Hélène Ferrer

Guillermo Owen

Fabrice Valognes

Stable syndicates of factor owners and distribution of social output: A Shapley value approach. *

Keywords: Shapley value, Increasing returns to scale, Syndicate of factor owners

The purpose of this paper is to examine the incentive of a player to join a syndicate in an environment of team production and payoff distribution according to Shapley value. We consider an economy in which a single output is produced by an increasing returns to scale production function using two inputs: labor and capital. By assuming that syndicates of factor owners can form, we are interested in their stability, i.e., the willingness of the members of the syndicate to stay in the syndicate. Our analysis, based on the Shapley value, allows us to find a fair imputation of the gains of cooperation and the conditions under which syndicates are stable.

Introduction

It seems intuitively obvious that the formation of binding agreements between the members of a group of economic agents improves their bargaining power. Trade unions, syndicates of property owners or of consumers provide natural examples of this phenomenon. The purpose of this paper is to give a rationale for this in a cooperative game setting. The problem is the following: What is the influence of binding agreements on the outcome of bargaining processes? Under which conditions are such binding agreements stable?

Following [START_REF] Gabszewicz | Syndicates of Traders in an Exchange Economy[END_REF], we will term a syndicate a group of identical agents who delegate to a single decision unit the task of representing their economic interests. This leads to assimilate a syndicate to a single agent. As remarked by [START_REF] Guesnerie | Monopoly, Syndicate, and Shapley Value: About Some Conjectures[END_REF], the study of syndicates so defined raises the question of their stability, i.e., the willingness of the members of the syndicate to stay in the syndicate. This stability will depend upon the advantages the syndicate gives to its members and different comparisons seem relevant for evaluating them. The first one is the comparison of the situation of the syndicate members in the presence, or absence, of the syndicate; this is the notion of total stability defined by [START_REF] Gabszewicz | Syndicates of Traders in an Exchange Economy[END_REF]. The second one is the comparison of the situation of syndicated and nonsyndicated agents of the same type when there is a syndicate; this is the notion of marginal stability defined by [START_REF] Gabszewicz | Syndicates of Traders in an Exchange Economy[END_REF]. We introduce a third and new comparison which consists in assuming that a syndicate of one type has already formed and we are interested in the reaction of the agents of the other type. This raises the following question: Will the formation of a syndicate of one type induce the formation of a syndicate of another type?

The stability of syndicates has been studied in the context of a productive economy using the core solution concept [START_REF] Hansen | Collusion of Factor Owners and Distribution of Social Output[END_REF]). Hansen and Gabszewicz consider the case of a productive economy in which the production possibilities are characterized by constant returns to scale. Then, one may wonder what happens when returns are increasing. In that context, we consider economies of scale since production is cheaper when scale is larger. This is for example the case when we consider the economy of software engineering where some programmers can concentrate more on particular kinds of programming, and get better at them. [START_REF] Gabszewicz | L'imputation du produit social: Un exemple de négociation collective[END_REF] recognized that core solution concept is rather limited when returns are increasing. First of all, the equal-treatment property does not hold in the core, i.e., agents of the same type can receive a different amount. 1 Therefore, the equal treatment property, which is satisfied in a productive economy characterized by constant returns to scale, is no longer satisfied when returns are increasing. Second, generally the larger the degree of increasing returns, the larger the core will be and the more the core solution will be unsuitable. Indeed, it is intuitively clear that the more pronounced is the degree of increasing returns, the more difficult it will be for proper subgroups to do better by their own means and therefore the more we could depart from the proportional imputation while remaining in the core.

So the core solution concept is not a very appropriate one, in a productive economy characterized by increasing returns to scale, due to the large core and hence the need for a deterministic allocation. Another solution seems more appropriate, namely the Shapley value. Indeed, the Shapley value is a solution concept which attempts to describe a reasonable, or "fair", way to divide the gains from cooperation and a powerful tool for evaluating the power structure in a coalitional game. The criterion of fairness to which value theory adheres is egalitarianism: the aim is to distribute the gains from cooperation equally. Moreover, given a game, the Shapley value assigns to it a single outcome -in contrast, the core solution assigns a set of outcomes. This is the reason why we investigate in this paper the stability of syndicates when the allocation of the gains of cooperation is governed by the Shapley value.

The paper is organized as follows. In section 2, we state our assumptions and we determine the Shapley values according to various states of competition. Section 3 contains the results. We show that a monopoly syndicate can increase the bargaining power of its members in terms of the Shapley value. Sufficient conditions are given for a monopoly to be stable.

The model

Assumptions

We consider an economy in which a single output is produced by an increasing returns to scale production function using two inputs or factors: labor and capital. Following [START_REF] Hansen | Collusion of Factor Owners and Distribution of Social Output[END_REF], we assume that these inputs are initially distributed among a given set of factor owners and that any coalition of owners has initial access to its aggregate input endowment.

agent owns initially a single unit of the input, the total output which can be produced in the economy is then 25. Let x be an imputation defined by , 3, 5, 7, 9) We can easily check that this imputation, which assigns a different amount to identical agents, belongs to the core!

x = (x1, x2, x3, x4, x5) = (1
The economy consists of a set I 1 of workers -each owning initially a single unit of labor -and a set I 2 of capitalists -each owning initially a single unit of capital. There are two types of agents, namely: We assume that the production function of the economy is a Cobb-Douglas function:

I 1 = {11, . . . ,
F (z 1 , z 2 ) = z α 1 • z β 2
, where α > 0, β > 0, α + β > 1 and z i , the units of factors.

The total output which can be produced in the economy is:

F (n, n) = n α+β .
We are then interested in the imputation of this amount among the factor owners. In other words, the problem is the following: How can we impute the total output among the workers and the capitalists? By definition, an imputation is a 2n-tuple of numbers (x 11 , . . . , x 1j , . . . , x 1n ; x 21 , . . . , x 2j , . . . , x 2n ) satisfying:

n j=1 x 1j + n j=1 x 2j = n α+β ,
where x 1j denotes the amount received by a worker j, and x 2j the amount obtained by a capitalist j.

Interesting imputations can be obtained through a collective decision mechanism. Indeed, some coalitions can form in order to improve their bargaining power. Consider a given coalition S composed of some workers (i.e., s 1 = |S ∩ I 1 |) and some capitalists (i.e., s 2 = |S ∩ I 2 |). The coalition S can produce, on its own means, an amount of output F (s 1 , s 2 ). This defines the characteristic function, v, by:

v (S) = F (s 1 , s 2 ) = s α 1 • s β 2 , for all S ⊂ N .
To obtain these imputations, we will define the Shapley value associated with this characteristic function.

Following Gabszewicz and Drèze [1972] and [START_REF] Gabszewicz | L'imputation du produit social: Un exemple de négociation collective[END_REF], we first assume that any group of factor owners is free to form a coalition, and also to disband it. A factor owner is then free to join or leave any coalition.

What happens if some factor owners decide to act together as one unit relative to the rest of the agents? Let us introduce the notion of syndicate. We define a syndicate as a group of identical agents, i.e., a group of agents of the same type, who agree that no proper subset of them can act independently from the group itself. In this case, it is no longer true that the formation of any coalition is possible: any proper subset of these groups is no longer a feasible coalition, only those coalitions which either contain all the members of the group or contain none of them are admissible. The set of permissible coalitions is then restricted. Moreover, as members of the syndicate are identical players, it is natural to suppose that all members of the group will accept an equal treatment, i.e., an equal sharing of the gains of cooperation. Consider then the syndicates that factor owners can form: Let us call J i a syndicate of agents of type i, where

J i ⊂ I i , i = 1, 2.
By assuming that syndicates of factor owners can form, we are interested in their stability, i.e., the willingness of the members of the syndicate to stay in the syndicate. So, the question is: Under what conditions will stable syndicates exist? Different definitions of stability might explain the formation of syndicates. The first one compares the amount a syndicate member can obtain with what he would have received if the syndicates did not exist; this is the total stability of [START_REF] Aumann | Disadvantageous Monopolies[END_REF], called A-stability.

Definition 1 A syndicate J i is A-stable if x ij > x ′ ij , ∀ij ∈ J i
, where x ij is the amount received by a syndicate member ij when J i = ∅, and x ′ ij the amount received by a syndicate member ij when J i = ∅ (every player ij is independent).

The second definition consists in comparing the situations of a syndicated and nonsyndicated agent of the same type, when there is a syndicate. This is the marginal stability of Gabzsewicz and Drèze [1971], called B-stability.

Definition 2 A syndicate J i is B-stable if x ij > x ik , ∀ij ∈ J i and ∀ik ∈ I i \J i ,
where x ij is the amount received by a syndicate member ij and x ik the amount received by a nonsyndicate member ik of the same type when J i = ∅.

Finally, we introduce an alternative definition of stability which we call C-stability. This definition rests on the assumption that a syndicate of some type is formed and we are interested in the reaction of the agents of the other type: Do they form a syndicate or decide to remain separate? In order to answer this question, we compare the situation of a syndicate member when the two types of syndicates exist, and when its own syndicate does not exist.

Definition 3 A syndicate J 1 (resp. J 2 ) is C-stable if x 1j > x ′′ 1j , ∀1j ∈ J 1 , resp. x 2j > x ′′ 2j , ∀2j ∈ J 2
, where x 1j (resp. x 2j ) is the amount received by a syndicate member 1j (resp. 2j) when J 1 , J 2 = ∅ and x ′′ 1j resp. x ′′ 2j the amount received by a syndicate member 1j (resp. 2j) when J 1 = ∅ (every player 1j is independent), J 2 = ∅ (resp. J 1 = ∅, J 2 = ∅) .

Payoff Distribution by the Shapley value

In order to study the incentive of a player to join a syndicate, it is necessary to determine the imputation each factor owner can obtain according to various states of competition. The solution concept we are dealing with is the Shapley value defined by [START_REF] Shapley | A Value for N-Person Games[END_REF].

Shapley value with no syndicate

In an economy with no syndicate, agents of the same type are not constrained to behave identically. Then there are 2n independent players in the game.

The value of the game, to a given player, can be described intuitively as his average marginal worth over all possible coalitions. Hence, for a situation of perfect competition, we get the following expression for the Shapley value of factor owner j of type i:

ϕ i = S⊂N ij / ∈S s! (2n -s -1)! (2n)! [v (S ∪ {ij}) -v (S)]
, for all S ⊂ N , ∀j = 1, . . . , n and i = 1, 2 and where s denotes the number of players in S, s = |S|.

