August 2007, Vol. 11, p. 385–411 www.edpsciences.org/ps
DOI: 10.1051/ps:2007026
HOMOGENIZATION OF A SEMILINEAR PARABOLIC PDE WITH LOCALLY PERIODIC COEFFICIENTS: A PROBABILISTIC APPROACH
Abdellatif Bench´ erif-Madani
1and ´ Etienne Pardoux
2Abstract. In this paper, a singular semi-linear parabolic PDE with locally periodic coefficients is homogenized. We substantially weaken previous assumptions on the coefficients. In particular, we prove new ergodic theorems. We show that in such a weak setting on the coefficients, the proper statement of the homogenization property concerns viscosity solutions, though we need a bounded Lipschitz terminal condition.
Mathematics Subject Classification. 35B27, 60H30, 60J60, 60J35.
Received March 28, 2006. Revised December 6, 2006.
1. Introduction
In this paper we continue investigating locally periodic homogenization in the case of the following semi-linear parabolic PDE defined on the product set [0, T]×Rd, T >0,
⎧⎨
⎩
∂tu(t, x) + Γ x
, x, u(t, x),∇u(t, x), ∂2u(t, x)
= 0, u(T, x) =g(x)
(1.1)
where Γ
x
, x, u(t, x),∇u(t, x), ∂2u(t, x)
= 1 2
d i,j=1aij
x , x
∂x2ixju(t, x) +d
i=1
1
bi+bi x , x
∂xiu(t, x) +1
e x
, x, u(t, x)
+f x
, x, u(t, x),∇u(t, x)σ x
, x
, for all >0 andxinRd. The functiong(x) is Lipschitz continuous and bounded. All the coefficients are periodic with respect to the first variable with period one in each direction ofRd. The matrixσ(x1, x2) = [σij(x1, x2)]
Keywords and phrases. Homogenization, nonlinear parabolic PDE, Poisson equation, diffusion approximation, backward SDE.
1 Universit´e Ferhat Abbas, Fac. Sciences, D´ept. Maths., S´etif 19000, Algeria;lotfi madani@yahoo.fr
2CMI, LATP – CNRS and Universit´e de Provence, 39, rue F. Joliot Curie, 13453 Marseille Cedex 13, France;pardoux@latp.unimrs.fr c EDP Sciences, SMAI 2007
Article published by EDP Sciences and available at http://www.esaim-ps.org or http://dx.doi.org/10.1051/ps:2007026
for x1 and x2 in Rd satisfiesσσ∗(x1, x2) = a(x1, x2) where the matrixa(x1, x2) = [aij(x1, x2)] is supposed to be uniformly elliptic. That is∃λstrictly positive and finite s.t. for allx1,x2and ξinRd
(a(x1, x2)ξ, ξ)≥λξ2. (1.2)
Our previous (probabilistic) arguments in [2] and [3] in the case of linear coefficients e(x/, x, u(t, x)) = e(x/, x)u(t, x) and vanishing non-linear term f = 0 were based on the Feynman-Kac formula. It is well known that a natural extension of this technique to the non-linear case turns out to be the theory of backward stochastic differential equations first discovered by Pardoux and Peng, see [13], and the references therein, for a complete and profound account. The operator acting onxin Rd
L= 1 2
d i,j=1aij
x , x
∂x2ixj+d i=1
1
bi+bi x , x
∂xi (1.3)
which appears in (1.1) is the infinitesimal generator of the following (forward) SDE, built on some probability space (Ω,F,Ft, Bt, P), whereBtis under P anFt-Brownian motion,
Xs,t,x=x+
s
t
1
b+b Xr,t,x , Xr,t,x
dr+
s
t
σ
Xr,t,x , Xr,t,x
dBr, (1.4)
for all 0≤t≤s≤T andxin Rd; whereas the non-linear term is the coefficient in the backward SDE, carried by the above probability space, see the notation below,
Ys,t,x=g(XT,t,x) +
T
s
1
e+f Xr,t,x
, Xr,t,x, Yr,t,x, Zr,t,x
dr− T
s
Zr,t,xdBr; (1.5) the process (Y.,t,x, Z.,t,x) isF.B-adapted and subject to the condition
E
t≤s≤Tsup
Ys,t,x2 +E
T
t
Zr,t,x2dr <∞, (1.6)
for all >0. Note thatu(t, x) =Yt,t,xis deterministic and solves (1.1) given our assumptions on the coefficients.
