Numerical simulation of BSDEs with drivers of quadratic growth

0.27 MB·English

by  Adrien Richou

Numerical simulation of BSDEs with drivers of quadratic growth
AdrienRichou
IRMAR, UniversitéRennes 1
CampusdeBeaulieu, 35042RENNES Cedex, France
e-mail: [email protected]
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August24, 2010
1
0
2

g Abstract
u
A This article deals with the numerical resolution of Markovian backward stochastic differential
equations (BSDEs) with drivers of quadratic growth with respect to z and bounded terminal con-
3
ditions. We first show some bound estimates on the process Z and we specify the Zhang’s path
2
regularitytheorem. Thenwegiveanewtimediscretizationschemewithanonuniformtimenetfor

] suchBSDEsandweobtainanexplicitconvergencerateforthisscheme.
R
P
. 1 Introduction
h
t
a
m Since the early nineties, there has been an increasing interest for backward stochastic differential equa-
tions (BSDEs for short). These equations have a wide range of applications in stochastic control, in
[
finance or in partial differential equation theory. A particular class of BSDEis studied since few years:
3
BSDEswithdriversofquadratic growthwithrespecttothevariable z. Thisclassarises, forexample,in
v
1 the context of utility optimization problems with exponential utility functions, or alternatively in ques-
0
tions related to risk minimization for the entropic risk measure (see e.g. [13]). Many papers deal with
4
existence anduniqueness ofsolution forsuchBSDEs: werefer thereaderto[17,18]whentheterminal
0
. conditionisboundedand[3,4,9]fortheunboundedcase. Ourconcernisratherrelatedtothesimulation
1
0 ofBSDEsand moreprecisely timediscretization ofBSDEscoupled withaforward stochastic differen-
0 tialequation(SDEforshort). Actually,thedesignofefficientalgorithmswhichareabletosolveBSDEs
1
in any reasonable dimension has been intensively studied since the first work of Chevance [6], see for
:
v
instance[19,1,11]. Butinalltheseworks,thedriveroftheBSDEisaLipschitzfunctionwithrespectto
i
X z and this assumption plays akeyrole intheirs proofs. Inarecent paper, Cheridito and Stadje [5]stud-
r ied approximation of BSDEs by backward stochastic difference equations which are based on random
a
walksinsteadofBrownianmotions. Theyobtainaconvergenceresultwhenthedriverhasasubquadratic
growthwithrespecttozandtheygiveanexamplewherethisapproximationdoesnotconvergewhenthe
driver has a quadratic growth. To the best of our knowledge, the only work where the time approxima-
tionofaBSDEwithaquadratic growthwithrespect toz isstudied istheoneofImkellerandReis[14].
Let notice that, when the driver has a specific form1, it is possible to get around the problem by using
anexponential transformation method (see[15])orbyusing results onfully coupled forward-backward
differential equations (see[7]).
1Roughlyspeaking,thedriverisasumofaquadratictermz 7→C|z|2andafunctionthathasalineargrowthwithrespect
toz.
1
1 INTRODUCTION 2
Toexplainideasofthispaper,letusintroduce(X,Y,Z)thesolutiontotheforwardbackwardsystem
t t
X = x+ b(s,X )ds+ σ(s)dW ,
t s s
Z0 Z0
T T
Y = g(X )+ f(s,X ,Y ,Z )ds Z dW ,
t T s s s s s
−
Zt Zt
where g is bounded, f is locally Lipschitz and has a quadratic growth with respect to z. A well-known
result is that when g is a Lipschitz function with Lipschitz constant K , then the process Z is bounded
g
by C(K +1) (see Theorem 3.1). So, in this case, the driver of the BSDE is a Lipschitz function with
g
respecttoz. Thereby,asimpleideaistodoanapproximationof(Y,Z)bythesolution(YN,ZN)tothe
BSDE
T T
YN = g (X )+ f(s,X ,YN,ZN)ds ZNdW ,
t N T s s s − s s
Zt Zt
where g is a Lipschitz approximation of g. Thanks to bounded mean oscillation martingale (BMO
N
martingale in the sequel) tools, we have an error estimate for this approximation: see e.g. [14, 2] or
−α
Proposition4.2. Forexample,ifgisα-Hölder,weareabletoobtaintheerrorboundCK1−α (seePropo-
gN
sition 4.10). Moreover, we can have an error estimate for the time discretization of the approximated
BSDE thanks to any numerical scheme for BSDEs with Lipschitz driver. But, this error estimate de-
pends on KgN: roughly speaking, this error is CeCKg2Nn−1 with n the number of discretization times.
