On the uniqueness of solutions to quadratic BSDEs with convex generators and unbounded terminal conditions

www.imstat.org/aihp 2011, Vol. 47, No. 2, 559–574

DOI:10.1214/10-AIHP372

© Association des Publications de l’Institut Henri Poincaré, 2011

On the uniqueness of solutions to quadratic BSDEs with convex generators and unbounded terminal conditions

Freddy Delbaen

, Ying Hu

and Adrien Richou

aDepartment of Mathematics, ETH-Zentrum, HG G 54.3, CH-8092 Zürich, Switzerland. E-mail:delbaen@math.ethz.ch bIRMAR, Université Rennes 1, Campus de Beaulieu, 35042 RENNES Cedex, France.

E-mails:ying.hu@univ-rennes1.fr;adrien.richou@univ-rennes1.fr Received 8 October 2009; revised 20 April 2010; accepted 23 April 2010

Abstract. In [Probab. Theory Related Fields141(2008) 543–567], the authors proved the uniqueness among the solutions of quadratic BSDEs with convex generators and unbounded terminal conditions which admit every exponential moments. In this paper, we prove that uniqueness holds among solutions which admit some given exponential moments. These exponential moments are natural as they are given by the existence theorem. Thanks to this uniqueness result we can strengthen the nonlinear Feynman–

Kac formula proved in [Probab. Theory Related Fields141(2008) 543–567].

Résumé. Les auteurs de l’article [Probab. Theory Related Fields141(2008) 543–567] ont prouvé un résultat d’unicité pour les so- lutions d’EDSRs quadratiques de générateur convexe et de condition terminale non bornée ayant tous leurs moments exponentiels finis. Dans ce papier, nous prouvons que ce résultat d’unicité reste vrai pour des solutions qui admettent uniquement certains mo- ments exponentiels finis. Ces moments exponentiels sont reliés de manière naturelle à ceux présents dans le théorème d’existence.

À l’aide de ce résultat d’unicité nous pouvons améliorer la formule de Feynman–Kac non linéaire prouvée dans [Probab. Theory Related Fields141(2008) 543–567].

MSC:60H10

Keywords:Backward stochastic differential equations; Generator of quadratic growth; Unbounded terminal condition; Uniqueness result;

Nonlinear Feynman–Kac formula

1. Introduction

In this paper, we consider the following quadratic backward stochastic differential equation (BSDE in short for the remaining of the paper)

Y_t=ξ− _T

t

g(s, Y_s, Z_s)ds+ _T

t

Z_sdWs, 0≤t≤T , (1.1)

where the generator−gis a continuous real function that is concave and has a quadratic growth with respect to the variablez. Moreover,ξ is an unbounded random variable (see, e.g., [8] for the case of quadratic BSDEs with bounded terminal conditions). Let us recall that, in the previous equation, we are looking for a pair of processes(Y, Z)which is required to be adapted with respect to the filtration generated by the R^d-valued Brownian motionW. In [4], the authors prove the uniqueness among the solutions which satisfy for anyp >0,

E

e^{psup0≤t≤T|Yt|}

<∞.

The main contribution of this paper is to strengthen their uniqueness result. More precisely, we prove the uniqueness among the solutions satisfying:

∃p >γ ,¯ ∃ε >0 E

e^{psup0≤t≤T(Yt−+}

t

0α¯sds)+e^{εsup0≤t≤TYt+}

<+∞,

whereγ >¯ 0 and(α¯t)t∈[0,T]is a progressively measurable nonnegative stochastic process such that,P-a.s.,

∀(t, y, z)∈ [0, T] ×R×R^1×d g(t, y, z)≤ ¯α_t+ ¯β|y| +γ¯ 2|z|².