Our aim in this paper is to establish, under weak conditions on the coefficients, the pointwise convergence as →0 ofutowards the solution of equation (3.6) below. For example in [12], who deals with the totally periodic case, the coefficientesatisfies a kind of an algebraic sub-linear growth iny, namely e(x, y) =e0(x, y) +e1(x)y where e0(x, y) is bounded besides taking f = 0. Note also that although Delarue [8] deals with a quasi-linear equation, he considers only periodic coefficients and his conditions concerningσ,bandeare much stronger than ours. Moreover, the functions that are homogenized possess weak second derivatives. However, his treatment of the homogenization process is more complete due to the presence of more regularity. On the other hand, although we don’t allow for a quadratic growth in the gradient, our work does considerably relax some of the regularity hypotheses of [4], where homogenization is studied by purely analytical methods.
Some facts from [3] will be used here, sometimes without special warnings.
The idea is still to freeze the slow component in (1.3) and consider the following family of operators indexed byx2 (since the coefficientb plays no asymptotic role it is omitted),
Lx2 =1 2
d
i,j=1aij(x1, x2)∂x21
ix1j +d
i=1bi(x1, x2)∂x1
i. (1.7)
These operators generate the following diffusions with transition densitiespt(x1, x1, x2) Xtx2 =x1+
t
0
b(Xrx2, x2)dr+
t
0
σ(Xrx2, x2)dBr,
which may rather be thought of as diffusions on the compact torusTd, i.e.
X· x2
t =x1+
t
0
b(X·
x2
r , x2)dr+
t
0
σ(X·
x2
r , x2)dBr, with transition densities
p·t(x1, x1, x2) =
k1...kd
pt
x1, x1+ d i=1
kiei, x2
,
ei being the canonical basis of Rd and ki integers. In what follows we shall drop the dots when no ambi- guity arises. These diffusions possess invariant probability measures µ(dx1, x2) with densities p∞(x1, x2), see Pardoux [12]. It is crucial to impose on the singular coefficientsbandethe following centering condition for all x2 andy. As this relation appears several times below and to avoid unnecessary repetitions, we shall use the general notationhforb,eand other functions of interest, namely we have
Tdh(x1, x2, y)µ(dx1, x2) = 0. (1.8) We can then solve the Poisson equation
Lx2h(x1, x2, y) =−h(x1, x2, y), (1.9) and carry on with the usual line of proof (seee.g. the introduction in [2]), provided we have sufficient regularity.
Recall that in the linear case we first establish a tightness result for the family of processesX, >0, in the spaceC( [0, T], Rd) endowed with the sup-norm and proceed to identify the limitviaan ergodic theorem and a martingale problem formulation. In the non-linear case however, it seems difficult to work out tightness results for the processY(and the related martingaleM, see (2.17)) in C( [0, T], Rd) endowed with the sup-norm and it turns out that the weaker topology of Jakubowski [9] onD( [0, T], Rd+1) is convenient, see also [11] where a tightness criterion is established (actually relaxed by Kurtz [10]). Moreover, it is important to note that given our formal assumptions on the coefficients, a natural stability argument, first devised in [5] and used below with a slight modification, seems to be necessary since the family of processes Z, > 0, does not seem to converge. In particular, Pardoux’s weak convergence scheme can’t be carried out as such. Instead, we simply begin by guessing the form of the limit PDE (see (3.6)) and then prove that convergence takes place. The whole procedure below for homogenizing our PDE (1.1) should not be too surprising since there seems indeed to be a gap in the bridge between viscosity solutions and BSDEs. This is well accounted for ine.g. [1].