The exponential term results from the use of Gronwall’s inequality. Finally, when g is α-Hölder and
K = N,theglobalerrorboundis
gN
1 eCN2
C + . (1.1)
N1−αα n !
So, whenN increases, n−1 willhave tobecome small very quickly and the speed of convergence turns
out to be bad: if we take N = C logn 1/2 with 0 < ε < 1, then the global error bound becomes
ε
−α
Cε(logn)2(1−α). The same drawb(cid:0)ack app(cid:1)ears in the work ofImkeller and Reis [14]. Indeed, their idea
istodoanapproximation of(Y,Z)bythesolution(YN,ZN)tothetruncated BSDE
T T
YN = g(X )+ f(s,X ,YN,h (ZN))ds ZNdW ,
t T s s N s − s s
Zt Zt
where h : R1×d R1×d is a smooth modification of the projection on the open Euclidean ball of
N
→
radiusN about0. Thankstoseveralstatementsconcerningthepathregularityandstochasticsmoothness
of the solution processes, the authors show that for any β > 1, the approximation error is lower than
C N−β. So,theyobtaintheglobalerrorbound
β
1 eCN2
C + , (1.2)
β Nβ n
!
and,consequently, thespeedofconvergence alsoturnsouttobebad: ifwetakeN = C logn 1/2 with
ε
0< ε < 1,thentheglobalerrorboundbecomesC (logn)−β/2.
β,ε (cid:0) (cid:1)
Another idea is to use an estimate of Z that does not depends on K . So, we extend a result of [8]
g
whichshows
M
Z 6 M + 2 , 0 6 t < T. (1.3)
t 1
| | (T t)1/2
−
2 PRELIMINARIES 3
Let us notice that this type of estimation is well known in the case of drivers with linear growth as a
consequence oftheBismut-Elworthy formula: seee.g. [10]. Butinourcase, wedonotneedtosuppose
thatσ isinvertible. Then, thanks tothisestimation, weknowthat, whent < T,f(t,.,.,.) isaLipschitz
functionwithrespecttozandtheLipschitzconstantdependsont. Soweareabletomodifytheclassical
uniformtimenettoobtainaconvergence speedforamodifiedtimediscretization schemeforourBSDE:
the idea is to put more discretization points near the final time T than near 0. The same idea is used
by Gobet and Makhlouf in [12] for BSDEs with drivers of linear growth and a terminal function g not
Lipschitz. Butduetotechnical reasons weneedtoapplythismodifiedtimediscretization schemetothe
approximated BSDE:
T T
YN,ε = g (X )+ fε(s,X ,YN,ε,ZN,ε)ds ZN,εdW ,
t N T s s s − s s
Zt Zt
with
fε(s,x,y,z) := s<T−εf(s,x,y,z)+ s>T−εf(s,x,y,0).
1 1
Thanks to the estimate (1.3), we obtain a speed convergence for the time discretization scheme of this
approximated BSDE(seeTheorem4.8). Moreover, BMOtoolsgiveusagainanestimate oftheapprox-
imation error (see Proposition 4.2). Finally, if we suppose that g is α-Hölder, we prove that we can
− 2α
choose properly N and ε to obtain the global error estimate Cn (2−α)(2+K)−2+2α (see Theorem 4.13)
whereK > 0dependsonconstantM definedinequation(1.3)andconstantsrelatedtof. Letusnotice
2
thatsuchaspeedofconvergence whereconstants relatedtof,g,bandσ appearinthepowerofnisun-
usual. Evenifwehaveanerrorfarbetterthan(1.1)or(1.2),thisresultisnotveryinteresting inpractice
becausethespeedofconvergence stronglydependsonK. But,whenbisbounded,weprovethatwecan
take M assmall aswewant in(1.3). Finally, weobtain aglobal error estimate lowerthan C n−(α−η),
2 η
forallη > 0(seeTheorem4.16).
