BSDEs with Default Jump

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BSDEs with Default Jump

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(eds.) Computation and Combinatorics in Dynamics, Stochastics and Control: The Abel

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BSDEs with default jump

Roxana Dumitrescu, Miryana Grigorova, Marie-Claire Quenez, Agn s Sulem

To cite this version:

BSDEs with default jump

Roxana Dumitrescu, Miryana Grigorova, Marie-Claire Quenez, Agn`es Sulem

Abstract We study (nonlinear) Backward Stochastic Differential Equations (BSDEs) driven by a Brownian motion and a martingale attached to a default jump with intensity process λ = (λt).

The driver of the BSDEs can be of a generalized form involving a singularoptionalfinite variation process. In particular, we provide a comparison theorem and a strict comparison theorem. In the special case of a generalized λ-linear driver, we show an explicit representation of the solution, involving conditional expectation and an adjoint exponential semimartingale; for this representation, we distinguish the case where the singular component of the driver is predictable and the case where it is only optional. We apply our results to the problem of (nonlinear) pricing of European contingent claims in an imperfect market with default. We also study the case of claims generating intermediate cashflows, in particular at the default time, which are modeled by a singularoptionalprocess. We give an illustrating example when the seller of the European option is a large investor whose portfolio strategy can influence the probability of default.

1 Introduction

The aim of the present paper is to study BSDEs driven by a Brownian motion and a compensated default jump process with intensity processλ= (λt). The applications we have in mind are the

pric-ing and hedgpric-ing of contpric-ingent claims in an imperfect financial market with default. The theory of BSDEs driven by a Brownian motion and a Poisson random measure has been developed extensively by several authors (cf., e.g., Barles, Buckdahn and Pardoux [2], Royer [22], Quenez and Sulem [21], Delong [10]). Several of the arguments used in the present paper are similar to those used in the previous literature. Nevertheless, it should be noted that BSDEs with a default jump do not corre-spond to a particular case of BSDEs with Poisson random measure. The treatment of BSDEs with a default jump requires some specific arguments and we present here a complete analysis of these

Roxana Dumitrescu

King’s College London, Department of Mathematics, Strand, London, WC2R 2LS, United Kingdom, e-mail: [email protected]

Miryana Grigorova

Centre for Mathematical Economics, University Bielefeld, PO Box 10 01 31, 33501 Bielefeld, Germany, e-mail: miryana [email protected]

M. Grigorova acknowledges financial support from the German Science Foundation through SFB1283.

Marie-Claire Quenez

Laboratoire de probabilit´es, statistiques et mod´elisations (CNRS/Sorbonne Universit´e/Universit´e Paris Diderot), Uni-versit´e Paris 7, 75013 Paris, France, e-mail: [email protected]

Agn`es Sulem

INRIA Paris, 2 rue Simone Iff, CS 42112, 75589 Paris Cedex 12, France, and Universit´e Paris-Est, e-mail: [email protected]

BSDEs, which is particularly useful in default risk modeling in finance. To our knowledge, there are few works on nonlinear BSDEs with default jump. The papers [6] and [1] are concerned only with the existence and the uniqueness of the solution, which are established under different assumptions. In this paper, we first provide some a priori estimates, from which the existence and uniqueness result directly follows. Moreover, we allow the driver of the BSDEs to have a singular component, in the sense that the driver is allowed to be of the generalized formg(t,y,z,k)dt+dDt, whereD

is an optional (not necessarily predictable) right-continuous left-limited (RCLL) process with finite variation. We stress that the case of a singularoptionalprocessDhas not been considered in the literature on BSDEs, even when the filtration is associated with a Brownian motion and a Poisson random measure. Moreover, these BSDEs are useful to study the nonlinear pricing problem in im-perfect markets with default. Indeed, in this type of markets, the contingent claims often generate intermediate cashflows - in particular at the default time - which can be modeled via an optional singular processD(see e.g. [4, 3, 8, 7]). We introduce the definition of aλ-lineardriver, whereλ

refers to the intensity of the jump process, which generalizes the notion of a linear driver given in the literature on BSDEs to the case of BSDEs with default jump andgeneralized driver. Whengisλ -linear, we provide an explicit solution of the BSDE associated with thegeneralizedλ-linear driver g(t,y,z,k)dt+dDtin terms of a conditional expectation and an adjoint exponential semimartingale.

We note that this representation formula depends on whether the singular processDis predictable or just optional. Under some suitable assumptions ong, we establish a comparison theorem, as well as a strict comparison theorem. We emphasize that these comparison results are shown for optional (not necessarily) predictable singular processes, which requires some specific arguments. We then give an application in mathematical finance. We consider a financial market with a defaultable risky asset and we study the problems of pricing and hedging of a European option paying a payoffξ at maturity

T and intermediate dividends (or cashflows) modeled by a singular processD. The option possibly generates a cashflow at the default time, which implies that the dividend processDis not necessarily predictable. We study the case of a market with imperfections which are expressed via the nonlin-earity of the wealth dynamics. Our framework includes the case of different borrowing and lending treasury rates (see e.g. [17] and [8]) and ”repo rates”1_{, which is usual for contracts with intermediate} dividends subjected to default (see [7]). We show that the price of the option is given byX_·g_,T(ξ,D),

whereX_·g_,T(ξ,D)is the solution of the nonlinear BSDE with default jump (solved under the primi-tive probability measureP) withgeneralized driver g(t,y,z,k)dt+dDt, terminal timeTand terminal conditionξ. This leads to a non linear pricing system(ξ,D)7→X_·g_,T(ξ,D), for which we establish some properties. We emphasize that the monotonicity property (resp. no arbitrage property) requires some specific assumptions on the driverg, which are due to the presence of the default. Furthermore, for each drivergand each (fixed) singular processD, we define the(g,D)-conditional expectation byEg,D

t,T(ξ):=X

g

t,T(ξ,D), forξ ∈L2(GT). In the case whereD=0, it reduces to theg-conditional

expectationEg _{(in the case of default). We also introduce the notion of}Eg,D_{-martingale, which is a}

useful tool in the study of nonlinear pricing problems: more specifically, those of American options and game options with intermediate dividends (cf. [13], [14]).

The paper is organized as follows: in Section 2, we present the properties of BSDEs with default jump andgeneralized driver. More precisely, in Section 2.1, we present the mathematical setup. In Section 2.2, we state some a priori estimates, from which we derive the existence and the uniqueness of the solution. In Section 2.3, we show the representation property of the solution of the BSDE associated with thegeneralized driver g(t,y,z,k)dt+dDt in the particular case whengisλ-linear.

We distinguish the two cases: the case when the singular processDis predictable and the case when it is just optional. In Section 2.4, we establish the comparison theorem and the strict comparison theorem. Section 3 is devoted to the application to the nonlinear pricing of European options with dividends in an imperfect market with default. The properties of the nonlinear pricing system as well as those of the(g,D)-conditional expectation are also studied in this section. As an illustrative example of market imperfections, we consider the case when the seller of the option is a large investor whose hedging strategy (in particular the cost of this strategy) has impact on the default probability.

