Second-order backward stochastic differential equations with

arXiv:1201.1050v2 [math.PR] 10 Jan 2012

Second Order Backward Stochastic Differential Equations with Quadratic Growth∗ Dylan Possamai†

Chao Zhou‡

January 11, 2012

Abstract We prove the existence and uniqueness of a solution for one-dimensionnal second order backward stochastic differential equations introduced by Soner, Touzi and Zhang [33], with a bounded terminal condition and a generator which is continuous with quadratic growth in z. We also prove a Feyman-Kac formula and a probabilistic representation for fully nonlinear PDEs in this setting.

Key words: Second order backward stochastic differential equation, quadratic growth, r.c.p.d., Feyman-Kac, fully nonlinear PDEs AMS 2000 subject classifications: 60H10, 60H30

∗ Research supported by the Chair Financial Risks of the Risk Foundation sponsored by Société Générale, the Chair Derivatives of the Future sponsored by the Fédération Bancaire Fran¸caise, and the Chair Finance and Sustainable Development sponsored by EDF and Calyon. † CMAP, Ecole Polytechnique, Paris, [email protected]. ‡ CMAP, Ecole Polytechnique, Paris, [email protected].

1

1

Introduction

Backward stochastic differential equations (BSDEs for short) appeared in Bismuth [3] in the linear case, and then have been widely studied since the seminal paper of Pardoux and Peng [26]. On a filtered probability space (Ω, F, {Ft }0≤t≤T , P) generated by an Rd -valued Brownian motion B, a solution to a BSDE consists of a pair of progressively measurable processes (Y, Z) such that Yt = ξ +

Z

T

fs (Ys , Zs )ds − t

Z

T

Zs dBs , t ∈ [0, T ], P − a.s. t

where f (called the driver) is a progressively measurable function and ξ is an FT -measurable random variable. Pardoux and Peng proved existence and uniqueness of the above BSDE provided that the function f is uniformly Lipschitz in y and z and that ξ and fs (0, 0) are square integrable, but also that if the randomness in f and ξ is induced by the current value of a state process defined by a forward stochastic differential equation, then the solution to the BSDE could be linked to the solution of a semilinear PDE by means of a generalized Feynman-Kac formula. Since their pioneering work, many efforts have been made to relax the assumptions on the driver f ; for instance, Lepeltier and San Martin [19] have proved the existence of a solution when f is only continuous in (y, z) with linear growth. More recently, motivated by applications in financial mathematics and probabilistic numerical methods (see [8], [15], [30] and [32]), Cheredito, Soner, Touzi and Victoir [9] introduced a more general notion of Second order BSDEs (2BSDEs), which are now connected to the larger class of fully nonlinear PDEs. Then, Soner, Touzi and Zhang [33] provided a complete theory of existence and uniqueness for 2BSDEs under uniform Lipschitz conditions similar to those of Pardoux and Peng. The crux of their approach was to impose that the 2BSDE must hold P − a.s. for every probability measure P in a non-dominated class of mutally singular measures (see Section 2 for precise definitions). Possamai in [31] extended their results to the case of a continuous coefficient with linear growth. Motivated by a robust utility maximization problem under volatility uncertainty (see the accompanying paper [23]), our aim here is to go beyond the results of [31] to prove an existence and uniqueness result for 2BSDEs whose generator has quadratic growth in z. The question of existence and uniqueness of solutions to these quadratic equations in the classical case was first examined by Kobylanski [18], who proved existence and uniqueness of a solution by means of approximation techniques borrowed from the PDE litterature, when the generator is continuous and has quadratic growth in z and the terminal condition ξ is bounded. Then, Tevzadze [37] has given a direct proof for the existence and uniqueness of a bounded solution in the Lipschitz-quadratic case, proving the convergence of the usual Picard iteration. Following those works, Briand and Hu [6] have extended the existence result to unbounded terminal condition with exponential moments and proved uniqueness for a convex coefficient [7]. Finally, Barrieu and El Karoui [2] recently adopted a completely different approach, embracing a forward point of view to prove existence under conditions 2

similar to those of Briand and Hu. The rest of the paper is organized as follows. We introduce the notions in 2, then we prove a representation formula for the solution of a quadratic 2BSDE in Section 3 which in turns implies uniqueness. Section 4 is devoted to the proof of existence. Indeed, we use the method introduced by Soner, Touzi and Zhang [33] to construct the solution to the quadratic 2BSDE path by path. Finally, in Section 5, we extend the results of Soner, Touzi and Zhang on the connections between fully non-linear PDEs and 2BSDEs to the quadratic case.

2

Preliminaries

Let Ω := ω ∈ C([0, 1], Rd ) : ω0 = 0 be the canonical space endowed with the uniform norm kωk∞ := sup0≤t≤T |ωt |, B the canonical process, P0 the Wiener measure, F := {Ft }0≤t≤T the filtration generated by B, and F+ := Ft+ 0≤t≤T the right limit of F. We first recall the notations introduced in [33].

2.1

The Local Martingale Measures

A probability measure P is said to be a local martingale measure if the canonical process B is a local martingale under P. By results of Karandikar [16], it is known that we can have a pathwise definition of Z t 1 T Bs dBsT and b at := lim sup hBit − hBit−ǫ , hBit := Bt Bt − 2 ǫց0 ǫ 0 where

T

denotes the transposition and the lim sup is componentwise.

Let P W denote the set of all local martingale measures P such that hBit is absolutely continuous in t and b a takes values in S>0 d , P − a.s.

(2.1)

where S>0 d denotes the space of all d × d real valued positive definite matrices.

As in [33], we concentrate on the subclass P S ⊂ P W consisting of all probability measures Pα := P0 ◦ (X α )−1 where Xtα :=

Z

t 0

α1/2 s dBs , t ∈ [0, 1], P0 − a.s.

(2.2)

RT for some F-progressively measurable process α satisfying 0 |αs | ds < +∞. We recall from [34] that every P ∈ P S satisfies the Blumenthal zero-one law and the martingale representation property. Notice that the set P S is bigger that the set PeS introduced in [31], which is defined by eS := Pα ∈ P S , a ≤ α ≤ a P ¯, P0 − a.s. , (2.3)

for fixed matrices a and a ¯ in S>0 d . Finally, we recall

Definition 2.1. We say that a property holds PH -quasi surely (PH − q.s. for short) if it holds P − a.s. for all P ∈ PH . 3

2.2

The non-linear Generator

We consider a map Ht (ω, y, z, γ) : [0, T ] × Ω × R × Rd × DH → R, where DH ⊂ Rd×d is a given subset containing 0 and its corresponding conjugate w.r.t. γ defined by 1 Ft (ω, y, z, a) := sup Tr(aγ) − Ht (ω, y, z, γ) for a ∈ Sd>0 2 γ∈DH Fbt (y, z) := Ft (y, z, b at ) and Fbt0 := Fbt (0, 0).

We denote by DFt (y,z) the domain of F in a for a fixed (t, ω, y, z). As in [33] we restrict the probability measures in PH ⊂ P S Definition 2.2. PH consists of all P ∈ P S such that a≤a ¯P , dt × dP − a.s. for some aP , a ¯P ∈ Sd>0 , and b a ∈ DFt (y,z) . aP ≤ b

We now state our main assumptions on the function F which will be our main interest in the sequel Assumption 2.1.

(i) The domain DFt (y,z) = DFt is independent of (ω, y, z).

