BSDEs with polynomial growth generators

In this paper, we give existence and uniqueness results for backward stochastic differential equations when the generator has a polynomial growth in 
the state variable. We deal with the case of a fixed terminal time, as well 
as the case of random terminal time. The need for this type of extension 
of the classical existence and uniqueness results comes from the desire to 
provide a probabilistic representation of the solutions of semilinear partial 
differential equations in the spirit of a nonlinear Feynman-Kac formula. 
Indeed, in many applications of interest, the nonlinearity is polynomial, 
e.g, the Allen-Cahn equation or the standard nonlinear heat and Schrodinger equations.


Introduction
It is by now well-known that there exists a unique, adapted and square integrable solution to a backward stochastic differential equation (BSDE for short) of type T T Yt--t-/ f(s, Ys, Zs)ds-J ZsdWs, O<_t<_T,   provided that the generator is Lipschitz in both variables y and z.We refer to the original work of E. Pardoux and S. Peng [13,14] for the general theory and to On the other hand, one of the most promising field of applications for the theory of BSDEs is the analysis of elliptic and parabolic partial differential equations (PDEs for short) and we refer to E. Pardoux [12] for a survey of their relationships.Indeed, as it was revealed by S. Peng [17] and by E. Pardoux, S. Peng [14] (see also the contributions of G. Barles, R. Buckdahn, E. Pardoux [1], Ph.Briand [3], E. Pardoux, F. Pradeilles, Z. Rao [15], E. Pardoux, S. Zhang [16] among others), BSDEs provide a probabilistic representation of solutions (viscosity solutions in the most general case) of semilinear PDEs.This provides a generalization to the nonlinear case of the well known Feynman-Kac formula.In many examples of semilinear PDEs, the nonlinearity is not of a linear growth (as implied by a global Lipschitz condition) but instead, it is of a polynomial growth, see e.g. the nonlinear heat equation analyzed by M. Escobedo, O. Kavian and H. Matano in [7]) or the Allen-Cahn equation (G.Barles, H.M. SoBer, P.E.Souganidis [2]).If one attempts to study these semilinear PDEs by means of a nonlinear version of the Feynman-Kac formula, alluded to above, one has to deal with BSDEs whose generators with a nonlinear (through polynomial) growth.Unfortunately, existence and uniqueness results for the solutions of BSDEs of this type were not available when we first started this investigation and filling this gap in the literature was at the origin of this paper..In order to overcome the difficulties introduced by the polynomial growth of the generator, we assume that the generator satisfies a kind of monotonicity condition in the state variable.This condition is very useful in the study of BSDEs with random terminal time.See the papers by S. Peng [17], R.W.R. Darling, E. Pardoux [5], Ph.Briand, Y. Hu [4] for attempts in the spirit of our investigation.Even though it looks rather technical at first, it is especially natural in our context: indeed, it is plain to check that it is satisfied in all the examples of semilinear PDEs quoted above.
The rest of the paper is organized as follows.In the next section, we introduce some notation, state our main assumptions, and prove a technical proposition which will be needed in the sequel.In Section 3, we deal with the case of BSDEs with fixed terminal time" we prove an existence and uniqueness result and establish some a priori estimates for the solutions of BSDEs in this context.In Section 4, we consider the case of BSDEs with random terminal times.BSDEs with random terminal times play a crucial role in the analysis of the solutions of elliptic semilinear PDEs.They were first introduced by S. Peng [17] and then studied in a more general framework by R.W.R. Darling, E. Pardoux [5].These equations are also considered in [12].Let (f,,P) be a probability space carrying a d-dimensional Brownian motion (Wt)t > 0, and (t)t > 0 be the filtration generated by (Wt) > 0" As usual, we assume that ech a-field -has been augmented with the P-null-sets to make sure that (t)t >0 is right continuous and complete.For y E Nk, we denote by yl its Eucli&an norm and if z belongs to N k x d, I I I I denotes {tr(zz*)} 1/2.For q > 1, we define the following spaces of processes: q progressively measurable; t e Rk; I I I I }q 4 sup I q < q progressively measurable; Ct e R k X d; I I II g: e f I I t I[ 2at < 0 and we consider the Banach space Jq-5q x J{q endowed with the norm I(/