2 BSDEs with default jump

2.1 Probability setup

Let(Ω,G_,P_{)be a complete probability space equipped with two stochastic processes: a} unidimen-sional standard Brownian motionW and a jump processNdefined byNt=1ϑ≤t for anyt∈[0,T],

whereϑ is a random variable which models a default time. We assume that this default can appear after any fixed time, that is P(ϑ ≥t)>0 for anyt _≥0. We denote byG_={G_t,t≥0}the aug-mented filtrationgenerated byW andN(in the sense of [9, IV-48]). In the following,P_{denotes the}

G_{-predictableσ-algebra onΩ×[0,T]. We suppose thatW is a}G_{-Brownian motion.}

Let(Λt)be theG_{-predictable compensator of the non decreasing process(Nt). Note that(Λt∧ϑ)} is then theG_{-predictable compensator of(Nt∧ϑ) = (Nt). By uniqueness of the}G_-predictable com-pensator,Λt∧ϑ =Λt,t≥0 a.s. We assume thatΛ is absolutely continuous w.r.t. Lebesgue’s

mea-sure, so that there exists a nonnegativeG_{-predictable process(λt), called the intensity process, such} that Λt =R0tλsds,t ≥0. SinceΛt∧ϑ =Λt, the processλ vanishes after ϑ. We denote byM the

G_{-compensated martingale given by}

Mt=Nt− Z t

0

λsds. (1)

LetT >0 be the finite horizon. We introduce the following sets:

• S2

T (also denoted byS2) is the set ofG-adapted right-continuous left-limited (RCLL) processes

ϕsuch thatE_[sup

0≤t≤T|ϕt|2]<+∞.

• A2

T (also denoted byA2) is the set of real-valued finite variational RCLLG-adapted (thus op-tional) processesAwith square integrable total variation process and such thatA0=0.

• A2

p,T (also denoted byAp2) is the set ofpredictableprocesses belonging toA2.

• H2

T (also denoted byH2) is the set ofG-predictable processes withkZk2:=E

h

RT 0 |Zt|2dt

i <∞.

• H2

λ,T :=L

2_{(Ω×[0,T],P,λ}

tdP⊗dt) (also denoted by H2_λ), equipped with scalar product

hU,V_iλ:=EhRT

0 UtVtλtdt

i

, for allU,V inH2

λ. For allU∈H2λ, we setkUk2λ :=E h

RT

0 |Ut|2λtdt

i

For eachU_∈H2

λ, we havekUk2λ=E h

RT∧ϑ

0 |Ut|2λtdt

i

because theG_{-intensityλ vanishes afterϑ.}

Note that, without loss of generality, we may assume thatUvanishes afterϑ2_.

Moreover,T _{is the set of stopping timesτsuch thatτ∈[0,T]a.s. and for eachSin}T_,T_{Sis the}

set of stopping timesτsuch thatS_≤τ≤T a.s.

We recall the martingale representation theorem in this framework (see [18]):

Lemma 1 (Martingale representation).Let m= (mt)0≤t≤T be aG-local martingale. There exists a unique pair ofG-predictable processes(zt,lt)3such that

mt=m0+ Z t

0

zsdWs+ Z t

0

lsdMs, ∀t_∈[0,T] a.s. (2)

If m is a square integrable martingale, then z_∈H2_{and l∈H}2

λ.

We now introduce the following definitions.

Definition 1 (Driver,λ-admissibledriver).

• A function g is said to be adriverifg:Ω×[0,T]×R3_{→R; (ω,t,y,z,k)7→g(ω,t,y,z,k)is}

P_⊗B₍R3_{)−measurable, and such thatg(.,0,0,0)∈H}2_.

2_{Indeed, eachUin}H2

λ(=L2(Ω×[0,T],P,λtdP⊗dt))can be identified withU1t≤ϑ, sinceU1t≤ϑis aG-predictable

process satisfyingUt1t≤ϑ=Ut λtdP⊗dt-a.s.

• A drivergis called aλ-admissible driverif moreover there exists a constantC_≥0 such that for

dP_⊗dt-almost every(ω,t), for all(y1,z1,k1),(y2,z2,k2),

|g(ω,t,y1,z1,k1)−g(ω,t,y2,z2,k2)| ≤C(|y1−y2|+|z1−z2|+

p

λt(ω)|k1−k2|). (3)

A non negative constantCsuch that (3) holds is called aλ-constantassociated with driverg.

Note that condition (3) implies that for each(y,z,k), we haveg(t,y,z,k) =g(t,y,z,0),t>ϑdP_⊗dt -a.e. Indeed, on the set{t>ϑ},gdoes not depend onk, sinceλt=0.

Remark 1.Note that adriver gsupposed to be Lipschitz with respect to(y,z,k)is not generallyλ -admissible. Moreover, adriver gsupposed to beλ-admissible is not generally Lipschitz with respect to(y,z,k)since the process(λt)is not necessarily bounded.

Definition 2.Letgbe aλ-admissibledriver, letξ _∈L2(G_T).

• A process(Y,Z,K)inS2_×H2_×H2

λ is said to be a solution of the BSDE with default jump

associated with terminal timeT, drivergand terminal conditionξ if it satisfies:

−dYt=g(t,Yt,Zt,Kt)dt−ZtdWt−KtdMt; YT =ξ. (4)

• LetD_∈A2_{. A process(Y,Z,K)inS}2_×H2_×H2

λ is said to be a solution of the BSDE with

default jump associated with terminal timeT,generalizedλ-admissible driver g(t,y,z,k)dt+dDt

and terminal conditionξ if it satisfies:

−dYt=g(t,Yt,Zt,Kt)dt+dDt−ZtdWt−KtdMt; YT=ξ. (5)

Remark 2.LetD= (Dt)0≤t≤T be a finite variational RCLL adapted process such thatD0=0, and with integrable total variation. We recall thatDadmits at most a countable number of jumps. We also recall that the processDhas the following (unique) canonical decomposition:D=A₋A′,whereA

andA′are integrable non decreasing RCLL adapted processes withA0=A′₀=0, and such thatdAt anddA′_t are mutually singular (cf. Proposition A.7 in [11]). IfDis predictable, thenAandA′ are predictable.

Moreover, by a property given in [14], for eachD_∈A2_{, there exist a unique (predictable) process}

D′belonging toA2

p and a unique (predictable) processηbelonging toIHλ2such that for allt∈[0,T],

Dt=D′t+ Z t

0 ηs

dNs a.s.

IfDis non decreasing, thenD′is non decreasing andηϑ ≥0 a.s on{ϑ≤T}.