Notation 1.1. Letξdenote any coefficienta, b, b, eandf. Even if the coefficientsa,bandb do not depend on y we still allow for the notationξ(r, r, r, r)meaning respectivelyξ(Xr, Xr),ξ(Xr, Xr, Yr)andξ(Xr, Xr, Yr, Zr).
There should be no confusion with space variables since we use different letters for these variables. However, in a solution u(r, r)of a parabolic equation the first r obviously refers to the time variable. We will also drop unambiguous superscripts like e.g. (t, x). The quantity ζ(t)−ζ(s) is shortened, using the difference operator, as ∆s,tζ(.). The linear space of Rd valued continuous, respectively c`adl`ag, functions on [0, T] is denoted by C( [0, T], Rd), respectively D( [0, T], Rd). Ifu(x) is a function of xin Rd, we shall write ∂xu(x) to denote the d-dimensional vector whose i-th coordinate is ∂xiu(x); similarly ∂x2u(x) will denote ad×dmatrix and so on.
Unimportant constants will invariably be designated by c the value of which may vary from line to line while proofs are in process but when there are many constants within a string of relations, we will use c,c ...
1.1. Condition A
Our standing assumptions are in this section, next to (1.2) in whichλis fixed,
Tdb(x1, x2)µ(dx1, x2) = 0,
Tde(x1, x2, y)µ(dx1, x2) = 0, for all (x1, x2, y), andg∈W1,∞, the conditions:
1.1.1. A1 Regularity
The coefficients σ, b, and b satisfy a global Lipschitz condition, i.e. there exists a constant c s.t. for any ξ=σ, b, andb
ξ(x1, x2)−ξ(x1, x2)≤c(x1−x1+x2−x2).
Moreover, for ξ = a and b, the partial derivatives ∂x2ξ(x1, .) belong to Wloc1,p(Rd), for some large p and all x1∈Td, and
sup
x2∈Rd
∂x22ξ(., x2)
Lp(Td)<∞.
The first order partials of the coefficiente(x1, x2, y) exist and satisfy for somec>0 and all triples (x1, x2, y) (∂x1e+∂x2e)(x1, x2, y)≤c(1 +|y|),
|∂ye|(x1, x2, y)≤c.
The second partial derivatives ofewith respect to (x2, x2), (x2, y) and (y, y) exist and satisfy ∂x22e(., x2, y)
Lp(Td)+∂x22ye(., x2, y)
Lp(Td)≤c, ∂y2e(., x2, y)
Lp(Td)≤c 1 (1 +|y|), for some largepandc>0.
The coefficient f(x1, x2, y, v) admits first order partials, which satisfy the condition ∂x1f(x1, x2, y, v)+∂x2f(x1, x2, y, v)≤c(1 +|y|+v),
∂yf(x1, x2, y, v)≤c, ∂vf(x1, x2, y, v)≤c,
for somec >0 and all (x1, x2, y, v) uniformly with respect tox1inTd andx2 inRd and moreover it is jointly continuous.
1.1.2. A2 Growth
The coefficientsσ, bandb are bounded,i.e. there exists a constantK s.t. for anyξ=σ,b,b ξ(x1, x2)≤K,
and the coefficients eandf satisfy a sub-linear growth in (y, v),
(e+f)(x1, x2, y, v)≤K(1 +|y|+v), for someK and allx1 inTd,x2 in Rd,y in Randv inRd.
Note that for every >0 our conditions guarantee the existence and uniqueness of solutions of the above sys- tem of forward-backward SDEs subject to condition (1.6) and that a viscosity solutionu(t, x) to equation (1.1) exists and is unique, see Theorem 12.3 p. 54 in [1].
1.2. Technical facts
We begin with the following simple fact, actually of independent interest, which will be needed below.
Lemma 1.2. Let (Ω,F)be a measurable space carrying two probability measuresP andP s.t. P is absolutely continuous with respect to P. If a family of continuous processes {ξsn}n, t ≤s ≤T, is P-tight, relative to a fixed metric on C[t, T], then it is also P-tight with respect to this same metric. If moreover the density ζ=ddPP is in L2(P)then for some absolute constantc >0we have for any non-negative random variable θ
E(θ) ≤c E(θ2).
