = (b -e λ ) 4 -1-λ α [(b -e λ ) 4 (e λ -a) + 4 5 (b -e λ ) 5 ] + λ β [(c -e λ )(t 4 -(b -e λ ) 4 ) -4 5 (t 5 -(b -e λ ) 5 )] + λ α [(e λ -a)((c -e λ ) 4 - t 4 ) -4 5 ((c -e λ ) 5 -t 5 )] + λ α [ 1 5 (e λ -a) 5 + 4 5 (c -e λ ) 5 -(c -e λ ) 4 (e λ -a)].
We end this Subsection by establishing some of their properties.

Proposition 3. Let ξ be a fuzzy variable, µ and ν two real numbers such that µ ≥ 0 and λ ∈ [0, 1]. We have:

1) S λ [µξ + ν] = µ 3 S λ [ξ]. 2) K λ [µξ + ν] = µ 4 K λ [ξ].
Proof : The proof of this Proposition is similar to the proof of the second result of Proposition 2.

In the following Subsection, we introduce semi-variance and semi-kurtosis of a fuzzy variable ξ such that E λ [ξ] = e λ and λ ∈ [0, 1].

Semi-variance and semi-kurtosis

Definition 4. 1) The semi-variance of ξ is defined by:

V S λ [ξ] = E λ [[(ξ -e λ ) -] 2 ] = ∫ +∞ 0 m λ {[(ξ -e λ ) -] 2 ≥ r}dr. ( 10
)
2) The semi-kurtosis of ξ is defined by:

K S λ [ξ] = E λ [[(ξ -e λ ) -] 4 ] = ∫ +∞ 0 m λ {[(ξ -e λ ) -] 4 ≥ r}dr. ( 11
)
Example 6.

1. For a trapezoidal fuzzy variable ξ = (a, b, c, d), we obtain: a) d -c) .

V S λ [ξ] = λ (e λ -a) 3 3(b -a) -λ max[0; (e λ -b) 3 ] 3(b -a) + (1 -λ) max[0; (e λ -c) 3 ] 3(
If λ = 1 2
, we obtain Huang's result on semi-variance [START_REF] Huang | Mean-semivariance models for fuzzy portfolio selection[END_REF].

b) d -c) .

K S λ [ξ] = λ (e λ -a) 5 5(b -a) -λ max[0; (e λ -b) 5 ] 5(b -a) + (1 -λ) max[0; (e λ -c) 5 ] 5(
If λ = 1 2
, we obtain Sadefo et al.'s result on semi-kurtosis [START_REF] Sadefo | Moments and semimoments for fuzzy portfolio selection[END_REF].

2. For a triangular fuzzy variable ξ = (a, b, c), we obtain: a)

V S λ [ξ] = λ (e λ -a) 3 3(b -a) -λ max[0; (e λ -b) 3 ] 3(b -a) + (1 -λ) max[0; (e λ -b) 3 ] 3(c -b) . If λ = 1 2
, we obtain Huang's result on semi-variance [START_REF] Huang | Mean-semivariance models for fuzzy portfolio selection[END_REF].

b)

K S λ [ξ] = λ (e λ -a) 5 5(b -a) -λ max[0; (e λ -b) 5 ] 5(b -a) + (1 -λ) max[0; (e λ -b) 5 ] 5(c -b) . If λ = 1 2
, we obtain Sadefo et al.'s result on semi-kurtosis [START_REF] Sadefo | Moments and semimoments for fuzzy portfolio selection[END_REF].

The following result justifies that the risk's level of a trapezoidal fuzzy variable (defined by semi-variance or semi-kurtosis) depend on the optimistic parameter and on the expected return position with respect to the support of the variable. Its proof is given in the Appendix.

1. If a ≤ e λ ≤ b or b ≤ e λ ≤ c then V S λ [ξ] and K S λ [ξ]
increases with respect to λ.

If c ≤ e λ ≤ d then:

-

V S λ [ξ] increases with respect to λ when λ ≤ 3 4 . -K S λ [ξ]
increases with respect to λ when λ ≤ 5 6 .