Our method is different from that in [4] where the authors apply the so-calledθ-difference method, i.e. estimating Y¹−θ Y², for θ∈(0,1), and then lettingθ→1. Whereas in this paper, we apply a verification method: first we define a stochastic control problem and then we prove that the first component of any solution of the BSDE is the optimal value of this associated control problem. Thus the uniqueness follows immediately. Moreover, using this representation, we are able to give a probabilistic representation of the following PDE:

∂tu(t, x)+Lu(t, x)−g

t, x, u(t, x),−σ^∗∇xu(t, x)

=0, u(T ,·)=h,

wherehandghave a “not too high” quadratic growth with respect to the variablex. We remark that the probabilistic representation is also given by [4] under the condition thathandgare subquadratic, i.e.:

∀(t, x, y, z)∈ [0, T] ×R^d×R×R^1×d h(x)+g(t, x, y, z)≤f (t, y, z)+C|x|^p withf ≥0,C >0 andp <2.

The paper is organized as follows. In Section2, we prove an existence result in the spirit of [3] and [4]: here we work with generators−gsuch thatg⁻has a linear growth with respect to variablesyandz. As in part 5 of [3], this assumption allows us to reduce hypothesis of [4]. Section3is devoted to the optimal control problem from which we get as a byproduct a uniqueness result for quadratic BSDEs with unbounded terminal conditions. Finally, in the last section we derive the nonlinear Feynman–Kac formula in this framework.

Let us close this introduction by giving the notations that we will use in all the paper. For the remaining of the paper, let us fix a nonnegative real numberT >0. First of all,(W_t)_t∈[0,T]is a standard Brownian motion with values inR^d defined on some complete probability space(Ω,F,P).(Ft)t≥0 is the natural filtration of the Brownian motionW augmented by theP-null sets ofF. The sigma-field of predictable subsets of[0, T] ×Ωis denoted byP.

As mentioned before, we will deal only with real valued BSDEs which are equations of type (1.1). The function

−gis called the generator andξ the terminal condition. Let us recall that a generator is a random function[0, T] × Ω×R×R^1×d→Rwhich is measurable with respect toP⊗B(R)⊗B(R^1×d)and a terminal condition is simply a realFT-measurable random variable. By a solution to the BSDE (1.1) we mean a pair(Yt, Zt)t∈[0,T]of predictable processes with values inR×R^1×dsuch thatP-a.s.,t→Y_tis continuous,t→Z_tbelongs toL²(0, T ),t→g(t, Y_t, Z_t) belongs toL¹(0, T )andP-a.s.(Y, Z)verifies (1.1). We will sometimes use the notation BSDE(ξ, f) to say that we consider the BSDE whose generator isf and whose terminal condition isξ.

For any realp≥1,S^pdenotes the set of real-valued, adapted and càdlàg processes(Yt)t∈[0,T]such that Y_S^p:=E sup

0≤t≤T

|Y_t|^p1/p

<+∞.

M^pdenotes the set of (equivalent class of) predictable processes(Zt)t∈[0,T]with values inR^1×d such that ZM^p:=E

_T

0

|Z_s|²ds

p/21/p

<+∞.

Finally, we will use the notationY^∗:=sup_0≤t≤T|Y_t| and we recall thatY belongs to the class (D) as soon as the family{Yτ: τ ≤T stopping time}is uniformly integrable.

2. An existence result

In this section, we prove a slight modification of the existence result for quadratic BSDEs obtained in [4] by using a method applied in Section 5 of [3]. We consider here the case whereg⁻has a linear growth with respect to variablesy andz. Let us assume the following on the generator.

Assumption A.1. There exist three constants β ≥0, γ >0 and r ≥0 together with two progressively measur- able nonnegative stochastic processes(α¯t)0≤t≤T,(α_t)0≤t≤T and a deterministic continuous nondecreasing function φ:R⁺→R⁺withφ(0)=0such that,P-a.s.:

(1) for allt∈ [0, T],(y, z)→g(t, y, z)is continuous;

(2) monotonicity iny:∀(t, y, z)∈ [0, T] ×R×R^1×d, y

g(t,0, z)−g(t, y, z)

≤β|y|²;

(3) growth condition:∀(t, y, z)∈ [0, T] ×R×R^1×d,

−α_t−r

|y| + |z|

≤g(t, y, z)≤ ¯αt+φ

|y| +γ

2|z|². Theorem 2.1. LetA.1hold.If there existsp >1such that

E

exp

γe^βTξ⁻+γ _T

0

¯ α_te^βtdt

+

ξ⁺p

+ _T

0

α_tdt p

<+∞

then the BSDE(1.1)has a solution(Y, Z)such that

−1 γlogE

exp

γe^{β(T−t )}ξ⁻+γ _T

t

¯

α_re^{β(r−t )}dr Ft

≤Y_t≤Ce^CT

E ξ⁺p

+ _T

t

α_rdr pFt

1/p

, withCa constant that does not depend onT.