The paper is organized as follows. In the introductory Section 2 we recall some of the well known
results concerning SDEsand BSDEs. In Section 3 we establish some estimates concerning the process
Z: weshowafirstuniformboundforZ,thenatimedependentboundandfinallywespecifytheclassical
pathregularitytheorem. InSection4wedefineamodifiedtimediscretization schemeforBSDEswitha
nonuniformtimenetandweobtainanexpliciterrorbound.
2 Preliminaries
2.1 Notations
Throughout thispaper, (Wt)t>0 willdenote ad-dimensional Brownianmotion, defined onaprobability
space (Ω, ,P). For t > 0, let denote the σ-algebra σ(W ;0 6 s 6 t), augmented with the P-null
t s
F F
sets of . The Euclidian norm on Rd will be denoted by .. The operator norm induced by . on the
F || ||
spaceoflinearoperator isalsodenotedby .. Forp > 2,m N,wedenotefurther
|| ∈
p(Rm), or p when no confusion ispossible, the space of alladapted processes (Y ) with
t t∈[0,T]
• S S
valuesinRmnormedby Y = E[(sup Y )p]1/p; ∞(Rm),or ∞,thespaceofbounded
k kSp t∈[0,T]| t| S S
measurable processes;
p(Rm), or p, the space of all progressively measurable processes (Z ) with values in
t t∈[0,T]
• M M
Rm normedby Z = E[( T Z 2ds)p/2]1/p.
k kMp 0 | s|
In the following, we keep the sameRnotation C for all finite, nonnegative constants that appear in our
computations: they may depend on known parameters deriving from assumptions and on T, but not on
2 PRELIMINARIES 4
anyoftheapproximation anddiscretization parameters. Inthesamespirit, wekeepthesamenotation η
forallfinite,positiveconstantsthatwecantakeassmallaswewantindependently oftheapproximation
anddiscretization parameters.
2.2 Someresults on BMOmartingales
In our work, the space of BMO martingales play a key role for the a priori estimates needed in our
analysis of BSDEs. We refer the reader to [16] for the theory of BMO martingales and we just recall
t
the properties that wewilluse in the sequel. Let Φ = φ dW , t [0,T] be areal square integrable
t 0 s s ∈
martingalewithrespecttotheBrownianfiltration. ThenΦisaBMOmartingaleif
R
T 1/2
Φ = sup E[ Φ Φ ]1/2 = sup E φ2ds < + ,
k kBMO h iT −h iτ|Fτ s |Fτ ∞
τ∈[0,T] τ∈[0,T] (cid:20)Zτ (cid:21)
wherethesupremum istaken overallstopping timesin[0,T]; Φ denotes thequadratic variation ofΦ.
h i
Inourcase,theveryimportantfeatureofBMOmartingales isthefollowinglemma:
Lemma2.1. LetΦbeaBMOmartingale. Thenwehave:
1. Thestochastic exponential
t 1 t
(Φ) = = exp φ dW φ 2ds , 0 6t 6 T,
t t s s s
E E − 2 | |
(cid:18)Z0 Z0 (cid:19)
isauniformly integrable martingale.
2. ThankstothereverseHölderinequality,thereexistsp > 1suchthat Lp. Themaximalpwith
T
E ∈
thisproperty canbeexpressed intermsoftheBMOnormofΦ.
n
3. n N∗,E T φ 2ds 6 n! Φ 2n .
∀ ∈ 0 | s| k kBMO
h(cid:16) (cid:17) i
R
2.3 The backward-forward system
Given functions b, σ, g and f, for x Rd we will deal with the solution (X,Y,Z) to the following
∈
systemof(decoupled) backward-forward stochastic differential equations: fort [0,T],
∈
t t
X = x+ b(s,X )ds+ σ(s)dW , (2.1)
t s s
Z0 Z0
T T
Y = g(X )+ f(s,X ,Y ,Z )ds Z dW . (2.2)
t T s s s s s
−
Zt Zt
Forthefunctions thatappearintheabovesystemofequations wegivesomegeneral assumptions.
(HX0). b : [0,T] Rd Rd, σ : [0,T] Rd×d are measurable functions. There exist four positive
× → →
constants M ,K ,M andK suchthat t,t′ [0,T], x,x′ Rd,
b b σ σ
∀ ∈ ∀ ∈
b(t,x) 6 M (1+ x ),
b
| | | |
b(t,x) b(t′,x′) 6 K ( x x′ + t t′ 1/2),
b
− − −
σ(t) 6 M ,
(cid:12) (cid:12) σ(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) | (cid:12)| (cid:12) (cid:12) (cid:12) (cid:12)
σ(t) σ(t′) 6 K t t′ .