I I I I
We now introduce the generator of our BSDEs.We assume that f is a function de- fined on flx[0, T]xN/xNxd, with values in N in such a way that the process (f(t'Y'Z))t e I0,T] is progressively measurable for each (y,z)in N/Cx N x d.Further- more, we make the following assumption.
Our contribution is merely to remark that his proof requires only an estimate of y. f(t, y,z) and thus that the result should still hold true in our context.We include a proof for the sake of completeness.
Proposition 2.1" Let ((Yt, Zt))t e [,T] E 2 be a solution of the BSDE (1).Let us assume moreover that for each t, y, z, Then, .for each > O, we havre, setting 0<t<T Proofi Let us fix t E [0, T]; fl will be chosen later in the proof.
we get

BSDEs with Fixed Terminal Times
The goal of this section is to study BSDE (1) for fixed (deterministic) terminal time T under assumptions (A1) and (A2).We first prove uniqueness, then we prove an a priori estimate and finally we turn to the existence.

Uniqueness and A Priori Estimates
This subsection is devoted to the proof of uniqueness and to the study of the integra- bility properties of the solutions of the BSDE (1).Theorem 3.1: If (A1) (1)-( 2) hold, the BSDE (1) has at most one solution in the space %2" Proof: Suppose that we have two solutions in the space %2, say (Y1,Z1) and (Y2, Z2).Setting 5Y-yl_ y2 and 5Z-Z 1-Z 2 for notational convenience, for each real number a and for each t E [0, T], taking expectations in It's formula gives: =E (f(s, Yls,Zls)f(s, r2s, Z2s) -a 6Y s 12}ds 1 .
The vanishing of the expectation of the stochastic integral is easily justified in view of BurkhSlder's inequality.Using monotonicity of f and the Lipschitz assumption, we We conclude the proof of uniqueness by choosing a 272-2# + 1.