Proof. Fix a topology onC[0, T] and designate the Radon-Nikodym derivative byζ. As the family of processes ξns isP-tight, for anyδ >0 there exists a compactKδ in C[0, T] s.t. P(ξn. ∈Kδc)< δfor alln. We also have
P(ξ n. ∈Kδc) =
{ξn.∈Kcδ}
ζ(ω)P(dω),
which implies the first result immediately sinceζis integrable.
The inequality follows from Cauchy-Schwarz.
On the other hand, letαandβ be two positive numbers. The following inequality will be used several times below,
2αβ≤ρα2+ρ−1β2, (1.10)
whereρis a small positive number which may depend on the particular productαβ.
1.3. A growth lemma
We need to control the growth in xandy of partial derivatives ofbande. We have the Lemma 1.3. There are constantsc0, c,c,c andc s.t. for all(x1, x2, y)inTd×Rd+1
b+∂x1b+∂x2b+∂x22b+∂2x1x2b
(x1, x2)≤c0, and on the other hand
(|∂ye|+∂x21ye+∂x22ye+∂x31x2ye)(x1, x2, y)≤c, (∂x31yye+∂y2e)(x1, x2, y)≤c 1
(1 +|y|) and
(|e|+∂x1e+∂x2e+∂x21e+∂x21x2e+∂x22e+∂x31x2x2e)(x1, x2, y)≤c(1 +|y|).
Finally
∂x2p∞(x1, x2)≤c. Moreover, all these partial derivatives are continuous.
Proof. It suffices to use the same manipulations as in Lemma 1 in [3] and take derivatives in the following Poisson equation
Lx2e(x1, x2, y) =−e(x1, x2, y),
for (x1, x2, y) ∈ Td×Rd+1 and to notice that the right hand-side of the corresponding Poisson equation is always majorized by either an absolute constant or byc(1 +|y|). Let use.g. deal with the estimate of∂x31x2x2e.
We have by our notation convention
∂x2Lx2e+Lx2∂x2e=−∂x2e.
Since equation (1.9) for eimplies thate(., x2, y) is inW2,∞(Td) for all (x2, y)∈Rd+1 and the coefficient ais Lipschitz, then ∂x2e(., x2, y) is also in W2,∞(Td) for all (x2, y)∈ Rd+1 with a norm majorized by c(1 +|y|).
Next we have
∂x22Lx2e+ 2∂x2Lx2∂x2e+Lx2∂2x2e=−∂2x2e, Condition A ona,bandethen gives immediately the result.
The other estimates are carried out in a similar way and are left to the reader.
The estimate on the derivatives with respect to x2 of the invariant probability density p∞(x1, x2) follows
from Lemma 18 in [2].
2. Removing the singularities
We want to get rid of the factor −1 in both the forward and backward SDEs above but keeping as few conditions on the coefficients as possible. Our method of attack, in fact Freidlin’s (see the introduction in [2]), consists in applying the joint Itˆo formula on the functionsb and e, when possible. In that respect, it follows from Lemma 1.3 thatb is inWloc2,∞(Td×Rd) and thateis inW2,p(Td×Rd+1) for some largepwhich allows the use of the Itˆo-Krylov formula.
2.1. Treatment of the forward process
Let us first defineF(x1, x2) = (∂x1bb+∂x2bb+T r∂x21x2ba)(x1, x2),
G(x, y) = [(I+∂x1b)σ](x1, x2), (2.1)
where, see the notation above, the quantityT r∂x21x2bastands for the vector whose components areT r∂2x1x2bia, 1≤i≤d. Recall that 0≤t≤s≤T. We have, see [3], the decomposition
Xs=Xs+Rs, (2.2)
where
Xs=x+
s
t
F(r, r)dr+
s
t
G(r, r)dBr, and
Rs=
s
t
∂x2bb+1
2T r∂x22ba
(r, r)dr+
s
t
∂x2bσ(r, r)dBr+ (b(t, t)−b(s, s))
. Note that sinceb,b andaare bounded we have
E(sup
s≤TRs4)≤c4, (2.3)
whence the tightness (in the sup-norm) sufficient condition (which was proved in [3] under weaker conditions).