σ
− −
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
3 SOMEUSEFULESTIMATESOFZ 5
(HY0). f : [0,T] Rd R R1×d R, g : Rd R are measurable functions. There exist
× × × → →
five positive constants M , K , K , K and M such that t [0,T], x,x′ Rd, y,y′ R,
f f,x f,y f,z g
∀ ∈ ∀ ∈ ∀ ∈
z,z′ R1×d,
∀ ∈
f(t,x,y,z) 6 M (1+ y + z 2),
f
| | | | | |
f(t,x,y,z) f(t,x′,y′,z′) 6 K x x′ +K y y′ +(K +L (z + z′ )) z z′ ,
f,x f,y f,z f,z
− − − | | −
g(x) 6 M .
(cid:12) (cid:12) g (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) | (cid:12)| (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
Wenext recall some results onBSDEswith quadratic growth. Fortheir original version and their proof
wereferto[17],[2]and[14].
Theorem2.2. Under(HX0),(HY0),thesystem(2.1)-(2.2)hasauniquesolution(X,Y,Z) 2 ∞
∈S ×S ×
2. Themartingale Z W belongstothespaceofBMOmartingales and Z W onlydepends
M ∗ k ∗ kBMO
onT,M andM . Moreover, thereexistsr > 1suchthat (Z W) Lr.
g f
E ∗ ∈
3 Some useful estimates of Z
3.1 A firstbound forZ
Theorem3.1. Supposethat(HX0),(HY0)holdandthatgisLipschitzwithLipschitzconstantK . Then,
g
thereexistsaversionofZ suchthat, t [0,T],
∀ ∈
Z 6 e(2Kb+Kf,y)TM (K +TK ).
t σ g f,x
| |
Proof. Firstly,wesupposethatb,gandf aredifferentiable withrespecttox,yandz. Then(X,Y,Z)
isdifferentiable withrespect toxand( X, Y, Z)issolution of
∇ ∇ ∇
t
X = I + b(s,X ) X ds, (3.1)
t d s s
∇ ∇ ∇
Z0
T
Y = g(X ) X Z dW (3.2)
t T T s s
∇ ∇ ∇ − ∇
Zt
T
+ f(s,X ,Y ,Z ) X + f(s,X ,Y ,Z ) Y + f(s,X ,Y ,Z ) Z ds,
x s s s s y s s s s z s s s s
∇ ∇ ∇ ∇ ∇ ∇
Zt
t
where X = (∂Xi/∂xj) , Y = (∂Y /∂xj) R1×d, Z = (∂Zi/∂xj) and
∇ t t 16i,j6d ∇ t t 16j6d ∈ ∇ t t 16i,j6d
T
Z dW means
t ∇ s s
T
R ( Z )idWi
∇ s s
16i6dZt
X
with( Z)i denoting the i-th line of the d dmatrix process Z. Thanks tousual transformations on
∇ × ∇
theBSDEweobtain
T
eR0t∇yf(s,Xs,Ys,Zs)ds Yt = eR0T∇yf(s,Xs,Ys,Zs)ds g(XT) XT eR0s∇yf(u,Xu,Yu,Zu)du ZsdW˜s
∇ ∇ ∇ − ∇
Zt
T
+ eR0s∇yf(u,Xu,Yu,Zu)du xf(s,Xs,Ys,Zs) Xsds,
∇ ∇
Zt
3 SOMEUSEFULESTIMATESOFZ 6
withdW˜ = dW f(s,X ,Y ,Z )ds. Wehave
s s z s s s
−∇
. 2 T
f(s,X ,Y ,Z )dW = sup E f(s,X ,Y ,Z ) 2ds
z s s s s z s s s τ
∇ |∇ | F
(cid:13)Z0 (cid:13)BMO τ∈[0,T] (cid:20)Zτ (cid:12) (cid:21)
(cid:13) (cid:13)
(cid:12)
(cid:13) (cid:13) T
(cid:12)
(cid:13) (cid:13) 6 C 1+ sup E Z 2ds
s τ
| | F
τ∈[0,T] (cid:20)Zτ (cid:12) (cid:21)!
