4.2 An existence and uniqueness result
4.2.6 Existence and uniqueness for a small terminal condition
The aim of this Section is to obtain an existence and uniqueness result for BSDEJ with quadratic growth when the terminal condition is small enough. However, we will need more assumptions for our proof to work. First, we assume from now on that we have the following martingale representation property. We need this assumption since we will rely on the existence results in [4] or [81] which need the martingale representation.
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Assumption 4.2.2 Any local martingale M has the predictable representation property, that is to say, that there exists a unique predictable process H and a unique predictable function U such that (H, U)∈ Z × U and
Mt=M0+ Z t
0
HsdBs+ Z t
0
Z
E
Us(x)µ(dx, ds),e P−a.s.
Remark 4.2.3 This martingale representation property holds for instance when the compensator ν does not depend on ω, i.e when ν is the compensator of the counting measure of an additive process in the sense of Sato [107]. It also holds when ν has the particular form described in Chapter 5, in which caseν depends on ω.
Of course, we also need to assume more properties for our generator g.
Assumption 4.2.3 [Lipschitz assumption]
Let Assumption 4.2.1(i),(ii) hold and assume furthermore that (i) g is uniformly Lipschitz iny.
gt(ω, y, z, u)−gt(ω, y0, z, u) ≤C
y−y0
for all (ω, t, y, y0, z, u).
(ii)∃ µ >0 and φ∈H2BMO such that for all(t, y, z, z0, u) gt(ω, y, z, u)−gt(ω, y, z0, u)−φt.(z−z0)
≤µ z−z0
|z|+ z0
. (iii) ∃µ >0 andψ∈J2BMOsuch that for all (t, x)
C1(1∧ |x|)≤ψt(x)≤C2(1∧ |x|),
whereC2>0,C1 ≥ −1 +δ whereδ >0. Moreover, for all (ω, t, y, z, u, u0)
gt(ω, y, z, u)−gt(ω, y, z, u0)−< ψt, u−u0>t
≤µ
u−u0 L2(νt)
kukL2(νt)+ u0
L2(νt)
, where< u1, u2 >t:=R
Eu1(x)u2(x)νt(dx) is the scalar product inL2(νt).
Remark 4.2.4 Let us comment on the above assumptions. The first one concerning Lipschitz con-tinuity in the variable y is classical in the BSDE theory. The two others may seem a bit complicated, but they are almost equivalent to saying that the function g is locally Lipschitz in z and u. In the case of the variable z for instance, those two properties would be equivalent if the process φ were bounded. Here we allow something a bit more general by letting φbe unbounded but inH2BMO. Once again, since these assumptions allow us to apply the Girsanov property of Proposition 4.2.3, we do not need to bound the processes and BMO type conditions are sufficient. Moreover, Assumption4.2.3 also implies a weaker version of Assumption 4.2.1. Indeed, it implies clearly that
|gt(y, z, u)−gt(0,0,0)−φt.z−< ψt, u >t| ≤C|y|+µ
|z|2+kuk2L2(νt)
.
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Then, for anyu∈L2(ν)∩L∞(ν) and for anyγ >0, we have using the mean value Theorem γ
2e−γkukL∞(ν)kuk2L2(νt)≤ 1
γjt(±γu)≤ γ
2eγkukL∞(ν)kuk2L2(νt).
Therefore we deduce using the Cauchy-Schwartz inequality and the trivial inequality 2ab≤a2+b2 gt(y, z, u)−gt(0,0,0)≤ |φt|2
Hence, we have obtained a growth property which is similar to(4.2.7), the only difference being that the constants appearing in the quadratic term in z and the term involving the function j cannot, a priori, be the same. This prevents us from recovering the structure already mentioned in Remark 4.2.1.
We now show that if we can solve the BSDEJ (4.2.2) for a generatorg satisfying Assumption4.2.3 with φ = 0 and ψ = 0, we can immediately obtain the existence for general φ and ψ. This will simplify our subsequent proof of existence. Notice that the result relies essentially on the Girsanov Theorem of Proposition 4.2.3.
Lemma 4.2.2 Define
gt(ω, y, z, u) :=gt(ω, y, z, u)−< φt(ω), z >Rd −< ψt(ω), u >L2(νt).
Then(Y, Z, U) is a solution of the BSDEJ with generatorg and terminal condition ξ under P if and only if (Y, Z, U) is a solution of the BSDEJ with generatorg and terminal conditionξ underQwhere
dQ
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Now, by our BMO assumptions onφandψand the fact that we assumed thatψ≥ −1 +δ, we can apply Proposition4.2.3andQis well defined. Then by Girsanov Theorem, we know thatdBs−φsds and eµ(dx, ds)−ψs(x)ν(dx)dsare martingales underQ. Hence the desired result.
Remark 4.2.5 It is clear that if gsatisfies Assumption4.2.3, thengdefined above satisfies Assump-tion 4.2.3with φ=ψ= 0.
Following Lemma 4.2.2 we assume for the time being thatg(0,0,0) =φ=ψ= 0. Our first result is the following
Theorem 4.2.1 Assume that
kξk∞≤ 1 2√
15√
2670µe32CT ,
where C is the Lipschitz constant of g in y, and µ is the constant appearing in Assumption 4.2.3.