Corollary 2.1. There exists a constantc s.t. for all 0≤α≤β≤T and >0 E(Xβ −Xα4)≤c[(β−α)2+4].
Note that we also have the following estimate which will be needed below E(Xβ−Xα2)≤c[(β−α) +2].
2.2. Treatment of the backward process
In order to remove the singularity of the backward process, we shall use a kind of stopping argument. Let us take a fine enough equidistant subdivision of the interval [0, T] by means of the pointsti, i= 0, ...,[T /∆t] =N, where t0 = 0 and ∆t = ti−ti−1. We denote byt∗ the largest ti below t, byt∗ the least ti abovet. On the last interval [T∗, T] however, we make the convention that forr∈(T∗, T] we haver∗=T (instead ofT∗+ ∆t).
We also note thatt∗=t whent=ti. We define in particular the subdivisions ∆tk=k2, wherekis a positive integer, and call ∆t1the neutral subdivision. We add a superscript (subscript)kto indicate which subdivision is involved. Recall that we have fors≤T
Ys=g(XT) +
T
s
1 e+f
(r, r, Yr, Zr)dr− T
s
ZrdBr.
Define fors≤T the discontinuous c`adl`ag adapted process Ysk,=Ys−e
Xs
, Xs, Ys∗(k)
,
that is the processYsk, is continuous on each (tki−1, tki) and undergoes a jump at eachtki,tk1≤tki ≤T∗(k)(there should be no ambiguity with these inequalities). We have thanks to the Itˆo formula,
T
s
1
e(r, r, Yr∗(k))dr=[e(s, s, Ys∗(k))−e(T, T, YT∗(k))] +
T
s
[∂x1e(r, r, r∗(k))b(r, r) +∂x2e(r, r, r∗(k))b(r, r) + Tr∂x21x2e(r, r, r ∗(k))a(r, r)
+∂x2e(r, r, r∗(k))b(r, r) +
2Tr∂2x2e(r, r, r∗(k))a(r, r)]dr +
T
s
(∂x1e+∂x2e)(r, r, r∗(k))σ(r, r)dBr
+
tki=T∗(k)
tki>s
∆tk
i−1,tkie(tki, tki, Y.).
Hence we have the representation
Ysk,= [g(XT)−e(T, T, YT∗(k)] +
T
s
Urk,dr−
T
s
Zrk,dBr+Jsk,, (2.4)
where
U1,rk,=1
∆r∗(k),re(r, r, Y.),
U2,rk,= [∂x1e(r, r, r∗(k))b(r, r) +∂x2e(r, r, r∗(k))b(r, r) + Tr∂x21x2e(r, r, r∗(k))a(r, r)],
U3,rk,= [∂x2e(r, r, r∗(k))b(r, r) +
2Tr∂x22e(r, r, r∗(k))a(r, r)], Urk,=
3 i=1
Ui,rk,+f(r, r, Yr, Zr) (2.5)
and Zrk,=Zr−(∂x1e+∂x2e)(r, r, r∗(k))σ(r, r) (2.6)
and finally
Jsk,=
T∗(k)
tki>s
∆tk
i−1,tkie(tki, tki, Y.).
2.3. Establishing tightness
Having thus produced a non singular processYsk,, which depends onkand is asymptotically close toYs, let us turn to the study of the tightness problem. Consider the formula (2.4). It is important to notice that in the first time interval [s, s∗(k)) in the Lebesgue (respectively the stochastic) integral the process Urk, (respectively the processZrk,) has the component Ys∗(k) which stays lagging below s. The same phenomenon arises in the first jump term inJsk,. Therefore estimating the second moment of the processYsk,and the first moment of the related process
T s
Zrk,2drby the usual methods needs a special care. This will allow us to handle supremums that run backwards in time. Of all the terms in (2.5), it is the first one which requires attention since it has exactely the reverse behaviour of the jump termJsk, in (2.4).