Remark 3.By Remark 2 and equation (5), the processY admits at most a countable number of jumps. It follows thatYt=Yt−, 0≤t≤T dP⊗dt-a.e. Moreover, we haveg(t,Yt,Zt,Kt) =g(t,Yt−,Zt,Kt),

0≤t_≤T dP_⊗dt-a.e.

2.2 Properties of BSDEs with default jump

We first show some a priori estimates for BSDEs with a default jump, from which we derive the existence and uniqueness of the solution. Forβ >0,φ∈IH2, andl_∈IH_λ2, we introduce the norms

kφk2

β:=E[

RT

0 eβsφs2ds],andkkk2λ,β:=E[

RT

0 eβsk2sλsds].

2.2.1 A priori estimates for BSDEs with default jump

Proposition 1.Let ξ1_,ξ2 _∈L2_(G

T). Let g1 and g2 be twoλ-admissible drivers. Let C be a λ

For i=1,2, let(Yi_,Zi_,Ki_{)be a solution of the BSDE associated with terminal time T ,generalized}

drivergi_{(t,y,z,k)dt+dD}

t and terminal conditionξi. Letξ¯:=ξ1−ξ2. For s in[0,T], denoteY¯s:= Y_s1₋Y_s2, Z¯s:=Zs1−Z2s andK¯s:=Ks1−Ks2.

Letη,β >0be such thatβ≥ 3

η+2C andη≤C12. For each t∈[0,T], we have

eβt(Y¯t)2≤E[eβTξ¯2|Gt] +ηE[

Z T

t

eβsg(s)¯ 2ds_|G_{t] a .s., (6)}

whereg(s)¯ :=g1(s,Y_s2,Z_s2,K_s2)−g2(s,Y_s2,Z_s2,K_s2). Moreover,

kY¯_k2_{β ≤}T[eβT_E[ξ¯2_{] +ηkg¯k}2

β]. (7)

Ifη< 1

C2, we have

kZ¯_k2_β+kK¯_k2_λ,β≤ 1 1−ηC2[e

βT_E[ξ¯2_{] +ηkg¯k}2

β]. (8)

Remark 4.IfC=0, then (6) and (7) hold for allη,β>0 such thatβ≥η3, and (8) holds (withC=0)

for allη>0.

Proof. By Itˆo’s formula applied to the semimartingale(eβs_Y¯2

s)betweentandT, we get

eβtY¯_t2+β

Z T

t

eβsY¯_s2ds+

Z T

t

eβsZ¯_s2ds+

Z T

t

eβsK¯_s2λsds

=eβTY¯_T2+2

Z T

t

eβsY¯s(g1_(s,Y1

s,Zs1,Ks1)−g2(s,Ys2,Z2s,Ks2))ds

−2

Z T

t

eβsY¯sZ¯sdWs−

Z T

t

eβs(2 ¯Y_s−K¯s+K¯s2)dMs. (9)

Taking the conditional expectation givenG_{t, we obtain}

eβtY¯_t2+E

β

Z T

t

eβsY¯_s2ds+

Z T

t

eβs(Z¯_s2+K¯_s2λs)ds|Gt

≤Eh_eβTY¯_T2_|G_ti₊₂E

Z T

t

eβsY¯s(g1(s,Y_s1,Z_s1,K_s1)−g2(s,Y_s2,Z2_s,K_s2))ds|G_t

. (10)

Now,g1(s,Y_s1,Z1_s,K_s1)−g2(s,Y_s2,Z_s2,K_s2) =g1(s,Y_s1,Z_s1,K_s1)−g1(s,Y_s2,Z_s2,K_s2) +g¯s.

Sinceg1satisfies condition (3), we derive that

|g1(s,Y_s1,Z1_s,K_s1)−g2(s,Y_s2,Z_s2,K_s2)| ≤C_|Y¯s|+C|Z¯s|+C|K¯s|

√

λs+|g¯s|.

Note that, for all non negative numbersλ,y,z,k,gandε>0, we have 2y(Cz+Ck√λ+g)≤_εy22+ε2(Cz+Ck

√

λ+g)2_≤y2

ε2+3ε2(C2y2+C2k2λ+g2). Hence,

eβtY¯_t2+E

β

Z T

t

eβsY¯_s2ds+

Z T

t

eβs(Z¯_s2+K¯_s2λs)ds|G_t

≤EheβTY¯_T2_|G_ti₊

+E

(2C+ 1

ε2)

Z T

t

eβsY¯_s2ds+3C2ε2

Z T

t

eβs(Z¯2_s+K¯_s2λs)ds+3ε2

Z T

t

eβsg¯2_sds_|G_t

. (11)

Let us make the change of variableη=3ε2_{. Then, for eachβ,η>0 chosen as in the proposition,} this inequality leads to (6). By integrating (6), we obtain (7). Using (7) and inequality (11), we derive (8).

Remark 5.By classical results on the norms of semimartingales, one similarly shows thatkY¯_kS2 ≤ K E_[ξ¯2_{] +kg¯k}

2.2.2 Existence and uniqueness result for BSDEs with default jump

By the representation property ofG_{-martingales (Lemma 1) and the a priori estimates given in} Propo-sition 1, we derive the existence and the uniqueness of the solution associated with ageneralized

λ-admissible driver.

Proposition 2.Let g be aλ-admissible driver, letξ _∈L2(G_{T), and let D be an (optional) process} belonging toA2_{. There exists a unique solution(Y,Z,K)inS}2_×H2_×H2

λ of BSDE(5).

Remark 6.Suppose thatD=0. Suppose also thatξ isG_{ϑ∧T-measurable and thatgis replaced by} g1t≤ϑ (which is aλ-admissible driver). Then, the solution(Y,Z,K)of the associated BSDE (4) is

equal to the solution of the BSDE with random terminal timeϑ_∧T, drivergand terminal condition

ξ, as considered in [6]. Note also that in the present paper, contrary to papers [6, 12], we do not suppose that the default intensity processλ is bounded (which is interesting since this is the case in some models with default).

Proof. Let us first consider the case wheng(t)does not depend on(y,z,k). Then the solutionY is given byYt=E[ξ+

RT

t g(s)ds+DT−Dt|Gt]. The processesZandKare obtained by applying the

rep-resentation property ofG_{-martingales to the square integrable martingaleE[ξ+}RT

0 g(s)ds+DT|Gt].

Hence, there thus exists a unique solution of BSDE (5) associated with terminal condition ξ ∈

L2(F_{T) and generalized driver g(t)dt+dDt. Let us now turn to the case with a general λ}

-admissible driver g(t,y,z,k). Denote by IH_β2 the space S2_×IH2_×IH2

λ equipped with the norm

kY,Z,K_k2_β:=kY_k2_β+kZ_k2_β+kK_k2_λ,β. We define a mappingΦ fromIH_β2into itself as follows. Given

(U,V,L)∈IH_β2, let(Y,Z,K) =Φ(U,V,L)be the solution of the BSDE associated withgeneralized driver g(s,Us,Vs,Ls)ds+dDsand terminal conditionξ. Let us prove that the mappingΦis a

contrac-tion fromIH2_βintoIH2_β. Let(U′_,V′_,L′_{)be another element ofIH}2

βand let(Y′,Z′,K′):=Φ(U′,V′,L′),

that is, the solution of the BSDE associated with thegeneralized driver g(s,U_s′,V_s′,L_s′)ds+dDs and

terminal conditionξ.