Proof. We will fit the proof of Proposition 4 in [3] to our situation. Without loss of generality, let us assume thatris an integer. For each integern≥r, let us consider the function

gn(t, y, z):=inf

g(t, p, q)+n|p−y| +n|q−z|, (p, q)∈Q^1+d .

gn is well defined and it is globally Lipschitz continuous with constantn. Moreover,(gn)n≥r is increasing and con- verges pointwise tog. Dini’s theorem implies that the convergence is also uniform on compact sets. We have also, for alln≥r,

h(t, y, z):= −α_t−r

|y| + |z|

≤g_n(t, y, z)≤g(t, y, z).

Let(Yⁿ, Zⁿ)be the unique solution inS^p×M^pto BSDE(ξ,−g_n). It follows from the classical comparison theorem that

Yⁿ⁺¹≤Yⁿ≤Y^r. Let us prove that for eachn≥r

Y_tⁿ≥ −1 γlogE

exp

γe^{β(T−t )}ξ⁻+γ _T

t α¯_re^{β(r−t )}dr Ft

:=X_t.

Let (Y˜ⁿ,Z˜ⁿ)be the unique solution in S^p×M^p to BSDE(−ξ⁻,−g_n⁺). It follows from the classical comparison theorem thatY˜ⁿ≤YⁿandY˜ⁿ≤0. Then, according to Proposition 3 in [4], we haveY˜ⁿ≥Xand soYⁿ≥Xfor all

n≥r. We setY =infn≥rYⁿand, arguing as in the proof of Proposition 3 in [4] or Theorem 2 in [3] with a localization argument, we construct a processZsuch that(Y, Z)is a solution to BSDE(ξ,−g). For the upper bound, let(Y ,¯ Z)¯ be the unique solution inS^p×M^pto BSDE(ξ⁺,−h). Then the classical comparison theorem gives us thatY ≤Yⁿ≤ ¯Y and we apply a classical a priori estimate forL^psolutions of BSDEs in [2] toY¯. Corollary 2.2. LetA.1hold.We suppose thatξ⁻+T

0 α¯_tdt has an exponential moment of orderγe^βT and there existsp >1such thatξ⁺+T

0 α_tdt∈L^p.

• Ifξ⁻+_T

0 α¯tdthas an exponential moment of orderqe^βT withq > γ then the BSDE(1.1)has a solution(Y, Z) such thatE[e^qA∗]<+∞withA_t :=Y_t⁻+t

0α¯_sds.

• Ifξ⁺+T

0 α_tdt has an exponential moment of orderεCe^CT,withC given in Theorem2.1,then the BSDE(1.1) has a solution(Y, Z)such thatE[e^ε(Y+)∗]<+∞.

Proof. Let us apply the existence result: BSDE (1.1) has a solution(Y, Z)and we have A_t=Y_t⁻+

_t

0

¯

α_sds≤ 1 γlogE

exp

γe^βT

ξ⁻+

_T

0

¯ α_rdr

Ft

:=Mt

.

So e^qAt ≤(Mt)^q/γ withq/γ >1. SinceM^q/γ is a submartingale, we are able to apply the Doob’s maximal inequality to obtain

E e^qA∗

≤CqE

e^{qeβT(ξ−+0Tα¯sds)}

<+∞. To prove the second part of the corollary, we define

Nt:=E ξ⁺p

+ _T

0

α_sds ^pFt

.

We set q >1. There exists C_C,ε,p,q ≥ 0 such that x →e^CeCTx1/pε/q is convex on [C_C,ε,p,q,+∞[. We have e^ε/qYt+≤e^{CeCT(CC,ε,p,q+Nt)1/pε/q}.Since e^{(CC,ε,p,q+N )1/pε/q} is a submartingale, we are able to apply the Doob’s maxi- mal inequality to obtain

E e^ε(Y+)∗

≤CE

e^{εCeCT(CC,ε,p,q+(ξ+)p+(}

T

0 α_sds)^p)^1/p

≤CE

e^εCeCT(ξ++

T 0 α_sds)

<+∞.