(ii) For fixed (y, z, γ), F is F-progressively measurable. (iii) F is uniformly continuous in ω for the || · ||∞ norm. (iv) F is continuous in z and has the following growth property. There exists (α, β, γ) ∈ R+ × R+ × R∗+ such that 2 γ |Ft (ω, y, z, a)| ≤ α + β |y| + a1/2 z , for all (t, y, z, ω, a). 2 (v) F is C 1 in y and C 2 in z, and there are constants r and θ such that for all (t, ω, y, z, a), |Dy Ft (ω, y, z, a)| ≤ r, |Dz Ft (ω, y, z, a)| ≤ r + θ a1/2 z , 2 |Dzz Ft (ω, y, z, a)| ≤ θ.

Remark 2.1. Let us comment on the above assumptions. Assumptions 2.1 (i) and (iii) are taken from [33] and are needed to deal with the technicalities induced by the quasisure framework. Assumptions 2.1 (ii) and (iv) are quite standard in the classical BSDE litterature. Finally, Assumption 2.1 (v) was introduced by Tevzadze in [37] for quadratic BSDEs and is essential in our framework to prove existence of the solution to 2BSDEs in Section 4. Remark 2.2. When the terminal condition is small enough, Assumption 2.1 (v) can be replaced by a weaker one, as shown by Tevzadze in [37]. We will therefore sometimes consider 4

Assumption 2.2.

(i) The domain DFt (y,z) = DFt is independent of (ω, y, z).

(ii) For fixed (y, z, γ), F is F-progressively measurable. (iii) F is uniformly continuous in ω for the || · ||∞ norm. (iv) F is continuous in z and has the following growth property. There exists (α, β, γ) ∈ R+ × R+ × R∗+ such that |Ft (ω, y, z, a)| ≤ α + β |y| +

γ 1/2 2 a z , for all (t, y, z, ω, a). 2

(v) We have the following ”local Lipschitz” assumption in z, ∃µ > 0 and a progressively ′ measurable process φ ∈ BMO(PH ) such that for all (t, y, z, z , ω, a), ′ ′ ′ ′ Ft (ω, y, z, a) − Ft (ω, y, z , a) − φt .a1/2 (z − z ) ≤ µa1/2 z − z a1/2 z + a1/2 z .

(vi) We have the following uniform Lipschitz-type property in y ′ ′ ′ Ft (ω, y, z, a) − Ft (ω, y , z, a) ≤ C y − y , for all (y, y , z, t, ω, a).

Furthermore, we observe that our subsequent proof of uniqueness of a solution of our quadratic 2BSDE only use Assumption (2.2).

Remark 2.3. By Assumption 2.1(v), we have that Fbt0 is actually bounded, so the strong integrability condition   Z T κ2 κ b0 EP  Ft dt  < +∞, 0

with a constant κ ∈ (1, 2] introduced in [33] is not needed here. Moreover, this implies that b a always belongs to DFt and thus that PH is not empty.

2.3

The Spaces and Norms

We now recall from [33] the spaces and norms which will be needed for the formulation of the second order BSDEs and add some particular spaces which are particularly linked to our quadratic growth framework. For p ≥ 1, LpH is the space of all FT -measurable scalar r.v. ξ which verify kξkpLp := sup EP [|ξ|p ] < +∞. H

P∈PH

In the case p = +∞ we define similarly the space of random variables which are bounded quasi-surely and take as a norm kξkL∞ := sup kξkL∞ (P) . H

P∈PH

5

HpH is the space of all F+ -progressively measurable Rd -valued processes Z which verify "Z p # T

kZkpHp := sup EP H

P∈PH

0

1/2

|b at Zt |2 dt

2

< +∞.

BMO(PH ) is the space of all F+ -progressively measurable Rd -valued processes Z which verify

Z .

kZkBMO(PH ) := sup Zs dBs < +∞,

P∈PH

0

BMO(P)

where k·kBMO(P) is the usual BMO(P) norm under P.

DpH is the space of all F+ -progressively measurable R-valued processes Y which verify # " PH − q.s. c` adl` ag paths, and kY kpDp := sup EP H

P∈PH

sup |Yt |p < +∞.

0≤t≤T

In the case p = +∞ we define kY kD∞ := sup kYt kL∞ . H

H

0≤t≤T

For each ξ ∈ L1H , P ∈ PH and t ∈ [0, T ] denote EH,P t [ξ] :=

P P Et ′ + P ∈PH (t ,P)

ess sup

′

o n ′ ′ [ξ] where PH (t+ , P) := P ∈ PH : P = P on Ft+ .

Here EPt [ξ] := E P [ξ|Ft ]. Then we define for each p ≥ 1, LpH

n

:= ξ ∈

LpH

: kξkLp < +∞ H

o

where

kξkpLp H

:= sup E

P

P∈PH

"

# p H,P ess sup Et [|ξ|] . P

0≤t≤T

In the case p = +∞ the natural generalization of the norm LpH is the norm L∞ H introduced above. Therefore, we will use the latter in order to be consistent with the notations of [33]. Finally, we denote by UCb (Ω) the collection of all bounded and uniformly continuous maps ξ : Ω → R with respect to the k·k∞ -norm, and we let LpH := the closure of UCb (Ω) under the norm k·kLp , for every p ≥ 1. H

2.4

Some properties of the space BMO(PH )

We present here some results on the space BMO(PH ) which generalize the already known results on the spaces BMO(P) for a given P. Lemma 2.1. Let M be a continuous P-local martingale for all P ∈ PH such that M ∈ BMO(PH ). Then there exists r > 1, such that E(M ) ∈ LrH . Proof. By Theorem 3.1 in [17], we know that if kM kBMO(P) ≤ Φ(r) for some one-to-one function Φ from (1, +∞) to R∗+ , then E(M ) is in Lr (P). Here, since M ∈ BMO(PH ), the same r can be used for all the probability measures. ✷ 6

Lemma 2.2. M be a continuous P-local martingale for all P ∈ PH such that M ∈ BMO(PH ). Then there exists r > 1, such that for all t ∈ [0, T ] " 1 # E(M )t r−1 P < +∞. sup Et E(M )T P∈PH Proof. This is a direct application of Theorem 2.4 in [17] for all P ∈ PH .

✷

We emphasize that the two previous Lemmas are absolutely crucial to our proof of uniqueness and existence. Besides, they will also play a major role in the following chapter.

2.5

Formulation

We shall consider the following second order BSDE (2BSDE for short), which was first defined in [33] Yt = ξ +

Z

T t

Fbs (Ys , Zs )ds −

Z

T

Zs dBs + KT − Kt , 0 ≤ t ≤ T, PH − q.s.

(2.4)

t

2 Definition 2.3. We say (Y, Z) ∈ D∞ H × HH is a solution to 2BSDE (2.4) if :

• YT = ξ, PH − q.s. • For all P ∈ PH , the process K P defined below has non-decreasing paths P − a.s. Z t Z t P b Zs dBs , 0 ≤ t ≤ T, P − a.s. (2.5) Fs (Ys , Zs )ds + Kt := Y0 − Yt − 0

0

• The family K P , P ∈ PH satisfies the minimum condition

′i ′ h KtP = ′ ess inf P EPt KTP , 0 ≤ t ≤ T, P − a.s., ∀P ∈ PH .

P ∈PH (t+ ,P)

(2.6)

Furthermore if the family K P , P ∈ PH can be aggregated into a universal process K, we call (Y, Z, K) a solution of 2BSDE (2.4).