Set ¯U=U₋U′, ¯V=V₋V′, ¯L=L₋L′, ¯Y =Y₋Y′, ¯Z=Z₋Z′, ¯K=K₋K′.

Set ∆gt :=g(t,Ut,Vt,Lt)−g(t,U_t′,V_t′,L′_t). By Remark 4 applied to the driver processes g1(t):=

g(t,Ut,Vt,Lt)4_andg

2(t):=g(t,Ut′,Vt′,L′t), we derive that for allη,β>0 such thatβ≥η3, we have

kY¯_k2_β+kZ¯_k2_β+kK¯_k2_{λ,β ≤}η(T+1)k∆g_k2_β.

Since the drivergisλ-admissiblewithλ-constantC, we get

kY¯_k2_β+kZ¯_k2_β+kK¯_k2_λ,β≤η(T+1)3C2(kU¯_k2_β+kV¯_k2_β+kL¯_k2_λ,β),

for allη,β >0 withβ ≥ 3

η. Choosing η= (T+11)6C2 andβ ≥ _η3 =18(T+1)C2, we derive that

k(Y,Z,K)k2

β≤ 12k(U,V,K)k 2

β.Hence, forβ ≥18(T+1)C2,Φ is a contraction fromIH2β intoIH2β

and thus admits a unique fixed point(Y,Z,K)in the Banach spaceIH2_β, which is the (unique) solution of BSDE (4).

2.3

λ

-

linear

BSDEs with default jump

We introduce the notion ofλ-linearBSDEs in our framework with default jump.

Definition 3 (λ-lineardriver).A drivergis calledλ-linearif it is of the form:

g(t,y,z,k) =δty+βtz+γtkλt+ϕt, (12)

4_{Note that the driver processesg}

where(ϕt)∈H2_{, and(δ}

t),(βt)and(γt)areR-valued predictable processes such that(δt),(βt)and (γt

√

λt)are bounded. By extension,

(δty+βtz+γtkλt)dt+dDt,

whereD_∈A2_{, is called ageneralizedλ-linear driver.}

Remark 7.Note thatggiven by (12) can be rewritten as

g(t,y,z,k) =ϕt+δty+βtz+νtk

p

λt, (13)

whereνt :=γt

√

λt is a bounded predictable process.5From this remark, it clearly follows that a

λ-lineardriver isλ-admissible.

We will now prove that the solution of aλ-linearBSDE (or more generally ageneralizedλ-linear driver) can be written as a conditional expectation via an exponential semimartingale. We first show a preliminary result on exponential semimartingales.

Let(βs)and(γs)be two real-valuedG-predictable processes such that the stochastic integrals R·

0βsdWsandR0·γsdMsare well-defined. Let(ζs)be the process satisfying the forward SDE: dζs=ζs−(βsdWs+γsdMs); ζ0=1. (14)

Remark 8.Recall that the process(ζs)satisfies the so-called Dol´eans-Dade formula, that is

ζs=exp{ Z s

0

βrdWr−

1 2

Z s

0

β2

rdr}exp{− Z s

0

γrλrdr}(1+γϑ1{s≥ϑ}), s≥0 a.s.

Hence, ifγϑ≥ −1 (resp.>−1) a.s, thenζs≥0 (resp.>0) for alls≥0 a.s.

Remark 9.The inequalityγϑ≥ −1 a.s. is equivalent to the inequalityγt≥ −1,λtdt⊗dP-a.s. Indeed,

we haveE_[1γϑ<−1]=E[

R+∞

0 1γr<−1dNr] =E[R0+∞1γr<−1λrdr], because the process(R0tλrdr)is the

G_{-predictable compensator of the default jump process}N.

Proposition 3.Let T >0. Suppose that the random variableRT

0(βr2+γr2λr)dr is bounded. Then, the process(ζs)0≤s≤T, defined by(14), is a martingale and satisfiesE[sup0≤s≤Tζs2]<+∞. Proof. By definition, the process(ζs)is a local martingale. LetT>0. Let us show that

E_[sup0

≤s_≤Tζs2]<+∞.By Itˆo’s formula applied toζs2, we getdζs2=2ζs−dζs+d[ζ,ζ]s. We have d[ζ,ζ]s=ζ2

s−β

2

sds+ζs2−γ

2

sdNs.

Using (1), we thus derive that

dζs2=ζs2−[2βsdWs+ (2γs+γs2)dMs+ (βs2+γs2λs)ds].

It follows thatζ2_{is an exponential semimartingale which can be written:}

ζs2=ηsexp{ Z s

0(β 2

r +γr2λr)dr}, (15)

whereηis the exponential local martingale satisfying

dηs=ηs−[2βsdWs+ (2γs+γs2)dMs],

withη0=1. By equality (15), the local martingaleηis non negative. Hence, it is a supermartingale, which yields thatE_{[ηT]≤1. Now, by assumption,}RT

0(βr2+γr2λr)dris bounded. By (15), it follows

that

E_[ζ2

T]≤E[ηT]K≤K,

whereK is a positive constant. By martingale inequalities, we derive thatE_[sup

0≤s≤Tζs2]<+∞.

Hence, the process(ζs)0≤s≤T is a martingale.

Remark 10.Note that, under the assumption from Proposition 3, one can prove by an induction argument (as in the proof of Proposition A.1 in [21]) that for all p_≥2, we haveE_[sup

0≤s≤Tζ p s]<

+∞.

We now show a representation property of the solution of ageneralizedλ-linearBSDE when the finite variational processDis supposed to be predictable.

Theorem 1 (Representation result forgeneralizedλ-linearBSDEs withDpredictable).Let(δt), (βt)and(γt)beR-valued predictable processes such that(δt),(βt)and(γt

√

λt)are bounded. Letξ ∈L2(G_{T)and let D be a process belonging to}A2

p, that is, a finite variational RCLLpredictable process with D0=0and square integrable total variation process.

Let(Y,Z,K)be the solution inS2_×IH2_×IH2

λ of the BSDE associated withgeneralizedλ-linear

driver(δty+βtz+γtkλt)dt+dDt and terminal conditionξ, that is

−dYt= (δtYt+βtZt+γtKtλt)dt+dDt−ZtdWt−KtdMt; YT =ξ. (16)

For each t_∈[0,T], let(Γt,s)s≥t (called theadjoint process) be the unique solution of the following forward SDE

dΓt,s=Γt,s−[δsds+βsdWs+γsdMs]; Γt,t=1. (17)

The process(Yt)satisfies

Yt =E[Γt,Tξ+

Z T

t

Γt,s−dDs|Gt], 0≤t≤T, a.s. (18)

Remark 11.From Remark 8, it follows that the process(Γt,s)s≥t, defined by (17), satisfies

Γt,s=e Rs

tδrdr_exp{

Z s

t

βrdWr−

1 2

Z s

t

β2

rdr}e− Rs

t γrλrdr₍₁₊γ_ϑ1

{s_≥ϑ>t_}) s≥t a.s.