Remark 2. 10

Some properties of fuzzy variables characteristics

For the cases where a ≤ e λ ≤ b or b ≤ e λ ≤ c, the trapezoidal fuzzy variable describes almost either lower expected returns or average expected returns. That is the reason why the more the investor is confident (λ increases), the more he is exposed to a high risk's level (semi-variance and semi-kurtosis are greater).

For c ≤ e λ ≤ d, the trapezoidal fuzzy variable describes almost either higher expected returns. Thus, the risk remains increasing for an optimistic maximum's level ( 3 4 for semi-variance and 5 6 for semi-kurtosis). For λ > 3 4 (resp. λ > 5 6 ), the monotonicity of the semi-variance of the trapezoidal fuzzy variable (resp. its semi-kurtosis) depends on α, β, γ = c-b. That means, the investor can feel himself secured with high values of returns. In that, risk cannot remain increasing with respect to the optimistic factor, it will necessarily depend on returns' spread (α, β, γ).

In the following, we establish some properties or relationships of characteristics of a fuzzy variable ξ such that E λ [ξ] = e λ and λ ∈ [0, 1].

Some properties of fuzzy variables characteristics

The following result justifies that variance (respectively kurtosis) is greater than semi-variance (respectively semi-kurtosis). In addition, it justifies that they are equal for symmetric fuzzy variables. Its proof is given in the Appendix.

Proposition 5. 1) We have:

0 ≤ V S λ [ξ] ≤ V λ [ξ] and 0 ≤ K S λ [ξ] ≤ K λ [ξ]. 2) Furthermore, if ξ is symmetric with respect to e λ , that is, ∀r ∈ R, µ(e λ - r) = µ(e λ + r), then { V S λ [ξ] = V λ [ξ] K S λ [ξ] = K λ [ξ]
.

The following result establishes necessary and sufficient conditions under which Variance and Kurtosis are null for some values of λ.

Proposition 6. If λ ∈ [ 1 2 , 1[, then { V λ [ξ] = 0 ⇔ m λ {ξ = e λ } = 1 K λ [ξ] = 0 ⇔ m λ {ξ = e λ } = 1 .
To prove this result, we need the following Lemma which is proved in the Appendix.

Lemma 1. Let A ⊂ X be an event and λ

∈ [0, 1]. 1) m λ (A) = 0 ⇒ m λ (A c ) =    0 if λ = 0 λ 1-λ or 1 if λ ∈]0, 1 2 ] 1 if λ ∈] 1 2 , 1] 2) Furthermore, for λ ∈ [ 1 2 , 1[, we have: m λ (A) = 0 ⇔ m λ (A c ) = 1.
m λ {(ξ -e λ ) 2 > r} ≤ {m λ {(ξ -e λ ) 2 > 0} ≤ {m λ {(ξ -e λ ) 2 ̸ = 0} = 0. This implies that m λ {(ξ -e λ ) 2 > r} = 0, ∀r > 0. Therefore, V λ [ξ] = ∫ +∞ 0 m λ {(ξ -e λ ) 2 ≥ r}dr = 0. (⇒): Let us assume that V λ [ξ] = 0. Since m λ takes values in [0;1], ∫ +∞ 0 m λ {(ξ -e λ ) 2 ≥ r}dr = 0 implies that m λ {(ξ -e λ ) 2 ≥ r} = 0, ∀r > 0, that is m λ {(ξ -e λ ) 2 ̸ = 0} = 0.
By Lemma 1, we have: m λ {(ξ -e λ ) 2 = 0} = 1 and we deduce that m λ {ξe λ = 0} = 1, that is, m λ {ξ = e λ } = 1. In a similar way, it is easy to prove that K λ [ξ] = 0 ⇔ m λ {ξ = e λ } = 1. In the following Section, we propose two models and implement them for the determination of optimal portfolios in Finance.