3

Representation and uniqueness of the solution

We follow once more Soner, Touzi and Zhang [33]. For any P ∈ PH , F-stopping time τ , and Fτ -measurable random variable ξ ∈ L∞ (P), we define (y P , z P ) := (y P (τ, ξ), z P (τ, ξ)) as the unique solution of the following standard BSDE (existence and uniqueness have been proved under our assumptions by Tevzadze in [37]) Z τ Z τ P P P b yt = ξ + Fs (ys , zs )ds − zsP dBs , 0 ≤ t ≤ τ, P − a.s. (3.1) t

t

First, we introduce the following simple generalization of the comparison Theorem proved in [37] (see Theorem 2). 7

Proposition 3.1. Let Assumptions 2.1 hold true. Let ξ1 and ξ2 ∈ L∞ (P) for some probability measure P, and V i , i = 1, 2 be two adapted, c` adl` ag non-decreasing processes null at i i ∞ 2 0. Let (Y , Z ) ∈ D (P) × H (P), i = 1, 2 be the solutions of the following BSDEs Yti

i

=ξ +

Z

t

T

Fbs (Ysi , Zsi )ds −

Z

T t

Zsi dBs + VTi − Vti , P − a.s., i = 1, 2,

respectively. If ξ1 ≥ ξ2 , P − a.s. and V 1 − V 2 is non-decreasing, then it holds P − a.s. that for all t ∈ [0, T ] Yt1 ≥ Yt2 . Proof. First of all, we need to justify the existence of the solutions to those BSDEs. Actually, this is a simple consequence of the existence results of Tevzadze [37] and for instance Proposition 3.1 in [21]. Then, the above comparison is a mere generalization of Theorem 2 in [37]. ✷ We then have similarly as in Theorem 4.4 of [33]. ∞ 2 Theorem 3.1. Let Assumptions 2.2 hold. Assume ξ ∈ L∞ H and that (Y, Z) ∈ DH × HH is a solution to 2BSDE (2.4). Then, for any P ∈ PH and 0 ≤ t1 < t2 ≤ T , ′

Yt1 = ess supP ytP1 (t2 , Yt2 ), P − a.s.

(3.2)

P′ ∈PH (t1 ,P)

2 Consequentlty, the 2BSDE (2.4) has at most one solution in D∞ H × HH .

Before proceeding with the proof, we will need the following Lemma which shows that in our 2BSDE framework, we still have a deep link between quadratic growth and the BMO spaces. ∞ 2 Lemma 3.1. Let Assumption 2.2 hold. Assume ξ ∈ L∞ H and that (Y, Z) ∈ DH × HH is a solution to 2BSDE (2.4). Then Z ∈ BMO(PH ).

Proof. Denote T0T the collection of stopping times taking values in [0, T ] and for each R. P ∈ PH , let (τnP )n≥1 be a localizing sequence for the P-local martingale 0 Zs dBs . By Itô’s ag process, for some ν > 0, we have for formula under P applied to e−νYt , which is a càdl` T every τ ∈ T0 ν2 2

Z

τnP

−νYt

e τ

Z −νYτ P 1/2 2 −νY τ n − e −ν at Zt dt = e b +ν

Z

τnP

τnP

e τ

e−νYt− Zt dBt −

τ

−νYt−

X

dKtP

−ν

Z

τnP τ

e−νYt Fbt (Yt , Zt )dt

e−νYs − e−νYs− + ν∆Ys e−νYs− .

τ ≤s≤τnP

P Since Y ∈ D∞ H , K is non-decreasing and since the contribution of the jumps is negative because of the convexity of the function x → e−νx , we obtain with Assumption 2.1(iv)

8

ν2 P E 2 τ

"Z

τnP

# 2 νkY kD∞ 1/2 H 1 + νT α + β kY kD∞ e−νYt b at Zt dt ≤ e H

τ

νγ P + E 2 τ

"Z

τnP

τ

# 2 1/2 at Zt dt . e−νYt b

By choosing ν = 2γ, we then have "Z P # 2 τn 1 2γkY kD∞ 1/2 H at Zt dt ≤ e EPτ e−2γYt b 1 + 2γT α + β kY kD∞ . H γ τ Finally, by monotone Z T P Eτ τ

convergence and Fatou’s lemma we get that 1 4γkY kD∞ 1/2 2 H 1 + 2γT α + β kY kD∞ , at Zt dt ≤ e b H γ

which provides the result by arbitrariness of P and τ .

✷

Proof. [Proof of Theorem 3.1] The proof follows the lines of the proof of Theorem 4.4 in [33], but we have to deal with some specific difficulties due to our quadratic growth assumption. First (3.2) implies that ′

Yt = ess supP ytP (T, ξ), t ∈ [0, T ], P − a.s. for all P ∈ PH , P′ ∈PH (t+ ,P)

and thus is unique. Then, since we have that d hY, Bit = Zt d hBit , PH − q.s., Z is also unique. We now prove (3.2) in three steps. ′

(i) Fix 0 ≤ t1 < t2 ≤ T and P ∈ PH . For any P ∈ PH (t+ 1 , P), we have Yt = Yt2 +

Z

′

t2 t

Fbs (Ys , Zs )ds −

Z

t2 t

′

′

′

Zs dBs + KtP2 − KtP , t1 ≤ t ≤ t2 , P − a.s.

′

Since K P is nondecreasing, P −a.s., we can apply the comparison Theorem 3.1 under ′ ′ ′ ′ P to obtain Yt1 ≥ ytP1 (t2 , Yt2 ), P − a.s. Then, we also have P = P on Ft+ , therefore ′

we get Yt1 ≥ ytP1 (t2 , Yt2 ), P − a.s. and thus ′

Yt1 ≥ ess supP ytP1 (t2 , Yt2 ), P − a.s. P′ ∈PH (t+ 1 ,P)

(ii) We now prove the reverse inequality. Fix P ∈ PH . Let us assume for the time being that ′ h ′ ′ p i P P P P E K − K CtP,p := ess sup < +∞, P − a.s., for all p ≥ 1. t t t 1 2 1 1 P′ ∈PH (t+ 1 ,P)

9

′

For every P ∈ PH (t+ , P), we define ′

′

δY := Y − y P (t2 , Yt2 ) and δZ := Z − z P (t2 , Yt2 ). By the Lipschitz Assumption 2.2(vi) and the local Lipschitz Assumption 2.2(v), there exist a bounded process λ and a process η with ′ 1/2 1/2 P′ |ηt | ≤ µ b at zt , P − a.s. at Zt + b such that Z Z t2 1/2 λs δYs + (ηs + φs )b as δZs ds − δYt =

t2

t

t

Define for t1 ≤ t ≤ t2

Z

Mt := exp

t

t1

′

′

′

δZs dBs + KtP2 − KtP , t ≤ t2 , P − a.s.

′ λs ds , P − a.s. ′

Now, 3.1, we know that the P -exponential martingale R since φ ∈ BMO(PH), by Lemma ′ −1/2 . E 0 (φs + ηs )b as dBs is a P -uniformly integrable martingale (see Theorem 2.3 in ′

the book by Kazamaki [17]). Therefore we can define a probability measure Q , which ′ is equivalent to P , by its Radon-Nykodym derivative ′

dQ =E dP′

Z

T

ηs ))b a−1/2 dBs s

(φs +

0

.

