Mean-Variance Hedging on Uncertain Time Horizon in a Market with a Jump

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Mean-Variance Hedging on Uncertain Time Horizon in a Market with a Jump

Idris Kharroubi, Thomas Lim, Armand Ngoupeyou

To cite this version:

Idris Kharroubi, Thomas Lim, Armand Ngoupeyou. Mean-Variance Hedging on Uncertain Time Hori- zon in a Market with a Jump. Applied Mathematics and Optimization, Springer Verlag (Germany), 2013, 68, pp.413 - 444. �10.1007/s00245-013-9213-5�. �hal-01103691�

Mean-Variance Hedging on Uncertain time Horizon in a Market with a Jump

Idris Kharroubi^∗

CEREMADE, CNRS UMR 7534, Universit´e Paris Dauphine kharroubi @ ceremade.dauphine.fr

Thomas Lim^†

Laboratoire d’Analyse et Probabilit´es, Universit´e d’Evry and ENSIIE,

lim @ ensiie.fr

Armand Ngoupeyou

Laboratoire de Probabilit´es et Mod`eles Al´eatoires, Universit´e Paris 7

armand.ngoupeyou @ univ-paris-diderot.fr

Abstract

In this work, we study the problem of mean-variance hedging with a random horizon T∧τ, whereT is a deterministic constant andτ is a jump time of the underlying asset price process. We first formulate this problem as a stochastic control problem and relate it to a system of BSDEs with a jump. We then provide a verification theorem which gives the optimal strategy for the mean-variance hedging using the solution of the previous system of BSDEs. Finally, we prove that this system of BSDEs admits a solution via a decomposition approach coming from filtration enlargement theory.

Keywords: Mean-variance hedging, Backward SDE, random horizon, jump processes, pro- gressive enlargement of filtration, decomposition in the reference filtration.

AMS subject classifications: 91B30, 60G57, 60H10, 93E20.

1 Introduction

In most financial markets, the assumption that the market is complete fails to be true. In particular, investors cannot always hedge the financial products that they are interested in. One possible approach to deal with this problem is mean-variance hedging. That is, for a given financial product with terminal value H at a fixed horizon time T and an initial

∗The research of the author benefited from the support of the French ANR research grant LIQUIRISK (ANR-11-JS01-0007).

†The research of the author benefited from the support of the “Chaire Risque de Cr´edit”, F´ed´eration Bancaire Fran¸caise.

capital x, we need to find a strategy π^∗ such that the value V^x,π∗ of the portfolio with initial amount x and strategyπ^∗ minimizes the mean square error

E h

V_T^x,π−H

2i

over all possible investment strategiesπ.

In this paper, we are concerned with the mean-variance hedging problem over a random horizon. More precisely, we consider a random time τ and a contingent claim with a gain at timeT ∧τ of the form

H = H^b1T <τ+H_τ^a1T≥τ, (1.1)

where T < ∞ is a fixed deterministic terminal time. We then study the mean-variance hedging problem over the horizon [0, T∧τ] defined by

infπ E h

V_T^x,π_∧τ−H

2i

. (1.2)

Financial products with gains of the form (1.1) naturally appear on financial markets, see e.g. Examples 2.1, 2.2 and 2.3 presented in Subsection 2.3.

The mean-variance hedging problem with deterministic horizonT is one of the classical problems from mathematical finance and has been considered by several authors via two main approaches. One of them is based on martingale theory and projection arguments and the other considers the problem as a quadratic stochastic control problem and describes the solution using BSDE theory.

The bulk of the literature primarily focuses on the continuous case where both ap- proaches are used (see e.g. Delbaen and Schachermayer [6], Gouri´erouxet al. [10], Laurent and Pham [23] and Schweizer [25] for the first approach, and Lim and Zhou [21] and Lim [20] for the second one).

In the discontinuous case, the mean-variance hedging problem is considered by Arai [2], Lim [22] and Jeanblanc et al [14]. In [2], the author uses the projection approach for general semimartingale price processes model whereas in [22] the problem is considered from the point of view of stochastic control for the case of diffusion price processes driven by Brownian motion and Poisson process. The author provides under a so-called “martingale condition” the existence of solutions to the associated BSDEs. In the recent paper [14], the authors combine tools from both approaches, which allows them to work in a general semimartingale model and to give a description of the optimal solution to the mean-variance hedging via the BSDE theory. More precisely the authors prove that the value process of the mean-variance hedging problem has a quadratic structure and that the coefficients appearing in this quadratic expression are related to some BSDEs. Then, they provide an equivalence between the existence of an optimal strategy and the existence of a solution to a BSDE associated to the control problem. They have also shown in some specific examples, via the control problem, the existence of solutions for BSDEs of interest. However the problem is still open in the general case.

In this paper, we study the mean-variance hedging with horizon T ∧τ given by (1.2).

We use a stochastic control approach and describe the optimal solution by a solution to a system of BSDEs.