VI
We close this section with the derivation of some a priori estimates in the space %2p" These estimates give short proofs of existence an uniqueness in the Lipschitz context.They were introduced in a "L p framework" by E. E1 Karoui, S. Peng, M.-C.
simply remark that (2) gives For the second assertion, we T v / e'16Ys 12ds 0 <_ eT 65 + 2 ecs 6Ys P. BRIAND and R. CARMONA which completes the proof using the first part of the proposition already shown and keeping in mind that if a > (')'1)2/-2#1 then v > 0.
El Corollary 3.3: Under the assumptions and with the notation of the previous pro- position, there exists a constant K, depending only on p,T, #1 and 71 such that El Remark: It is easy to verify that assumptions (A1) (3)-( 4) are not needed in the above proofs of the results of Proposition 3.2 and its corollary.
Corollary 3.4: Let ((Yt, Zt))o < < T E 62p be a solution of BSDE (1) and let us assume thai L 2p and assumalso lhal lhere exists a process (ft)o<t< T 2p( k) such thai V(s, y, z) E [0, T] x I k x I/ x d, y.f(,y,z) lyl'lfsl-lyl2+lyl "llzll.", which depends only Then, if 0 < < 1 and a > 72/ 2#, there exists a constanl K p on p and on such that: This inequality is exactly the same as inequality (2).As a consequence, we can complete the proof of this corollary as that of Proposition 3.2.
Our proof is based on the following strategy: first, we solve the problem when the function f does not depend on the variable z and then we use a fixed point argument using the a priori estimate given in subsection 3.1, Proposition 3.2 and Corollary 3.3.
The following proposition gives the first step.Proposition 3.5: Let assumptions (A1) and (A2) hold.Given a process (Vt)o <t<_ T in the space 2p, there exists a unique solution ((Yt, Zt))t [O,T] in the space-2p to the BSDE 0 <_ t _< T. (4) Proof: We shall write in the sequel h(s,y) in place of f(s,y, V s).Of course, h satisfies assumption (A1) with the same constants as f and (h(.,0)) belongs to 2p since f is Lipschtiz with respect to z and the process V belongs to :E2p.What we would like to do is to construct a sequence of Lipschitz (globally in y uniformly with respect to (w,s)) functions h n which approximate h and which are monotone.
However, we only manage to construct a sequence for which each h n is monotone in a given ball (the radius depends on n).As we will see later in the proof, this "local" monotonicity is sufficient to obtain the result.This is mainly due to Proposition 2.1 whose key idea can be traced back to a work of S. Peng [18, Theorem 2.2].
We shall use an approximate identity.Let p:Rk--+ be a nonnegative ( function with the unit ball for support and such that fp(u)du-1 and define for each integer n >_ 1, pn(u) nkp(nu).We denote also, for each integer n, by O n a e function from k to + such that0_<O n_<l,On(u)-l for ul _<nandOn(u)-0 as soon as ul _> n + 1.We set, moreover, and, if otherwise, / I h(s, y) if h(,, 0) < hn(s' Y) I _n_ .h(s,y) otherwise.
Hence, we deduce that, since the function hn(s is monotone (recall that #-0 in this section) and in view of the growth assumption on f, we have" V(s, y) x [0, T], y. h,(s, y) 5 (n A h(s, 0)] + 2C) y l. (6) This estimate will turn out to be very useful in the sequel.Indeed, we can apply Proposition 2.1 to BSDE (5) to show that, for each n, choosing c l/T, sup Y <_ (n + 2C)el/2v/l + T2.It is worth noting that, thanks to h(s,O) <_ f(s,O,O) / I I V s II, the right-hand side of the previous inequality is finite.We want to prove that the sequence ((.Yn, Zn))Nconverges towards the solution of BSDE (4) and in order to do that we first show-that the sequence ((Yn, Zn))N is a Cauchy sequence in the space %2" This fact relies mainly on the following property: h n satisfies the monotonicity condition in the ball radius q(n).Indeed, fix n G N and let us pick y,y' such that Yl <_ q(n) and y'l G q(n).We have: (y y')" (hn(8, y hn(8, y')) (y y').fln(U)Oq(n) + l(y t)hn(8, y u)du --(y--y')" Pn(U)Oq(n)+l(Y'-U)hn(s,y'-u)du.But, since Yl, Y'I < q(n) and since the support of Pn is included in the unit ball, we get from the fact that Oq(n)+ l(X) 1 as soon as Ix G q(n) + 1, ( V). ((, ) (, V)) ] (u)( V). ((, ) (, V-))d.
Thus, the sequence ((Yn, Zn)) N converges towards a progressively measurable process (Y,Z) in the space %2" Moreover, since (Yn, Zn)) N is bounded in %2p (see (8)), Fatou's lemma implies that (Y,Z) belongs also to the space %2p" It remains to verify that (Y,Z) solves BSDE (4) which is nothing but P. BRIAND and R. CARMONA Yt T T Of course, we want to pass to the limit in BSDE (5.Let us first notice that n in L 2p and that, for each t e [0, T], f TtzydWs---f[ZsdWs, since Z n converges to Z in the space 2( kx d).Actually, we only need to prove that for t E [0, T], T T i hn(s'Yy)ds--'i h(s, Ys)ds, as ncxz.
I/ h,,( The first term of the right-hand side of the previous inequality tends to 0 as n goes to cxz by the same argument we use earlier in the proof to see that _[fTo 16Ys I" Ihm(s,Y)-hn(s,Y)lds goes to 0. For the second term, we shall firstly prove that there exists a converging subsequne.nce.Indeed, since yn converges to Y is the space 3'2, there exists a subsequence (Y J) such that P-a.s., nj vt 6 [o, T], Yt -Yt.
n Since h(t,.)iscontinuous (P-a.s., Vt), e-a.s.(Vt, h(t, Yt')---h(t, Yt)).Moreover, since Y E 3'2p and (Yn)N is bounded in 3'2p ((8)), it is not hard to check that the growth assumption on f that I/ suph(s, n 2ds Ys J) h(s, Y.)l < , jeN o and then the result follows by uniform integrability of the sequence.Actually, the convergence hold for the whole sequence since each subsequence has a converging sub- sequence.Finally, we can pass to the limit in BSDE ( 5) and the proof is complete.V1 With the help of this proposition, we can now construct a solution (Y, Z) to BSDE (1).We claim the following result: Theorem 3.6: Under assumptions (A1) and (A2), BSDE (1) has a unique solution (Y, Z) in the space aJJ32p.
Proof: The uniqueness part of this statement is already proven in Theorem 3.1.
The first step in the proof of the existence is to show the result when T is sufficiently small.According to Theorem 3.1 and Proposition 3.5, let us define the following function (I) from %2p into itself.For (U, V)6 2p, O(U, V)-(Y,Z) where (Y,Z)is the unique solution in %2p of the BSDE: T T Yt + / f(s, Ys, Vs)ds-/ ZsdWs, O _ t _ T.
Next we prove that (I) is a strict contraction provided that T is small enough.
Indeed, if (U1,V1) and (U2, V. 2) .areboth elements of the space %2p, we have, If(s, Y2 s, V) f(s, Y2s, V2s ds 0 where 5Y--yl_y2, 5Z =_ Z 1 -Z 2 and K, is a constant depending only on p Using the Lipschitz assumption on f, (A1)ll), and H61der's inequality, we get the sup Yt I I zt [I 2dr   O<t<T 0 <_ Kp72pTp[[: inequality Hence, if T is such that Kp')'2pT p < 1, (I) is a strict contraction and thus (I) has a uni- que fixed point in the space '2p which is a unique solution of BSDE (1).The general case is treated by subdividing the time interval [0, T] into a finite number of intervals whose lengths are small enough and using the above existence and unique- ness result in each of the subintervals. !-1 4. The Case of Random Terminal Times In this section, we briefly explain how to extend the results of the previous section to the case of a random terminal time.