Theorem 1 In a productive economy with increasing returns to scale, the Shapley value, denoted by ϕ = (ϕ 1 , ϕ 2 ), and the asymptotic Shapley value, denoted by ϕ(n) ∼ (ϕ 1 (n), ϕ 2 (n)), are given by:

(i) symmetric agents: α = β ⇒ ϕ = ( 1 2 n 2α-1 , 1 2 n 2α-1 ), ∀n.
(ii) asymmetric agents:

α = β ⇒ ϕ ∼ ( α α+β n α+β-1 , β α+β n α+β-1
), for n large enough.

Proof. With increasing returns to scale, production functions are quasiconcave but not concave. Thus, the Shapley value does not converge to the classical "competitive equilibrium" solution when the number of agents is increased3 .

(i) With symmetric agents, the proof is immediate. Since the Shapley value gives the same value to all the identical players, the total amount of output, F (n, n), is equally shared between the 2n players, i.e.:

ϕ 1 = ϕ 2 = F (n, n) 2n ⇔ ϕ 1 = ϕ 2 = 1 2 n 2α-1 , ∀n.
(ii) With asymmetric agents, the proof is less immediate. Suppose that there is a large finite number n of players, each of whom is individually insignificant. We know from Aumann and [START_REF] Aumann | Values of Non-Atomic Games[END_REF] that there exists another way of defining the value: The infinite-person game is approximated by games with finitely many players (with the aid of sequences of increasingly fine partitions of the players space) and it is proved that the value for finite games (called the "asymptotic value") coincides with the value for nonatomic games.

As a first step in the proof, we define the value for non-atomic games. Let x 1 be the fraction of workers owning a single unit of labor and zero unit of capital, i.e., (1, 0) ; and let x 2 be the fraction of capitalists owning a single unit of capital and zero unit of labor, i.e., (0, 1). Thus, the function

F (x 1 , x 2 ) = x α 1 • x β
2 determines the output which can be produced in this normalized game. Then, the Shapley value is given by the following diagonal formula:

ϕ = 1 0 (∇F ) t(1,1) dt,
where (∇F ) denotes the gradient of the function F (x 1 , x 2 ).

We obtain ϕ = (α, β) 1 α + β And we deduce the asymptotic value for our finite game as

ϕ 1 (n) ∼ α α+β n α+β-1 , for n large enough. ϕ 2 (n) ∼ β α+β n α+β-1 , for n large enough.
We can note that the game associated with the characteristic function v (S) = s α 1 • s β 2 , for all S ⊂ N, is convex if α ≥ 1 and β ≥ 1. Therefore, the Shapley value, ϕ i , belongs to the core if the marginal productivities with respect to all inputs are increasing. Then, in a productive economy with no syndicate and characterized by increasing returns to scale, the Shapley value is a "stable" imputation since it cannot be disrupted by the actions of any coalition of agents.

Shapley value with one syndicate

Let us assume that k agents of type 1, k ≤ n, decide to establish a syndicate, denoted by J 1 , J 1 ⊂ I 1 . As a consequence, a new game is played which has formally (since the syndicate acts as a single agent) (1 + 2nk) players.

The Shapley value is represented by three numbers (x 1s , x1ns , x2 ) where x1s denotes the value of a syndicated worker; x1ns the value of a nonsyndicated worker and x2 , the value of a capitalist. The purpose of this section is to determine this Shapley value.

First, by definition, the value of a syndicated worker, x1s , is given by:

x1s = 1 k S⊂N J 1 / ∈S s! (2n -k -s)! (1 + 2n -k)! [v (S ∪ {J 1 }) -v (S)]
We can note that the marginal contribution of syndicate J 1 in a given coalition S will depend on the number of nonsyndicated workers in S i.e. s 1 = |S ∩ I 1 \J 1 | , and on the number of capitalists in S i.e.

s 2 = |S ∩ I 2 |. Then x1s = 1 k n-k s 1 =0 n s 2 =0 (s 1 + s 2 )! (2n -k -s 1 -s 2 )! (1 + 2n -k)! n-k s 1 n s 2 s β 2 ((s 1 + k) α -s α 1 )
Next, we note that the value of a nonsyndicated worker, x1ns , is given by

x1ns = S⊂N 1j / ∈S s! (2n -k -s)! (1 + 2n -k)! [v (S ∪ {1j}) -v (S)]
We shall distinguish between two cases: (1) the syndicate J 1 belongs to the coalition S (first term of the following expression), (2) the syndicate J 1 does not belong to the coalition S (second term). Then

x1ns = n-k-1 s 1 =0 n s 2 =0 (s 1 +s 2 +1)!(2n-k-s 1 -s 2 -1)! (1+2n-k)! n-k-1 s 1 n s 2 s β 2 ((s 1 + k + 1) α -(s 1 + k) α ) + n-k-1 s 1 =0 n s 2 =0 (s 1 +s 2 )!(2n-k-s 1 -s 2 )! (1+2n-k)! n-k-1 s 1 n s 2 s β 2 ((s 1 + 1) α -s α 1 )
Finally, we deduce the value of a capitalist, x2 from the property of efficiency of the Shapley value:

x2 = 1 n n α+β -kx 1s -(n -k) x1ns
Similarly, the symmetric case is obtained if we assume the formation of a syndicate of capitalists, J 2 , J 2 ⊂ I 2 , composed of k members. The Shapley value is then described by three numbers (ŷ 1 , ŷ2s , ŷ2ns ) where ŷ1 is the value of a worker; ŷ2s , the value of a syndicated capitalist and ŷ2ns , the value of a nonsyndicated capitalist.

Shapley value with two syndicates

We now assume that k agents of type 1, k ≤ n, decide to establish a syndicate, denoted by J 1 , J 1 ⊂ I 1 , and that k agents of type 2, k ≤ n, decide to establish a syndicate, denoted by J 2 , J 2 ⊂ I 2 . As a consequence, a new game is played which has 2 (nk + 1) players.

The Shapley value is then represented by four numbers (ẑ 1s , ẑ1ns , ẑ2s , ẑ2ns ) where ẑ1s denotes the value of a syndicated worker; ẑ1ns , the value of a nonsyndicated worker; ẑ2s , the value of a syndicated capitalist and ẑ2ns , the value of a nonsyndicated capitalist. So, we want to determine successively these four values.

First, the value of a syndicated worker, ẑ1s , is given by

ẑ1s = 1 k S⊂N J 1 / ∈S s! (2 (n -k + 1) -s -1)! (2 (n -k + 1))! [v (S ∪ {J 1 }) -v (S)]
Let s 1 be the number of nonsyndicated workers in a given coalition S; and let s 2 be the number of nonsyndicated capitalists in the coalition S. We shall again consider two cases depending on whether the syndicate J 2 belongs or not to the coalition S. Then we have

ẑ1s = 1 k n-k s 1 =0 n-k s 2 =0 (s 1 +s 2 +1)!(2(n-k)-s 1 -s 2 )! (2(n-k+1))! n-k s 1 n-k s 2 (s 2 + k) β ((s 1 + k) α -s α 1 ) + 1 k n-k s 1 =0 n-k s 2 =0 (s 1 +s 2 )!(2(n-k)-s 1 -s 2 +1)! (2(n-k+1))! n-k s 1 n-k s 2 s β 2 ((s 1 + k) α -s α 1 )
We now define the value of a nonsyndicated worker as

ẑ1ns = S⊂N 1j / ∈S s! (2 (n -k + 1) -s -1)! (2 (n -k + 1))! [v (S ∪ {1j}) -v (S)]
We shall distinguish between four cases: (1) the two syndicates belong to the coalition S (first term of the following expression), (2) the syndicate J 1 belongs to the coalition S (second term), (3) the syndicate J 2 belongs to the coalition S (third term), (4) the two syndicates do not belong to the coalition S (fourth term). Then

ẑ1ns = n-k-1 s 1 =0 n-k s 2 =0 (s 1 +s 2 +2)!(2(n-k)-s 1 -s 2 -1)! (2(n-k+1))! n-k-1 s 1 n-k s 2 (s 2 + k) β ((s 1 + k + 1) α -(s 1 + k) α ) + n-k-1 s 1 =0 n-k s 2 =0 (s 1 +s 2 +1)!(2(n-k)-s 1 -s 2 )! (2(n-k+1))! n-k-1 s 1 n-k s 2 s β 2 ((s 1 + k + 1) α -(s 1 + k) α ) + n-k-1 s 1 =0 n-k s 2 =0 (s 1 +s 2 +1)!(2(n-k)-s 1 -s 2 )! (2(n-k+1))! n-k-1 s 1 n-k s 2 (s 2 + k) β ((s 1 + 1) α -s α 1 ) + n-k-1 s 1 =0 n-k s 2 =0 (s 1 +s 2 )!(2(n-k)-s 1 -s 2 +1)! (2(n-k+1))! n-k-1 s 1 n-k s 2 s β 2 ((s 1 + 1) α -s α 1 )
Similarly, the value of a syndicated capitalist, ẑ2s , and the value of a nonsyndicated capitalist, ẑ2ns , are determined.

Results

We first assume that all agents of type 1 form a syndicate whereas the agents of the other type remain unorganized.

Assumption 1 J 1 = I 1 and J 2 = ∅.

Proposition 1 Let x1s (n) be the asymptotic Shapley value of a syndicated worker under assumption 1 and let ϕ 1 (n) be the asymptotic Shapley value of a worker in an economy without syndicate, then:

x1s (n) > ϕ 1 (n)if and only if α < 1, ∀β > 0,
where x1s (n) ∼ 1 1 + β n α+β-1 and ϕ 1 (n) ∼ α α + β n α+β-1 , for n large enough.

Proof. Let x1s be the Shapley value of a syndicated worker. Since k = n and s 1 = 0, we have:

x1s = 1 n + 1 n s 2 =0 n α-1 s β 2
Therefore, we obtain the asymptotic Shapley value:

x1s (n) ∼ 1 1 + β n α+β-1 , for n large enough.
The stability of syndicate J 1 depends on the comparison of the members of the syndicate with the members of the unorganized coalition, thus we compare x1s (n) to ϕ 1 (n). Since ϕ 1 (n) ∼ α α + β n α+β-1 , we obtain the desired result.

Proposition 1 implies that, under assumption 1, the imputation x1s (n) is strictly preferred by the members of the syndicate to the imputation ϕ 1 (n). Hence, the syndicate J 1 is stable (A-stable) when the parameter α is less than one and the syndicate J 1 is not stable when α is greater than one. The critical value, α = 1, corresponds to the case where the marginal productivity of labor equals the average productivity of labor, i.e., P m z 1 = P M z 1 . So, if α < 1, the marginal productivity of labor is decreasing and we have P m z 1 < P M z 1 which means that a worker's average contribution is always greater than a worker's marginal contribution.