(cid:12)
= C 1+ Z W 2 . (cid:12)
k ∗ kBMO
(cid:16) (cid:17)
.
Since Z W belongs to the space of BMO martingales, f(s,X ,Y ,Z )dW < + .
∗ . 0∇z s s s s BMO ∞
Lemma2.1givesusthat ( f(s,X ,Y ,Z )dW ) isauniformly integrable martingale, soweare
E 0∇z s s s s t (cid:13)R (cid:13)
able to apply Girsanov’s theorem: there exists a probabilit(cid:13)y Q under which (W˜ ) (cid:13)is a Brownian
R t∈[0,T]
motion. Then,
eR0t∇yf(s,Xs,Ys,Zs)ds Yt = EQ eR0T∇yf(s,Xs,Ys,Zs)ds g(XT) XT
∇ ∇ ∇
hT
+ eR0s∇yf(u,Xu,Yu,Zu)du xf(s,Xs,Ys,Zs) Xsds t ,
∇ ∇ F
Zt (cid:12) (cid:21)
(cid:12)
and (cid:12)
Y 6 e(Kb+Kf,y)T(K +TK ), (3.3)
t g f,x
|∇ |
because Xt 6 eKbT. Moreover,thankstotheMalliavincalculus, itisclassicaltoshowthataversion
|∇ |
of(Z ) isgivenby( Y ( X )−1σ(t)) . Soweobtain
t t∈[0,T] t t t∈[0,T]
∇ ∇
Z 6 eKbTM Y 6 e(2Kb+Kf,y)TM (K +TK ), a.s.,
t σ t σ g f,x
| | |∇ |
because X−1 6 eKbT.
∇ t
Whenb,g andf arenotdifferentiable, wecanalsoprovetheresultbyastandardapproximation and
(cid:12) (cid:12)
stability(cid:12)resultsf(cid:12)orBSDEswithlineargrowth.
⊓⊔
3.2 A timedependent estimateofZ
Wewillintroduce twoalternativeassumptions.
(HX1). b is differentiable with respect to x and σ is differentiable with respect to t. There exists
λ R+ suchthat η Rd
∈ ∀ ∈
tησ(s)[tσ(s)t b(s,x) tσ′(s)]η 6 λ tησ(s) 2. (3.4)
∇ −
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(HX1’). σ isinvertible a(cid:12)nd t [0,T], σ(t)−1 6 M .(cid:12)
σ−1
∀ ∈
(cid:12) (cid:12)
Example. Assumption (HX1) is verifie(cid:12)d when,(cid:12) s [0,T], b(s,.) commutes with σ(s) and A :
∀ ∈ ∇ ∃
[0,T] Rd×d bounded suchthatσ′(t) = σ(t)A(t).
→
Theorem3.2. Supposethat(HX0),(HY0)holdandthat(HX1)or(HX1’)holds. Moreover,supposethat
g is lower (or upper) semi-continuous. Then there exists a version of Z and there exist two constants
C,C′ R+ thatdependonlyinT,M ,M ,K ,K ,K andL suchthat, t [0,T[,
g f f,x f,y f,z f,z
∈ ∀ ∈
Z 6 C +C′(T t)−1/2.
t
| | −
3 SOMEUSEFULESTIMATESOFZ 7
Proof. Inafirsttime,wewillsupposethat(HX1)holdsandthatf,garedifferentiable withrespectto
x,y andz. Then(Y,Z)isdifferentiable withrespect toxand( Y, Z)isthesolutionoftheBSDE
∇ ∇
T
Y = g(X ) X Z dW
t T T s s
∇ ∇ ∇ − ∇
Zt
T
+ f(s,X ,Y ,Z ) X + f(s,X ,Y ,Z ) Y + f(s,X ,Y ,Z ) Z ds.