We shall consider a model of diffusion price process driven by a Brownian motion and a random jump time τ. We follow the progressive enlargement approach initiated by Jacod, Jeulin and Yor (see [15] and [16]), which leads to considering an enlargement of the initial information given by the Brownian motion to makeτ a stopping time. We note that this approach allows to work under wide class of assumptions, in particular, on contrary to the Poisson case, no a priori law is fixed for the random timeτ.

Following the quadratic form obtained in [14], we use a martingale optimality principle to obtain an associated system of nonstandard BSDEs. We then establish a verification result (Theorem 3.2) which provides an explicit optimal investment strategy via the solution to the associated system of BSDEs. Our contribution is twofold.

•We link the mean-variance hedging problem on a random horizon with a system of BSDE, in a general filtration progressive enlargement setup which allows us to work without a priori knowledge of the law of jump part. We show that, under wide assumptions, the mean-variance hedging problem admits an optimal strategy described by the solution of the associated system of BSDEs.

•We prove that the associated system of BSDEs, which is nonstandard, admits a solution.

The main difficulty here is that the obtained system of BSDEs is nonstandard since it is driven by a Brownian motion and a jump martingale and has generators with quadratic growth in the variable z and are undefined for some values of the variable y. To solve these BSDEs we follow a decomposition approach inspired by the result of Jeulin (see Proposition 2.1) which allows to consider BSDEs in the smallest filtration (see Theorem 4.3). Then using BMO properties, we provide solutions to the decomposed BSDEs which lead to the existence of a solution to the BSDEs in the enlarged filtration.

We notice that, for the problem at handi.e. mean-variance hedging with horizonT∧τ, the interest of our approach is that it provides a solution to the associated BSDEs, without supposing any additional specific assumptions to the studied BSDEs unlike in [22] where to prove existence of a solution to the BSDE the author introduces the “martingale condition”

or in [14] where the existence of a solution to the BSDE is given in specific cases.

The paper is organized as follows. In Section 2, we present the details of the probabilistic model for the financial market, and setup the mean-variance hedging on random horizon.

In Section 3, we show how to construct the associated BSDEs via the martingale optimality principle and we state the two main theorems of this paper. The first one concerns the existence of a solution to the associated system of BSDEs and the second one is a verification theorem which gives an optimal strategy via the solution of the BSDEs. Then, Section 4 is dedicated to the proof of the existence of solution to the associated system of BSDEs.

Finally, some technical results are relegated to the appendix.

2 Preliminaries and market model

2.1 The probability space

Let (Ω,G,P) be a complete probability space. We assume that this space is equipped with a one-dimensional standard Brownian motion W and we denote by F:= (F_t)t≥0 the right continuous complete filtration generated by W. We also consider on this space a random time τ, which represents for example a default time in credit risk or in counterparty risk, or a death time in actuarial issues. The random timeτ is not assumed to be anF-stopping time. We therefore use in the sequel the standard approach of filtration enlargement by considering Gthe smallest right continuous extension of Fthat turns τ into aG-stopping time (see e.g. [15, 16]). More preciselyG:= (G_t)t≥0 is defined by

G_t := \

ε>0

G˜_t+ε ,

for all t≥0, where ˜G_s:=F_s∨σ(1τ≤u , u∈[0, s]), for all s≥0.

We denote by P(F) (resp. P(G)) the σ-algebra of F (resp. G)-predictable subsets of Ω×R+, i.e. theσ-algebra generated by the left-continuousF(resp. G)-adapted processes.

We now introduce a decomposition result forP(G)-measurable processes.

Proposition 2.1. Any P(G)-measurable process X= (Xt)t≥0 is represented as X_t = X_t^b1t≤τ+X_t^a(τ)1t>τ ,

for allt≥0, where X^b is P(F)-measurable and X^a isP(F)⊗ B(R+)-measurable.

This result is proved in Lemma 4.4 of [15] for bounded processes and is easily extended to the case of unbounded processes. For the sake of completeness, we detail its proof in the appendix.

Remark 2.1. In the case where the studied process X depends on another parameter x evolving in a Borel subset X of R^p, and if X is P(G) ⊗ B(X), then, decomposition given by Proposition 2.1 is still true but where X^b is P(F)⊗ B(X)-mesurable and X^a is P(F)⊗ B(R₊)⊗ B(X)-measurable. Indeed, it is obvious for the processes generating P(G)⊗ B(X) of the formXt(ω, x) =Lt(ω)R(x), (t, ω, x)∈R+×Ω× X, whereL isP(G)- measurable and R is B(X)-measurable. Then, the result is extended to any P(G)⊗ B(X) -measurable process by the monotone class theorem.

We then impose the following assumption, which is classical in the filtration enlargement theory.

(H) The processW remains aG-Brownian motion.

We notice that under(H), the stochastic integralRt

0 X_sdW_sis well defined for allP(G)- measurable processesX such thatRt

0|X_s|²ds <∞.