Application for portfolio selection 4.1 Optimization models

Let us consider an investor who likes to invest his capital in n securities in the proportion x 1 , x 2 ,..., x n such that ∀i ∈ {1, 2, ..., n}, x i ∈ [0, 1] and ∑ n i=1 x i = 1. It is well-known that an investment of a part x i of the capital in the i th security generates a return denoted by x i ξ i which is not currently known. Making up such investment consists on constituting a portfolio ((x i , ξ i )) 1≤i≤n where the n fuzzy variables x 1 ξ 1 , ..., x n ξ n are future returns of the n securities and the fuzzy variable ξ = ξ 1 x 1 + ξ 2 x 2 + ... + ξ n x n is the total future return or the portfolio future return. With respect to the credibility measure, Huang [START_REF] Huang | Mean-semivariance models for fuzzy portfolio selection[END_REF], Li et al. [START_REF] Li | Mean-variance-skewness model for portfolio selection with fuzzy returns[END_REF], Sadefo et al. [START_REF] Sadefo | Moments and semimoments for fuzzy portfolio selection[END_REF] assumed that ξ i is a fuzzy variable and they proposed models based on parameters (mean, variance and semi-variance for Huang; mean, variance and skewness for Li et al.; mean, variance, skewness and kurtosis for Sadefo et al.) in order to determine optimal portfolios. In this Section, we propose two models based on parameters with respect to m λ to solve the same question of determination of best portfolios. Those new models give a family of solutions which are optimal portfolios and generalize more recent models proposed by Sadefo et al. [START_REF] Sadefo | Moments and semimoments for fuzzy portfolio selection[END_REF] in the particular case where λ = 1 2 . For that, we consider the first family of seven assets returns described by the following triangular fuzzy variables proposed by Huang [START_REF] Huang | Mean-semivariance models for fuzzy portfolio selection[END_REF] and used by many authors: ξ 1 = (-0.3, 1.8, 2.3), ξ 2 = (-0.4, 2.0, 2.2), ξ 3 = (-0.5, 1.9, 2.7), ξ 4 = (-0.6, 2.2, 2.8), ξ 5 = (-0.7, 2.4, 2.7), ξ 6 = (-0.8, 2.5, 3.0) , ξ 7 = (-0.6, 1.8, 3.0) and the second family of ten assets returns described by the following trapezoidal fuzzy variables proposed by Hasuike et al. [START_REF] Hasuike | Portfolio selection problems with random fuzzy variable return[END_REF] (written as (a, b, c, d)): ξ 1 = (-0.362, -0.123, 0.005, 0.873), ξ 2 = (-0.37, -0.069, 0.069, 0.536), ξ 3 = (-0.329, -0.129, 0.025, 0.738), ξ 4 = (-0.193, 0.005, 0.177, 0.412), ξ 5 = (-0.299, -0.082, 0.114, 0.437), ξ 6 = (-0.342, -0.052, 0.108, 0.49) , ξ 7 = (-0.292, -0.056, 0.067, 0.436), ξ 8 = (-0.25, 0.060, 0.193, 0.649), ξ 9 = (-0.405, -0.093, 0.130, 0.756) , ξ 10 = (-0.554, 0.009, 0.236, 0.5). In addition, we set for each family of assets, three targets values t 1 , t 2 , t 3 considered respectively as the minimum benefit (expected value), the maximum risk (variance) and the minimum skewness that the investor can bear. The two proposed selection models for best portfolios of those assets are:

                   minimize K λ [x i 1 ξ i 1 + x i 2 ξ i 2 + ... + x i k ξ i k ] subject to E λ [x i 1 ξ i 1 + x i 2 ξ i 2 + ... + x i k ξ i k ] ≥ t 1 V λ [x i 1 ξ i 1 + x i 2 ξ i 2 + ... + x i k ξ i k ] ≤ t 2 S λ [x i 1 ξ i 1 + x i 2 ξ i 2 + ... + x i k ξ i k ] ≥ t 3 x i 1 + x i 2 + ... + x i k = 1 x i k ≥ 0, k ∈ {7; 10} (12) and                    minimize K S λ [x i 1 ξ i 1 + x i 2 ξ i 2 + ... + x i k ξ i k ] subject to E λ [x i 1 ξ i 1 + x i 2 ξ i 2 + ... + x i k ξ i k ] ≥ t 1 V λ [x i 1 ξ i 1 + x i 2 ξ i 2 + ... + x i k ξ i k ] ≤ t 2 S λ [x i 1 ξ i 1 + x i 2 ξ i 2 + ... + x i k ξ i k ] ≥ t 3 x i 1 + x i 2 + ... + x i k = 1 x i k ≥ 0, k ∈ {7; 10} (13)
In the following paragraph, we implement the two models by using Huang's data.