Notation and Assumptions
Let us recall that (Wt) > 0 is a d-dimensional Brownian motion defined on a probabi- lity space (f,,[P) ant that (5t)t> 0 is the complete r-algebra generated by (Wt)t>o.
Let-7 be a stopping time with respect to (t)t > o and let us assume thatis finite : e"r](I 2p e("/2)S f(s,O,O) 12ds 0 For notational convenience, we will simply write throughout the remainder of the paper qP' and 3q p instead of qP' )and ), respectively.

Existence and Uniqueness
In this section, we deal with the existence and uniqueness of the solutions of BSDE (13).We state the following proposition.Proposition 4.1: Under assumptions (A3) and (A4), there exists at mosl one solution of BSDE (13) in the space ,r x 3.
Let us notice first that Y-Yt 2-if t>_v and Z-Z-0 on the set {t>v}.
s and then, sup ePt 6Yt ]2 + ePS ]1 6Zs I] 2ds eSY 5ZdW <_ ---E o < < r 0 0 which is finite, since (SY, SZ) belongs to the space Y'7. .Due to the inequality p > 72-2, we can choose such that 0 < < 1 and p > 72/-2#.Using inequality ( 14), we deduce that the expectation of the stochastic integral vanishing, in view of the above computation, for each t, is 7" E[eP( A ^7. [2 + (1-)/ e ps I I 5Z, 11 ads] <-O, which gives the desired result, rl Before proving the existence part of the result, let us introduce a sequence of pro- cesses whose construction is due to R.W.R. assumption (A4) and Theorem 3.6 ensure that (yn, Z n) belongs to the space %2p (on the interval [0, hi).In view of [12, Proposition 3.1], we have Yn(v A T") Yt, and, Z O on {t>-}.