In the sequel we denote by N the process1τ≤. and we suppose

(Hτ) The process N admits an F-compensator of the form R.∧τ

0 λtdt, i.e. N −R.∧τ 0 λtdt is a G-martingale, whereλis a bounded P(F)-measurable process.

We then denote by M theG-martingale defined by M_t := N_t−

Z t∧τ 0

λ_sds ,

for all t≥0. We also introduce the processλ^G which is defined by λ^G_t := (1−N_t)λ_t. 2.2 Financial model

We consider a financial market model on the time interval [0, T] where 0 < T < ∞ is a finite time horizon. We suppose that the financial market is composed by a riskless bond with zero interest rate and a risky asset S. The price process (S_t)t≥0 of the risky asset is modeled by the linear stochastic differential equation

S_t = S₀+ Z t

0

S_s⁻(µ_sds+σ_sdW_s+β_sdM_s), ∀t∈[0, T], (2.1) whereµ,σandβareP(G)-measurable processes andS₀ is a positive constant. For example S could be a Credit Default Swap on the firm whose default time is τ. We impose the following assumptions on the coefficients µ,σ and β.

(HS)

(i) The processes µand σ are bounded: there exists a constant C >0 such that

|µ_t|+|σ_t| ≤ C , ∀t∈[0, T], P−a.s.

(ii) The processσ is uniformly elliptic: there exists a constantC >0 such that

|σ_t| ≥ C , ∀t∈[0, T], P−a.s.

(iii) There exists a constant C such that

−1 ≤ β_t ≤ C , ∀t∈[0, T], P−a.s.

Under (HS), we know from e.g. Theorem 1 in [9] that the process S defined by (2.1) is well defined.

2.3 Mean-variance hedging

We consider investment strategies which are P(G)-measurable processesπ such that Z T∧τ

0

|π_t|²dt < +∞, P−a.s.

This condition and(HS) ensure that the stochastic integral Rt 0

πr

S_r−dSr is well defined for such a strategyπ andt∈[0, T∧τ]. The wealth processV^x,πcorresponding to a pair (x, π), wherex∈Ris the initial amount, is defined by the stochastic integration

V_t^x,π := x+ Z t

0

π_r

S_r⁻dS_r, ∀t∈[0, T ∧τ]. We denote by Athe set of admissible strategies π such that

E

hZ T∧τ 0

|π_t|²dti

< ∞.

Forx∈R, the problem of mean-variance hedging consists in computing the quantity

π∈Ainf E h

V_T^x,π_∧τ −H

2i

, (2.2)

whereH is a bounded G_T∧τ-measurable random variable of the form

H = H^b1T <τ +H_τ^a1T≥τ , (2.3)

where H^b is an F_T-measurable random variable valued in R and H^a is a c`ad-l`ag P(F)- measurable process also valued in Rand such that

H^b

_∞ < ∞, and sup

t∈[0,T]

H_t^a

_∞ < ∞, (2.4)

where we recall thatk.k∞ is defined by kXk_∞ := infn

C≥0 : P |X| ≤C

= 1o , for any random variableX.

Since the problem we are interested in uses the values of the coefficients µ, σ and β only on the interval [0, T ∧τ], we can assume by Proposition 2.1 that µ, σ and β are P(F)-measurable and we shall do that in the sequel.

Remark 2.2. For simplicity, we have supposed that the riskless interest rate is equal to zero. However, all the results can be extended to the case of a bounded P(G)-measurable interest rate processr. Indeed, for such an interest rate process the mean-variance hedging problem becomes

π∈Ainf E h

V˜_T^x,π_∧τ−H˜

2i ,

where ˜V^x,π and ˜H are the discounted values of V^x,π and H given by H˜ := Hexp

− Z T∧τ

0

r_sds and

V˜_t^x,π := V_t^x,πexp

− Z t

0

rsds

, t∈[0, T].

From the dynamic of V^x,π we see that ˜V^x,π satisfies V˜_t^x,π = x+

Z t 0

πs µ˜sds+ ˜σsdWs+ ˜βsdMs

where

˜

µt := e⁻

Rt

0rsds(µt−rt), σ˜t := e⁻

Rt

0rsdsσt and β˜t := e⁻

Rt 0rsdsβt

fort∈[0, T]. In particular, we get the same model but with coefficients ˜µ, ˜σ and ˜β instead of µ,σ and β. Since ˜µ, ˜σ and ˜β also satisfy (HS), we can extend the results to this model with new coefficients.

We end this section by two examples of financial products taking the form (2.3).

Example 2.1(Insurance contract). Consider a seller of an insurance policy which protects the buyer over the time horizon [0, T] from some fixed lossL. Then if we denote byτ the time at which the loss appears, the losses of the seller are of the form

H = −p1T <τ+ (L−p)1T≥τ ,

wherep denotes the premium that the insurance policy holder pays at time 0.