Implementation of the two models with Huang's data

In the following, we implement with Huang's data the two previous models for λ = 1 3 , λ = 1 2 and λ = 2 3 where the selected target values t 1 , t 2 , t 3 are given in Table 3. For that, we set: ∀i ∈ {1, ..., 7}, ξ i = (a i , b i , c i ) and the combination of those seven triangular fuzzy variables is also a triangular fuzzy variable denoted by ξ = ∑ 7

i=1 ξ i = (a, b, c) where a = ∑ 7 i=1 a i , b = ∑ 7 i=1 b i and c = ∑ 7 i=1 c i , α i = b i -a i , β i = c i -bi and α = ∑ 7 i=1 α i , β = ∑ 7 i=1 β i .
We recall that Sadefo et al. [START_REF] Sadefo | Moments and semimoments for fuzzy portfolio selection[END_REF] defined these characteristics in the particular case where λ = 1 2 . By implementing those formulas in Matlab through the previous proposed models, we obtain (i) Table 1 which presents in each line (from the second line) how the best portfolio is made up of in term of percentage of the seven securities for a given value of λ and for either K, either K S as objective function and, (ii) Table 2 gives characteristics (parameters) of best portfolios of Table 1.

Asset i 1 (%) 2 (%) 3 (%) 4 (%) 5 (%) 6 (%) 7 (%)

λ = 1 3 ( K) 0.
t 3 λ = 1 3 1.3 0.5 0 λ = 1 2 1.6 0.8 -0.6823 λ = 2 3 1.8 1.2 -2
Table 3: Selected target values depending on λ.

Table 1 gives strategies to invest in the seven assets in order to obtain best portfolios for each of the three values of the parameter. More precisely, no investment in ξ 3 and ξ 7 , greater investment in ξ 1 and ξ 4 for greater values of λ. We can observe the impact of the parameter λ variation through the following histogram. According to the results of Table 2, we can make the following observations: -The mean increases with respect to λ, it can be interpreted as follows: the more the optimistic parameter is greater (the more self confident the investor is), the greater is expected return which represents its benefit. In addition, risk measures such as variance, kurtosis, semi-variance, semikurtosis increase with respect to λ. It means that, when the investor is self confident in investment and he is looking for greater benefits, meanwhile he is exposed to greater risk.

-The skewness is decreasing with respect to λ. It means that returns' spread is more greater on left of the mean when λ increases. Thus, the investor is exposed to loss (negative returns) when he is more self confident.

-Notice that, for a given value of λ, it is not always possible to get investment proportions with some targets values. By varying λ as in Table 3, one can obtain models that converge to a unique solution with some given target values.

-For λ = 2 3 , the two proposed models coincide. For further research, it is interesting to determine different optimistic parameter's values or the minimum optimistic parameter's value such that the two models coincide. As we can observe from Tables 5 and2, optimal portfolios characteristics except the skewness (skewness neither describes benefit nor risk), vary in the same manner with respect to the parameter λ whatever data are described.

The following histogram provides for Tokyo stock exchange data, clear observations of those characteristics variations according to the model and the parameter λ.

Concluding Remarks

In this paper, we have determined two useful axioms of the m λ -measure of fuzzy events combining possibility and necessity measures with an optimistic parameter which describes investor attitude to make a decision in uncertain situations. By means of the m λ -measure and its axioms, we have defined and determined properties of some characteristics (four first moments and two first semi-moments) of a fuzzy variable. These moments and semi-moments have been determined and analyzed for trapezoidal and triangular fuzzy variables. The obtained results generalize those obtained earlier by Liu [START_REF] Liu | Expected value of fuzzy variable and fuzzy expected value models[END_REF], Li et al. [START_REF] Li | Mean-variance-skewness model for portfolio selection with fuzzy returns[END_REF], Huang et al. [START_REF] Huang | Mean-semivariance models for fuzzy portfolio selection[END_REF] and Sadefo et al. [START_REF] Sadefo | Moments and semimoments for fuzzy portfolio selection[END_REF]. Those theoretical results have illustrated in portfolio selection with fuzzy returns in Finance.

We have proposed and implemented two models which led to a huge variety of solutions in portfolio selection according to some given target values. We have displayed the impact of the variation of the optimistic parameter on the investment's risk and the benefit: the benefit and the risk increase with respect to the optimistic parameter.