2.3.1. Estimates on the increments of Y Let us begin with the following easy lemma.
Lemma 2.2. Under the above notations, there exists a constantc >0and an0>0 s.t. for anyk≥1,≤0 andr in[0, T], we have
Zr ≤c
1 +∆r∗(k),rY.+|Yr|+Zrk,
(2.7)
and Urk,≤c
1 +1
∆r∗(k),rY.+|Yr|+Zrk,
. Proof. The first inequality follows immediately from Lemma 1.3, when≤1 we have
Zr ≤c
1 +Yr∗(k)+Zrk,
≤c
1 +∆r∗(k),rY.+|Yr|+Zrk, .
This inequality will serve to recover the process Zrk,2 when we deal with Lebesgue integrals involving the processZr2.
The other inequality is proved in a similar way and follows from the first inequality (2.7).
This allows us to control powers of the increments of the processYs. We have the
Lemma 2.3. There exists an absolute constantc >0that depends only on the coefficients in Condition A and an 0>0 s.t., when≤0we have on each ∆tki,i= 1, ..., Nk+ 1,k≥1,
Yr−Ytk i−1
≤c
⎛
⎜⎝∆tk+1
r
tki−1
Yu−Ytk i−1
du+∆tk
i−1,rX.+Ytk i−1
∆tk i−1,rX. +
r
tki−1
|Yu|du+
r
tki−1
Zuk,du+
r
tki−1
Zuk,dBu
⎞
⎟⎠.
Proof. Notice first thateis Lipschitz with respect toy, thatbandbare bounded and that, thanks to Lemma 1.3, there is a constantc >0 s.t. for all≤1
∆tk
i−1,re(., ., tki−1)≤c
1 +Ytk i−1
∆tk
i−1,rX.,
then it suffices to use Lemma 2.2 in the Itˆo-Krylov decomposition (2.4).
Applying the Gronwall-Bellman inequality, we deduce the
Corollary 2.4. Under the same hypotheses as in the previous lemma we have Yr−Ytk
i−1
≤ceck
∆tk+∆tk
i−1,rX.+Ytk i−1
∆tk i−1,rX. +
r
tki−1
|Yu|du+
r
tki−1
Zuk,du+
r
tki−1
Zuk,dBu
⎞
⎟⎠
and the
Corollary 2.5. There exists an absolute constant c >0 which depends only on the coefficients in Condition A s.t., for any fixedk≥1 there is an 0(k)>0 s.t. when ≤0 we have on each∆tki,i= 1, ..., Nk+ 1,
E sup
r∈∆tki
Yr−Ytk i−1
2≤c
⎛
⎜⎝∆tk+
∆tki
E|Yu|2du+ ∆tkEYtk i−1
2+E
∆tki
Zuk,2du
⎞
⎟⎠.
Proof. Since the coefficients of the forward equation are bounded, it suffices to write thanks to the relation (2.2) and to Lemma 1.3
Ytk i−1
∆tk
i−1,rX.≤c(+ ∆tk)Ytk i−1
+
r
tki−1
Ytk i−1
G(u, u)dBu ,
where
G(u, u) = (I +∂x1b+∂x2b)(u, u).
By the Burkholder-Davis-Gundy inequality, we see that for any givenk, we can choose an0s.t. for all≤0 we have
E sup
r∈∆tki
Ytk i−1
∆tk
i−1,rX.2≤c(1 + ∆tk)∆tkEYtk i−1
2≤c∆tkEYtk i−1
2.
Using convexity, the inequalities of Cauchy-Schwarz and Burkholder-Davis-Gundy on the relation of the above corollary we arrive at the desired result, upon takingsmall enough.
Remark 2.6. Notice that in both the previous corollaries, the timercan take the extreme valuetki.