Therefore, as the number of agents is increased, a syndicate composed of all the agents of one type (the agents of the other type remaining unorganized) is A-stable if the decreasing marginal productivity property of the factor owned by this syndicate is satisfied, whatever be the marginal productivity of the other factor. In other words, a monopoly syndicate is stable when its marginal productivity is less than its average productivity.

We now assume that both agents of type 1 and type 2 form a monopoly syndicate.

Assumption 2 J 1 = I 1 and J 2 = I 2 .

Proposition 2 Let ẑ = (ẑ 1s , ẑ2s ) be the Shapley value under assumption 2 and let ϕ(n) ∼ (ϕ 1 (n), ϕ 2 (n)) be the asymptotic Shapley value in an economy without syndicate, then:

ẑ1s < ϕ 1 (n) ẑ2s > ϕ 2 (n)
, if and only if α > β,

where ẑ1s = ẑ2s = 1 2 n α+β-1 , ∀n, and ϕ

1 (n) ∼ α α + β n α+β-1 , ϕ 2 (n) ∼ β α + β n α+β-1 , for n large enough.
Proof. Let ẑ1s be the Shapley value of a syndicated worker. Since k = n and s 1 = s 2 = 0, we have:

ẑ1s = 1 2 n α+β-1 , ∀α, β > 0
Similarly, let ẑ2s be the Shapley value of a syndicated capitalist. Since k = n and s 1 = s 2 = 0, we have:

ẑ2s = 1 2 n α+β-1 , ∀α, β > 0
In order to analyze the stability of these two syndicates, we compare ẑ1s to ϕ 1 (n) and ẑ2s to ϕ 2 (n), and then we obtain the desired result.

J 2 = I 2 J 2 = ∅ J 1 = I 1 1 2 , 1 2 1 1 + β , β 1 + β J 1 = ∅ α 1 + α , 1 1 + α α α + β , β α + β
In the light of empirical studies, we are able to let α < 1, β < 1 and α > β (and, where applicable, α+β slightly greater than 1) (see, for example, [START_REF] Pendharkar | An empirical study of the Cobb-Douglass production function properties of software development effort[END_REF]). Which implies for the workers:

1 1 + β > α α + β > 1 2 > α 1 + α
and for the preferences we get:

(I 1 , ∅) ≻ I 1 (∅, ∅) ≻ I 1 (I 1 , I 2 ) ≻ I 1 (∅, I 2 ) (∅, I 2 ) ≻ I 2 (I 1 , I 2 ) ≻ I 2 (∅, ∅) ≻ I 2 (I 1 , ∅)

Conclusion

This paper analyses the stability of syndicates of factor owners in a productive economy characterized by increasing returns to scale. Our analysis, based on the Shapley value, allows us to find a fair allocation of the gains of cooperation and the conditions under which syndicates are stable.

We prove that the formation of a monopoly syndicate can increase the bargaining power of its members in terms of the Shapley value. We determine conditions for the monopoly to be stable: it turns out that the stability of a syndicate critically depends upon the marginal productivity of the factor that its members own.

Under the particular conditions where α, β < 1, α > β, in the presence of increasing returns to scale, we show that workers have never interest that capitalists form a syndicate; whereas capitalists still have interest to form it. Then, it remains to workers to form a syndicate so as to maximize their payoffs.

Under these circumstances it seems that the answer to the question to know whether the presence of a type of syndicate leads to the formation of a syndicate of the other type, is not trivial. Thus, it is true that workers have always interest in forming a syndicate if one exists in capitalist and vice versa. Alongside the capitalists still have an interest in forming a syndicate, in which case, the presence or absence of a labor syndicate cannot be considered as incentive. Finally, starting from a situation of bilateral monopoly, the workers are also willing not to form a syndicate if the capitalists do the same.

13

  denotes the set of labor owners with |I 1 | = n; I 2 , the set of capital owners with |I 2 | = n and N , the set of all factor owners with N = I 1 ∪ I 2 . 2

		1j, . . . , 1n}
	and	I 2 = {21, . . . , 2j, . . . , 2n},
	where I 1

Following Gabszewicz [1996], let us assume an economy in which a single output is produced by an increasing returns to scale production function using only one input, given by F (z) = z

. By assuming that the economy consists of five factor owners and that each

|I1| = |I2| allows us to avoid size effects.

[START_REF] Shapley | Value of Large Market Games-VII: A General Exchange Economy with Money[END_REF] proved that in a k-fold market with transferable utility and concave utility functions, the value converges to the competitive solution.

Given n, the players will share the average productivity y n = n α+β-1 according to the shares that are varying with the type of the market ((I1, I2) = bilateral oligopoly, (I1, ∅) = monopoly syndicate, (∅, I2) = monopsony, (∅, ∅) = competitive economy).

Proposition 2 shows that, under assumption 2, only the monopoly syndicate which has the weaker marginal productivity will be A-stable. In other words, since α and β measure the impact of a variation in inputs on the quantity of output, the syndicate J 2 will be A-stable if the impact on output from the variation of capital inputs is weaker than that of a variation in labor input, i.e. α > β.

We will now consider another definition of stability: C-stability. Without loss of generality, we can compare the situation of a syndicated worker when two monopoly syndicates are present and when there is only a syndicate of capitalists, i.e., ẑ1s to ŷ1 .

Proposition 3 Let ẑ1s be the Shapley value of a syndicated worker under assumption 2 and let ŷ1 (n) be the asymptotic Shapley value of a worker under assumption 3, then:

where ẑ1s = 1 2 n α+β-1 , ∀n, and ŷ1 (n) ∼ α 1 + α n α+β-1 , for n large enough.

Proof. In Proposition 2, we have shown that ẑ1s = 1 2 n α+β-1 when k = n and s 1 = s 2 = 0.

Let us consider the formation of a monopoly syndicate of capitalists. Some easy computations give the asymptotic Shapley value of a worker:

We compare ẑ1s to ŷ1 (n) and we obtain the desired result.

Proposition 3 implies that a syndicate of workers is C-stable when the parameter α is less than one. In other words, this means that the formation of a syndicate of capitalists leads to the formation of the syndicate of workers if the marginal productivity of labor is decreasing. Hence, the formation of a syndicate of one type can lead to the formation of a syndicate of the other type.

We can summarize our results in the following bi-matrix 4 :


Lijun Bo

Ying Jiao

Xuewei Yang

Credit derivatives pricing with default density term structure modelled by Lévy random fields

Keywords: default density, Lévy random field, credit derivatives pricing, parabolic integro-differential equation

We model the term structure of the forward default intensity and the default density by using Lévy random fields, which allow us to consider the credit derivatives with an after-default recovery payment. As applications, we study the pricing of a defaultable bond and represent the pricing kernel as the unique solution of a parabolic integro-differential equation. Finally, we illustrate by numerical examples the impact of the contagious jump risks on the defaultable bond price in our model.

Introduction

The term structure modelling in the interest rate and in the credit risk modelling has been widely adopted and extended since the original paper of Heath-Jarrow-Morton [START_REF] Heath | Bond pricing and the term structure of interest rates[END_REF]. Notably, there have appeared many important papers (e.g. [START_REF] Albeverio | A model of the term structure of interest rates based on Lévy fields[END_REF][START_REF] Björk | Bond market structure in the presence of marked point processes[END_REF][START_REF] Eberlein | Lévy term structure models: no-arbitrage and completeness[END_REF][START_REF] Eberlein | Term structure models driven by general Lévy processes[END_REF][START_REF] Filipović | Existence of Lévy term structure models[END_REF][START_REF] Filipović | Term structure model driven by Wiener process and Poisson measures: existence and positivity[END_REF]) incorporating jump diffusions to describe the family of bond prices or the forward curves as a generalization of the classic HJM model.

In the credit risk modelling, the conditional survival probability associated to the default time is an important quantity for measuring default risk and studying valuation of credit derivatives. Let τ be a nonnegative random variable defined on a complete probability space (Ω, F, P) equipped with a filtration F = (F t ) t 0 satisfying the usual conditions. The conditional survival probability (CSP) is defined as S t (θ) = P(τ > θ|F t ), t, θ 0. To describe the term structure of the CSP, we can use both the density and the intensity point of view. On the one hand, as in El Karoui et al [START_REF] Karoui | What happens after a default: the conditional density approach[END_REF], we assume that there exists a family of F t ⊗ B(R + )-measurable functions (ω, θ) → α t (ω, θ) such that the CSP has the following additive representation:

S t (θ) = ∞ θ α t (v)dv.
(1.1)

The family of random variables α t (•) is called the conditional density of the default time τ given F t . On the other hand, similarly to the definition of forward rate, we can use the "intensity" point of view and the following multiplicative representation:

S t (θ) = exp - θ 0 λ t (v)dv (1.2)
where the F t ⊗B(R + )-measurable function (ω, θ) → λ t (ω, θ) is called the forward intensity. It is equivalent to assume the existence of the density or the intensity for all positive t and θ. We have the relationship:

α t (θ) = S t (θ)λ t (θ). (1.3)
In the interest rate models, the time θ is always larger than t and the forward rate has no economic interpretation for θ < t. However, it is noted in [START_REF] Karoui | What happens after a default: the conditional density approach[END_REF] that to study what happens after a default event, we need the whole term structure of the conditional survival probability, that is, for all positive t and θ. One typical example is a defaultable bond where the recovery payment is effectuated at a given maturity later than the economic default date.

In this paper, we consider the whole term structure modelling of CSP and the applications to the credit derivative pricing. In the credit risk models, the default contagion phenomenon is often modelled by positive jumps in the intensity process. We take this point into modelling consideration and propose a forward intensity driven by Lévy random fields. In the existing Lévy term structure models, in Filipović et al. [START_REF] Filipović | Existence of Lévy term structure models[END_REF][START_REF] Filipović | Term structure model driven by Wiener process and Poisson measures: existence and positivity[END_REF], the authors consider forward curve evolutions as solutions of the infinite dimensional Musiela parametrization first-order hyperbolic stochastic differential equations driven by n-independent Lévy processes or driven by a Wiener process together with an independent Poisson measure. In [START_REF] Eberlein | Term structure models driven by general Lévy processes[END_REF], Eberlein and Raible present a class of bond price models that can be driven by a wide range of Lévy processes with finite exponential moments. This model was further applied to describe the defaultable Lévy term structure and explore ratings and restructuring of the defaultable market. The driving process of the Lévy term structure model in [START_REF] Eberlein | Term structure models driven by general Lévy processes[END_REF] was further extended to non-homogeneous Lévy processes in [START_REF] Eberlein | Lévy term structure models: no-arbitrage and completeness[END_REF].