For further research, we propose to tackle some interesting questions among which: (i) the determination of k-moments and 2k-moments (for k ∈ N) of a fuzzy variable (ii) the study of dominance on fuzzy variables based on the new measure and (iii) the study of partial lower moments of a fuzzy variable based on the new measure.

Appendix

Proof of Proposition 2 (linearity of the mean): According to the expressions recalled in relations ( 4) and ( 5) and the notions of optimistic and pessimistic functions respectively given by: X sup (α) = sup{r/P os{x ∈ X/ξ(x) ≥ r} ≥ α} and X sup (α) = sup{r/P os{x ∈ X/ξ(x) ≥ r} ≥ α} with α ∈ (0, 1], Liu and Liu [START_REF] Liu | Expected value operator of random fuzzy variable and random fuzzy expected models[END_REF] proved that:

1

. E[ξ] = ∫ 1 0 X sup (α)dα and E[ξ] = ∫ 1 0 X inf (α)dα. 2. E[ξ + η] = E[ξ] + E[η] and E[ξ + η] = E[ξ] + E[η]. 3. For a ≥ 0, E[aξ] = aE[ξ] and E[aξ] = aE[ξ]. 4. For a ≤ 0, E[aξ] = aE[ξ] and E[aξ] = aE[ξ].
Furthermore, according to relation ( 6), we have:

For λ ∈ R, E λ [ξ] = λE[ξ] + (1 -λ)E[ξ] and thus: E λ [ξ + η] = λE[ξ + η] + (1 -λ)E[ξ + η], that is, E λ [ξ + η] = [λE[ξ] + (1 -λ)E[ξ]] + [λE[η] + (1 -λ)E[η]] = E λ [ξ] + E λ [η].
On the other hand, we have for a nonnegative real number α:

E λ [αξ] = λE[αξ] + (1 -λ)E[αξ]=α(λE[ξ] + (1 -λ)E[ξ]]) = αE λ [ξ].
Proof of Proposition 4: Let ξ = (a, b, c, d) be a trapezoidal fuzzy variable with expected value e λ and λ ∈ [0, 1].

• If a ≤ e λ ≤ b then V S λ [ξ] = λ (e λ -a) 3 3α
and K S λ [ξ] = λ (e λ -a) 5

5

.

We have: (

V S ) ′ λ [ξ] = (e λ -a) 3 3α +λ e ′ λ (e λ -a) 2 α ≥ 0 and (K S ) ′ λ [ξ] = (e λ -a) 5 5α + λ e ′ λ (e λ -a) 4 α ≥ 0. • If b ≤ e λ ≤ c then V S λ [ξ] = λ (e λ -a) 3 3α -λ (e λ -b) 3 3α and K S λ [ξ] = λ (e λ -a) 5 5 - λ (e λ -b) 5 5α . We have: (V S ) ′ λ [ξ] = (e λ -a) 3 3α -(e λ -b) 3 3α + λ e ′ λ (e λ -a) 2 α -λ e ′ λ (e λ -b) 2 α ≥ 0 and (K S ) ′ λ [ξ] = (e λ -a) 5 5α -(e λ -a) 5 5α + λ e ′ λ (e λ -a) 4 α -λ e ′ λ (e λ -b) 4 α ≥ 0. • If c ≤ e λ ≤ d then V S λ [ξ] = λ (e λ -a) 3 3α -λ (e λ -b) 3 3α + (1 -λ) (e λ -c) 3 3β and K S λ [ξ] = λ (e λ -a) 5 5 -λ (e λ -b) 5 5α + (1 -λ) (e λ -c) 5 5β . We have: (V S ) ′ λ [ξ] = (e λ -a) 3 3α -(e λ -b) 3 3α + λ e ′ λ (e λ -a) 2 α -λ e ′ λ (e λ -b) 2 α - 18 (e λ -c) 3 3β +(1-λ) e ′ λ (e λ -c) 2 β and (K S ) ′ λ [ξ] = (e λ -a) 5 5α -(e λ -a) 5 5α +λ e ′ λ (e λ -a) 4 α - λ e ′ λ (e λ -b) 4 α -(e λ -c) 5 5α + (1 -λ) e ′ λ (e λ -c) 4 β . We easily check that: λ ≤ 3 4 ⇒ (V S ) ′ λ [ξ] ≥ 0 and λ ≤ 5 6 ⇒ (K S ) ′ λ [ξ] ≥ 0. Proof of Proposition 5 1) Let us prove that 0 ≤ V S λ [ξ] ≤ V λ [ξ]. Let x ∈ X and r ∈ R. We have: [(ξ -e λ ) -] 2 = { (ξ -e λ ) 2 if ξ ≤ e λ 0 if ξ > e λ
. This definition leads us to two cases: i) First case: ξ(x) ≤ e λ .