x s s s s y s s s s z s s s s
∇ ∇ ∇ ∇ ∇ ∇
Zt
Thankstousualtransformations weobtain
t
eR0t∇yf(s,Xs,Ys,Zs)ds Yt+ eR0s∇yf(u,Xu,Yu,Zu)du xf(s,Xs,Ys,Zs) Xsds =
∇ ∇ ∇
Z0
T
eR0T∇yf(s,Xs,Ys,Zs)ds g(XT) XT + eR0s∇yf(u,Xu,Yu,Zu)du xf(s,Xs,Ys,Zs) Xsds
∇ ∇ ∇ ∇
Z0
T
eR0s∇yf(u,Xu,Yu,Zu)du ZsdW˜s,
− ∇
Zt
withdW˜ = dW f(s,X ,Y ,Z )ds. Wecanrewriteitas
s s z s s s
−∇
T
Ft = FT eR0s∇yf(u,Xu,Yu,Zu)du ZsdW˜s (3.5)
− ∇
Zt
with
t
Ft := eR0t∇yf(s,Xs,Ys,Zs)ds Yt+ eR0s∇yf(u,Xu,Yu,Zu)du xf(s,Xs,Ys,Zs) Xsds.
∇ ∇ ∇
Z0
Z W belongstothespaceofBMOmartingalessoweareabletoapplyGirsanov’stheorem: thereexists
∗
a probability Q under which (W˜ ) is a Brownian motion. Thanks to the Malliavin calculus, it is
t∈[0,T]
possibletoshowthat( Y ( X )−1σ(t)) isaversionofZ. Nowwedefine:
t t t∈[0,T]
∇ ∇
t
αt := eR0s∇yf(u,Xu,Yu,Zu)du xf(s,Xs,Ys,Zs) Xsds( Xt)−1σ(t),
∇ ∇ ∇
Z0
Z˜t := Ft( Xt)−1σ(t) = eR0t∇yf(s,Xs,Ys,Zs)dsZt+αt, a.s.,
∇
F˜ := eλtF ( X )−1.
t t t
∇
Since d X = b(t,X ) X dt, then d( X )−1 = ( X )−1 b(t,X )dt and thanks to Itô’s for-
t t t t t t
∇ ∇ ∇ ∇ − ∇ ∇
mula,
dZ˜ = dF ( X )−1σ(t) F ( X )−1 b(t,X )σ(t)dt+F ( X )−1σ′(t)dt,
t t t t t t t t
∇ − ∇ ∇ ∇
and
d(eλtZ˜ ) = F˜(λId b(t,X ))σ(t)dt+F˜σ′(t)dt+eλtdF ( X )−1σ(t).
t t t t t t
−∇ ∇
Finally,
d eλtZ˜ 2 = d M +2 λ F˜σ(t) 2 F˜σ(t)[tσ(t)t b(t,X ) tσ′(t)]tF˜ dt+dM∗,
t h it t − t ∇ t − t t
(cid:12) (cid:12) (cid:20) (cid:12) (cid:12) (cid:21)
with Mt(cid:12)(cid:12):= 0t(cid:12)(cid:12)eλsdFs(∇Xs)−1σ((cid:12)(cid:12)s) and M(cid:12)(cid:12) t∗ a Q-martingale. Thanks to the assumption (HX1) weare
2
abletoconclRude that eλtZ˜t isaQ-submartingale. Hence,
(cid:12) (cid:12)
T (cid:12) (cid:12)2 2
EQ e2λ(cid:12)s Z˜ (cid:12) ds > e2λt Z˜ (T t)
s t t
F −
(cid:20)Zt (cid:12) (cid:12) (cid:12) (cid:21) (cid:12) (cid:12)
(cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) > e2λt(cid:12)(cid:12)eR0t(cid:12)(cid:12)∇yf(s,Xs,Ys,Zs)dsZt+αt 2(T t) a.s.,
−
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
3 SOMEUSEFULESTIMATESOFZ 8
whichimplies
Zt 2(T t) = e−2λte−2R0t∇yf(s,Xs,Ys,Zs)dse2λt eR0t∇yf(s,Xs,Ys,Zs)dsZt+αt αt 2(T t)
| | − − −
6 C e2λt eR0t∇yf(s,Xs,Ys,Zs)dsZt+(cid:12)(cid:12)(cid:12) αt 2+1 (T t) (cid:12)(cid:12)(cid:12)
−
(cid:18) (cid:12) (cid:12) (cid:19)
(cid:12) T 2 (cid:12)
6 C EQ (cid:12) e2λs Z˜ ds +(T (cid:12) t) a.s.,
s t
F −
(cid:18) (cid:20)Zt (cid:12) (cid:12) (cid:12) (cid:21) (cid:19)
withCaconstant thatonlydependsonT,(cid:12)K(cid:12),M(cid:12),K ,K andλ. Moreover, wehave,a.s.,
(cid:12) b(cid:12) σ(cid:12) f,x f,y
T 2 T
EQ e2λs Z˜ ds 6 CEQ Z 2+ α 2ds
s t s s t
F | | | | F
(cid:20)Zt (cid:12) (cid:12) (cid:12) (cid:21) (cid:20)Zt (cid:12) (cid:21)
(cid:12) (cid:12) (cid:12) 6 C Z 2 +(T t)(cid:12) .