Motivated by those existing Lévy term structure models and the random field models which are widely used to model various stochastic dynamics (e.g. [START_REF] Albeverio | A model of the term structure of interest rates based on Lévy fields[END_REF][START_REF] Dalang | Extending martingale measure stochastic integral with applications to spatially homogeneous S[END_REF][START_REF] Dalang | The stochastic wave equation in two spatial dimensions[END_REF][START_REF] Dalang | Stochastic integrals for SPDE's: a comparison[END_REF][START_REF] Goldstein | The term structure of interest rates as a random field[END_REF][START_REF] Kennedy | The term structure of interest rates as a Gaussian random field[END_REF][START_REF] Kennedy | Characterizing Gaussian models of the term structure of interest rates[END_REF]), we suppose that the Lévy random field in our model is a combination of a kernel-correlated Gaussian field and an independent (central) Poisson random measure. The jump component described as Poisson measure is similar to that used in [START_REF] Filipović | Term structure model driven by Wiener process and Poisson measures: existence and positivity[END_REF], but it is not necessary to assume the exponential integrability condition for the characteristic measure under our framework (see Section 2). The kernel-correlated Gaussian field is more flexible compared to the Gaussian components without kernel-correlation considered in [START_REF] Goldstein | The term structure of interest rates as a random field[END_REF][START_REF] Kennedy | The term structure of interest rates as a Gaussian random field[END_REF][START_REF] Kennedy | Characterizing Gaussian models of the term structure of interest rates[END_REF]. In fact, we can choose appropriate correlated-kernels of the Gaussian field so that the models considered in [START_REF] Eberlein | Term structure models driven by general Lévy processes[END_REF][START_REF] Filipović | Existence of Lévy term structure models[END_REF][START_REF] Filipović | Term structure model driven by Wiener process and Poisson measures: existence and positivity[END_REF] can be covered (see Remark 2.3). Note that it is not genetically tractable for pricing of defaultable bonds under infinite dimensional framework as in [START_REF] Filipović | Existence of Lévy term structure models[END_REF][START_REF] Filipović | Term structure model driven by Wiener process and Poisson measures: existence and positivity[END_REF]. Although we do not intend to consider the forward intensity under infinite dimensional framework as in [START_REF] Filipović | Existence of Lévy term structure models[END_REF][START_REF] Filipović | Term structure model driven by Wiener process and Poisson measures: existence and positivity[END_REF], it has a close relationship between the (infinite dimensional) Wiener process and the kernel-correlated Gaussian field. Indeed, the kernel-correlated Gaussian field can product a cylindrical Wiener process by establishing appropriate Hilbert spaces (see Proposition 2.5 in [START_REF] Dalang | Stochastic integrals for SPDE's: a comparison[END_REF]). We deduce the dynamics of the CSP and the associated density in this setting. In particular, we emphasize on a martingale condition, which can be viewed as an analogue of the non-arbitrage condition in the classical HJM model.

For the pricing of credit derivatives, we follow the standard general framework in Bielecki and Rutkowski [START_REF] Bielecki | Credit risk: modeling, valuation and hedging[END_REF]. The global market information contains both the default information and the "default-free" market information represented by the filtration F, which is obtained by an enlargement of filtration. We are particularly interested in an economic default case, that is, the default does not lead to the total bankruptcy of the underlying firm and a partial recovery value is repaid at the maturity date of the bond in case of default prior to the maturity. To evaluate this "after-default" payment, we use the density approach in [START_REF] Karoui | What happens after a default: the conditional density approach[END_REF] and obtain that the key quantities for the pricing of a defaultable bond are two pricing kernels, one depending on the interest rate and the default density, and the other depending on additionally the recovery rate. We assume that both the short interest rate and the default density are modelled by the Lévy random field model and are correlated between them. For the recovery rate, we analyze firstly the simple case where the recovery rate is deterministic and then the random recovery case. We show that the pricing kernel is related to the solution of a second-order parabolic integro-differential equation and we prove, based on a result of Garroni and Menaldi [START_REF] Garroni | Green functions for second order parabolic integro-differential problems[END_REF], the existence and the uniqueness of the solution to the equation.

The rest of the paper is organized as follows. We present our model setting in Section 2 and give the martingale condition. We then analyze the dynamics of the CSP and the conditional density in Section 3. In Section 4, we discuss the pricing of credit derivatives and in particular the defaultable zero-coupon bond. The two sections 5 and 6 focus on the pricing kernels. Finally, we present some numerical illustrations in the last section 7.

Forward intensity driven by Lévy random field

In this paper, we adopt a random field point of view to model the forward intensity λ t (θ) where both t and θ are positive. We consider a Lévy random field on R + × R d which is a combination of a Gaussian random field Y G and a compensated Poisson random measure Y P independent to Y G . Here R + denotes the time space and R d is considered as a parameter space.

We assume that the covariance of the Gaussian random field Y G is given by a kernel measure c on R d which has a continuous and symmetric density on R d \ {0} with respect to the Lebesgue measure and such that c({0}) > 0. Namely for

(φ 1 , φ 2 ) ∈ C ∞ 0 (R + × R d ) 2 , E[Y G (φ 1 )Y G (φ 2 )] = ∞ 0 R 2d φ 1 (t, ξ 1 )φ 2 (t, ξ 2 )c(ξ 1 -ξ 2 )dξ 1 dξ 2 dt,
where by abuse of language c(ξ 1 -ξ 2 )dξ 1 dξ 2 denotes the measure on R d × R d the inverse image of the measure c by the mapping from

R d × R d to R d which sends (ξ 1 , ξ 2 ) to ξ 1 -ξ 2 .
The Gaussian random field Y G defines a worthy martingale measure (see [20, p.289] and [6, p.190]). Let F G = (F G t ) t 0 be the filtration satisfying the usual conditions which is generated by

σ(Y G ([0, u] × A), u t, A ∈ B b (R d )), t 0,
where B b (R d ) denotes the set of all bounded Borel subsets of R d . Let P G be the predictable σ-algebra on Ω × R + associated to F G and Φ c be the linear space of all P G ⊗ B(R d )-measurable functions h such that

h c,T := E T 0 R 2d |h(t, ξ 1 )h(t, ξ 2 )|c(ξ 1 -ξ 2 )dξ 1 dξ 2 dt 1/2 < +∞
for any T > 0. The stochastic integral h•Y G is well defined for any h ∈ Φ c . When c is the Dirac distribution concentrated on the origin, the stochastic integral: Let F = (F t ) t 0 be the natural filtration generated by the Lévy random field, namely F := F G ∨ F P . We describe the forward intensity by using the Lévy random filed as the following additive HJM type model:

B(t 0 , . . . , t d ) = Y G ([0, t 0 ] × • • • × [0, t d ]), (t 0 , . . . , t d ) ∈ R d+1 + (2.
dλ t (θ) = µ t (θ)dt + R d σ t (θ, ξ)Y G (dt, dξ) + R d γ t-(θ, ξ)Y P (dt, dξ), (2.2)
where

(1) µ = (µ t (θ); (t, θ) ∈ R 2 + ) is P ⊗ B(R + )-measurable and T 0 E [|µ t (θ)|] dt < ∞,
where P is the predictable σ-algebra on Ω × R + associated to the filtration F,

(2) σ = (σ t (θ, ξ); (t, θ, ξ) ∈ R + × R + × R d ) is P G ⊗ B(R + × R d )-measurable and for any θ ∈ R + , σ • (θ, •) ∈ Φ c , ( 3
) γ = (γ t (θ, ζ) 0; (t, θ, ζ) ∈ R + × R + × R d ) is P P ⊗ B(R + × R d )-measurable and for any θ ∈ R + , γ • (θ, •) ∈ Ψ ν .
The model (2.2) can also be written in the integral form as

λ t (θ) = λ 0 (θ) + t 0 µ s (θ)ds + t 0 R d σ s (θ, ξ)Y G (ds, dξ) + t 0 R d γ s-(θ, ξ)Y P (ds, dξ) (2.3)
where both stochastic integrals with respect to Y G and Y P are F-martingales with mean zero, λ 0 (•) is a deterministic Borel function on R + .

Similarly to the classical HJM model, in the above Lévy field model (2.2), there exists a relationship between the drift coefficient µ and the diffusion coefficients σ and γ due to the fact that, for any θ 0, the conditional survival probability process S t (θ) = exp -θ 0 λ t (v)dv , t 0 should be an F-martingale. We call this relationship the martingale condition (MC). Let us introduce the following notation:

I µ (t, θ) := θ 0 µ t (v)dv, I σ (t, θ, ξ) := θ 0 σ t (v, ξ)dv, and I γ (t, θ, ξ) := θ 0 γ t (v, ξ)dv, where (t, θ, ξ) ∈ R + × R + × R d .
Theorem 2.1 For all θ 0 and T > 0, one has

T 0 E [|I µ (t, θ)|] dt < ∞, I σ (•, θ, •) ∈ Φ c , I γ (•, θ, •) ∈ Ψ ν , and e -Iγ (•,θ,•) -1 ∈ Ψ ν , (2.4)
Moreover, the process family S t (θ) = exp -θ 0 λ t (v)dv , t 0 is a family of F-martingales if and only if the following condition is satisfied:

(MC) ∀ θ 0, µ t (θ) = R 2d σ t (θ, ξ 1 )I σ (t, θ, ξ 2 )c(ξ 1 -ξ 2 )dξ 1 dξ 2 + R d γ t (θ, ξ)(1 -e -Iγ(t,θ,ξ) )ν(dξ).
(2.5)

Proof. The proofs for the first three statements in (2.4) are similar. We only provide the details for the third one. For any T > 0, we have

T 0 R d E|I γ (t, θ, ξ)| 2 ν(dξ)dt = T 0 R d E θ 0 γ t (v, ξ)dv 2 ν(dξ)dt = T 0 R d E θ 0 γ t (v 1 , ξ)dv 1 θ 0 γ t (v 2 , ξ)dv 2 ν(dξ)dt ≤ 1 2 T 0 θ 0 θ 0 R d E |γ t (v 1 , ξ)| 2 + |γ t (v 2 , ξ)| 2 ν(dξ)dv 1 dv 2 dt = θ θ 0 T 0 R d E |γ t (v, ξ)| 2 ν(dξ)dtdv, which is finite since γ • (v, •) ∈ Ψ ν for any v 0.
For the last assertion in (2.4), note that γ t (θ, ξ) 0 and thus e -Iγ(t,θ,ξ) -