3. A uniqueness result

To prove our uniqueness result for the BSDE (1.1), we will introduce a stochastic control problem. For this purpose, we use the following assumption ong:

Assumption A.2. There exist three constantsKg,y≥0,β¯≥0andγ >¯ 0 together with a progressively measurable nonnegative stochastic process(α¯t)t∈[0,T]such that,P-a.s.,

• for each(t, z)∈ [0, T] ×R^1×d, g(t, y, z)−g

t, y, z≤K_g,yy−y ∀ y, y

∈R²;

• for each(t, y, z)∈ [0, T] ×R×R^1×d, g(t, y, z)≤ ¯α_t+ ¯β|y| +γ¯

2|z|²;

• for each(t, y)∈ [0, T] ×R,z→g(t, y, z)is a convex function.

Sinceg(t, y,·)is a convex function we can define the Legendre–Fenchel transformation ofg:

f (t, y, q):= sup

z∈R^1×d

zq−g(t, y, z)

∀t∈ [0, T], q∈R^d, y∈R.

f is a function with values inR∪ {+∞}that verifies direct properties.

Proposition 3.1.

• ∀(t, y, y, q)∈ [0, T] ×R×R×R^d such thatf (t, y, q) <+∞, f

t, y, q

<+∞ and f (t, y, z)−f

t, y, z≤Kg,yy−y;

• f is a convex function inq;

• f (t, y, q)≥ − ¯αt− ¯β|y| +₂¹_γ¯|q|².

Forε >0 andp >γ¯ given we setN∈N^∗such that T

N <

1

¯ γ −1

p

1

β(1/p¯ +1/ε). (3.1)

Fori∈ {0, . . . , N}we definet_i:=^iT_N and, for all realFt_i+1-measurable random variableη, At_i,t_i+1(η):=

(q_s)_{s∈[ti,ti+1]}: _ti+1

t_i |q_s|²ds <+∞P-a.s.,E^Qi t_i+1

t_i |q_s|²ds

<+∞, M_tⁱ

t∈[ti,ti+1]is a martingale,E^Qi

|η| + _ti+1

ti

f (s,0, qs)ds

<+∞,

withM_tⁱ:=exp t

ti

q_sdWs−1 2

t ti

|q_s|²ds

anddQⁱ dP :=M_tⁱ

i+1

.

Let q be inAt_i,t_i+1(η), if this set is not empty. We define dW_t^q:=dWt −qtdt. Thanks to the Girsanov theorem, (W_t^q

i+h−W_t^q_i)h∈[0,T /N]is a Brownian motion under the probabilityQⁱ. So, we are able to apply Proposition 6.4 in [2]

to obtain this existence result:

Proposition 3.2. There exist two processes(Y^η,q, Z^η,q)such that(Y_t^η,q)_{t∈[ti,ti+1]}belongs to the class(D)underQⁱ, _ti+1

t_i |Z_s^η,q|²ds <+∞P-a.s.,_ti+1

t_i |f (s, Y_s^η,q, qs)|ds <+∞P-a.s.and Y_t^η,q=η+

_ti+1

t

f

s, Ys^η,q, qs

ds+ _ti+1

t

Z^η,qs dWs^q, ti≤t≤ti+1. We are now able to define the admissible control set:

A:=

(q_s)_s∈[0,T]: q_|[tN−1,T]∈At_N−1,T(ξ ),∀i∈ {N−2, . . . ,0}, q_|[ti,ti+1]∈At_i,t_i+1

Y_t^q

i+1

withY_t^q_i+1:=Y^Y

q

ti+2,q_|[ti+1,ti+2]

t_i+1 andY_T^q:=ξ

.

Ais well defined by a decreasing recursion oni∈ {0, . . . , N−1}. Forq∈Awe can define our cost functionalY^qon [0, T]by

∀i∈ {N−1, . . . ,0},∀t∈ [ti, ti+1] Y_t^q:=Y^Y

q

ti+1,q_{|[ti ,ti+1]}

t ,

and, similarly, we define the processZ^qassociated toY^qby

∀i∈ {N−1, . . . ,0},∀t∈(t_i, t_i+1) Z^q_t :=Z^Y

q

ti+1,q_{|[ti ,ti+1]}

t .