Hence, ifγϑ≥ −1 (resp.>−1) a.s. , we then haveΓt,s≥0 (resp.>0) for alls≥ta.s.

Note also that the process (eRtsδrdr)t_{≤s≤T is positive, and bounded since} δ _{is bounded. Using}

Proposition 3, sinceβ andγ√λ are bounded, we derive thatE_[sup

t≤s≤TΓt,2s]<+∞.

Proof. Fixt_∈[0,T]. Note first that sinceDis a finite variational RCLL process, here supposed to be

predictable, and since the processΓt,·admits only one jump at the totally inaccessible stopping time

ϑ, we get[Γt,·,D] =0. By applying the Itˆo product formula toYsΓt,s, we get

−d(YsΓt,s) =−Ys−dΓt,s−Γt,s−dYs−d[Y,Γ]s

=−YsΓt,s−δsds+Γt,s−[δsYs+βsZs+γsKsλs]ds+Γt,s−dDs (19)

−βsZsΓt,s−ds−Γt,s−γsKsλsds−Γt,s−(Ysβs+Zs)dWs−Γt,s−[Ks(1+γs) +Ys−γs]dMs.

Setting

dms=−Γt,s−(Ysβs+Zs)dWs−Γt,s−[Ks(1+γs) +Ys−γs]dMs,

we get

−d(YsΓt,s) =Γt,s−dDs−dms. (20)

By integrating betweentandT, we obtain

Yt=ξ Γt,T+

Z T

t Γt,s

By Remark 11, we have(Γt,s)t≤s_≤T ∈S2. Moreover,Y ∈S2,Z∈IH2,K∈IH_λ2, andβ andγ

√

λ

are bounded. It follows that the local martingalem= (ms)t_≤s≤Tis a martingale. Hence, by taking the

conditional expectation in equality (21), we get equality (18).

When the finite variational processDis no longer supposed to be predictable (which is often the case in the literature on default risk6_{), the representation formula (18) does not generally hold. We} now provide a representation property of the solution in that case, that is, when the finite variational processDis only supposed to be RCLL and adapted, which is new in the literature on BSDEs.

Theorem 2 (Representation result for generalizedλ-linearBSDEs withDoptional ).Suppose that the assumptions of Theorem 1 hold, except that D is supposed to belong toA2_{instead ofA}2

p. Let D′_∈A2

p andη∈IHλ2be such that for all t∈[0,T],7

Dt=D′t+ Z t

0 ηs

dNs a.s. (22)

Let(Y,Z,K)be the solution inS2_×IH2_×IH2

λ of the BSDE associated withgeneralizedλ-linear

driver(δty+βtz+γtkλt)dt+dDtand terminal conditionξ, that is BSDE(16). Then, a.s. for all t_∈[0,T],

Yt =E_[Γt,Tξ+

Z T

t

Γ_t,s−(dD′_s+(1+γs)ηsdNs)|Gt] =E[Γt,T ξ+

Z T

t

Γ_t,s−dD′_s+Γt,ϑηϑ1_{t<ϑ_≤T_}|Gt]

(23)

where(Γt,s)s∈[t,T]satisfies(17).

Proof. SinceDsatisfies (22), we getd[Γt,·,D]s=Γt,s−γsηsdNs. The computations are then similar

to those of the proof of Theorem 1, with Γt,s−dDs replaced byΓt,s−(dDs+γsηsdNs)in equations

(19), (20) and (21). We thus derive thatYt =E[Γt,Tξ+RtTΓt,s−(dDs+γsηsdNs)|Gt]a.s. From this

together with (22), the first equality of (23) follows. Now, we have a.s.

E_[

Z T

t

Γ_t,s−(1+γs)ηsdNs)|Gt] =E[Γt,ϑ−(1+γϑ)ηϑ1_{t<ϑ_≤T_}|Gt] =E[Γt,ϑηϑ1_{t<ϑ_≤T_}|Gt],

where the second equality is due to the fact thatΓt,ϑ−(1+γϑ) =Γt,ϑa.s. (cf. Remark 11). This yields

the second equality of (23).

Remark 12.By adapting the arguments of the above proof, this result can be generalized to the case of a BSDE driven by a Brownian motion and a Poisson random measure8_{, which provides a new} result in the theory of BSDEs in this framework.

2.4 Comparison theorems for BSDEs with default jump

We now provide a comparison theorem and a strict comparison theorem for BSDEs withgeneralized

λ-admissible driversassociated with finite variational RCLL adapted processes.

Theorem 3 (Comparison theorems).Letξ1andξ2∈L2(GT). Let g1and g2be twoλ-admissible

drivers. Let D1and D2be two (optional) processes inA2. For i=1,2, let(Yi_,Zi_,Ki_{)be the solution inS}2_×IH2_×IH2

λ of the following BSDE

−dY_ti=gi(t,Y_ti,Z_ti,K_ti)dt+dDi_t−Z_tidWt−KtidMt; YTi =ξi.

6_{In the case of a contingent claim or a contract subjected to default,∆D}

ϑ represents the cashflow generated by the

claim at the default timeϑ(see Section 3). It is sometimes called ”rebate” (cf. [16, 4]).

7_{see Remark 2.}

(i)(Comparison theorem). Assume that there exists a predictable process(γt)with

(γt

p

λt) bounded and γt≥ −1, dP⊗dt−a.e. (24)

such that

g1(t,Yt2,Zt2,Kt1)−g1(t,Yt2,Zt2,Kt2)≥γt(Kt1−Kt2)λt, t∈[0,T], dP⊗dt− a.e. (25)

Suppose thatξ1≥ξ2a.s. , that the processD¯:=D1−D2is non decreasing, and that

g1(t,Yt2,Zt2,Kt2)≥g2(t,Yt2,Zt2,Kt2), t∈[0,T], dP⊗dt− a.e. (26)

We then have Y_t1_≥Y_t2for all t_∈[0,T]a.s.

(ii)(Strict Comparison Theorem). Suppose moreover thatγϑ >−1a.s.

If Y_t1 0 =Y

2

t0 a.s. for some t0∈[0,T], thenξ

1_=ξ2_{a.s. , and the inequality(26)is an equality on}

[t0,T]. Moreover,D¯=D1−D2is constant on[t0,T]and Y1=Y2on[t0,T].

Remark 13.We stress that the above comparison theorems hold even in the case when the generalized drivers are associated with non-predictable finite variational processes, which thus may admit a jump at the default timeϑ. This is important for the applications to nonlinear pricing of contingents claims. Indeed, in a market with default, contingent claims often generate a cashflow at the default time (see Section 3.3 for details).