1 ≤ |I γ (t, θ, ξ)| , for all (t, θ, ξ) ∈ R + × R + × R d .
We now prove that the condition (MC) is equivalent to the martingale condition for (S t (θ), t 0). In fact

dS t (θ) S t-(θ) = -I µ (t, θ)dt - R d I σ (t, θ, ξ)Y G (dt, dξ) + 1 2 R 2d I σ (t, θ, ξ 1 )I σ (t, θ, ξ 2 )c(ξ 1 -ξ 2 )dξ 1 dξ 2 dt + R d (e -Iγ (t -,θ,ξ) -1)Y P (dt, dξ) + R d (e -Iγ (t,θ,ξ) -1 + I γ (t, θ, ξ))ν(dξ)dt, (2.6)
so the martingale condition of (S t (θ), t 0) is thus equivalent to the following equality

I µ (t, θ) = 1 2 R 2d I σ (t, θ, ξ 1 )I σ (t, θ, ξ 2 )c(ξ 1 -ξ 2 )dξ 1 dξ 2 + R d (e -Iγ (t,θ,ξ) -1 + I γ (t, θ, ξ))ν(dξ), which is equivalent to (MC). 2
Remark 2.2 Consider the particular case where d = 0, c is the Dirac measure, and ν = 0. The condition (MC) becomes

∀ θ 0, µ t (θ) = σ t (θ) θ 0 σ t (v)dv.
This corresponds to the non-arbitrage condition in the classical HJM model where the forward intensity is driven by a standard Brownian motion.

Remark 2.3 There exist random field models in the literature. We make below some comparisons. The forward intensity model (2.2) can be extended to the following form:

dλ t (θ) = µ t (θ)dt + R d σ t (θ, ζ)Y G (dt, dζ) + 0<|ξ| 1 γ t-(θ, ξ)Y P (dt, dξ) + |ξ|>1 γt-(θ, ξ)(Y P (dt, dξ) + ν(dξ)dt), (2.7)
where

σ • (θ, •) ∈ Φ c , γ • (θ, •)1l {|•| 1} and γ• (θ, •)1l {|•|>1} ∈ Ψ ν ,
for each fixed θ 0. Under the model (2.7), the corresponding martingale condition (MC) will be changed accordingly. We next consider a special form of the predictable random filed with separable variables:

σ t (θ, ζ) = σ t (θ) φ(ζ), γ t (θ, ξ) = γt (θ, ξ) = γ t (θ), ξ , ζ ∈ R d , ξ ∈ R d + where ( σ t (θ); (t, θ) ∈ R 2 + ) is a real-valued predictable random field, ( γ t (θ) = ( γ 1 t (θ), . . . , γ d t (θ)); (t, θ) ∈ R 2 + ) is a R d + -valued predictable field and φ(ζ) is a deterministic measurable function on R d .
In this case, the extended model (2.7) can be rewritten as

dλ t (θ) = (µ t (θ) -a)dt + σ t (θ)Y G (dt, φ(⋆)) + γ t (θ), dL t , (2.8)
where a ∈ R, •, • denotes the inner-product on R d and

dL t = adt + 0<|ξ| 1 ξY P (dt, dξ) + |ξ|>1 ξ(Y P (dt, dξ) + ν(dξ)dt) is a non-Gaussian Lévy process if the characteristic measure ν is a Lévy measure. If φ ≡ 1, then Y G (1l [0,t] × φ(⋆)
) becomes a Brownian motion when the correlated-kernel is Dirac. Choose appropriate smooth function φ as in the proof of Proposition 2.5 in [START_REF] Dalang | Stochastic integrals for SPDE's: a comparison[END_REF], then Y G (1l [0,t] × φ(⋆)) becomes a cylindrical Wiener process. Thus we recover the Lévy interest rate term structure models considered in [START_REF] Eberlein | Term structure models driven by general Lévy processes[END_REF][START_REF] Filipović | Existence of Lévy term structure models[END_REF][START_REF] Filipović | Term structure model driven by Wiener process and Poisson measures: existence and positivity[END_REF], if the Lévy measure ν satisfies the exponential integrability condition. We next give a comparison of our Lévy random field Y G + Y P introduced previously in this section with existing Lévy fields in literature.

1. As in (2.1), the field Y G + Y P can be reduced to a Brownian sheet in Walsh [START_REF] Walsh | An introduction to stochastic partial differential equations[END_REF], when the kernel c is Dirac and the characteristic measure ν = 0 (hence Y P = 0);

2. the field Y G + Y P becomes a so-called "colored" space-time white noise model established by [START_REF] Dalang | Extending martingale measure stochastic integral with applications to spatially homogeneous S[END_REF], when the kernel c(ξ) = |ξ| -α with 0 < α < d and ν = 0;

3. the fractional space-time white noise (fractional in space and time in white) used in [START_REF] Nualart | Regularization of quasilinear heat equations by a fractional noise[END_REF] corresponds to the field

Y G + Y P with the kernel c(ξ) = h(2h -1)|ξ| 2h-2 with 1 2 < h < 1, d = 1 and ν = 0; 4. the Poisson sheet in [1] corresponds to the field Y G + Y P + ν(dξ)dt with c = 0 and ν(ξ) = zδ 1 (dξ)
where z > 0 is single point and δ 1 is the Dirac measure concentrated at 1. The Gamma sheet in [START_REF] Albeverio | A model of the term structure of interest rates based on Lévy fields[END_REF] is the field Y G + Y P + ν(dξ)dt with c = 0 and ν(dξ) = e -ξ ξ 1l {ξ>0} zdξ where d = 1 and z > 0 is a single point.

Conditional survival probability and density

In this section, we concentrate on the conditional survival probability (S t (θ), t 0) and the conditional density (α t (θ), t 0). Here we specify a càdlàg version of the martingale (S t (θ), t 0) for any θ 0.

In fact, to show that the integral θ 0 λ t (v)dv defines a càdlàg process, we need a stronger assumption on the process λ(θ) in order to apply Lebesgue's theorem. The càdlàg version of S(θ), if well defined, should have a universal version of its predictable projection as follows:

S t-(θ) = S (p) t (θ) = exp - θ 0 λ t-(v)dv . Thus S t (θ) := lim q∈Q + q↓t exp - θ 0 λ q-(v)dv (3.1)
defines a universal càdlàg version of the martingale S(θ).

We observe from the equality (2.6) that, under the condition (MC), the conditional survival probability admits the following dynamics:

dS t (θ) S t-(θ) = - R d I σ (t, θ, ξ)Y G (dt, dξ) + R d (e -Iγ (t -,θ,ξ) -1)Y P (dt, dξ), (3.2)
where S 0 (θ) = exp(-θ 0 λ 0 (v)dv). For θ 0, we denote by M (θ) the martingale defined as

dM t (θ) = - R d I σ (t, θ, ξ)Y G (dt, dξ) + R d (e -Iγ (t-,θ,ξ) -1)Y P (dt, dξ), M 0 (θ) = 0. (3.3)
With this notation, S(θ)/S 0 (θ) is the Doléans-Dade exponential of the martingale M (θ). Moreover, denote by m(θ) the martingale defined by the dynamics:

dm t (θ) = - R d σ t (θ, ξ)Y G (dt, dξ) - R d γ t-(θ, ξ)e -Iγ (t-,θ,ξ) Y P (dt, dξ), m 0 (θ) = 0. (3.4)
Observe that the following relation holds

M t (θ) = θ 0 m t (u)du.
We then consider the dynamics of the conditional density of default given in (1.1).

Proposition 3.1

The conditional density process α(θ) admits the dynamics:

dα t (θ) = α t-(θ)dM t (θ) -S t-(θ)dm t (θ) (3.5) or equivalently dα t (θ) α t-(θ) = dM t (θ) - 1 λ t-(θ) dm t (θ).
Proof. Keep the martingale condition (MC) in mind. The dynamics (3.5) is derived by employing Itô's formula to α(θ) = λ(θ)S(θ) for each positive θ fixed. 2

An important property in the credit analysis is the immersion property, or the so called (H)-hypothesis, which means that an F-martingale remains a G-martingale. The (H)-hypothesis is satisfied if and only if α t (θ) = α θ (θ) or equivalently λ t (θ) = λ θ (θ) for any t θ. In the random field setting, by (2.3), this is equivalently to

t θ R d σ s (θ, ξ)Y G (ds, dξ) = t θ R d γ s-(θ, ξ)Y P (ds, dξ) = 0
for t θ, or equivalently σ t (θ, ξ) = 0 and γ t (θ, ξ) = 0 ν(dξ)-a.e.

for t > θ. Note that the martingale condition (MC) then implies that µ t (θ) = 0 for t > θ.

We recall that the F-intensity process λ of the default time τ coincides with the diagonal forward intensity, i.e. λ t = λ t (t). It is closely related to the Azéma supermartingale:

S t = S t (t) = P(τ > t | F t ),
which is also called the survival process of τ . Proposition 3.2 Let M be the F-martingale having the dynamics

dM t = - R d I σ (t, t, ξ)Y G (dt, dξ) + R d e -Iγ(t-,t,ξ) -1 Y P (dt, dξ). Then S t = exp - t 0 λ s ds E(M ) t ,
where E(M ) is the Doléans-Dade exponential of M .