(Y^q, Z^q)is also well defined by a decreasing recursion oni∈ {0, . . . , N−1}. Finally, the stochastic control problem consists in minimizingY^q among all the admissible controlsq∈A. Our strategy to prove the uniqueness is to prove that given a solution(Y, Z), the first component is the optimal value.

Theorem 3.3. We suppose that there exists a solution(Y, Z)of the BSDE(1.1)verifying

∃p >γ ,¯ ∃ε >0 E exp

pA^∗ +exp

ε Y⁺_∗

<+∞, withA_t :=Y_t⁻+_t

0α¯_sds.Then we haveY =ess infq∈AY^q,and there existsq^∗∈Asuch thatY =Y^q∗.Moreover, this implies that the solution(Y, Z)is unique among solutions verifying such condition.

Proof. Let us first prove that for anyqadmissible, we haveY ≤Y^q. To do this, we will show thatY_|[ti,ti+1]≤Y_|[^q_t

i,t_i+1]

by decreasing recurrence on i∈ {0, . . . , N−1}. Firstly, we have Y_T =Y_T^q =ξ. Then we suppose that Y_t ≤Y_t^q,

∀t∈ [t_i+1, T]. We sett∈ [t_i, t_i+1]and we define τ_nⁱ:=inf

s≥t: sup s

t

|Z_u|²du, s

t

Z_u^q2du, s

t

|q_u|²du

> n

∧t_i+1, h(s, y, z):= −g(s, y, z)+zq_s, and

hs:=

_h(s,Yq

s,Z_s)−h(s,Y_s,Z_s)

Ys^q−Ys ifY_s^q−Y_s=0,

0 otherwise.

We observe that|h_s| ≤K_g,y. Then, by applying Itô formula to the process(Y_s^q−Y_s)e^tshuduwe obtain

Y_t^q−Y_t=e

_{τ in}

t h_sds Y^q

τ_nⁱ−Y_τi n

+ τ_nⁱ

t

e

s t h_udu

f

s, Y_s^q, q_s

−h

s, Y_s^q, Z_s ds+

τ_nⁱ t

e

s t h_udu

Z_s^q−Z_s dWs^q. By definition,f (s, Y_s^q, q_s)−h(s, Y_s^q, Z_s)≥0, so

Y_t^q−Yt≥E^Qi

e^{tτ inhsds} Y^q

τ_nⁱ−Y_τi nFt

.

Since(Y^q

τ_nⁱe^{tτ inhsds})ntends toY_t^q_i+1e^tti+1hsds almost surely and is uniformly integrable underQⁱ, we have

n→+∞lim E^Qi

e^{tτ inhsds}Y^q

τ_nⁱFt

=E^Qi

e^tti+1hsdsY_t^q

i+1Ft

.

Moreover,|Y_τi

ne^{tτ inhsds}| ≤(Y⁺)^∗e^{T Kg,y}+(Y⁻)^∗e^{T Kg,y}. Let us recall a useful inequality: from xy≤exp(x)+y

log(y)−1

∀(x, y)∈R×R⁺, we deduce

xy=pxy

p≤exp(px)+y

p(logy−logp−1). (3.2)

Thus E^Qi

Y⁻_∗

=E M_tⁱ

i+1

Y⁻_∗

≤E exp

p Y⁻_∗

+1 pE

M_tⁱ

i+1

logM_tⁱ

i+1−logp−1

≤Cp+ 1 2pE^Qi

t_i+1 ti

|qs|²ds

<+∞, and, in the same manner,

E^Qi Y⁺_∗

≤C_ε+ 1 2εE^Qi

_ti+1

t_i

|q_s|²ds

<+∞. So, by applying the dominated convergence theorem we obtain

n→+∞lim E^Qi

e^{tτ inhsds}Y_τi nFt

=E^Qi

e^tti+1hsdsY_ti+1Ft

.