As seen in the proof below, the treatment of the case of non-predictable finite variational processes requires some additional arguments, compared to the case of predictable ones.

Proof. Setting ¯Ys=Ys1−Ys2; ¯Zs=Zs1−Z2s ; ¯Ks=Ks1−Ks2,we have

−dY¯s=hsds+dD¯s−Z¯sdWs−K¯sdMs; Y¯T =ξ1−ξ2,

wherehs:=g1(s,Y_s1−,Zs1,Ks1)−g2(s,Y_s2−,Zs2,Ks2).

Setδs:=

g1(s,Y_s1−,Zs1,Ks1)−g1(s,Y_s2−,Z1s,Ks1)

¯

Y_s− if ¯

Y_s−6=0, and 0 otherwise.

Setβs:=

g1(s,Y_s2−,Zs1,Ks1)−g1(s,Y_s2−,Z2s,Ks1)

¯

Zs

if ¯Zs6=0, and 0 otherwise.

By definition, the processesδ andβ arepredictable. Moreover, sinceg1satisfies condition (3), the processesδ andβ are bounded. Now, we have

hs=δsY¯s−+βsZ¯s+g1(s,Y_s2−,Zs2,Ks1)−g1(s,Y_s2−,Zs2,Ks2) +ϕs,

whereϕs:=g1(s,Y_s2−,Z2s,Ks2)−g2(s,Y_s2−,Zs2,Ks2).9

Using the assumption (25) and the equality ¯Y_s−=Y¯sdP⊗ds-a.e. (cf. Remark 3), we get

hs≥δsY¯s+βsZ¯s+γsK¯sλs+ϕs dP⊗ds−a.e. (27)

Fixt_∈[0,T]. LetΓt,.be the process defined by (17). Sinceδ,βandγ

√

λare bounded, it follows from Remark 11 thatΓt,.∈S2. Also, sinceγs≥ −1, we haveΓt,.≥0 a.s. Let us first consider the simpler case when the processesD1andD2are predictable. By Itˆo’s formula and similar computations to those of the proof of Theorem 1, we derive that

−d(Y¯sΓt,s) =Γt,s(hs−δsY¯s−βsZ¯s−γsK¯sλs)ds+Γt,s−dD¯s−dms,

wheremis a martingale (becauseΓt,.∈S2, ¯Y ∈S2, ¯Z ∈IH2, ¯K∈IHλ2andβ,γ

√

λ are bounded). Using inequality (27) together with the non negativity ofΓ, we thus get−d(Y¯sΓt,s)≥Γt,sϕsds+

Γt,s−dD¯s−dms. By integrating betweentandTand by taking the conditional expectation, we obtain

9_{Note that, by Remark 3, we haveϕ}

¯

Yt ≥E[Γt,T(ξ1−ξ2) +

Z T

t Γt,s−(ϕs

ds+dD¯s)|Gt], 0≤t≤T, a.s. (28)

By assumption (26), ϕs≥0dP⊗ds-a.e. Moreover, ξ1−ξ2≥0 and ¯Dis non decreasing, which, together with the non negativity ofΓt,·, implies that ¯Yt=Yt1−Yt2≥0 a.s. Since this inequality holds

for allt_∈[0,T], the assertion (i) follows. Suppose moreover thatY_t1

0 =Y 2

t0 a.s. and thatγ·>−1. Sinceγϑ>−1 a.s. , we haveΓt,s>0 a.s. for alls_≥t. From this, together with (28) applied witht=t0, we getξ1=ξ2a.s. andϕt=0,t∈[t0,T]

dP_⊗dt-a.e. On the other hand, set ˜Dt:= Rt

t0Γt0,s−dD¯s,for eacht∈[t0,T]. By assumption, ˜DT≥0 a.s. By (28), we thus getE_[D˜T |Gt0] =0 a.s. Hence ˜DT =0 a.s. Now, sinceΓt0,s>0, for alls≥t0a.s. , we can write ¯DT−D¯t0=

RT t0 Γ

−1

t0,s−d ˜

Ds. We thus get ¯DT =D¯t0 a.s. The proof of (ii) is thus complete.

Let us now consider the case when the processesD1andD2are not predictable. By Remark 2, for

i=1,2, there existD′i_∈A2

p andηi∈IHλ2such that

Di_t=D_t′i+ Z t

0

ηi

sdNs a.s.

Since ¯D:=D1_−D2_{is non decreasing, we derive that the process ¯D}′_:=D′1_−D′2_{is non decreasing} and thatη1

ϑ ≥ηϑ2 a.s. on{ϑ ≤T}. By Itˆo’s formula and similar computations to those of the proof

of Theorems 1 and 2, we get

−d(Y¯sΓt,s) =Γt,s(hs−δsY¯s−βsZ¯s−γsK¯sλs)ds+Γt,s−[dD¯s+ (ηs1−ηs2)γsdNs]−dms,

wheremis a martingale. Using inequality (27) and the equality ¯Dt=D¯t′+ Rt

0(ηs1−ηs2)dNsa.s. , we

thus derive that

¯

Yt ≥E[Γt,T(ξ1−ξ2) +

Z T

t Γt,s−(ϕs

ds+dD¯′_s+ (η1

s −ηs2)(1+γs)dNs)|Gt], 0≤t≤T, a.s. (29)

Sinceη1

ϑ≥ηϑ2a.s. on{ϑ≤T}andγϑ≥ −1 a.s. , we have(ηϑ1−ηϑ2)(1+γϑ)≥0 a.s. on{ϑ≤T}.

Hence, using the other assumptions made in (i), we derive that ¯Yt =Yt1−Yt2≥0 a.s. Since this

inequality holds for allt_∈[0,T], the assertion (i) follows. Suppose moreover thatY_t1

0=Y 2

t0 a.s. and thatγϑ>−1 a.s. By the inequality (29) applied witht=t0, we derive thatξ1_=ξ2_{a.s. ,ϕ}

s=0dP⊗ds-a.e. on[t0,T],η_ϑ1=η_ϑ2a.s. on{t0<ϑ≤T}. Moreover, ¯

D′=D′1₋D′2is constant on the time interval[t0,T]. Hence, ¯Dis constant on[t0,T]. The proof is thus complete.

Remark 14.By adapting the arguments of the above proof, this result can be generalized to the case of BSDEs driven by a Brownian motion and a Poisson random measure (since the jumps times associated with the Poisson random measure are totally inaccessible). This extends the comparison theorems given in the literature on BSDEs with jumps (see [21, Theorems 4.2 and 4.4]) to the case of generalized drivers of the formg(t,y,z,k)dt+dDt, whereDis a finite variational RCLL adapted

process (not necessarily predictable).

When the assumptions of the comparison theorem (resp. strict comparison theorem) are violated, the conclusion does not necessarily hold, as shown by the following example.