Example 2.2 (Credit Default Swap with counterparty risk). Consider a protection seller who sells a CDS against a credit event to a protection buyer for a nominal N against a premium payments p with a maturity T. If the reference entity defaults, the protection seller pays the buyer the nominal N and the CDS contract is terminated. Moreover, both the buyer and seller of credit protection take on counterparty risk:

– the buyer takes the risk that the seller of credit protection may default, if the seller defaults the buyer loses its protection against default by the reference entity,

– the seller takes the risk that the buyer may default on the contract, depriving the seller of the expected revenue stream.

Denote byτ the first default time, and byξ the random variable such thatξ= 1 if the first default is the reference entity one and ξ = 0 otherwise. The losses of the seller are of the form

H = −pN T1T <τ +N1τ≤T,ξ=1−pNX^T

k=0

k1k≤τ <k+1

1τ≤T .

Example 2.3 (Credit contract). Consider a bank which lends an amount Ato a company over the period [0, T]. Suppose that the time horizon [0, T] is divided on n subintervals [k^T_n,(k+1)^T_n],k= 0, . . . , n−1, and that the interest rate of the loan over a time subinterval is r. The company has then to pay ^(1+r)_n ⁿA to the bank at each time k^T_n,k= 1, . . . , n. If we denote byτ the company default time, then the losses of the bank are given by

H = −((1 +r)ⁿ−1)A1T <τ +H_τ^a1T≥τ , where the functionH^a is given by

H_t^a = −

n−1

X

k=1

k(1 +r)ⁿ n −1

A1_k^T

n<t≤(k+1)^T

n , t∈[0, T].

3 Solution of the mean-variance problem by BSDEs

3.1 Martingale optimality principle

To find the optimal value of the problem (2.2), we follow the approach initiated by Hu et al. [12] to solve the exponential utility maximization problem in the pure Brownian case.

More precisely, we look for a family of processes n

J_t^π

t∈[0,T] : π∈ Ao satisfying the following conditions

(i) J_T^π_∧τ =

V_T^x,π_∧τ −H

2, for allπ ∈ A.

(ii) J₀^π1 =J₀^π2, for all π1, π2 ∈ A.

(iii) J_t^π

t∈[0,T] is aG-submartingale for all π∈ A.

(iv) There exists some π^∗ ∈ Asuch that J_t^π∗

t∈[0,T] is aG-martingale.

Under these conditions, we have

J₀^π∗ = inf

π∈AE h

V_T^x,π_∧τ−H

2i .

Indeed, using (i), (iii) and Doob’s optional stopping theorem, we have J₀^π ≤ E

J_T^π_∧τ

= E h

V_T^x,π_∧τ−H

2i

, (3.5)

for all π∈ A. Then, using (i), (iv) and Doob’s optional stopping theorem, we have J₀^π∗ = E

h

V_T^x,π_∧τ^∗−H

2i

. (3.6)

Therefore, from (ii), (3.5) and (3.6), we get for anyπ∈ A E

h

V_T^x,π_∧τ^∗−H

2i

= J₀^π∗ = J₀^π ≤ E h

V_T^x,π_∧τ−H

2i . We can see that

J₀^π∗ = inf

π∈AE h

V_T^x,π_∧τ−H

2i .

3.2 Related BSDEs

We now construct a family{(J_t^π)_t∈[0,T], π∈ A}satisfying the previous conditions by using BSDEs as in [12]. To this end, we define the following spaces.

– S^∞

G is the subset ofR-valued c`ad-l`agG-adapted processes (Yt)t∈[0,T]essentially bounded kYk_S∞ :=

sup

t∈[0,T]

|Y_t|

_∞ < ∞.

– S^∞,+

G is the subset of S^∞

G of processes (Yt)_t∈[0,T] valued in (0,∞), such that

1 Y

_S∞ < ∞. – L²

G is the subset of R-valued P(G)-measurable processes (Z_t)_t∈[0,T] such that kZk_L2 :=

E hZ T

0

|Z_t|²dti¹₂

< ∞.

– L²(λ) is the subset ofR-valued P(G)-measurable processes (U_t)_t∈[0,T]such that kUk_L2(λ) :=

E

hZ T∧τ 0

λs|U_s|²ds i¹₂

< ∞.