Finally,

Y_t^q−Y_t≥ lim

n→+∞E^Qi e

_{τ in}

t hsds Y^q

τ_nⁱ−Y_τi nFt

=E^Qi e

_ti+1

t hsds

Y_t^q

i+1−Y_ti+1Ft

≥0,

becauseY_t^q_i+1≥Yt_i+1by the recurrence’s hypothesis.

Now we set^tq_s^∗∈∂_zg(s, Y_s, Z_s)with∂_zg(s, Y_s, Z_s)the subdifferential ofz→g(s, Y_s, z)atZ_s. We recall that for a convex functionl:R^1×d→R, the subdifferential ofl atx₀is the nonempty convex compact set ofu∈R^1×dsuch that

l(x)−l(x₀)≥u^t(x−x₀) ∀x∈R^1×d.

We havef (s, Y_s, q_s^∗)=Z_sq_s^∗−g(s, Y_s, Z_s)for alls∈ [0, T], so g(s, Ys, Zs)≤Zsq_s^∗− 1

2γ¯q_s^∗2+ ¯β|Ys| + ¯αs

≤1 2

2γ¯|Z_s|²+|q_s^∗|² 2γ¯

− 1

2γ¯q_s^∗2+ ¯β|Y_s| + ¯α_s,

|q_s^∗|²

4γ¯ ≤ −g(s, Ys, Zs)+ ¯γ|Zs|²+ ¯β|Ys| + ¯αs, and finally,_T

0 |q_s^∗|²ds <+∞,P-a.s. Moreover,∀t, t∈ [0, T], Y_t=Y_t+

t t

f

s, Y_s, q_s^∗ ds+

t t

Z_s

dWs−q_s^∗ds .

Thus, we just have to show thatq^∗is admissible to prove thatq^∗is optimal, i.e.Y =Y^q∗. For this, we must prove that (q_s^∗)_{s∈[ti,ti+1]}∈At_i,t_i+1(Y_ti+1)fori∈ {0, . . . , N−1}. We define

M_tⁱ:=exp t

t_i

q_s^∗dWs−1 2

_t

t_i

q_s^∗2ds

, dQ^∗,i dP :=M_tⁱ

i+1, τ_nⁱ=inf

t∈ [t_i, t_i+1]: sup t

t_i

q_s^∗2ds, _t

t_i |Z_s|²ds

> n

∧t_i+1, dQ^∗n^,i

dP :=Mⁱ

τ_nⁱ. Let us show the following lemma:

Lemma 3.4. (Mⁱ

τ_nⁱ)_nis uniformly integrable.

Proof. We apply inequality (3.2) to obtain E^Q∗n,i

A^∗

=E M_τⁱ_i

nA^∗

≤E exp

pA^∗ +1

pE M_τⁱ_i

n

logM_τⁱ_i

n−logp−1

≤C_p+ 1 2pE^Q∗n,i

τ_nⁱ ti

q_s^∗2ds

, and, in the same manner,

E^Q∗n,i Y⁺_∗

≤C_ε+ 1 2εE^Q∗n,i

τ_nⁱ t_i

q_s^∗2ds

.

Sinceg(s, Ys, Zs)=Zsq_s^∗−f (s, Ys, q_s^∗)and(Mⁱ

t∧τ_nⁱ)t∈[t_i,t_i+1]is a martingale, we can apply the Girsanov theorem and we obtain

E^Q∗n,i

Y_τi

n+

_τⁱ

n

ti

f

s, Ys, q_s^∗ ds

=E^Q∗n,i[Yt_i] =E M_τⁱ_i

nYt_i

=E[Yt_i].

Moreover,f (t, y, q)≥ ₂¹_γ¯|q|²− ¯β|y| − ¯α_t andY_τi

n≥ −Y⁻

τ_nⁱ, so E[Y_ti] ≥ −E^Q∗n,i

Y⁻

τ_nⁱ

−E^Q∗n,i τ_nⁱ

t_i α¯_sds

+ 1 2γ¯E^Q∗n,i

τ_nⁱ t_i

q_s^∗2ds

− ¯βE^Q∗n,i τ_nⁱ

t_i |Y_s|ds

≥C−E^Q∗n,i A^∗

+ 1 2γ¯E^Q∗n,i

_τⁱ

n

t_i

q_s^∗2ds

−T N

β¯E^Q∗n,i Y⁻_∗

+ Y⁺_∗

≥C_p,ε+1 2

1

¯ γ −1

p −T N

β¯ p +β¯

ε

>0

E^Q∗,in _τnⁱ

t_i

q_s^∗2ds

.