We have:

[(ξ(x) -e λ ) -] 2 = (ξ(x) -e λ ) 2 . Therefore, [(ξ(x) -e λ ) -] 2 ≥ r ⇔ (ξ(x) -e λ ) 2 ≥ r. ii) Second case: ξ(x) > e λ . We have: [(ξ(x)-e λ ) -] 2 = 0 which implies that (ξ(x)-e λ ) 2 ≥ [(ξ(x)-e λ ) -] 2 . Thus, [(ξ(x) -e λ ) -] 2 ≥ r implies (ξ(x) -e λ ) 2 ≥ r. It follows that: ∀(x, r) ∈ X ×R, {x/[(ξ(x)-e λ ) -] 2 ≥ r} ⊆ {x/(ξ(x)-e λ ) 2 ≥ r}.
By the fact that m λ is monotone, we obtain: . That is, λm λ (A c ) = 0.

∀r ∈ R, m λ {[(ξ(x) -e λ ) -] 2 ≥ r} ≤ m λ {(ξ(x) -e λ ) 2 ≥ r}. Finally, V λ [ξ] = ∫ +∞ 0 m λ {(ξ -e λ ) 2 ≥ r}dr ≥ ∫ +∞ 0 m λ {[(ξ -e λ ) -] 2 ≥ r}dr = V S λ [ξ]. In the same manner, one can prove that 0 ≤ k S λ [ξ] ≤ K λ [ξ].
• If λ ∈]0, 1 2 ], then we obtain by axiom 4:

   (1 -λ)m λ (A c ) = λ or λm λ (A c ) = λ which implies that    m λ (A c ) = λ 1-λ or m λ (A c ) = 1 .
Furthermore, λ ∈]0, 1 2 ] ⇒ 0 < m λ (A c ) = λ 1-λ ≤ 1.

• If λ ∈]

  , b] and decreasing on [c, d] and (iv) ∀r ∈ [b, c], µ(r) = 1. Thus, we denote it by ξ = (a, b, c, d). In the particular case where µ is a straight line on [a, b] and [c, d], then ξ = (a, b, c, d) is the usual and wellknown trapezoidal fuzzy number. Then, we deduce analytical expressions of the membership functions of ξ

Figure 1 :

 1 Figure 1: On right trapezoidal fuzzy variable (1, 2, 3, 4) and on left triangular fuzzy variable (1, 3.5, 4).

Proposition 4 .

 4 Let ξ = (a, b, c, d) be a trapezoidal fuzzy variable with expected value e λ and λ ∈ [0, 1].

11 We now justify the Proposition. Proof of Proposition 6 :

 116 Let ξ be a fuzzy variable withE λ [ξ] = e λ and λ ∈ [ 1 2 , 1[. Let us prove that V λ [ξ] = 0 ⇔ m λ {ξ = e λ } = 1. (⇐): Let us assume that m λ {ξ = e λ } = 1. It is obvious that m λ {ξ -e λ = 0} = 1 if and only if m λ {(ξ -e λ ) 2 } = 1.By Lemma 1, we obtain m λ {(ξ -e λ ) 2 ̸ = 0} = 0. Let r > 0, we have:

Figure 2 :

 2 Figure 2: Illustration of fuzzy variables characteristics from different models with Huang's data.

Figure 3 :

 3 Figure 3: Illustration of fuzzy variables characteristics from different models with Tokyo stock exchange data.

2 )