(cid:12) (cid:12) (cid:12) k kBMO(Q) − (cid:12)
(cid:16) (cid:17)
But Z doesnotdependonK because(Y,Z)isasolutionofthefollowingquadratic BSDE:
k kBMO(Q) g
T T
Y = g(X )+ (f(s,X ,Y ,Z ) Z f(s,X ,Y ,Z ))ds Z dW˜ . (3.6)
t T s s s s z s s s s s
− ∇ −
Zt Zt
Finally Z 6 C 1+(T t)−1/2 a.s..
t
| | −
When σ is invertible, the inequality (3.4) is verified with λ := M (M K + K ). Since this
(cid:0) (cid:1) σ−1 σ b σ
λ does not depend on b and σ′, we can prove the result when b(t,.) and σ are not differentiable by a
∇
standardapproximationandstabilityresultsforBSDEswithlineargrowth. So,weareallowedtoreplace
assumption(HX1)by(HX1’).
When f is not differentiable and g is only Lipschitz we can prove the result by a standard approxi-
mationandstabilityresultsforlinearBSDEs. ButwenoticethatourestimationonZ doesnotdependon
K . Thisallowsustoweakenthehypothesisongfurther: whengisonlyloweroruppersemi-continuous
g
theresultstaystrue. TheproofisthesameastheproofofProposition 4.3in[8].
⊓⊔
Remark3.3. Theprevious proof givesusamoreprecise estimation foraversion ofZ whenf isdiffer-
entiablewithrespecttoz: t [0,T],
∀ ∈
T 1/2
Z 6 C +C′EQ Z 2ds (T t)−1/2.
t s t
| | | | F −
(cid:20)Zt (cid:12) (cid:21)
Remark3.4. Whenassumptions(HX1)or(HX1’)arenot(cid:12)verified,theprocessZ mayblowupbeforeT.
(cid:12)
Zhanggivesanexampleofsuchaphenomenonindimension1: wereferthereadertoexample1in[20].
3.3 Zhang’s pathregularity Theorem
Let 0 = t < t < ... < t = T be any given partition of [0,T], and denote δ the mesh size of this
0 1 n n
partition. Wedefineasetofrandom variables
1 ti+1
Z¯ = E Z ds , i 0,...,n 1 .
ti t t s Fti ∀ ∈ { − }
i+1− i (cid:20)Zti (cid:12) (cid:21)
ThenweareabletopreciseTheorem3.4.3in[21]: (cid:12)
(cid:12)
Theorem3.5. Supposethat(HX0),(HY0)holdandgisaLipschitzfunction,withLipschitzconstantK .
g
Thenwehave
n−1E ti+1 Z Z¯ 2dt 6 C(1+K2)δ ,
t− ti g n
i=0 (cid:20)Zti (cid:21)
X (cid:12) (cid:12)
whereCisapositive constantindependent(cid:12)ofδ and(cid:12) K .
n g
3 SOMEUSEFULESTIMATESOFZ 9
Proof. We will follow the proof of Theorem 5.6., in [14]: we just need to specify how the estimate
dependsonK . Firstly,itisnotdifficulttoshowthatZ¯ isthebest -measurable approximation ofZ
g ti Fti
in 2([t ,t ]),i.e.
i i+1
M
E ti+1 Z Z¯ 2dt = inf E ti+1 Z Z 2dt .