This inequality explains why we takeN verifying (3.1). Finally we get that

2E Mⁱ

τ_nⁱlogMⁱ

τ_nⁱ

=E^Q∗n,i τ_nⁱ

t_i

q_s^∗2ds

≤C_p,ε. (3.3)

Then we conclude the proof of the lemma by using the de La Vallée Poussin lemma.

Thanks to this lemma, we have thatE[M_tⁱ

i+1] =1 and so(M_tⁱ)_{t∈[ti,ti+1]}is a martingale. Moreover, applying Fatou’s lemma and inequality (3.3), we obtain

2E

M_tⁱ_i+1logM_tⁱ_i+1

=E^Q∗,i t_i+1

ti

q_s^∗2ds

≤lim inf

n E^Q∗n,i τ_nⁱ

ti

q_s^∗2ds

<+∞. (3.4)

So, by using this result and inequality (3.2) we easily show thatE^Q∗,i[(Y⁺)^∗+(Y⁻)^∗]<+∞. To conclude we have to prove thatE^Q∗,i[_ti+1

ti |f (s,0, q_s^∗)|ds]<+∞:

E^Q∗,i t_i+1

ti

f

s,0, q_s^∗ds

≤E^Q∗,i t_i+1

ti

f

s, Ys, q_s^∗+Kg,y|Ys|ds

≤E^Q∗,i t_i+1

ti

f

s, Ys, q_s^∗ds+Kg,yT Y⁺_∗

+ Y⁻_∗

≤C+E^Q∗,i t_i+1

ti

f⁺

s, Y_s, q_s^∗ +f⁻

s, Y_s, q_s^∗ ds

.

Firstly, E^Q∗,i

t_i+1 t_i

f⁻

s, Ys, q_s^∗ ds

≤E^Q∗,i t_i+1

t_i α¯s+ ¯β|Ys|ds

<+∞. Moreover, thanks to the Girsanov theorem we have

E^Q∗,i[Y_ti] =E^Q∗,i

Y_τi

n+

_τnⁱ

ti

f

s, Y_s, q_s^∗ ds

, so

E^Q∗,i _τnⁱ

t_i

f⁺

s, Y_s, q_s^∗ ds

≤E^Q∗,i[Y_ti−Y_τi

n] +E^Q∗,i _τnⁱ

t_i

f⁻

s, Y_s, q_s^∗ ds

≤C+E^Q∗,i ti+1

t_i

f⁻

s, Y_s, q_s^∗ ds

≤C.

Finally,E^Q∗,i[t_i+1

t_i f⁺(s, Ys, q_s^∗)ds]<+∞andE^Q∗,i[t_i+1

t_i |f (s,0, q_s^∗)|ds]<+∞. Thus, we prove thatq^∗is opti- mal, i.e.Y^q∗=Y.

The uniqueness ofY is a direct consequence of the fact thatY =Y^q∗=ess infq∈AY^q. The uniqueness ofZfollows

immediately.

Remark 3.5. If we have g(t, y, z)≤g(t,0, z), thenf (t, y, q)≥f (t,0, q)≥ ₂¹_γ¯|q|²− ¯α_t and we do not have to introduceNin the proof of Lemma3.4.So we have a simpler representation theorem:

Yt= ess inf

q∈A0,T(ξ )Y_t^q ∀t∈ [0, T].

For example,whengis independent ofy,we obtain Y_t= ess inf

q∈A0,T(ξ )E^Q

ξ+ T

t

f (s, q_s)dsFt

∀t∈ [0, T].

4. Application to quadratic PDEs

In this section we give an application of our results concerning BSDEs to PDEs which are quadratic with respect to the gradient of the solution. Let us consider the following semilinear PDE

∂tu(t, x)+Lu(t, x)−g

t, x, u(t, x),−σ^∗∇xu(t, x)

=0 u(T ,·)=h, (4.1)