Proof. The Azéma supermartingale S has a multiplicative decomposition of the form S t = L t exp(-t 0 λ s ds) (see [START_REF] Karoui | What happens after a default: the conditional density approach[END_REF]Proposition 4.1]), where L is an F-martingale having the following dynamics

dL t = exp t 0 λ s ds d L t , with L t = - t 0 α t (u) -α u (u)du.
By Proposition 3.1, together with (3.3) and (3.4),

d L t = - R d t 0 A(t, θ, ξ)dθY G (dt, dξ) - R d t 0 B(t, θ, ξ)dθY P (dt, dξ), where A(t, θ, ξ) = -α t-(θ)I σ (t, θ, ξ) + S t-(θ)σ t (θ, ξ), B(t, θ, ξ) = α t-(θ)(e -Iγ (t-,θ,ξ) -1) + S t-(θ)γ t-(θ, ξ)e -Iγ (t-,θ,ξ) .
By integration by part, we obtain

- t 0 A(t, θ, ξ)dθ = -S t-(t)I σ (t, t, ξ), - t 0 B(t, θ, ξ)dθ = S t-(t)(e -Iγ (t-,t,ξ) -1).
Moreover, the Doob-Meyer decomposition of S is given by

S t = 1 + L t - t 0 α u (u)du, which implies that dL t L t- = d L t S t- = dS t S t- + λ t dt.
By (3.1), one has S t-= S t-(t). Hence the martingale L is the Doléans-Dade exponential of M and the assertion follows. 2

The pricing of defaultable bonds

In this section, we focus on the pricing of credit derivatives. In general, a credit sensitive contingent claim can be represented by a triplet (C, G, R) (see Bielecki and Rutkowski [START_REF] Bielecki | Credit risk: modeling, valuation and hedging[END_REF]) where the F T -measurable random variable C T represents the maturity payment if no default occurs before the maturity T , and G is an F-adapted continuous process of finite variation such that G 0 = 0 and represents the coupon payment. Differently from the case where the default payment occurs at τ immediately, we assume that in the economic default case, the default (or the recovery) payment takes place, after a period of legal proceedings, at the maturity date T later than the economic default date τ and admits the form R T (τ ) where R T (•) is

F T ⊗ B(R + )-measurable.
The global market information is described by the filtration G = (G t ) t 0 , G t = F t ∨ σ(τ ∧ t), which is made to satisfy the usual conditions. The value at time t T of the contingent claim (C, G, R) is given by the following G t -conditional expectation:

V t = E Q C T 1l {τ >T } + T t 1l {τ >u} e -u t rsds dG s + 1l {τ T } R T (τ ) e -T t rsds G t , (4.1)
where Q denotes a risk-neutral pricing probability measure and the interest rate r = (r t ; t 0) is an Fadapted process. The following result computes V t using F t -conditional expectations. The first two terms result from [START_REF] Bielecki | Credit risk: modeling, valuation and hedging[END_REF] and the third one from [START_REF] Karoui | What happens after a default: the conditional density approach[END_REF]. With an abuse of notation, we denote in the following the Fconditional density of τ under the risk-neutral probability Q by (α t (θ), t 0). The general result on the density under a change of probability measure is given in [10, Theorem 6.1]. Proposition 4.1 We suppose that the economic default time τ admits a conditional density w.r.t. the filtration F, denoted by α t (•) under the risk-neutral probability measure Q. Then the value of the credit sensitive contingent claim (C, G, R) is given by

V t = 1l {τ >t} B t S t E Q C T S T + T t R T (u)α T (u)du B -1 T + T t S u B -1 u dG u F t + 1l {τ t} B t E Q R T (θ) α T (θ) α t (θ) B -1 T |F t θ=τ (4.2)
where

S t = Q(τ > t|F t ) = ∞ t α t (θ)dθ and B t = exp( t 0 r s ds).
Proof. The G t -measurable random variable V t can be decomposed in two parts

V t = 1l {τ >t} V t +1l {τ ≤t} V t (τ ) where V t is F t -measurable and V t (•) is F t ⊗ B(R + )-measurable.
On the set {τ > t}, we use Jeulin-Yor's lemma (see [START_REF] Bielecki | Credit risk: modeling, valuation and hedging[END_REF]) and the conditional density to obtain

V t = 1 S t E Q C T 1l {τ >T } + 1l {t<τ T } R T (τ ) e -T t rsds + T t 1l {τ >u} e -u t rsds dG s F t = 1 S t E Q C T S T + T t R T (θ)α T (θ)dθ e -T t rsds + T t
S u e -u t rsds dG s F t .

On the set {τ t}, by [10, Thm 3.1], we have

V t (τ ) = E Q R T (θ) α T (θ) α t (θ) exp - T t r s ds F t θ=τ ,
which complete the proof. 2

Note that for the pricing of the two "before default" payment terms (C, G), the quantity

S u S t = exp - u t λ s ds E(M ) u E(M ) t , u > t
and hence the intensity λ play an important role. However, for the default recovery payment R (which depends on τ ), the "after-default" density α t (θ) where t θ is needed. This point has been discussed in [START_REF] Karoui | What happens after a default: the conditional density approach[END_REF]. In the following of this paper, we adopt the density approach for both the before-default and afterdefault pricing.

We consider in particular a defaultable zero-coupon bond of maturity T with C = 1 and G = 0. Its price at t T is given by

P (t, T ) = E Q 1l {τ >T } + 1l {τ T } R T (τ ) exp - T t r s ds G t . (4.3)
The following result is a direct consequence of the previous proposition. We first introduce the following price kernels:

K 1 (t, θ) = 1 S t E Q α T (θ) exp - T t r s ds F t (4.4) K 2 (t, θ) = 1 α t (θ) E Q R T (θ)α T (θ) exp - T t r s ds F t , (4.5)
where t T , and θ 0.

Corollary 4.2

Using the conditional density of τ under Q, the price (4.3) of the defaultable zero-coupon bond at time t T has the following representation:

P (t, T ) = 1l {τ >t} ∞ T K 1 (t, θ)dθ + T t K 2 (t, θ) α t (θ) S t dθ + 1l {τ t} K 2 (t, τ ). (4.6)
We will identify the above price kernels in the next two sections with different settings.

The first pricing kernel

In this section, we study in detail the pricing kernels (4.4) and (4.5) when the random interest rate is described as an extended Vasicek model. We suppose in this section the after-default recovery payment is deterministic. The case where the after-default recovery payment is random will be considered in the next section.

Firstly we recall the forward intensity model (2.2) and assume that the F-predictable random fields (µ, σ, γ) are deterministic in (2.2). We then express the instantaneous interest rate process r = (r t , t ≥ 0) as the following extended Vasicek model under the risk-neutral pricing measure Q :

dr t = κ(δ -r t )dt + R d ρ t (ξ)Y G (dt, dξ) + R d φ t (ξ)Y P (dt, dξ), (5.1)
where κ > 0, δ > 0, and ρ • (•) and φ • (•) are deterministic volatility functions, assumed to belong to Φ c and Ψ ν respectively. In the particular case where d = 0, φ t (ξ) ≡ 0 and the volatility function ρ • (•) ≡ ρ > 0 is constant, the interest rate r satisfies the classical Brownian-driven Vasicek model:

dr t = κ(δ -r t )dt + ρdW t , (5.2)
where W is a standard Brownian motion. Similarly to the solution form of the Ornstein-Uhlenbeck stochastic differential equation, the extended Vasicek model (5.1) also admits an explicit expression as follows:

r t = r 0 e -κt + δ(1 -e -κt ) + t 0 R d e -κ(t-u) ρ u (ξ)Y G (du, dξ) + t 0 R d e -κ(t-u) φ u (ξ)Y P (du, dξ), (5.3)
where r 0 > 0 denotes the deterministic initial interest rate value.

Next we compute the first pricing kernel in (4.4). For θ 0, we introduce the following integrodifferential operator A θ acting on functions with three variables t, x and y which are differential in t and second-order differentiable in (x, y) :

A θ K(t, x, y) = κ( δ t (θ) -x) ∂K ∂x (t
where

δ t (θ) = δ + κ -1 R 2d ρ t (ξ)I σ (t, θ, ξ)c(ζ -ξ)dζdξ + κ -1 R d φ t (ξ)(e -Iγ (t,θ,ξ) -1)ν(dξ), a(t, θ) = µ t (θ) - R d σ t (θ, ζ)I σ (t, θ, ξ)c(ζ -ξ)dζdξ - R d γ t -(θ, ζ)(1 -e -Iγ (t,θ,ξ) )ν(dξ), a 11 (t) = 1 2 R 2d ρ t (ξ 1 )ρ t (ξ 2 )c(ξ 1 -ξ 2 )dξ 1 dξ 2 , a 22 (t, θ) = 1 2 R 2d σ t (θ, ξ 1 )σ t (θ, ξ 2 )c(ξ 1 -ξ 2 )dξ 1 dξ 2 , a 12 (t, θ) = R 2d σ t (θ, ξ 1 )ρ t (ξ 2 )c(ξ 1 -ξ 2 )dξ 1 dξ 2 .
Remark 5.1 Recall the martingale condition (MC) given by (2.5) which has been assumed throughout the paper. We have the coefficient a(t, θ) = 0 for the partial derivative ∂K ∂y under (MC). We introduce the following assumptions where we have fixed θ 0.

Assumption 5.2 (1)

There exists q ∈ (0, 1) such that (i) the functions a 11 (•), a 22 (•, θ) and a 12 (•, θ) are q 2 -Lipschitz on [0, T ], (ii) there exists a Borel function J q on R d (which could depend on θ) such that

max |φ t (ξ) -φ s (ξ)|, |γ t (θ, ξ) -γ s (θ, ξ)| ≤ J q (ξ)|t -s| q/2 and R d J q (ξ) 2 1 + J q (ξ) ν(dξ) < ∞.
(2) |φ t (ξ)| and |γ t (θ, ξ)| are uniformly bounded from above by a Borel function J 0 (ξ) such that

R d J 0 (ξ) 2 1 + J 0 (ξ)
ν(dξ) < +∞.

(3) There exists a constant β(θ) > 0 such that, for any (x, y) ∈ R 2 and any t ∈ [0, T ], one has a 11 (t)x 2 + 2a 12 (t, θ)xy + a 22 (t, θ)y 2 β(θ)(x 2 + y 2 ).

Then we have the main result of this section.

Theorem 5.3 Let θ 0 be fixed. Under Assumption 5.2, the Cauchy problem

∂K ∂t (t, x, y) -xK(t, x, y) + A θ K(t, x, y) = 0, K(T, x, y) = y (5.5)
has a unique solution K, where the integro-differential operator A θ is defined in (5.4). Moreover, the following equality holds

E Q α T (θ) exp - T t r s ds F t = S t (θ) K(t, r t , λ t (θ)), (5.6)
where S t (θ) = Q(τ > θ|F t ) is CSP and λ t (θ) is the corresponding forward intensity under the pricing measure Q.