 2 Let ξ be a symmetric fuzzy variable. Let us prove that V S λ [ξ] = V λ[ξ]. Let us write:V S λ [ξ] = E λ [[(ξ -e λ ) -] 2 ] = ∫ +∞ 0 ⇕ λ {[(ξ -e λ ) -] 2 ≥ r}dr = ∫ +∞ 0 ⇕ λ {(ξ -e λ ) -∈] -∞; -√ r] ∪ [ √ r; +∞[}dr and V λ [ξ] = E λ [(ξ -e λ ) 2 ] = ∫ +∞ 0 ⇕ λ {(ξ -e λ ) 2 ≥ r}dr = ∫ +∞ 0 ⇕ λ {(ξ -e λ ) ∈ ] -∞; -√ r] ∪ [ √ r;+∞[}dr. If µ and ν are respectively membership functions of fuzzy variables ξ -e λ and (ξ -e λ ) -, then we have ν = { µ if ξ -e λ ≤ 0 0 otherwise . Let us observe that ξ is symmetric with respect to e λ if and only if ξ -e λ is symmetric with respect to 0. We have, ∀r > 0:M λ {(ξ-e λ ) ∈]-∞; -√ r]∪[ √ r; +∞[} = λ sup x∈]-∞;- √ r]∪[ √ r;+∞[ µ(x)+(1λ)(1-sup x∈]- √ r;-√ r[ µ(x)) = λ max(sup x∈]-∞;- √ r] µ(x), sup x∈[ √ r;+∞[ µ(x))+ (1 -λ)[1 -max(sup x∈]- √ r;0] µ(x), sup x∈[0; √ r[ µ(x))].Let us notice that by fact that ξ is symmetric, we have:sup x∈]-∞;- √ r] µ(x) = sup x∈[ √ r;+∞[ µ(x) and sup x∈]- √ r;0] µ(x) = sup x∈[0; √ r[ µ(x). so, we have:⇕ λ {(ξ -e λ ) ∈] -∞; -√ r] ∪ [ √ r; +∞[} = λ sup x∈]-∞;- √ r] µ(x) + (1 -λ)[1sup x∈]- √ r;0] µ(x)). However, sup x∈]-∞;- √ r] µ(x) = sup x∈]-∞;- √ r] ν(x) and sup x∈]- √ r;0] µ(x) = sup x∈]- √ r;0] ν(x), that is: ⇕ λ {(ξ -e λ ) ∈] -∞; -√ r] ∪ [ √ r; +∞[} = λ sup x∈]-∞;- √ r] ν(x) + (1 -λ)[1sup x∈]- √ r;0] ν(x)).19 By the fact that ν is null on the intervals [ √ r; +∞[ and [0; √ r[, we finally have:⇕ λ {(ξ -e λ ) ∈]-∞; -√ r]∪[ √ r; +∞[} = λ sup x∈]-∞;- √ r]∪[ √ r;+∞[ ν(x)+(1λ)[1 -sup x∈]- √ r;-√ r[ ν(x)) = ⇕ λ {(ξ -e λ ) -∈] -∞; -√ r] ∪ [ √ r; +∞[}. Therefore, we get V S λ [ξ] = V λ [ξ]. By a similar way, one can prove that KS λ [ξ] = K λ [ξ].Proof ofLemma 1: 1) Let us consider A ⊂ X such that M λ (A) = 0 and λ ∈ [0, 1]. Let us recall the third axiom of the measure M λ (A):    λm λ (A) + (1 -λ)m λ (A c ) = λ or (1 -λ)m λ (A) + λm λ (A c ) = λ • If λ = 0, then we obtain by Axiom 3:    m λ (A c ) = 0 or 0 = 0

1 2 ,( 1 1 . 1 . 2 ).

 21112 [START_REF] Dai | Optimal strategies for carbon capture, utilization and storage based on an inexact m λ -measure fuzzy chance-constrained programming[END_REF][, then we obtain by Axiom 4:  -λ)m λ (A c ) = λ or λm λ (A c ) = λ which implies that    m λ (A c ) = λ 1-λ or m λ (A c ) = However, λ ∈] 1 2 , 1[⇒ m λ (A c ) = λ 1-λ > 1, which is wrong. So, we have m λ (A c ) = 1.Finally, for λ = 1, we obtain by Axiom 3: However, 0 = 1 is wrong. So,λm λ (A c ) = 1. Let us consider A ⊂ X and λ ∈ [ 1 2 , 1[. ⇒) If m λ (A) = 0, then according to Lemma 1, we have m λ (A c ) = 1. ⇐) If m λ (A c ) = 1, then by Axiom 3, we have:    λm λ (A) + (1 -λ) = λ or (1 -λ)m λ (A) + λ = λ However, λ ∈ [ 1 2 , 1[⇒ m λ (A) = 2λ-1 λ< 0, which is wrong. So, we have m λ (A) = 0. Hence the result.