3.eppt y 12p
Moreover, since p > 2,, we can find c such that 0 < c < 1 and p > 72/e-2#.Applying Proposition 3.2 (actually a mere extension to deal with bounded stopping times as terminal times), we get 12dm + llz 112 We have YnnA3" ----yn_n -,(nA r)F{e,Xrln ^r} and then we deduce immediately that, since p/2-, > 0 and due to Jensen's inequality But the expectation is over the set {n < v} and coming back to the definition of (Yn, Zn) for t > n, it is enough to verify that 0 in order to get inequality (16) of the lemma and thus to complete the proof, since, in view of the definition of r, the previous inequality is nothing but inequality (17).But, for each n, ((, r/) solves the following BSDE: We have already seen (of.( 18)) that f_[e p(nA r) l( n A r 2p] --< [P 12p] and thus the proof of this rather technical lemma is complete.El With the help of this useful lemma, we can construct a solution to BSDE (13).
This is the objective of the following theorem.
Theorem 4.3: Under assumptions (A3) and (A4), BSDE (13) has a unique solution (Y,Z) in the space f,r ] which satisfies sup e po(t ^r) Yt 2P epS Ys 288 + epS I I Z s I I 2ds <_ K(, f).>_o 0 0 Proofi The uniqueness part of this claim is already proven in Proposition 4.1.We concentrate ourselves on the existence part.We split the proof into the two following steps: first we show that the sequence ((Yn, ZU)) N is a Cauchy sequence in the space , r x ] and then we shall prove that the limiting process is indeed a solution.
Using inequality ( 14) with this e, we deduce from the previous inequality that " Y-.f(s, Y, Z)lds LnAv _ V/p i 2a -eP AY-f(s, Yy, zy) d and we have already proven that the right-hand side tends to 0 (see the definition of Fn).It remains to study the term f '^7 .f(s,Yy, Zr)ds.But, since f is Lipschitz in z, we have E elf(s, Yns, Zr)-f(s, yn,, t^and thus this term goes to 0 with n.So now it suffices to show that eo f(s, yr, Z s f(s, Y s, Z) ds -0, 0 to control the limit in the equation.We prove this by showing that each subsequence has a subsequence for which the above convergence holds.Indeed, if we pick a Oe#(t A -) y n12 subsequence (still denoted by yn), since we have E[supt > -Yt ]--0, there exist a subsequence still under the same notation sucti that F-a.s.(Vt, YrYt).By the continuity of the function f, -.s. (Vt, f(t,Y,Zt)f(t, Yt, Zt)).If we prove that supE e" f(s, Y, Z s)-f(s, Y s, Z,) uds < , N o then the sequence f(',Yn.,z.)f(.,Y.,Z.)l will be a uniformly integrable sequence for the finite measure eaSls < 7.ds(R)d (remember that c < 0) and thus converge in Ll(eaSl s < rds (R)dP), whicl7 is the desired result.But from the growth assumption on f, we h-ave and the inequality 271 Ys I I Zs I I _< ((ya)/z)I Y 12 + I[ z ][ the notation K p for a constant depending only on p and whose value could be changing from line to line.Due to the inequality ab <_ a2/2 + b2/2E sup {e(Pa/)tlYt]P} eUS I I az I I d o<t<T 0 which yields the inequality, using one more time the inequality ab <_ a2/2 -k-b2/the upper bound established for [V[suPo < < TePatlYtl2P],

F
set 6Y ym_ yn, 6Z Z m-Zn.It follow from the definition of fn, mA" ep(t A -)lSy A 2 -I-/ e" I I I I tAr mA" ,O,O) / I I zZ I I a / I I z I I }d 0 2.2 A First A Priori Estimate Darling and E.Pardoux [5, pp. 1148-  1149].Let us set ,-72/2-# and let (n,n) be a unique solution of the classical (the terminal time is deterministic) BSDE on [0, hi: Then, we have, with the