Then, by Itô’s formula, we obtain, as in [33], that ′

δYt1 = ′

EQ t1

Z

t2

t1

Mt dKtP

′

′

≤

EQ t1

′

sup t1 ≤t≤t2

(Mt )(KtP2

′

−

KtP1

) ,

since K P is non-decreasing. Then, since λ is bounded, we have that M is also bounded and thus for every p ≥ 1 ′ ′ P p Et1 sup (Mt ) ≤ Cp , P − a.s. (3.3) t1 ≤t≤t2

−1/2

is in BMO(PH ), we know by Lemma 2.1 that Since (η+φ)b as R there exists r > 1, inde −1/2 T pendent of the probability measure considered, such that E 0 (φs + ηs )b as dBs ∈ LrH . Then it follows from the Hölder inequality and Bayes Theorem that r i 1 ′ h R r −1/2 T q 1 as dBs EPt1 E 0 (φs + ηs )b ′ ′ ′ q q P P P h R i Et1 sup Mt Kt2 − Kt1 δYt1 ≤ ′ −1/2 T t1 ≤t≤t2 EPt1 E 0 (φs + ηs )b as dBs 1 1 ′h ′ ′ i 4q 4q P P P E . K − K ≤ C CtP,4q−1 t1 t2 t1 1

′

By the minimum condition (2.6) and since P ∈ PH (t+ , P) is arbitrary, this ends the proof. 10

(iii) It remains to show that the estimate for CtP,p holds for p ≥ 1. By definition of the 1 P family K , P ∈ PH , we have E

P

′

h

′

KtP2

′

−

KtP1

p i

≤ C 1 + kY

kpD∞ H

Z

′

+ CEPt1

+

t2

Zt dBt

t1

′

kξkpL∞ H

+ EPt1

p

Z

t2 t1

.

p 1/2 2 at Zt dt b

Thus by the energy inequalities for BMO martingales and by Burkholder-DavisGundy inequality, we get that ′ h ′ ′ p i p EP KtP2 − KtP1 ≤ C 1 + kY kpD∞ + kξkpL∞ + kZk2p BMOH + kZkBMOH . H

H

Therefore, we have proved that sup

EP

′

P′ ∈PH (t+ 1 ,P)

h

′

′

KtP2 − KtP1

p i

< +∞.

Then we proceed exactly as in the proof of Theorem 4.4 in [33].

✷ We conclude this section by showing some a priori estimates which will be useful in the sequel. Theorem 3.2. Let Assumption 2.2 hold. ∞ 2 (i) Assume that ξ ∈ L∞ H and that (Y, Z) ∈ DH × HH is a solution to 2BSDE (2.4). Then, there exists a constant C such that kY kD∞ + kZk2BMO(PH ) ≤ C 1 + kξkL∞ H H i h p P P p P ∀p ≥ 1, sup Eτ (KT − Kτ ) ≤ C 1 + kξkL∞ . H

P∈PH , τ ∈T0T

i i ∞ 2 (ii) Assume that ξ i ∈ L∞ H and that (Y , Z ) ∈ DH × HH is a corresponding solution to 2BSDE (2.4), i = 1, 2. Denote δξ := ξ 1 − ξ 2 , δY := Y 1 − Y 2 , δZ := Z 1 − Z 2 and δK P := K P,1 − K P,2 . Then, there exists a constant C such that

kδY kD∞ ≤ C kδξkL∞ H

H

≤ C kδξkL∞ 1 + ξ 1 L∞ + ξ 2 L∞ H H H # " p p

p

p ∀p ≥ 1, sup EP sup δKtP ≤ C kξkL2∞ 1 + ξ 1 L2∞ + ξ 2 L2∞ . kδZk2BMO(PH )

P∈PH

Proof.

H

0≤t≤T

11

H

H

(i) By Theorem 3.1 we know that for all P ∈ PH and for all t ∈ [0, T ] we have ′

Yt =

ess sup ytP .

P′ ∈PH (t+ ,P)

Then by Lemma 1 in [6], we know that for all P ∈ PH 1 eβT − 1 P P βT + γe x . yt ≤ log Et [ψ(|ξ|)] , where ψ(x) := exp γα γ β Thus, we obtain

eβT − 1 P + eβT kξkL∞ , yt ≤ α H β and by the representation recalled above, the estimate of kY kD∞ is obvious. H

By the proof of Lemma 3.1, we have now CkY kD∞ H kZk2BMO(PH ) ≤ Ce 1 + kY kD∞ ≤ C 1 + kξkL∞ . H

H

Finally, we have for all τ ∈ T0T , for all P ∈ PH and for all p ≥ 1, by definition (KTP − KτP )p =

Yτ − ξ −

Z

T

τ

Fbt (Yy , Zt )dt +

Therefore, by our growth Assumption 2.1(iv) EPτ

h

(KTP

−

KτP )p

i

≤C

1+

+ CEPτ

kξkpL∞ H

"Z

+ kY

kpD∞ H

Zt dBt

p #

T τ

+ EPτ

Z

T

Zt dBt τ

"Z

τ

T

p

.

p 1/2 2 at Zt dt b

#!

p + kZk ≤ C 1 + kξkpL∞ + kZk2p BMO(PH ) BMO(PH ) H ≤ C 1 + kξkpL∞ , H

where we used again the energy inequalities and the BDG inequality. This provides the estimate for K P by arbitrariness of τ and P. (ii) With the same notations and calculations as in step (ii) of the proof of Theorem 3.1, it is easy to see that for all P ∈ PH and for all t ∈ [0, T ], we have δytP = EQ t [MT δξ] ≤ C kδξkL∞ , H

since M is bounded and we have (3.3). By Theorem 3.1, the estimate for δY follows. Now apply Itô’s formula under a fixed P ∈ PH to |δY |2 between a given stopping time τ ∈ T0T and T Z T Z T 2 1/2 2 2 P P 1 1 2 2 b b Eτ |δYτ | + δYt Ft (Yt , Zt ) − Ft (Yt , Zt ) dt at δZt dt ≤ Eτ |δξ| + 2 b τ τ Z T P P − 2Eτ δYt− d(δKt ) . τ

12

Then, we have by Assumption 2.1(iv) and the estimates proved in (i) above ! Z T 2 2 X

i

i 1/2 P

Y ∞ + Z Eτ at δZt dt ≤ C kδY kD∞ 1 + b D BMO(P ) H

τ

H

H

i=1

i h + kδξk2L∞ + 2 kδY kD∞ EPτ KTP,1 − KτP,1 + KTP,2 − KτP,2 H H

≤ C kδξkL∞ 1 + ξ 1 L∞ + ξ 2 L∞ , H

H

H

which implies the required estimate for δZ.

Finally, by definition, we have for all P ∈ PH and for all t ∈ [0, T ] Z t Z t 1 2 2 1 P b b δZs dBs . Fs (Ys , Zs ) − Fs (Ys , Zs )ds + δKt = δY0 − δYt − 0

0

By Assumptions 2.2(v) and (vi), it follows that Z T 1/2 1 1/2 2 P 1/2 as Zs + b as Zs ds sup δKt ≤ C kδY kD∞ + as δZs 1 + b b H 0≤t≤T 0 Z t + sup δZs dBs , 0≤t≤T

0

and by Cauchy-Schwarz, BDG and energy inequalities, we see that " "Z # #1 p 2 2 p 2 T 2 EP sup δKtP ≤ CEP as1/2 Zs1 + b 1 + b a1/2 ds s Zs 0≤t≤T

0

× EP

"Z

T

0

+C ≤C

2 p 1/2 as δZs ds b

kδξkpL∞ H

p/2 kδξkL∞ H

+E

P

" Z

0

T

#1

2

2 p/2 1/2 b a δZ s s ds

p/2 p/2 1 + ξ 1 L∞ + ξ 2 L∞ . H

#!