To construct a family {(J_t^π)_t∈[0,T], π∈ A}satisfying the previous conditions, we set J_t^π = Yt

V_t∧τ^x,π− Y_t

2+ Υt, t∈[0, T], where¹ (Y, Z, U) is solution inS^∞,+

G ×L²_G×L²(λ) to Yt = 1 +

Z T∧τ t∧τ

f(s, Ys, Zs, Us)ds− Z T∧τ

t∧τ

ZsdWs− Z T∧τ

t∧τ

UsdMs, t∈[0, T], (3.7) (Y,Z,U) is solution inS^∞

G ×L²

G×L²(λ) to Y_t = H+

Z T∧τ t∧τ

g(s,Y_s,Z_s,U_s)ds− Z T∧τ

t∧τ

Z_sdW_s− Z T∧τ

t∧τ

U_sdM_s, t∈[0, T], (3.8) and (Υ,Ξ,Θ) is solution inS^∞

G ×L²

G×L²(λ) to Υt =

Z T∧τ t∧τ

h(s,Υs,Ξs,Θs)ds− Z T∧τ

t∧τ

ΞsdWs− Z T∧τ

t∧τ

ΘsdMs, t∈[0, T]. (3.9) Remark 3.3. We notice that the jump components U, U and Θ are also bounded since Y,Y and Υ are inS^∞

G . Indeed, letC be a constant such that

kYk_S∞ ≤ C . (3.10)

Then since Y.−+U isG-predictable, we have E

hZ T 0

1|Y_t−+Ut|>Cλ^G_tdti

= E

hZ T 0

1|Y_t−+Ut|>CdN_ti

= E

h

1|Y_τ−+Uτ|>C,τ≤T

i

= E

h

1|Yτ|>C,τ≤T

i

= 0.

Therefore, we have |Y_.−+U| ≤ C inL²(λ). From (3.10) we get |U| ≤2C in L²(λ). The same argument can be applied for U and Θ.

1As commonly done for the integration w.r.t. jump processes, the integralRb

a stands forR

(a,b].

In these terms, we are bound to choose three functions f, g and h for which J^π is a submartingale for all π ∈ A, and there exists a π^∗ ∈ A such that J^π∗ is a martingale. In order to calculate f, g and h, we write J^π as the sum of a (local) martingale M^π and an (not strictly) increasing processK^π that is constant for someπ^∗ ∈ A.

To alleviate the notation we writef(t) (resp. g(t),h(t)) forf(t, Y_t, Z_t, U_t) (resp. g(t,Y_t,Z_t,U_t), h(t,Υt,Ξt,Θt)) fort∈[0, T].

Define for eachπ ∈ Athe processX^π by

X_t^π := V_t∧τ^x,π− Y_t, t∈[0, T]. From Itˆo’s formula, we get

dJ_t^π = dM_t^π+dK_t^π , (3.11)

whereM^π and K^π are defined by dM_t^π := n

2X_t^π−(π_tβ_t− U_t)(Y_t⁻+U_t) +|π_tβ_t− U_t|²(Y_t⁻+U_t) +|X_t^π−|²U_t+ Θ_to dM_t +n

2Y_tX_t^π(π_tσ_t− Z_t) +Z_t|X_t^π|²+ Ξ_to dW_t,

dK_t^π := n Y_t

2X_t^π(π_tµ_t+g(t)) +|π_tσ_t− Z_t|²

− |X_t^π|²f(t) + 2X_t^πZ_t(π_tσ_t− Z_t) + 2λ^G_tX_t^πUt(πtβt− U_t) +λ_t^G|π_tβt− U_t|²(Ut+Yt)−h(t)

o dt . We then writedK^π in the following form

dK_t^π = Kt(πt)dt , whereK is defined by

Kt(π) := At|π|²+Btπ+Ct, π ∈R, t∈[0, T], with

A_t := |σ_t|²Y_t+λ_t^G|β_t|²(U_t+Y_t),

B_t := 2X_t^π(µ_tY_t+σ_tZ_t+λ^G_tβ_tU_t)−2σ_tY_tZ_t−2λ^G_tβ_tU_t(Y_t+U_t),

C_t := −f(t)|X_t^π|²+ 2X_t^π(Y_tg(t)−Z_tZ_t−λ^G_tU_tU_t) +Y_t|Z_t|²+λ^G_t|U_t|²(U_t+Y_t)−h(t), for all t ∈ [0, T]. To ensure that K^π is nondecreasing for any π ∈ A and that K^π∗ is constant for some π^∗ ∈ A, we take Kt such that minπ∈RKt(π) = 0. Using Y ∈ S^∞,+

G and (HS) (ii), we then notice that A_t>0 for allt∈[0, T]. Indeed, we have

0 = E[[Y_τ]⁻1τ≤T] = E[[Y_τ⁻+U_τ]⁻1τ≤T] = E hZ T

0

[Y_s⁻+U_s]⁻dN_si , therefore we get that

E hZ T

0

[Y_s⁻+Us]⁻dMs+ Z T

0

[Ys+Us]⁻λ^G_sds i

= 0.