Then under Assumption 4.2.3 with φ = 0, ψ = 0 and g(0,0,0) = 0, there exists a unique solution (Y, Z, U)∈ S∞×H2BMO×J2BMO∩L∞(ν) of the BSDEJ (4.2.2).
Remark 4.2.6 Notice that in the above Theorem, we do not need Assumption4.2.1(iii) to hold. This is linked to the fact that, as discussed in Remark 4.2.4, Assumption 4.2.3 implies a weak version of Assumption 4.2.1(iii), which is sufficient for our purpose here.
Proof. We first recall that we have with Assumption 4.2.3 wheng(0,0,0) =φ=ψ= 0
|gt(y, z, u)| ≤C|y|+µ|z|2+µkuk2L2(νt). (4.2.14) Consider now the mapΦ : (y, z, u)∈ S∞×H2BMO×J2BMO∩L∞(ν)→(Y, Z, U) defined by
Yt=ξ+ Z T
t
gs(Ys, zs, us)ds− Z T
t
ZsdBs− Z T
t
Z
E
Us(x)µ(dx, ds).e (4.2.15) The above is nothing more than a BSDE with jumps whose generator depends only on Y and is Lipschitz. Besides, since(z, u)∈H2BMO×J2BMO∩L∞(ν), using (4.2.14) and the energy inequalities (4.2.3) we clearly have
EP
"
Z T 0
|gs(0, zs, us)|ds 2#
<+∞.
Hence, the existence of (Y, Z, U) ∈ S2×H2 ×J2 is ensured by the results of Barles, Buckdahn and Pardoux [4] or Li and Tang [81] for Lipschitz BSDEs with jumps. Of course, we could have let the generator in (4.2.15) depend on (ys, zs, us) instead. The existence of(Y, Z, U) would then have been a consequence of the predictable martingale representation Theorem. However, the form that we have chosen will simplify some of the following estimates.
Step 1: We first show that(Y, Z, U)∈ S∞×H2BMO×J2BMO∩L∞(ν).
Recall that by the Lipschitz hypothesis in y, there exists a bounded process λsuch that gs(Ys, zs, us) =λsYs+gs(0, zs, us).
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Let us now apply Itô’s formula toeRtsλudu|Ys|. We obtain easily from Assumption 4.2.3
ThereforeY is bounded and consequently, since its jumps are also bounded, we know that there is a version of U such that
kUkL∞(ν) ≤2kYkS∞. And finally, choosingε= 1/2
kYk2S∞+kZk2
Our problem now is that the norms for Z and U in the left-hand side above are to the power 2, while they are to the power4on the right-hand side. Therefore, it will clearly be impossible for us to prevent an explosion if we do not first start by restricting ourselves in some ball with a well chosen radius. This is exactly what we are going to do. Define therefore R = 1
2√
2670µeνT, and assume that kξk∞≤ √ R
15e12ηT and that
kyk2S∞+kzk2
H2BMO+kuk2
J2BMO+kuk2L∞(ν)≤R2.
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Denote Λ :=kYk2S∞+kZk2 Step 2: We show thatΦis a contraction in this ball of radius R.
Fori= 1,2 and(yi, zi, ui)∈ BR, we denote (Yi, Zi, Ui) := Φ(yi, zi, ui) and δy:=y1−y2, δz :=z1−z2, δu:=u1−u2, δY :=Y1−Y2 δZ :=Z1−Z2, δU :=U1−U2, δg :=g(0, z1, u1)−g(0, z2, u2).
Arguing as above, we obtain easily kδYk2S∞+kδZk2
From these estimates, we get kδYk2S∞+kδZk2
Therefore Φis a contraction which has a unique fixed point.
Then, from Lemma 4.2.2, we have immediately the following Corollary
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Corollary 4.2.1 Assume that
where C is the Lipschitz constant of g in y, and µ is the constant appearing in Asssumption 4.2.3.
Then under Assumption 4.2.3 with g(0,0,0) = 0, there exists a unique solution (Y, Z, U) ∈ S∞×
where C is the Lipschitz constant ofg in y, µis the constant appearing in Asssumption4.2.3 andD is a large enough positive constant. Then under Assumption 4.2.3, there exists a solution(Y, Z, U)∈ S∞×H2BMO×J2BMO∩L∞(ν) of the BSDEJ (4.2.2).
Proof. By Corollary 4.2.1, we can show the existence of a solution to the BSDEJ with generator egt(y, z, u) :=gt(y−Rt
0gs(0,0,0)ds, z, u)−gt(0,0,0)and terminal conditionξ:=ξ+RT
0 gt(0,0,0)dt.
Indeed, even though g is not null at (0,0,0), it is not difficult to show with the same proof as in Theorem 4.2.1that a solution(Y , Z, U)exists (the same type of arguments are used in [117]). More precisely, eg still satisfies Assumption4.2.3(i) and whenφand ψ in Assumption4.2.3are equal to 0, we have the estimate
which is the counterpart of (4.2.14). Thus, since the constant term in the above estimate is assumed to be small enough, it will play the same role askξk∞in the first Step of the proof of Theorem4.2.1.
For the Step2, everything still work thanks to the following estimate
it is clear that it is a solution to the BSDEJ with generator g and terminal conditionξ.
Remark 4.2.7 We emphasize that the above proof of existence extends readily to a terminal condition which is in Rn for any n≥2.