H

✷ Remark 3.1. Let us note that the proof of (i) only requires that Assumption 2.2(iv) holds true, whereas (ii) also requires Assumption 2.2(v) and (vi).

4

Proof of existence

In the article [33], the main tool to prove existence of a solution is the so-called regular conditional probability distributions of Stroock and Varadhan [36], which allows to construct a solution to the 2BSDE when the terminal condition belongs to the space UCb (Ω). In this section we will generalize their approach to the quadratic case.

13

4.1

Notations

For the convenience of the reader, we recall below some of the notations introduced in [33]. • For 0 ≤ t ≤ T , we denote by Ωt := ω ∈ C [t, 1], Rd , w(t) = 0 the shifted canonical space, B t the shifted canonical process, Pt0 the shifted Wiener measure and Ft the filtration generated by B t . • For 0 ≤ s ≤ t ≤ T and ω ∈ Ωs , we define the shifted path ω t ∈ Ωt by ωrt := ωr − ωt , ∀r ∈ [t, T ]. • For 0 ≤ s ≤ t ≤ T and ω ∈ Ωs , ω e ∈ Ωt we define the concatenated path ω ⊗t ω e ∈ Ωs by (ω ⊗t ω e )(r) := ωr 1[s,t) (r) + (ωt + ω er )1[t,T ] (r), ∀r ∈ [s, T ].

• For 0 ≤ s ≤ t ≤ T and a FTs -measurable random variable ξ on Ωs , for each ω ∈ Ωs , we define the shifted FTt -measurable random variable ξ t,ω on Ωt by ξ t,ω (e ω ) := ξ(ω ⊗t ω e ), ∀e ω ∈ Ωt .

Similarly, for an Fs -progressively measurable process X on [s, T ] and (t, ω) ∈ [s, T ] × n o t,ω Ωs , the shifted process Xr , r ∈ [t, T ] is Ft -progressively measurable.

• For a F-stopping time τ , the r.c.p.d. of P (noted Pωτ ) naturally induces a probability τ (ω) which in particular satisfies measure Pτ,ω (that we also call the r.c.p.d. of P) on FT that for every bounded and FT -measurable random variable ξ ω

EPτ [ξ] = EP

τ,ω

[ξ τ,ω ] .

• We define similarly as in Section 4 the set P¯St , by restricting to the shifted canonical t . space Ωt , and its subset PH • Finally, we define our ”shifted” generator Fbst,ω (e ω , y, z) := Fs (ω ⊗t ω e , y, z, b ats (e ω )), ∀(s, ω e ) ∈ [t, T ] × Ωt .

Then note that since F is assumed to be uniformly continuous in ω under the L∞ norm, then so is Fbt,ω .

4.2

Existence when ξ is in UCb (Ω)

When ξ is in UCb (Ω), we know that there exists a modulus of continuity function ρ for ξ and F in ω. Then, for any 0 ≤ t ≤ s ≤ T, (y, z) ∈ [0, T ] × R × Rd and ω, ω ′ ∈ Ω, ω ˜ ∈ Ωt ,

bt,ω t,ω t,ω ′ ′ t,ω ′ b

ω , y, z) ≤ ρ ω − ω ′ t , ω, y, z) − Fs (˜ ω) − ξ (˜ ω ) ≤ ρ ω − ω t and Fs (˜ ξ (˜ 14

where kωkt := sup0≤s≤t |ωs | , 0 ≤ t ≤ T . To prove existence, as in [33], we define the following value process Vt pathwise: Vt (ω) := sup YtP,t,ω (T, ξ) , for all (t, ω) ∈ [0, T ] × Ω,

(4.1)

t P∈PH

t1 where, for any (t1 , ω) ∈ [0, T ]×Ω, P ∈ PH , t2 ∈ [t1 , T ], and any Ft2 -measurable η ∈ L∞ (P), P,t1 ,ω P,t1 ,ω we denote Yt1 (t2 , η) := yt1 , where y P,t1 ,ω , z P,t1 ,ω is the solution of the following BSDE on the shifted space Ωt1 under P Z t2 Z t2 P,t1 ,ω P,t1 ,ω t1 ,ω P,t1 ,ω t1 ,ω b zrP,t1 ,ω dBrt1 , s ∈ [t1 , t2 ] , P − a.s. (4.2) dr − yr , zr Fr ys =η + s

s

We recall that since the Blumenthal zero-one law holds for all our probability measures, t . Therefore, the process V is well YtP,t,ω (1, ξ) is constant for any given (t, ω) and P ∈ PH defined. This is why it is absolutely crucial to use r.c.p.d. in this framework. Indeed, they allow us to define the process V pathwise, which in turn allows us to avoid the problems raised by working simultaneaously under mutually singular probability measures. However, we still do not know anything about the measurability of V . The following Lemma answers this question and justifies retrospectively our regularity assumptions on ω. Lemma 4.1. Let Assumptions 2.1 hold true and let ξ be in UCb (Ω). Then |Vt (ω)| ≤ C 1 + kξkL∞ , for all (t, ω) ∈ [0, T ] × Ω. H

Furthermore

Vt (ω) − Vt ω ′ ≤ Cρ ω − ω ′ , for all t, ω, ω ′ ∈ [0, T ] × Ω2 . t

In particular, Vt is Ft -measurable for every t ∈ [0, T ].

t , note that Proof. (i) For each (t, ω) ∈ [0, T ] × Ω and P ∈ PH

ysP,t,ω

=ξ

t,ω

−

Z

s

−

Z

s

T

T

h

atr Fbrt,ω (0) + λr yrP,t,ω + ηr b

zrP,t,ω dBrt , s ∈ [t, T ] , P − a.s.

1/2

atr zrP,t,ω + φr b

1/2

i zrP,t,ω dr

where λ is bounded and η satisfies 1/2 P,t,ω at zr , P − a.s. |ηr | ≤ µ b

Then proceeding exactly as in the second step of the proof of Theorem 3.1, we can define a bounded process M and a probability measure Q equivalent to P such that t,ω P,t,ω Q M ≤ E ξ . ≤ C 1 + kξk y t T t L∞ H 15

By arbitrariness of P, we get |Vt (ω)| ≤ C(1 + kξkL∞ ). H

(ii) The proof is exactly the same as above, except that we need to use uniform continuity in ω of ξ t,ω and Fbt,ω . In fact, if we define for (t, ω, ω ′ ) ∈ [0, T ] × Ω2 ′ ′ ′ ′ δy := y P,t,ω − y P,t,ω , δz := z P,t,ω − z P,t,ω , δξ := ξ t,ω − ξ t,ω , δFb := Fbt,ω − Fbt,ω ,

then we get with the same notations Z Q |δyt | = E MT δξ +

T

t

We get the result by arbitrariness of P.

b Ms δFs ds ≤ Cρ( ω − ω ′ t ).