From Remark 3.3, the predictable process [Y_.⁻+U]⁻ is bounded. Thus we get that the first integral on the left is a true martingale thus we have

E hZ T

0

[Y_s+U_s]⁻λ^G_sdsi

= 0, (3.12)

which gives (Y_s+U_s)λ^G_s ≥ 0 for s ∈[0, T]. Therefore, the minimum of K_t over π ∈R is given by

K_t := min

π∈R

Kt(π) = Ct−|B_t|² 4At

. We then obtain from the expressions ofA,B and C that

K_t = A_t|X_t^π|²+B_tX_t^π+C_t, with

A_t := −f(t)− |µ_tYt+σtZt+λ^G_tβtUt|²

|σ_t|²Y_t+λ^G_t|β_t|²(U_t+Y_t) , B_t := 2

n(µ_tY_t+σ_tZ_t+λ^G_tβ_tU_t)(λ^G_tβ_tU_t(Y_t+U_t) +σ_tY_tZ_t)

|σ_t|²Yt+λ_t^G|β_t|²(Ut+Yt) +g(t)Yt−ZtZ_t−λ^G_tUtU_to , C_t := −h(t) +|Z_t|²Y_t+λ^G_t(U_t+Y_t)|U_t|²−|σ_tY_tZ_t+λ^G_tβ_tU_t(U_t+Y_t)|²

|σ_t|²Y_t+λ^G_t |β_t|²(U_t+Y_t) .

For that the family (J^π)π∈A satisfies the conditions (iii) and (iv) we choosef,gandhsuch that

A_t = 0, B_t = 0 and C_t = 0,

for all t∈[0, T]. This leads to the following choice for the drivers f,g and h



























f(t, y, z, u) := − |µ_ty+σtz+λ^G_tβtu|²

|σ_t|²y+λ_t^G|β_t|²(u+y) , g(t, y, z, u) := _Y¹

t

h

Ztz+λ^G_tUtu−(µtYt+σtZt+λ^G_tβtUt)(σtYtz+λ^G_tβt(Ut+Yt)u)

|σ_t|²Yt+λ^G_t|β_t|²(Ut+Yt)

i ,

h(t, y, z, u) := |Z_t|²Yt+λ^G_t(Ut+Yt)|U_t|²−|σ_tYtZ_t+λ^G_tβtU_t(Ut+Yt)|²

|σ_t|²Yt+λ^G_t|β_t|²(Ut+Yt) .

We then notice that the obtained system of BSDEs is not fully coupled, which allows to study each BSDE alone as soon as we start from the BSDE (f,1)² and end with the BSDE (h,0). However the obtained generators are nonstandard since they involve the jump component and they are not Lipschitz continuous. Moreover, these generators are not defined on the whole space R×R×R. Using a decomposition approach based on Proposition 2.1, we obtain the following result whose proof is detailed in Section 4.

Theorem 3.1. The BSDEs (3.7), (3.8)and (3.9)admit solutions (Y, Z, U), (Y,Z,U) and (Υ,Ξ,Θ) in S^∞

G ×L²_G×L²(λ). MoreoverY ∈ S^∞,+

G .

2The notation BSDE (f, H) holds for the BSDE with generatorf and terminal conditionH.

3.3 A verification Theorem

We now turn to the sufficient condition of optimality. As explained in Subsection 3.1, a candidate to be an optimal strategy is a process π^∗ ∈ A such that J^π∗ is a martingale, which implies that dK^π∗ = 0. This leads to

π_t^∗ = arg min

π∈R

K_t(π), which gives the implicit equation inπ^∗

π_t^∗ = Y_t−−V_t^x,π− ^∗

µ_tY_t⁻+σ_tZ_t+λ^G_tβ_tU_t

|σ_t|²Y_t⁻+λ^G_t|β_t|²(U_t+Y_t⁻) +σ_tY_t⁻Z_t+λ^G_tβ_tU_t(Y_t⁻+U_t)

|σ_t|²Y_t⁻+λ^G_t|β_t|²(U_t+Y_t⁻) . Integrating each side of this equality w.r.t. _S^dSt

t− leads to the following SDE V_t^∗ = x+

Z _t

0

Y_r−−V_r^∗−

µrY_r⁻+σrZr+λ^G_rβrUr

|σ_r|²Y_r⁻+λ^G_r|β_r|²(U_r+Y_r⁻) dSr

S_r⁻ (3.13)

+ Z t

0

σrY_r⁻Z_r+λ^G_rβrU_r(Y_r⁻+Ur)

|σ_r|²Y_r⁻+λ^G_r|β_r|²(Ur+Y_r⁻) dSr

S_r⁻ , t∈[0, T ∧τ]. We first study the existence of a solution to SDE (3.13).