Before we work out the usuala priori estimates, let us first settle the problems that we mentionned at the beginning if this subsection. We formulate our treatment as an independent
Lemma 2.7. There exists a constantc >0 s.t., for any fixedk >0there is an0(k)>0s.t. for all ≤0and s≤T
E sup
s∗(k)≤r≤s∗(k)
∆s∗(k),rY.2≤c
1 + sup
s≤r≤TEYrk,2
. Proof. We can write by Corollary 2.5
E sup
s∗(k)≤r≤s∗(k)
∆s∗(k),rY.2≤c∆tk
⎛
⎜⎝1 + sup
s∗(k)≤r≤s∗(k)E|Yr|2+E
s∗(k)
s∗(k)
Zuk,2du
⎞
⎟⎠
≤c∆tk
⎛
⎜⎝1 + sup
s∗(k)≤r≤s∗(k)
E|Yr|2+E
s∗(k)
s∗(k)
Zu2du
⎞
⎟⎠,
since by definition ofZuk, it follows by Lemma 1.3 that for ≤1 anduin [s∗(k), s∗(k)] we have Zuk,2≤c
1 +Ys∗(k)2+Zu2
.
On the other hand, by Condition A and by a well known BSDE estimate, derived when starting from our crude and untreated BSDE on [s∗(k), s∗(k)],i.e.
Yr=Ys∗(k)+
s∗(k)
r
1 e+f
(u, u, Yu, Zu)du− s
∗(k)
r
ZudBu,
the right hand-side of the above inequality is majorized, when≤0(k), by c(1 +EYs∗(k)2),
since the factor 1 (more than) cancels out with ∆tk=k2.
Next we need to recover the processYsk,, that is we have to compare our former processYs with the new one Ysk,. Recall that we have for all s≤T
Ys=Ysk,+e(s, s, Ys∗(k)),
so that by well known inequalities when≤1
E|Ys|2≤c[EYsk,2+2(1 +EYs∗(k)2)]
≤c(1 +EYsk,2+2E∆s∗(k),s∗(k)Y.2+2EYs∗(k)2).
We also know from the above manipulation that whenis small enough E∆s∗(k),s∗(k)Y.2≤c(1 +EYs∗(k)2), so that
E|Ys|2≤c
1 +EYsk,2+2EYs∗(k)2 . Now
EYs∗(k)2≤ sup
s≤r≤TE|Yr|2, which gives for anys≤T
(1−c2) sup
s≤r≤TE|Yr|2≤c
1 + sup
s≤r≤TEYrk,2
, (2.8)
when≤0(k), which immediately implies our lemma.
Remark 2.8. The sharp estimate on E sup
s∗(k)≤r≤s∗(k)
∆s∗(k),rY.2 given by Corollary 2.5 is not really needed in order to prove the above lemma. A rough and well known estimate based on the untreated BSDE above is sufficient since we can controlkthanks to the parameter ρ1 as announced in equation (1.10), see below.
2.3.2. A priori estimates
We seek to establish the following relation, seee.g. [11],
≤sup0
⎛
⎝E sup
0≤s≤T|Ys|2+E
T
0
Zr2dr
⎞
⎠< c, (2.9)
for some fixedc >0 and0. This fundamental estimate will serve to imply that the family of processes (Ys, Ms) (see eq. (2.17) below for the definition of the martingale Ms) is tight in the JakubowskiS-topology. We will use parametersρ >0 which will be used as in equation (1.10).
By the Itˆo formula, we have fort≤s≤T Ysk,2+
T
s
Zrk,2dr+
T∗(k)
tki>s
(∆Ytk,k
i
)2= [g(XT)−e(T, T, YT∗(k))]2+
T
s
2Yrk,Urk,dr
−
T
s
2Yrk,Zrk,dBr−
T∗(k)
tki>s
2Yk,
tki−·∆Yk,
tki , (2.10) where ∆Ytk,k
i stands for the jump of the processYsk, at the instant tki, i.e.
∆Ytk,k i =Ytk,k
i+−Ytk,k i−
=[e(tki, tki, Ytk
i−1)−e(tki, tki, Ytk i)].