✷

Then, we show the same dynamic programming principle as Proposition 4.7 in [34] Proposition 4.1. Under Assumption 2.1 and for ξ ∈ UCb (Ω), we have for all 0 ≤ t1 < t2 ≤ T and for all ω ∈ Ω 1 ,ω (t2 , Vtt21 ,ω ). Vt1 (ω) = sup YtP,t 1 t

P∈PH1

The proof is almost the same as the proof in [34], but we give it for the convenience of the reader. Proof. Without loss of generality, we can assume that t1 = 0 and t2 = t. Thus, we have to prove V0 (ω) = sup Y0P (t, Vt ). P∈PH

Denote (y P , z P ) := (Y P (T, ξ), Z P (T, ξ)) (i) For any P ∈ PH , it follows from Lemma 4.3 in [34], that for P − a.e. ω ∈ Ω, the t . By Tevzadze [37], we know that when the terminal condition is small, r.c.p.d. Pt,ω ∈ PH quadratic BSDEs whose generator satisfies Assumption (2.2) (v) can be constructed via Picard iteration. Thus, it means that at each step of the iteration, the solution can be formulated as a conditional expectation under P. Then, for general terminal conditions, Tevzadze showed that if the generator satisfies Assumption (2.1) (v), the solution of the quadratic BSDE can be written as a sum of quadratic BSDEs with small terminal condtion. By the properties of the r.p.c.d., this implies that ytP (ω) = YtP

t,ω ,t,ω

(T, ξ), for P − a.e. ω ∈ Ω.

By definition of Vt and the comparison principle for quadratic BSDEs, we deduce that y0P ≤ Y0P (t, Vt ) and it follows from the arbitrariness of P that V0 (ω) ≤ sup Y0P (t, Vt ). P∈PH

(ii) For the other inequality, we proceed as in [34]. Let P ∈ PH and ǫ > 0. The idea is to use the definition of V as a supremum to obtain an ǫ-optimizer. However, since V depends 16

obviously on ω, we have to find a way to control its dependence in ω by restricting it in a small ball. But, since the canonical space is separable, this is easy. Indeed, there exists a partition (Eti )i≥1 ⊂ Ft such that kω − ω ′ kt ≤ ǫ for any i and any ω, ω ′ ∈ Eti . Now for each i, fix an ω bi ∈ Eti and let, as advocated above, Pit be an ǫ−optimizer of Vt (b ωi ). If we define for each n ≥ 1, Pn := Pn,ǫ by # " n X i h t,ω i n P Pt btn ), where E btn := ∪i>n Eti , P (E) := E E 1E 1Eti + P(E ∩ E i=1

then, by the proof of Proposition 4.7 in [34], we know that Pn ∈ PH and that n

Vt ≤ ytP + ǫ + Cρ(ǫ), Pn − a.s. on ∪ni=1 Eti . Let now (y n , z n ) := (y n,ǫ , z n,ǫ ) be the solution of the following BSDE on [0, t] Z t Z t h n i zrn dBr , Pn − a.s. Fbr (yrn , zrn )dr − ysn = ytP + ǫ + Cρ(ǫ) 1∪n Eti + Vt 1Eb n + i=1

t

(4.3)

s

s

Note that since Pn = P on Ft , the equality (4.3) also holds P − a.s. By the comparison theorem, we know that Y0P (t, Vt ) ≤ y0n . Using the same arguments and notations as in the proof of Lemma 4.1, we obtain i h n Pn Pn Q y0 − y0 ≤ CE ǫ + ρ(ǫ) + Vt − yt 1Eb n . t

Then, by Lemma 4.1, we have

i h Y0P (t, Vt ) ≤ y0n ≤ V0 (ω) + C ǫ + ρ(ǫ) + EQ Λ1Eb n . t

The result follows from letting n go to +∞ and ǫ to 0.

✷

Define now for all (t, ω), the F+ -progressively measurable process Vt+ :=

lim

Vr .

r∈Q∩(t,T ],r↓t

Lemma 4.2. Under the conditions of the previous Proposition, we have Vt+ =

lim

Vr , PH − q.s.

r∈Q∩(t,T ],r↓t

and thus V + is c` adl` ag PH − q.s. Proof. Actually, we can proceed exactly as in the proof of Lemma 4.8 in [34], since the theory of g-expectations of Peng has been extended by Ma and Yao in [21] to the quadratic case (see in particular their Corollary 5.6 for our purpose). ✷ Finally, proceeding exactly as in Steps 1 and 2 of the proof of Theorem 4.5 in [34], and in particular using the Doob-Meyer decomposition proved in [21] (Theorem 5.2), we can get the 17

existence of a universal process Z and a family of non-decreasing processes K P , P ∈ PH such that Z t Z t Zs dBs − KtP , P − a.s. ∀P ∈ PH . Fbs (Vs+ , Zs )ds + Vt+ = V0+ − 0

0

For the sake of completeness, we provide the representation (3.2) for V and V + , and that, as shown in Proposition 4.11 of [34], we actually have V = V + , PH − q.s., which shows that in the case of a terminal condition in U Cb (Ω), the solution of the 2BSDE is actually F-progressively measurable. This will be important in Section 5. Proposition 4.2. Assume that ξ ∈ U Cb (Ω) and that Assumption 2.1 hold. Then we have ′

′

Vt = ess supP YtP (T, ξ) and Vt+ = ess supP YtP (T, ξ), P − a.s., ∀P ∈ PH . P′ ∈PH (t+ ,P)

P′ ∈PH (t,P)

Besides, we also have for all t Vt = Vt+ , PH − q.s. Proof. The proof for the representations is the same as the proof of proposition 4.10 in [34], since we also have a stability result for quadratic BSDEs under our assumptions. For the equality between V and V + , we also refer to the proof of Proposition 4.11 in [34]. ✷ To be sure that we have found a solution to our 2BSDE, it remains to check that the family of non-decreasing processes above satisfies the minimum condition. Let P ∈ PH , t ∈ [0, T ] and P ∈ PH (t+ , P). From the proof of Theorem 3.1, we have with the same notations

δVt =

EQ t

′

Z

T

Mt dKtP

t

′

≥

EQ t EPt

=

′

′

inf

t≤s≤T

(Ms )(KTP

′

−

KtP

′

)

′ ′ RT −1/2 P P E 0 (φs + ηs )b as dBs inf (Ms )(KT − Kt ) t≤s≤T h i RT ′ −1/2 P Et E 0 (φs + ηs )b as dBs

R −1/2 t For notational convenience, denote Et := E 0 (φs + ηs )b as dBs . Let r be the number given by Lemma 2.2 applied to E. Then we estimate h ′ ′i Et KTP − KtP ≤ EPt

′

′ ET inf (Ms )(KTP Et t≤s≤T

"

 1 ′ ′ 2r−1  − KtP ) EPt 



ET  inf (Ms )Et

t≤s≤T

1 2(r−1)

 2(r−1) 2r−1

′ ′  (KTP − KtP )

r−1 1 #! i 2(2r−1) ′ ′ ′ ′ h 2 ET r−1 − r−1 P P P 4 P ≤ (δVt ) Et inf (Ms ) Et (KT − Kt ) t≤s≤T Et r−1 ′ ′ 4 2(2r−1) 1 P P (δVt ) 2r−1 . ≤ C Et KT 1 2r−1

EPt

′

r−1 2r−1

18

By following the arguments of the proof of Theorem 3.1 (ii) and (iii), we then deduce the minimum condition. Remark 4.1. In order to prove the minimum condition it is fundamental that the process M above is bounded from below. For instance, it would not be the case if we had replaced the Lipschitz assumption on y by a monotonicity condition as in [31].