Proposition 3.2. The SDE (3.13) admits a solution V^∗ which satisfies E

h sup

t∈[0,T∧τ]

|V_t^∗|²i

< ∞. (3.14)

Proof. To alleviate the notation we rewrite (3.13) under the form (

V₀^∗ = x ,

dV_t^∗ = (E_tV_t^∗−−F_t)(µ_tdt+σ_tdW_t+β_tdM_t), (3.15) whereE and F are defined by

E_t := − µ_tY_t⁻+σ_tZ_t+λ^G_tβ_tU_t

|σ_t|²Y_t⁻+λ^G_t|β_t|²(U_t+Y_t⁻) ,

F_t := −λ^G_tβ_tU_t(Y_t⁻+U_t) +µ_tY_t⁻Y_t−+λ^G_tβ_tU_tY_t−+σ_tZ_tY_t−+σ_tZ_tY_t⁻

|σ_t|²Y_t⁻+λ^G_t|β_t|²(Ut+Y_t⁻) , for all t∈[0, T]. We first notice that from (HS) (ii), and sinceY ∈ S^∞,+

G and λ^G(Y +U) is nonnegative, there exists a constantC >0 such that

|σ_t|²Yt+λ^G_t|β_t|²(Ut+Y_t⁻) ≥ C , P⊗dt−a.e.

Therefore, using (Y, Z, U), (Y,Z,U), (Υ,Ξ,Θ)∈ S^∞

G ×L²_G×L²(λ), Remark 3.3 and (HS), we get that E and F are square integrable

E hZ T

0

|E_t|²+|F_t|² dt

i

< ∞.

Using Itˆo’s formula, we obtain that the process V^∗ defined by

V_t^∗ := (x+ Ψ_t)Φ_t, t∈[0, T ∧τ), (3.16) and V_T^∗_∧τ = 1τ≤T

(1 +E_τβ_τ)V_τ^∗−−F_τβ_τ

+1τ >T(x+ Ψ_T)Φ_T , where

Φt := exp Z t

0

Es(µs−λ^G_sβs)−1

2|σ_sEs|² ds+

Z t 0

σsEsdWs

, and

Ψ_t := − Z t

0

F_s Φ_s h

µ_s−λ^G_sβ_s− |E_sσ_s|²i ds−

Z t 0

F_s

Φ_sσ_sdW_s, for all t∈[0, T], is solution to (3.13).

We now prove that V^∗ defined by (3.16) satisfies (3.14). We proceed in two steps.

Step 1: We prove that

E h

|V_T^∗_∧τ|²i

< ∞. (3.17)

Since V^∗ satisfies (3.15), we have V^∗ =V^x,π∗ whereπ^∗ is given by π_t^∗ = E_tV_t−^∗ −F_t, t∈[0, T].

We therefore have Y|V_.∧τ^∗ − Y|² = J^π∗−Υ and from (3.11) and the dynamics of Υ given by (3.9), we have

d Yt|V_t∧τ^∗ − Y_t|²

= dM_t^∗+dK_t^π∗−h(t)dt

where M^∗ is a locally square integrable martingale with M₀^∗ = 0. From the definition of K^π∗ and using the fact that

π_t^∗ = X_t^π−^∗(µ_tY_t⁻+σ_tZ_t+λ^G_tβ_tU_t) +σ_tY_t⁻Z_t+λ_t^Gβ_tU_t(Y_t⁻+U_t)

|σ_t|²Y_t⁻+λ^G_t|β_t|²(U_t+Y_t⁻) , we getK_t^π∗ = 0 for all t∈[0, T ∧τ]. Therefore, from the definition of h, we get

Y_T∧τ|V_T^∗_∧τ − Y_T∧τ|² = Y₀|x− Y₀|²+M_T^∗_∧τ + Z T∧τ

0

h|Z_t|²Y_t+λ_t^G(U_t+Y_t)|U_t|²

− |σ_tY_tZ_t+λ^G_tβ_tU_t(U_t+Y_t)|²

|σ_t|²Yt+λ^G_t|β_t|²(Ut+Yt) i

dt .

Since M^∗ is a local martingale, there exists an increasing sequence ofG-stopping times (ν_i)i∈Nsuch that ν_i→+∞ asi→ ∞ and

E

Y_T∧τ∧ν_i|V_T^∗_∧τ∧ν

i− Y_T∧τ∧νi|²

= Y₀|x− Y₀|²+E

Z _T∧τ∧νi

0

h|Z_t|²Y_t+λ_t^G(U_t+Y_t)|U_t|²

−|σ_tY_tZ_t+λ^G_tβ_tU_t(U_t+Y_t)|²

|σ_t|²Y_t+λ^G_t|β_t|²(U_t+Y_t) i

dt . (3.18)

Since Y ∈ S^∞,+

G , there exists a positive constantC such that E

|V_T^∗_∧τ∧ν

i − Y_T∧τ∧νi|²

≤ CE

Y_T∧τ∧ν_i|V_T^∗_∧τ∧ν

i− Y_T∧τ∧νi|² . Therefore, using (3.18), we get that

E

|V_T^∗_∧τ∧ν

i− Y_T∧τ∧νi|²

≤ C

Y₀|x− Y₀|²+E Z T

0

h|Z_t|²Y_t+λ_t^G(U_t+Y_t)|U_t|²i dt

. SinceY,U and U are uniformly bounded andZ ∈L²

G, there exists a constantC such that E

V_T^∗_∧τ∧νi− Y_T∧τ∧νi

2

≤ C . From Fatou’s lemma, we get that

E

V_T^∗_∧τ− Y_T∧τ

2

≤ lim

i→inf

∞ E

V_T^∗_∧τ∧νi− Y_T∧τ∧ν_i

2

≤ C . Which implies that

E V_T^∗_∧τ

2

≤ C+ 2E

V_T^∗_∧τY_T∧τ .