Proof. The first assertion comes from a general result of Garroni and Menaldi [14, Theorem II.3.1]. Let q be as in Assumption 5.2. We shall actually prove that the Cauchy problem (5.5) with a terminal condition 1 K(T,

•, •) = ψ ∈ C q (R 2
) has a unique solution in the Hölder space C 1+ q 2 ,2+q ([0, T ] × R 2 ) by constructing a contractible operator. The case of (5.5) with unbounded terminal function ϕ(t, x, y) = y will be treated by taking limits. We recall that C 1+ q 2 ,2+q ([0, T ] × R 2 ) denotes the vector subspace of 2 C 1,2 ([0, T ] × R 2 ) of functions f such that

f 1+ q 2 ,2+q := f 1,2 + 1≤a+b+2c≤2 ∂ c t ∂ a x ∂y b t, 1 2 (q+a+b+2c-1) + a+b+2c=2 ∂ c t ∂ a x ∂y b (x,y),q < +∞,
where for any function g : [0, T ] × R 2 → R and any β ∈ (0, 1),

g t,β := sup z∈R 2 sup s,t∈[0,T ] s =t |g(s, z) -g(t, z)| |s -t| β , g (x,y),β := t∈[0,T ] sup z,w∈R 2 z =w |g(t, z) -g(t, w)| |z -w| β .
1 The expression C q (R 2 ) denotes the vector space of all bounded functions f on R 2 which are Hölder continuous of order q (namely, such that f sup + f q < +∞), where

f q := sup z,w∈R 2 z =w |h(z) -h(w)| |z -w| q . 2 The expression C 1,2 ([0, T ] × R 2 ) denotes the vector space of all continuous functions f on [0, T ] × R 2 such that f 1,2 := a+b+2c≤2 ∂ c t ∂ a x ∂ b y f sup < +∞.
The vector space C 1,2 ([0, T ] × R 2 ) together with the norm • 1,2 form a Banach space.

The vector space C 1+ q 2 ,2+q ([0, T ] × R 2 ) together with the norm • 1+ q 2 ,2+q form a Banach space. Let I θ be the integro-differential operator defined as

(I θ K)(t, x, y) = R d K(t, x + φ t (ξ), y + γ t (θ, ξ)) -K(t, x, y) -φ t (ξ) ∂K ∂x -γ t (θ, ξ) ∂K ∂y ν(dξ).
For K ∈ C 1+ q 2 ,2+q ([0, T ] × R 2 ) and ψ ∈ C q (R 2 ), let Θ ψ (K) be the unique solution of the Cauchy problem

∂F ∂t -xF + A θ (F ) = I θ (K), F (T, x, y) = ψ(x, y), (5.7)
where A θ denotes the differential operator

a 11 (t) ∂ 2 ∂x 2 + a 12 (t, θ) ∂ 2 ∂x∂y + a 22 (t, θ) ∂ 2 ∂y 2 + κ( δ t (θ) -x) ∂ ∂x + a(t, θ) ∂ ∂y .
Denote by

C q 2 ,q ([0, T ] × R 2 ) the vector space of functions f on [0, T ] × R 2 such that f q 2 ,q := f sup + sup (x,y)∈R 2 f (•, x, y) q 2 + sup t∈[0,T ] f (t, •, •) q < +∞
which is a Banach space with respect to the norm

• q 2 ,q . Since K ∈ C 1+ q 2 ,2+q ([0, T ] × R 2 )
, by the Assumption 5.2 (1.ii) and ( 2), we obtain that [START_REF] Garroni | Green functions for second order parabolic integro-differential problems[END_REF]Lemma II.1.5]). Therefore the existence and uniqueness of the solution Θ ψ (K) ∈ C 1+ q 2 ,2+q ([0, T ] × R 2 ) to (5.4) comes from the classical theory of parabolic partial differential equations (e.g. [START_REF] Friedman | Partial differential equations[END_REF]). Moreover, the solution verifies the following Hölder estimate ([14, Theorem I.2.1])

I θ (K) ∈ C q 2 ,q ([0, T ]×R 2 ) (see
Θ ψ (K 1 ) -Θ ψ (K 2 ) 1+ q 2 ,2+q C 1 I θ (K 1 -K 2 ) q 2 ,q (5.8)
which holds for all

K 1 , K 2 ∈ C 1+ q 2 ,2+q ([0, T ] × R 2 ) such that K 1 (T, •, •) = K 2 (T, •, •) = ψ, where C 1 is a constant independent of ψ.
For arbitrary ε > 0, the following estimate holds for any

K ∈ C 1+ q 2 ,2+q ([0, T ] × R 2 ) (see [14, Lemma II.1.5]) I θ (K) q 2 ,q ε ∇ 2 (x,y) K q 2 ,q + C(ε) K q 2 ,q + ∇ (x,y) (K) q 2 ,q , (5.9)
where the constant C(ε) only depends on ε. Denote by C q ψ the subset of functions in C 1+ q 2 ,2+q ([0, T ] × R 2 ) whose restriction on {T } × R 2 coincides with ψ. By choosing ε > 0 small enough, we obtain from (5.8) and (5.9) that Θ ψ is a contracting operator on the complete metric space C q ψ , provided that T is sufficiently small. Hence for sufficiently small T , the operator Θ ψ has a unique fixed point and therefore the Cauchy problem ∂K ∂t (t, x, y) -xK(t, x, y)

+ A θ K(t, x, y) = 0, K(T, x, y) = ψ(x, y)
has a unique solution. For general T , it suffices to divide [0, T ] into a finite union of small intervals and resolve the Cauchy problem progressively.

For the terminal function ϕ(x, y) = y, we can take, for each integer n ≥ 1, a function ψ n ∈ C ∞ 0 (R 2 ) which coincides with ϕ on the ball B n of radius n centered at (0, 0). For any n 1, let K n be the unique solution of the equation (5.5) with terminal condition K n (T, x, y) = ψ n (x, y). By a maximum principle for the equation (5.5) (see [START_REF] Garroni | Green functions for second order parabolic integro-differential problems[END_REF]Theorem II.2.15]), for n m, K n coincides with K m on the ball B m . By taking K = K n on [0, T ] × B n we obtain a global solution to the Cauchy problem (5.5). The uniqueness of K also results from the maximum principle.

We now prove the second assertion that K satisfies the equality (5.6). To this end, we compute the denominator of the pricing kernel (4.4) by introducing a change of probability measure:

dQ θ dQ Ft = S t (θ) S 0 (θ) . (5.10)
By Bayes' formula and (1.

3), we have

E Q α T (θ) exp - T t r s ds F t = S t (θ)E Q θ λ T (θ) exp - T t r s ds F t .
Note that, by Girsanov's theorem (see [START_REF] Björk | Bond market structure in the presence of marked point processes[END_REF]Theorem 3.3]), under the probability measure

Q θ , Y G (dt, dξ) := Y G (dt, dξ) + R d I σ (t, θ, ξ)c(ζ -ξ)dζ dξdt
defines a Gaussian field with correlated kernel c on R d , and

Y P (dt, dξ) := Y P (dt, dξ) + (1 -e -Iγ (t,θ,ξ) )ν(dξ)dt
defines a compensated Poisson random measure with predictable compensator e -Iγ(t,θ,ξ) ν(dξ)dt. Then the dynamics (5.1) of the interest rate r can be rewritten as

dr t = κ( δ t (θ) -r t )dt + R d ρ t (ξ) Y G (dt, dξ) + R d φ t (ξ) Y P (dt, dξ),
where

δ t (θ) = δ + κ -1 R 2d ρ t (ξ)I σ (t, θ, ξ)c(ζ -ξ)dζdξ + κ -1 R d φ t (ξ)(e -Iγ (t,θ,ξ) -1)ν(dξ),
and the dynamics (2.2) of the forward intensity rate can be rewritten as

dλ t (θ) = µ t (θ)dt + R d σ t (θ, ξ) Y G (dt, dξ) + R d γ t-(θ, ξ) Y P (dt, dξ)
where

µ t (θ) = µ t (θ) - R d σ t (θ, ζ)I σ (t, θ, ξ)c(ζ -ξ)dζdξ - R d γ t-(θ, ξ)(1 -e -Iγ(t,θ,ξ) )ν(dξ).
Note that the forward intensity process λ(θ) is a (Q θ , F)-martingale for each θ fixed. Assume that

E Q θ λ T (θ) exp - T t r s ds F t = K(t, r t , λ t (θ)),
where the function K(t, x, y) is sufficiently regular. Then Itô's formula applied to the Conversely, if K is the solution to

(Q θ , F)-martingale exp - t 0 r s ds K(t, r t , λ t (θ)) yields -r t K(t,
∂K ∂t -xK + A θ K = 0, K(T, x, y) = y,
then one has

E Q α T (θ) exp - T t r s ds F t = S t (θ) K(t, r t , λ t (θ)).
Thus we complete the proof of the theorem. 2

Accordingly the pricing kernels (4.4) and (4.5) are given by

K 1 (t, θ) = S t (θ) S t K(t, r t , λ t (θ)) (5.11) K 2 (t, θ) = S t (θ) α t (θ) R T (θ) K(t, r t , λ t (θ)) (5.12)
where t T and θ 0. By Corollary 4.2, we obtain immediately the pricing formula for the defaultable zero-coupon bond.

Remark 5. [START_REF] Björk | Bond market structure in the presence of marked point processes[END_REF] Concerning the pricing kernel at the left side of the equality (5.6), one possible alternative way is to solve it directly by using the dynamics of the density α t (θ). However, in view of (3.5), the corresponding solution K(t, r t , S t (θ), λ t (θ)) will include three variables apart from time variable. The main advantage of the change of probability method (5.10) is that we obtain the solution function in the form K(t, r t , S t (θ), λ t (θ)) = S t (θ) Ǩ(t, r t , λ t (θ)). This indeed decreases the dimension of variables for our pricing kernel function and is important in the numerical computation. Remark 5.5 If the interest rate r is independent of the forward intensity, hence independent of the density, then the computation of the pricing kernels is easier. Denote by B(t, T ) the price of the standard zerocoupon bond, i.e. B(t, T ) = E Q [exp(-T t r s ds)|F t ]. Recall that we have assumed the recovery rate deterministic in this section. Then

K 1 (t, θ) = α t (θ)B(t, T ) S t , K 2 (t, θ) = R T (θ)B(t, T )
which implies that the time-t value (4.6) of defaultable zero-coupon bond has the following representation: .13) This quantity serves to measure the default risk including both the default probability and the loss given default. We also notice in (5.13) that the recovery corresponds to a "recovery of face value" since it can be written as the quotient between the defaultable bond and an equivalent default-free bond.