4.3

Main result

We are now in position to state the main result of this section Theorem 4.1. Let ξ ∈ L∞ H . Under Assumption 2.1, there exists a unique solution (Y, Z) ∈ ∞ 2 DH × HH of the 2BSDE (2.4). Proof. For ξ ∈ L∞ H , there exists ξn ∈ UCb (Ω) such that kξ − ξn k

→

n→+∞

0. Then, thanks

to the a priori estimates obtained in Proposition 3.2, we can proceed exactly as in the proof of Theorem 4.6 (ii) in [33] to obtain the solution as a limit of the solution of the 2BSDE (2.4) with terminal condition ξn . ✷

5

Connection with fully nonlinear PDEs

In this section, we place ourselves in the general case of Section 4, and we assume moreover that all the randomness in H only depends on the current value of the canonical process B (the so-called Markov property) Ht (ω, y, z, γ) = h(t, Bt (ω), y, z, γ), where h : [0, T ] × Rd × R × Rd × Dh → R is a deterministic map. Then, we define as in Section 4 the corresponding conjugate and bi-conjugate functions 1 Tr [aγ] − h(t, x, y, z, γ) γ∈Dh 2 1 b h(t, x, y, z, γ) := sup Tr [aγ] − f (t, x, y, z, a) a∈S>0 2 f (t, x, y, z, a) := sup

(5.1) (5.2)

d

We denote Ph := PH , and following [33], we strengthen Assumption 2.1 Assumption 5.1. (x, y, z).

(i) The domain Dft of the map a → f (t, x, y, z, a) is independent of

(ii) On Dft , f is uniformly continuous in t, uniformly in a. (iii) f is continuous in z and has the following growth property. There exists (α, β, γ) such that 2 γ |f (t, x, y, z, a)| ≤ α + β |y| + a1/2 z , for all (t, x, y, z, a). 2 19

(iv) f is C 1 in y and C 2 in z, and there are constants r and θ such that for all (t, x, y, z, a) |Dy f (t, x, y, z, a)| ≤ r, |Dz f (t, x, y, z, a)| ≤ r + θ a1/2 z 2 |Dzz f (t, x, y, z, a)| ≤ θ.

(v) f is uniformly continuous in x, uniformly in (t, y, z, a), with a modulus of continuity ρ which has polynomial growth. Remark 5.1. As mentioned in Remark 2.2, when the terminal condition is small enough, Assumption 5.1 (iv) can be replaced by the following weaker assumptions. ′

(iv’)[a] There exists µ > 0 and a bounded Rd -valued function φ such that for all (t, y, z, z , a) 1 ′ 1 1 ′ ′ 1 ′ 2 2 2 f (t, x, y, z, a) − f (t, x, y, z , a) − φ(t).a (z − z ) ≤ µa z − z a z + a 2 z .

(iv’)[b] f is Lipschitz in y, uniformly in (t, x, z, a).

Let now g : Rd → R be a Lebesgue measurable and bounded function. Our object of interest here is the following 2BSDE with terminal condition ξ = g(BT ) Yt = g(BT ) +

Z

T t

f (s, Bs , Ys , Zs , b a1/2 s )ds −

Z

T

t

Zs dBs + KTP − KtP , Ph − q.s.

(5.3)

The aim of this section is to generalize the results of [33] and establish the connection Yt = v(t, Bt ), Ph − q.s., where v is the solution in some sense to be specified later of the following fully nonlinear PDE  

∂v ∂t (t, x)



+b h t, x, v(t, x), Dv(t, x), D 2 v(t, x) = 0, t ∈ [0, T )

(5.4)

v(T, x) = g(x).

Following the classical terminology in the BSDE litterature, we say that the solution of the 2BSDE is Markovian if it can be represented by a deterministic function of t and Bt . In this subsection, we will construct such a function following the same spirit as in the construction in the previous section. With the same notations for shifted spaces, we define for any (t, x) ∈ [0, T ] × Rd Bst,x := x + Bst , for all s ∈ [t, T ]. Let now τ be an Ft -stopping time, P ∈ Pht and η a P-bounded Fτt -measurable random variable. Similarly as in (4.2), we denote (y P,t,x , z P,t,x ) := (Y P,t,x (τ, η), Z P,t,x (τ, η)) the unique solution of the following BSDE

ysP,t,x

=η+

Z

s

τ

atu )du f (u, But,x , yuP,t,x , zuP,t,x , b

−

20

Z

s

τ

zuP,t,x dBut,x , t ≤ s ≤ τ, P − a.s.

(5.5)

Then we define the following deterministic function (by virtue of the Blumenthal 0 − 1 law) u(t, x) := sup YtP,t,x (T, g(BTt,x )), for (t, x) ∈ [0, T ] × Rd .

(5.6)

P∈Pht

We then have the following Theorem, which is actually Theorem 5.9 of [33] in our framework Theorem 5.1. Let Assumption 5.1 hold, and assume that g is bounded and uniformly 2 continuous. Then the 2BSDE (5.3) has a unique solution (Y, Z) ∈ D∞ H × HH and we have Yt = u(t, Bt ). Moreover, u is uniformly continuous in x, uniformly in t and right-continuous in t. Proof. The existence and uniqueness for the 2BSDE follows directly from Theorem 4.1. Since ξ ∈ UCb (Ω), we have with the notations of the previous section Vt = u(t, Bt ). But, by Proposition 4.2, we know that Yt = Vt , hence the first result. Then the uniform continuity of u is a simple consequence of Lemma 4.1. Finally, the right-continuity of u in t can be obtained exactly as in the proof of Theorem 5.9 in [33]. ✷

5.1

Non-linear Feynman-Kac formula in the quadratic case

Exactly as in the classical case and as in Theorem 5.3 in [33], we have a non-linear version of the Feynaman-Kac formula. Notice however that the proof is more involved than in the classical case, mainly due to the technicalities introduced by the quasi-sure framework. Theorem 5.2. Let Assumption 5.1 hold true. Assume further that b h is continuous in its domain, that Df is independent of t and is bounded both from above and away from 0. Let 2 v ∈ C 1,2 ([0, T ), Rd ) be a classical solution of (5.4) with {(v, Dv)(t, Bt )}0≤t≤T ∈ D∞ H × HH . Then Z t

ks ds,

Yt := v(t, Bt ), Zt := Dv(t, Bt ), Kt :=

0

is the unique solution of the quadratic 2BSDE (5.3), where 1 h 1/2 i at ) and Γt := D 2 v(t, Bt ). kt := b h(t, Bt , Yt , Zt , Γt ) − Tr b at Γt + f (t, Bt , Yt , Zt , b 2 Proof. The proof follows line-by-line the proof of Theorem 5.3 in [33], so we omit it.

5.2

✷

The viscosity solution property

As usual when dealing with possibly discontinuous viscosity solutions, we introduce the following upper and lower-semicontinuous enveloppes u∗ (t, x) := b h∗ (ϑ) :=

lim

u(t′ , x′ ), u∗ (t, x) :=

(t′ ,x′ )→(t,x)

lim b h(ϑ′ ), b h∗ (ϑ) :=

(ϑ′ )→(ϑ)

lim

(t′ ,x′ )→(t,x)

h(ϑ′ ) lim b

u(t′ , x′ )

(ϑ′ )→(ϑ)

In order to prove the main Theorem of this subsection, we will need the following Proposition, whose proof (which is rather technical) is omitted, since it is exactly the same as the proof of Propositions 5.10 and 5.14 and Lemma 6.2 in [33]. 21

Proposition 5.1. Let Assumption 5.1 hold. Then for any bounded function g (i) For any (t,x) and arbitrary Ft -stopping times τ P , P ∈ Pht , we have u(t, x) ≤ sup YtP,t,x (τ P , u∗ (τ P , Bτt,x P )). P∈Pht

(ii) If in addition g is lower-semicontinuous, then u(t, x) = sup YtP,t,x (τ P , u(τ P , Bτt,x P )). P∈Pht

Now we can state the main Theorem of this section Theorem 5.3. Let Assumption 5.1 hold true. Then (i) u is a viscosity subsolution of −∂t u∗ − b h∗ (·, u∗ , Du∗ , D 2 u∗ ) ≤ 0, on [0, T ) × Rd .