Finally, using the Young inequality and noting thatY is uniformly bounded, it follows that there exists a constant C such that

E V_T^∗_∧τ

2

≤ C . Step 2: We prove that

E h

sup

t∈[0,T∧τ]

|V_t^∗|²i

< ∞.

For that we remark thatV_.∧τ^∗ is solution to the following linear BSDE V_t∧τ^∗ = V_T^∗_∧τ −

Z T∧τ t∧τ

µ_s σs

z_sds− Z T∧τ

t∧τ

z_sdW_s− Z T∧τ

t∧τ

u_sdM_s, t∈[0, T], (3.19) with

z_t := σ_t(Y_t−−V_t^∗−)(µtY_t⁻+σtZt+λ^G_tβtUt) +σtY_t⁻Z_t+λ^G_tβtU_t(Y_t⁻+Ut)

|σ_t|²Y_t⁻+λ^G_t|β_t|²(U_t+Y_t⁻) , u_t := β_t(Y_t−−V_t^∗−)(µ_tY_t⁻+σ_tZ_t+λ^G_tβ_tU_t) +σ_tY_t⁻Z_t+λ^G_tβ_tU_t(Y_t⁻+U_t)

|σ_t|²Y_t⁻+λ^G_t|β_t|²(U_t+Y_t⁻) , for all t∈[0, T]. Applying Itˆo’s formula to |V^∗|² we have

E|V_t∧τ^∗ |² = E|V_T^∗_∧τ|²−2E Z T∧τ

t∧τ

V_s∧τ^∗ µ_s σs

z_sds−E Z T∧τ

t∧τ

|z_s|²ds−E Z T∧τ

t∧τ

|u_s|²λ_sds , for allt∈ [0, T]. Using (3.17), (HS) and the Young inequality we obtain the existence of a constant C such that

E|V_t∧τ^∗ |²+E Z T∧τ

t∧τ

|z_s|²ds+E Z T∧τ

t∧τ

|u_s|²λsds ≤ C

1 +E Z T

t

|V_s∧τ^∗ |² .

We then deduce from the Gronwall inequality that sup

t∈[0,T]

E|V_t∧τ^∗ |²+E Z T∧τ

0

|z_s|²ds+E Z T∧τ

0

|u_s|²λsds < +∞. (3.20) Now from (3.19), we have

E h

sup

t∈[0,T]

|V_t∧τ^∗ |²i

≤ 3

|V₀^∗|²+E h

sup

t∈[0,T]

Z t∧τ 0

µ_s σ_sz_sds

2i

+E h

sup

t∈[0,T]

Z t∧τ 0

z_sdW_s+ Z t∧τ

0

u_sdM_s

2i .

From(HS) and the BDG inequality, there exists a constantC such that E

h sup

t∈[0,T]

|V_t∧τ^∗ |²i

≤ C

1 +E Z T∧τ

0

|z_s|²ds+E Z T∧τ

0

|u_s|²λsds

.

This last inequality with (3.20) gives (3.14).

As explained previously, we now consider the strategyπ^∗ defined by

π^∗_t = (Y_t⁻−V_t^∗−)(µtY_t⁻+σtZt+λ^G_tβtUt) +σtY_t⁻Z_t+λ^G_tβtU_t(Y_t⁻+Ut)

|σ_t|²Y_t⁻+λ^G_t|β_t|²(Ut+Y_t⁻) , (3.21) for all t∈[0, T]. We first notice from the expressions of π^∗ and V^∗ that

V_t^x,π∗ = V_t^∗ , (3.22)

for all t∈[0, T]. Using (3.14) and (3.22), we have E

h sup

t∈[0,T∧τ]

|V_t^x,π∗|²i

< ∞. (3.23)

We can now state our verification theorem which is the main result of this section.

Theorem 3.2. The strategy π^∗ given by (3.21) belongs to the setA and is optimal for the mean-variance problem (2.2). Thus we have

E h

V_T^x,π_∧τ^∗−H

2i

= min

π∈AE h

V_T^x,π_∧τ−H

2i

= Y₀|x− Y₀|²+ Υ₀ , where Y,Y and Υare solutions to (3.7)-(3.8)-(3.9).

To prove this verification theorem, we first need of the following lemma.

Lemma 3.1. For any π ∈ A, the process M_.∧τ^π defined by (3.11)is a G-local martingale.

Proof. Fix π ∈ A. Then from the definition of V^x,π, (HS) and the BDG inequality, we have

E h

sup

t∈[0,T]

V_t∧τ^x,π

2i

< ∞. (3.24)