Example 1.Suppose that the processλ is bounded. Letgbe aλ-linear driver (see (12)) of the form

g(ω,t,y,z,k) =δt(ω)y+βt(ω)z+γkλt(ω), (30)

whereγ is here a real constant. At terminal timeT, the associated adjoint processΓ0,·satisfies (see

(17) and Remark 11) :

Γ0,T =HTexp{−

Z T

0

γλrdr}(1+γ1_{T≥ϑ}). (31)

LetY be the solution of the BSDE associated with drivergand terminal condition

ξ:=1_{T≥ϑ}.

The representation property ofλ-linear BSDEs with default jump (see (18)) gives

Y0=E[Γ0,Tξ] =E[Γ0,T1_{T≥ϑ}].

Hence, by (31), we get

Y0=E[Γ0,T1_{T≥ϑ}] =E[HTe−γ RT

0 λsds_(1+γ1

{T≥ϑ})1{T≥ϑ}] = (1+γ)E[HTe−γ RT

0 λsds₁

{T≥ϑ}]. (32)

Equation (32) shows that under the additional assumptionP(T ≥ϑ)>0, whenγ<−1, we have

Y0<0 althoughξ≥0 a.s.

This example also gives a counter-example for the strict comparison theorem by takingγ=−1. Indeed, in this case, the relation (32) yields thatY0=0. Under the additional assumptionP(T≥ϑ)> 0, we haveP(ξ>0)>0, even thoughY0=0.

3 Nonlinear pricing in a financial market with default

3.1 Financial market with defaultable risky asset

We consider a complete financial market with default as in [5], which consists of one risk-free asset, with price processS0_satisfyingdS0

t =St0rtdtwithS00=1, and two risky assets with price processes

S1,S2evolving according to the equations:

dS1_t =S_t1[µ1

tdt+σt1dWt] with S10>0;

dS2_t =S_t2−[µt2dt+σt2dWt−dMt] with S20>0, (33)

where the process(Mt)is given by (1).

The processesσ1_,σ2_{, r,µ}1_,µ2_{are predictable (that is P-measurable). We setσ = (σ}1_,σ2₎′_,

where′denotes transposition.

We suppose thatσ1_,σ2_{>0, andr,µ}1_,µ2_,σ1_,σ2_,(σ1₎−1_,(σ2₎−1_{are bounded. Note that the} in-tensity process(λt)is not necessarily bounded, which is useful in market models with default where the intensity process is modeled by the solution (which is not necessarily bounded) of a forward stochastic differential equation.

Remark 15.By Remark 11, we have

S_t2=eR0tµr2dr_exp{

Z t

0

σ2

rdWr−

1 2

Z t

0

(σ2

r)2dr}e Rt

0λrdr₍₁₋₁

{t≥ϑ}), t≥0 a.s.

The second risky asset is thus defaultable with total default: we haveS2

t =0,t≥ϑ a.s.

We consider an investor who, at time 0, invests an initial amountx_∈Rin the three assets. For

i=1,2, we denote byϕi

t the amount invested in theithrisky asset. After timeϑ, the investor does not

invest in the defaultable asset since its price is equal to 0. We thus haveϕ2

t =0 ont>ϑ. A process

ϕ= (ϕ1_,ϕ2₎′ _{belonging to} H2_×H2

λ is called arisky assets strategy. LetCbe a finite variational

optional process belonging toA2_{, representing thecumulative cash amount withdrawnfrom the} portfolio.

The value at timet of the portfolio (orwealth) associated withx,ϕ andCis denoted byV_tx,ϕ,C. The amount invested in the risk-free asset at timetis then given byV_tx,ϕ,C₋(ϕ1

3.2 Pricing of European options with dividends in a perfect (linear) market with

default

In this section, we place ourselves in a perfect (linear) market model with default. In this case, by the self financing condition, the wealth processVx,ϕ,C(simply denoted byV) follows the dynamics:

dVt= (rtVt+ϕt1(µt1−rt) +ϕt2(µt2−rt))dt−dCt+ (ϕt1σt1+ϕt2σt2)dWt−ϕt2dMt = rtVt+ϕt′σtθt1−ϕt2θt2λt

dt₋dCt+ϕt′σtdWt−ϕt2dMt, (34)

whereϕt′σt=ϕt1σt1+ϕt2σt2, and

θ1

t :=

µ1

t −rt

σ1

t

; θ2

t :=−

µ2

t −σt2θt1−rt

λt

1_{λt6=0_}.

Suppose that the processesθ1_andθ2√_{λ are bounded.}

LetT >0. Letξ be aG_{T-measurable random variable belonging toL}2_{, and letDbe a finite} vari-ational optional process belonging toA2

T. We consider a European option with maturityT, which

generates aterminal payoffξ, and intermediate cashflows calleddividends, which are not necessarily positive (cf. for example [7, 8]). For eacht_∈[0,T],Dtrepresents the cumulative intermediate

cash-flows paid by the option between time 0 and timet. The processD= (Dt)is called thecumulative dividend process. Note thatDis not necessarily non decreasing.

The aim is to price this contingent claim. Let us consider an agent who wants to sell the option at time 0. With the amount the seller receives at time 0 from the buyer, he/she wants to be able to construct a portfolio which allows him/her to pay to the buyer the amountξ at timeT, as well as the intermediate dividends.

Now, setting

Zt:=ϕt′σt ; Kt:=−ϕt2, (35)

by (34), we derive that the process(V,Z,K)satisfies the following dynamics:

−dVt=−(rtVt+θt1Zt+θt2Ktλt)dt+dDt−ZtdWt−KtdMt.

We set for each(ω,t,y,z,k),

g(ω,t,y,z,k):=−rt(ω)y−θ1

t (ω)z−θt2(ω)kλt(ω). (36)

Since by assumption, the coefficientsr,θ1_,θ2√_{λ are predictable and bounded, it follows thatgis a}

λ-lineardriver (see Definition 3). By Proposition 2, there exists a unique solution(X,Z,K)∈S2_× H2_×H2

λ of the BSDE associated with terminal timeT,generalizedλ-linear driver g(t,y,z,k)dt+

dDt(withgdefined by (36)) and terminal conditionξ.

Let us show that the process(X,Z,K)provides a replicating portfolio. Letϕ be the risky-assets strategy such that (35) holds. Note that this defines a change of variablesΦ as follows:

Φ :H2_×H2

λ →H2×H2λ;(Z,K)7→Φ(Z,K):=ϕ, whereϕ= (ϕ1,ϕ2)is given by (35), which is

equivalent to

ϕt2=−Kt ; ϕt1=

Zt−ϕt2σt2

σ1

t

=Zt+σ

2

tKt

σ1

t

. (37)

and thatϕis thehedging risky-assets strategy. Similarly, for each timet_∈[0,T],Xt is thehedging priceat timetof the option, and is denoted byXt,T(ξ,D).