(ii) If in addition g is lower-semicontinuous and Df is independent of t, then u is a viscosity supersolution of −∂t u∗ − b h∗ (·, u∗ , Du∗ , D 2 u∗ ) ≥ 0, on [0, T ) × Rd .

Proof. The proof follows closely the proof of Theorem 5.11 in [33], with some minor modifications (notably when we prove (5.10)). We provide it for the convenience of the reader. (i) We proceed by contradiction. Assume that 0 = (u∗ − φ)(t0 , x0 ) > (u∗ − φ)(t, x) for all (t, x) ∈ [0, T ) × Rd \ {(t0 , x0 )} , for some (t0 , x0 ) ∈ [0, T ) × Rd and −∂t φ − b h∗ (·, φ, Dφ, D 2 φ) (t0 , x0 ) > 0,

(5.7)

(5.8)

for some smooth and bounded function φ (we can assume w.l.o.g. that φ is bounded since we are working with bounded solutions of 2BSDEs). Now since φ is smooth and since by definition b h∗ is upper-semicontinuous, there exists an open ball O(r, (t0 , x0 )) centered at (t0 , x0 ) with radius r, which can be chosen less than T − t0 , such that −∂t φ − b h(·, φ, Dφ, D 2 φ) ≥ 0, on O(r, (t0 , x0 )).

By definition of b h, this implies that for any α ∈ S>0 d

1 − ∂t φ − Tr αD 2 φ + f (·, φ, Dφ, α) ≥ 0, on O(r, (t0 , x0 )). 2 22

(5.9)

Let us now denote µ := −

max

(u∗ − φ).

∂O(r,(t0 ,x0 ))

By (5.7), this quantity is strictly positive. Let us now consider a sequence (tn , xn ) in O(r, (t0 , x0 )) such that (tn , xn ) → (t0 , x0 ) and u(tn , xn ) → u∗ (t0 , x0 ). Consider the following stopping time / O(r, (t0 , x0 )) . τn := inf s > tn , (s, Bstn ,xn ∈

Since r < T − t0 , we have τn < T and therefore (τn , Bτtnn ,xn ) ∈ ∂O(r, (t0 , x0 )). Hence, we have cn := (φ − u)(tn , xn ) → 0 and u∗ (τn , Bτtnn ,xn ) ≤ φ(τn , Bτtnn ,xn ) − µ. Fix now some Pn ∈ Phtn . By the comparison Theorem for quadratic BSDEs, we have YtPnn ,tn ,xn (τn , u∗ (τn , Bτtnn,xn )) ≤ YtPnn ,tn ,xn (τn , φ(τn , Bτtnn ,xn ) − µ). Then proceeding exactly as in the second step of the proof of Theorem 3.1, we can define a bounded process Mn , whose bounds only depend on T and the Lipchitz constant of f in y, and a probability measure Qn equivalent to Pn such that ′ n YtPnn ,tn ,xn (τn , φ(τn , Bτtnn ,xn ) − µ) − YtPnn ,tn ,xn (τn , φ(τn , Bτtnn ,xn )) = −EQ tn [Mτn µ] ≤ −µ ,

for some strictly positive constant µ′ which is independent of n. Hence, we obtain by definition of cn YtPnn ,tn ,xn (τn , u∗ (τn , Bτtnn ,xn )) − u(tn , xn ) ≤ YtPnn ,tn ,xn (τn , φ(τn , Bτtnn,xn )) − φ(tn , xn ) + cn − µ′ . (5.10) With the same arguments as above, it is then easy to show with Itô’s formula that Z τn Pn ,tn ,xn Qn n n tn ,xn Ms ψs ds , Ytn (τn , φ(τn , Bτn )) − φ(tn , xn ) = Etn − tn

where

1 t 2 ψsn := (−∂t φ − Tr b as D φ + f (·, Dφ, b ats ))(s, Bstn ,xn ). 2

But by (5.9) and the definition of τn , we know that for tn ≤ s ≤ τn , ψsn ≥ 0. Recalling (5.10), we then get YtPnn ,tn ,xn (τn , u∗ (τn , Bτtnn ,xn )) − u(tn , xn ) ≤ cn − µ′ . Since cn does not depend on Pn , we immediately get sup YtPnn ,tn ,xn (τn , u∗ (τn , Bτtnn ,xn )) − u(tn , xn ) ≤ cn − µ′ .

P∈Phtn

23

The right-hand side is strictly negative for n large enough, which contradicts Proposition 5.1(i). (ii) We also proceed by contradiction. Assuming to the contrary that 0 = (u∗ − φ)(t0 , x0 ) < (u∗ − φ)(t, x) for all (t, x) ∈ [0, T ) × Rd \ {(t0 , x0 )} , for some (t0 , x0 ) ∈ [0, T ) × Rd and −∂t φ − b h∗ (·, φ, Dφ, D 2 φ) (t0 , x0 ) < 0,

(5.11)

(5.12)

for some smooth and bounded function φ (we can assume w.l.o.g. that φ is bounded since we are working with bounded solutions of 2BSDEs). Now we have by definition b h∗ ≤ b h, hence

−∂t φ − b h(·, φ, Dφ, D 2 φ) (t0 , x0 ) < 0,

(5.13)

Unlike with the subsolution property, we do not know whether D 2 φ(t0 , x0 ) ∈ Dbh or not. If it is the case, then by the definition of b h, there exists some α ¯ ∈ S>0 d such that 1 2 −∂t φ − Tr α ¯ D φ + f (·, φ, Dφ, α) ¯ (t0 , x0 ) < 0, (5.14) 2 which implies in particular that α ¯ ∈ Df .

If D 2 φ(t0 , x0 ) ∈ / Dbh , we still have that ∂t φ(t0 , x0 ) is finite, and thus α ¯ ∈ Df and (5.13) holds. Now since φ is smooth and since Df does not depend on t, there exists an open ball O(r, (t0 , x0 )) centered at (t0 , x0 ) with radius r, which can be chosen less than T − t0 , such that 1 −∂t φ − Tr α ¯ D 2 φ + f (·, φ, Dφ, α) ¯ ≤ 0, on O(r, (t0 , x0 )). 2 Let us now denote

µ :=

min

∂O(r,(t0 ,x0 ))

(u∗ − φ).

By (5.11), this quantity is strictly positive. Let us now consider a sequence (tn , xn ) in O(r, (t0 , x0 )) such that (tn , xn ) → (t0 , x0 ) and u(tn , xn ) → u∗ (t0 , x0 ). Define the following stopping time / O(r, (t0 , x0 )) . τn := inf s > tn , (s, Bstn ,xn ∈

Since r < T − t0 , we have τn < T and therefore (τn , Bτtnn ,xn ) ∈ ∂O(r, (t0 , x0 )). Hence, we have cn := (φ − u)(tn , xn ) → 0 and u∗ (τn , Bτtnn ,xn ) ≥ φ(τn , Bτtnn ,xn ) + µ. ¯ n := Pα¯ induced by the constant diffusion Now for each n consider the probability measure P α ¯ from time tn onwards. It is clearly in Phtn . Then, arguing exactly as in (i), we prove that 24

¯ n ,tn ,xn

u(tn , xn ) − YtPn

¯ n − a.s. (τn , u∗ (τn , Bτtnn ,xn )) ≤ cn − µ′ , P

For n large enough, the right-hand side becomes strictly negative, which contradicts Proposition 5.1(ii). ✷

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