P (t, T ) B(t, T ) = 1l {τ >t} 1 - T t (1 -R T (θ))α t (θ)dθ S t + 1l {τ t} R T (τ ). ( 5
Remark 5. [START_REF] Dalang | The stochastic wave equation in two spatial dimensions[END_REF] The zero coupon price B(t, T ) := E Q exp -T t r s ds F t can be given in the form K(t, r t ) where K(•, •) is the unique solution to the following integro-differential equation:

-xK + ∂K ∂t + κ(δ -x) ∂K ∂x + a 11 (t) ∂ 2 K ∂x 2 + R d K(t, x + φ t (ξ)) -K(t, x) -φ t (ξ) ∂K ∂x (t, x) ν(dξ) = 0
with the terminal condition K(T, x) = 1. If there is no jumps in r t , i.e., φ t (ξ) ≡ 0, then the above equation becomes

-xK + ∂K ∂t + κ(δ -x) ∂K ∂x + a 11 (t) ∂ 2 K ∂x 2 = 0. Its unique solution is K(t, x) = exp 1 -e -κ(T -t) κ (δ -x) -δ(T -t) + T t a 11 (u) 1 -e -κ(T -u) κ 2 2 du ,
where a 11 (t) is given in (5.4). Thus we obtain the following equality B(t, T ) = K(t, r t ), which is similar to the classical case.

Random recovery rate and the second pricing kernel

In this section, we consider the general case for the pricing kernel (4.5), where the after-default recovery payment is random as an extension to the previous section. Bakshi et al. [START_REF] Bakshi | Understanding the role of recovery in default risk models: empirical comparisons and implied recovery rates[END_REF] assumed that the recovery rate is related to the underlying intensity as the following form: R t = w 0 + w 1 e -λt , w 0 , w 1 0, w 0 + w 1 1 and λ is the intensity process of default. In a similar manner, we assume that R T (θ) is of the form

R T (θ) = w 0 + w 1 e -f (λ T (θ)) , θ 0 (6.1)
where λ T (θ) is the forward intensity implied by (1.2) under the pricing measure Q, w 0 , w 1 satisfy the same condition as above and f is a non-negative function which is locally Hölder continuous of positive order. Proposition 6.1 Let θ 0 be fixed. Under the Assumption 5.2, the pricing kernel (4.5) is given by

K 2 (t, θ) = w 0 λ t (θ) K(t, r t , λ t (θ)) + w 1 λ t (θ) K(t, r t , λ t (θ)), (6.2)
where K and K are respectively solutions to the partial integro-differential equation:

∂K ∂t (t, x, y) -xK(t, x, y) + A θ K(t, x, y) = 0 (6.3)
under the terminal conditions K(T, x, y) = y and K(T, x, y) = ye -f (y) .

Proof. Similarly to Theorem 5.5, the equation (6.3) with the terminal condition K(T, x, y) = ye -f (y) admits a unique solution K. Moreover, by a change of probability measure we obtain

E Q α T (θ)e -f (λ T (θ)) exp - T t r s ds F t = S t (θ) K(t, r t , λ t (θ)).
Hence the formula (6.2) follows from the following relation (see (4.4), (4.5) and (6.1)) :

K 2 (t, θ) = w 0 S t α t (θ) K 1 (t, θ) + w 1 α t (θ) E α T (θ)e -f (λ T (θ)) exp - T t r s ds F t ,
where K 1 (t, θ) is the first price kernel (4.4).

Numerical illustrations

In this section, we illustrate our previous results by numerical examples. We are particularly interested in the contagion phenomenon. More precisely, we shall analyze in detail the the jump part in the default density dynamics and its impact on the defaultable bond pricing.

In the numerical example, we consider the dynamics of the default density described by (3.5) and we let the martingale m(θ) be given by

dm t (θ) = -σ t (θ)dW t + R + γ t-(θ)ξe -ξ θ 0 γ t-(v)dv Y P (dt, dξ), m 0 (θ) = 0, (7.1)
with W = (W t ; t 0) being a standard Brownian motion independent of the Poisson measure Y P . Compared with (3.4), the corrected kernel c of the Gaussian field Y G is the Dirac measure and d = 1, the volatility coefficient σ t (θ, ξ) = σ t (θ) does not depend on ξ and the jump amplitude coefficient is given by γ

t (θ, ξ) = γ t (θ)ξ1l {ξ>0} where γ t (θ) > 0. Recall in addition that M t (θ) = θ 0 m t (u)du and dα t (θ) = α t-(θ)dM t (θ) -S t-(θ)dm t (θ).
To illustrate the impact of the jump part on the defaultable bond price P (t, T ) given by (4.6), we first consider the case when the martingale m(θ) has no jumps, i.e., γ = 0. We then include the jump part in the density dynamics. We use the initial default density given by α 0 (θ) = λe -λθ with λ being a positive constant.

In the coming tests, we suppose that σ t (θ) and γ t (θ) are deterministic and we use the following forms of the coefficients and the characteristic measure in (7.1),

     σ t (θ) = σ(θ -t) + , σ > 0, γ t (θ) = b(θ -t) + , b > 0, ν(dξ) = ζ ̟ e -ξ/̟ 1l {ξ>0} dξ, ζ > 0, ̟ > 0.
We assume that both the recovery rate R ∈ [0, 1] and the interest rate r are constants and define B(t, T ) = e -r(T -t) for 0 t T . By Remark 5.5, the defaultable bond price P (t, T ) given by (4.6) admits an explicit form. Since the quotient P (t, T )/B(t, T ) equals the constant R on the set {τ t} in this case, we only study the pre-default part on {τ > t} in (5.13), which is denoted by P (t, T ) henceforth and is given by

P (t, T ) = B(t, T ) 1 -(1 -R) T t α t (θ)dθ ∞ t α t (θ)dθ . (7.2)
The main task is then to approximate the integral ∞ t α t (θ)dθ by a finite sum N/∆ i=t/∆+1 ∆ * α t (i * ∆). Here we choose ∆ = 1/100 and N = 10/λ. We perform 10 4 experiments to compute the F t -measurable random variable P (t, T ). In each experiment, we first generate the underlying Brownian motion and the central compound Poisson process. Then for each θ ∈ {i∆; i = 1, 2, . . . , N/∆}, we compute α t i (θ) on {t i = i∆ t ; i = 1, 2, . . . , t/∆ t } with ∆ t = 1/100.

The preferred parameter values are as follows: f h (x -P i (t, T )), (7.3) where P i (t, T ) is the price obtained in the i-th experiment, f h (x) = 1 √ 2πh exp -x 2 2h 2 , and h = 1.06s k k -1/5 is the bandwidth, with s k being the sample standard deviation. From Figure 1, we find that the existence of the jump risk will increases the decentrality of the price. The right tail of the price distribution becomes fatter and fatter as the mean jump size ̟ increasing. Figure 2 shows the mean of the price P e (0.5, 1) := E[P (0.5, 1)] as a function of ̟ for different values of λ. We observe that the defaultable bond price is a decreasing function of the intensity λ, and also of the mean jump size ̟. Hence, when there is larger default risk of the underlying asset itself (with larger λ), the corresponding bond price is smaller. Furthermore, when there is more significant counterparty risks, that is, when there is a larger contagious jump in the density (larger ̟), then the bond price will also decrease. Both observations correspond to the reality on the market.

Figure 3 shows the mean of the price P e (t, 1) := E[P (t, 1)] as a function of t. It is noted that the numerical illustration of the quantity P (t, T )/B(t, T ) discussed in Remark 5.5 is very similar to that of P (t, T ), since B(t, T ) here is a deterministic function B(t, T ) = e -r(T -t) which is close to 1. We observe similar results as in the previous test: the counterparty jump risks in the density will decrease the bond prices. In the last graph, we show the quoted bond price at the initial time t = 0 as a function of the maturity time T for different values of intensities. Again we observe that the bond price is decreasing when there is larger default risks and for long term bonds.

t = 0. 5 ,

 5 T = 1, r = 0.05, R = 0.4, b = 1, ζ = 10, λ = 0.1.

Figure 1

 1 Figure 1 plots the kernel estimations of the densities of P (t, T ) given by

Figure 1 :

 1 Figure 1: The (normal) kernel estimations of the price densities for ̟ = 0, 0.0002, 0.0006, 0.001, 0.002 and σ = 0.001.

Figure 2 :

 2 Figure 2: P e (0.5, 1) := E[P (0.5, 1)] as a function of ̟ with λ = 0.01, 0.03, 0.1, 0.3 and σ = 0.001.

  0

Figure 3 :

 3 Figure 3: P e (t, 1) := E[P (t, 1)] as a function of t. Right hand side is the relative price P e (t, 1) := E[P (t, 1)]/B(t, 1).

Figure 4 :

 4 Figure 4: P (0, T ) as a function of T .

  P be the predictable σ-algebra on Ω × R + associated to F P . Denote by Ψ ν the linear space of all P P ⊗ B(R d )-measurable functions such that The stochastic integral g • Y P is well defined for any g ∈ Ψ ν .

	and P g ν,T := E	0	T	R d	|g(t, ξ)| 2 ν(dξ)dt < +∞
	for any T > 0.
					1)
	defines a (d + 1)-parameter Brownian sheet. If in particular d = 0, it becomes a standard Brownian motion.
	Denote the intensity measure of the compensated Poisson field Y P by ν(dξ)dt, (t, ξ) ∈ R + × R d , where ν is a σ-finite measure on R d . Let F P = (F P t ) t 0 be the filtration satisfying the usual conditions generated
	by
	σ(Y P ([0, u] × A), u t, A ∈ B b (R d )), t 0

  r t , λ t (θ)) + ∂K ∂t (t, r t , λ t (θ)) + κ( δ t (θ) -r t )

						∂K ∂x	(t, r t , λ t (θ)) + µ t (θ)	∂K ∂y	(t, r t , λ t (θ))
	+ a 11 (t)	∂ 2 K ∂x 2 (t, r t , λ t (θ)) + a 22 (t, θ)	∂ 2 K ∂y 2 (t, r t , λ t (θ)) + a 12 (t, θ)	∂ 2 K ∂x∂y	(t, r t , λ t (θ))
	+	R d	K(t, r t + φ t (ξ), λ t (θ) + γ t (θ, ξ)) -K(t, r t , λ t (θ))
			-φ t (ξ)	∂K ∂x	(t, r t , λ t (θ)) -γ t (θ, ξ)	∂K ∂y	(t, r t , λ t (θ)) ν(dξ) = 0.

2

  Corollary 6.2 Under the Assumption 5.2, the price of the defaultable zero-coupon bond is given by K(t, r t , λ t (θ)) + w 1 K(t, r t , λ t (θ)) dθ+ 1l {τ t} 1 λ t (τ ) w 0 K(t, r t , λ t (τ )) + w 1 K(t, r t , λ t (τ )) ,where K and K are given in Proposition 6.1 respectively.

	P (t, T ) =1l {τ >t}	∞ T	S t (θ) S t	K(t, r t , λ t (θ))dθ +	t	T	S t (θ) S t	w 0

Funding: NSF of China (No. 11101223)