Suppose that thecumulative dividend process Dis predictable. Since the driverggiven by (36) is

λ-linear, the representation property of the solution of ageneralizedλ-linearBSDE (see Theorem 1) yields

Xt,T(ξ,D) =E[e− RT

t rsds_ζ

t,Tξ+

Z T

t

e−Rtsrudu_ζ

t,s−dDs|Gt] a.s. , (38)

whereζ satisfies

dζt,s=ζt,s−[−θs1dWs−θs2dMs]; ζt,t=1.

Suppose now thatθ2

t <1 dP⊗dt-a.e. By Proposition 3 and Remark 8, the processζ0,.is a square integrable positive martingale. LetQbe the probability measure which admitsζ0,T as density with

respect toPonG_T.10_{By the equality (38), we have}

Xt,T(ξ,D) =EQ[e− RT

t rsds_ξ+

Z T

t e−

Rs trudu_dD

s|Gt] a.s. (39)

When thecumulative dividend process Dis not predictable and thus admits a jump at timeϑ, the representation formulas (38) and (39) for the no-arbitrage price of the contingent claim do not generally hold. In this case, by Remark 2, there exist a (unique) processD′_∈A2

p and a (unique)

processη_∈IH_λ2such that for allt_∈[0,T],

Dt=D′t+ Z t

0 ηs

dNs a.s. (40)

The random variableηϑ (sometimes called ”rebate” in the literature) represents the cash flow

gen-erated by the contingent claim at the default time ϑ (see e.g. [7, 4, 8, 16] for examples of such contingent claims). By Theorem 2, we get

Xt,T(ξ,D) =E[e− RT

t rsds_ζ

t,Tξ+

Z T

t e−

Rs t rudu_ζ

t,s−dD′_s+e−

Rϑ t rsds_ζ

t,ϑηϑ1_{t<ϑ_≤T_}|Gt] a.s. ,

or equivalently

Xt,T(ξ,D) =EQ[e− RT

t rsds_ξ+

Z T

t

e−Rtsrudu_dD′

s+e− Rϑ

t rsds_η

ϑ1{t<ϑ≤T}|Gt] a.s.

We thus recover the risk-neutral pricing formula of [4, 16], which we have established here by work-ing under the primitive probability measure, uswork-ing BSDE techniques.

We note that the pricing system (for a fixed maturityT):(ξ,D)7→X_·,T(ξ,D)is linear.

3.3 Nonlinear pricing of European options with dividends in an imperfect

market with default

From now on, we assume that there are imperfections in the market which are taken into account via thenonlinearityof the dynamics of the wealth. More precisely, we suppose that thewealthprocess

V_tx,ϕ,C (or simplyVt) associated with an initial wealth x, a risky-assets strategy ϕ = (ϕ1,ϕ2)in

H2_×H2

λ and a cumulative withdrawal processC∈A2satisfies the following dynamics:

10 _{Note that the discounted price process(e}−Rt

0rsds_S1

t)0≤t≤T (resp.(e−

Rt

0rsds_S2

t)0≤t≤T) is a martingale (resp. local martingale) underQ. Suppose now thatE[eqR0Tλrdr_]<+

∞for someq>2. Using Remark 15, we show thate−

RT

0rsds_S2

T

∈L2_Q, which, by martingale inequalities, implies that(e−R0trsds_S2

−dVt=g(t,Vt,ϕt′σt,−ϕt2)dt−ϕt′σtdWt+dCt+ϕt2dMt;V0=x, (41)

wheregis a nonlinearλ-admissibledriver (see Definition 1). Equivalently, settingZt=ϕt′σt and Kt=−ϕt2, we have

−dVt=g(t,Vt,Zt,Kt)dt−ZtdWt+dCt−KtdMt;V0=x. (42)

Let us consider a European option with maturity T, terminal payoff ξ ∈L2(G_{T), and}

divi-dend process D_∈A2 _{(with a possible jump at the default time ϑ) in this market model. Let}

(X_·g_,T(ξ,D),Z_·g_,T(ξ,D),K_·g_,T(ξ,D)),simply denoted by (X,Z,K), be the solution of BSDE associ-ated with terminal time T,generalized driver g(t,y,z,k)dt+dDt and terminal conditionξ, that is

satisfying

−dXt=g(t,Xt,Zt,Kt)dt+dDt−ZtdWt−KtdMt; XT=ξ.

The processX =X_·g_,T(ξ,D)is equal to the wealth process associated with initial valuex=X0, strategyϕ=Φ(Z,K)(see (37)) and cumulative amountDof cash withdrawals, that isX=VX0,ϕ,D_. Its initial valueX0=X₀g_,T(ξ,D)is thus a sensible price (at time 0) of the option for the seller since this amount allows him/her to construct a risky-assets strategyϕ, calledhedgingstrategy, such that the value of the associated portfolio is equal toξ at time T, and such that the cash withdrawals perfectly replicate the dividends of the option. We callX0=X₀g_,T(ξ,D)thehedging priceat timetof the option. Similarly, for eacht_∈[0,T],Xt=Xtg,T(ξ,D)is thehedging priceat timetof the option.

Thus, for each maturityS_∈[0,T]and for each pairpayoff-dividend(ξ,D)∈L2(G_S)×A2

S,the

process X_·g_,S(ξ,D) is called thehedging price process of the option with maturity S and payoff-dividend(ξ,D). This leads to apricingsystem

Xg :(S,ξ,D)7→X_·g_,S(ξ,D), (43) which is generallynonlinearwith respect to(ξ,D).

We now give some properties of thisnonlinear pricingsystemXg which generalize those given in [15] to the case with a default jump and dividends.

• Consistency.By the flow property for BSDEs, the pricing systemXgisconsistent. More precisely, letS′_∈[0,T],ξ∈L2(G

S′),D∈AS2′,and letS∈[0,S′]. Then, thehedging priceof the option

asso-ciated with payoffξ, cumulative dividend processDand maturityS′coincides with thehedging priceof the option associated with maturityS, payoffX_Sg_,S′(ξ,D)and dividend process(Dt)t≤S

(still denoted byD), that is

X_·g_,S′(ξ,D) =X

g

·,S

X_Sg_,S′(ξ,D),D

.

• Wheng(t,0,0,0) =011, then the price of the European option with null payoff and no dividends is equal to 0, that is, for allS_∈[0,T],X_·g_,S(0,0) =0.

Due the presence of the default, thenonlinear pricingsystemXg is not necessarily monotone with respect to the payoff and the dividend. We introduce the following assumption.

Assumption 4 Assume that there exists a map

γ:Ω×[0,T]×R4_→R;(ω,t,y,z,k1,k2)7→γty,z,k1,k2(ω)

P_⊗B₍R4)-measurable, satisfying dP⊗dt-a.e. , for each(y,z,k1,k2)∈R4,

|γy,z,k1,k2 t

p

λt| ≤C and γty,z,k1,k2≥ −1, (44)

and