Table 1 :

 1 Optimal selection from each model.

	00	63.39	0.00	0.00	36.61	0.00	0.00
	λ = λ = λ = 1 1 3 1 ( K S ) ( K) 2 ( K S ) 20.00 0.05 0.6471 0.00 20.04 0.00 0.00 0.00 0.00 2 λ = 2 ( K) 42.50 0.00 0.00 3 λ = 2 3 (K S ) 42.50 0.00 0.00	0.05 79.89 80.00 57.50 57.50	31.33 0.00 0.00 0.00 0.00	3.85 0.07 0.00 0.00 0.00	0.00 0.00 0.00 0.00 0.00
	Mean Variance	SV	Skewness	Kurtosis	SK
	λ = 1 3 ( K) 1.3005 λ = 1 3 ( K S ) 1.3004 λ = 1 1.60 2 ( K) λ = 1 1.60 2 ( K S ) λ = 2 1.80 3 ( K) λ = 2 3 (K S ) 1.80	0.4571 0.4570 0.7018 0.7019 1.0510 1.0510	0.2482 0.2482 -1.5336 × 10 -5 -0.0014 0.6140 -0.6823 0.6141 -0.6823 1.0404 -1.6573 1.0404 -1.6573	0.6593 0.6594 1.7290 1.7291 3.2213 3.2213	0.4880 0.4875 1.6873 1.6872 3.2202 3.2202

Table 2 :

 2 Comparison of the characteristics of different optimal portfolios.

Target values Mean:t 1 Variance: t 2 Skewness:

Table 4 :

 4 Optimal selection from each model.

	Mean Variance	SV	Skewness Kurtosis	SK
	λ = 1 4 ( K) λ = 1 4 ( K S ) 0.0201 0.02 λ = 1 0.07 3 ( K) λ = 1 0.07 3 ( K S ) λ = 1 0.09 2 ( K) λ = 1 2 (K S ) 0.09	0.0262 0.0264 0.0285 0.0286 0.04 0.04	0.0047 0.0047 0.0114 0.0114 0.0389 0.0385	0.012 0.0122 0.0141 0.0142 0.01 0.01	0.0049 0.0050 0.0058 0.0058 0.0083 0.0081	1.6745 × 10 -4 1.7094 × 10 -4 6.6001 × 10 -4 6.6330 × 10 -4 0.0037 0.0036

Table 5 :

 5 Comparison of the characteristics of different optimal portfolios. Target values Mean:t 1 Variance: t 2 Skewness: t 3

	λ = λ = λ =	1 4 1 3 1 2	0.02 0.07 0.09	0.04 0.04 0.04	0.01 0.01 0.01

Table 6 :

 6 Selected target values depending on λ.

* The work has been done under the research Grant No 17-497RG/M AT HS/AF/ACG-F R3240297728 offered by The World Academy of Sciences (TWAS) to the Applied Mathematics to Social Sciences Research Group of the Laboratory of Mathematics-University of Douala-Cameroon. The authors (members of the Laboratory of Mathematics) sincerely thanks TWAS.

Implementation of the two models with Tokyo stock exchange data

To end this Section, we implement the two proposed models by using real data as Tokyo stock exchange data.

4.3

Implementation of the two models with Tokyo stock exchange data

We implement Tokyo stock exchange data through the two previous models for λ = 1 4 , λ = 1 3 and λ = 1 2 . The target values t 1 , t 2 , t 3 are given in Table 6. Different optimal investment proportions and optimal characteristics are respectively given in Tables 4 and5.

Asset i 1 (%) 2 (%) 3 (%) 4 (%) 5 (%) 6 (%) 7 (%) 8 (%) (%) [START_REF] Tassak | Characterization of order dominances on fuzzy variables for portfolio selection with fuzzy returns[END_REF]