(cid:20)Zti t− ti (cid:21) Zi∈L2(Ω,Fti) (cid:20)Zti | t− i| (cid:21)
(cid:12) (cid:12)
Inparticular, (cid:12) (cid:12)
E ti+1 Z Z¯ 2dt 6 E ti+1 Z Z 2dt .
t− ti | t− ti|
(cid:20)Zti (cid:21) (cid:20)Zti (cid:21)
(cid:12) (cid:12)
In the same spirit as previous proo(cid:12)fs, we su(cid:12)ppose in a first time that b, g and f are differentiable with
respecttox,y andz. So,
Z Z = Y ( X )−1σ(t) Y ( X )−1σ(t ) = I +I +I , a.s.,
t− ti ∇ t ∇ t −∇ ti ∇ ti i 1 2 3
with I = Y ( X )−1(σ(t) σ(t )), I = Y (( X )−1 ( X )−1)σ(t ) and I = (Y
1 ∇ t ∇ t − i 2 ∇ t ∇ t − ∇ ti i 3 ∇ t −
Y )( X )−1σ(t ). Firstly,thankstotheestimation(3.3)wehave
ti ∇ ti i
I 2 6 Y 2e2KbTK2 t t 2 6 C(1+K2)δ2.
| 1| |∇ t| σ| i+1− i| g n
Weobtainthesameestimationfor I because
2
| |
t
( X )−1 ( X )−1 6 ( X )−1 b(s,X )ds 6 K eKbT t t .
∇ t − ∇ ti ∇ s ∇ s b | − i|
(cid:12)Zti (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
Lastly,|I3| 6(cid:12)MσeKbT |∇Yt−∇Yti|.(cid:12)So,(cid:12)(cid:12) (cid:12)(cid:12)
n−1 ti+1 n−1
E I 2dt 6 Cδ E esssup Y Y 2 .
| 3| n |∇ t−∇ ti|
i=0 (cid:20)Zti (cid:21) i=0 "t∈[ti,ti+1] #
X X
ByusingtheBSDE(3.2),(HY0),theestimateon X andtheestimate(3.3),wehave
s
∇
Y Y 2
|∇ t−∇ ti|
t 2 t 2
6 C (C(1+K )+ f(s,X ,Y ,Z ) Z )ds +C Z dW .
g z s s s s s s
|∇ ||∇ | ∇
(cid:18)Zti (cid:19) (cid:18)Zti (cid:19)
Theinequalities ofHölderandBurkholder-Davis-Gundy giveus
n−1
E esssup Y Y 2
|∇ t−∇ ti|
i=0 "t∈[ti,ti+1] #
X
n−1 ti+1 2 ti+1
6 C(1+K2)+C E f(s,X ,Y ,Z ) Z ds +CE Z 2ds
g |∇z s s s ||∇ s| |∇ s|
i=0 (cid:18)Zti (cid:19) (cid:18)Zti (cid:19)
X
T 2 T
6 C(1+K2)+CE f(s,X ,Y ,Z ) Z ds + Z 2ds
g |∇z s s s ||∇ s| |∇ s|
"(cid:18)Z0 (cid:19) Z0 #
T T T
6 C(1+K2)+CE (1+ Z 2)ds Z 2ds + Z 2ds
g | s| |∇ s| |∇ s|
(cid:20)(cid:18)Z0 (cid:19)(cid:18)Z0 (cid:19) Z0 (cid:21)
T p 1/p T q 1/q
6 C(1+K2)+C 1+E Z 2ds E Z 2ds ,
g  | s|  |∇ s|
"(cid:18)Z0 (cid:19) # "(cid:18)Z0 (cid:19) #
 
4 CONVERGENCEOFAMODIFIEDTIMEDISCRETIZATIONSCHEMEFORTHEBSDE 10
for all p > 1 and q > 1 such that 1/p+1/q = 1. But, ( Y, Z) is solution of BSDE(3.2), so, from
∇ ∇
Corollary9in[2],thereexistsq thatonlydependson Z W suchthat
k ∗ kBMO
T q 1/q
E Z 2ds 6 C(1+K2).
|∇ s| g
"(cid:18)Z0 (cid:19) #
Moreover,wecanapplyLemma2.1toobtaintheestimate
T p 1/p
E Z 2ds 6 C Z 2 6 C.
| s| k kBMO
"(cid:18)Z0 (cid:19) #
Finally,
n−1 ti+1
E I 2dt 6 C(1+K2)δ
| 3| g n
i=0 (cid:20)Zti (cid:21)
X
and
n−1E ti+1 Z Z¯ 2dt 6 n−1E ti+1 I 2+ I 2+ I 2 dt
t− ti | 1| | 2| | 3|
Xi=0 (cid:20)Zti (cid:12) (cid:12) (cid:21) Xi=0 (cid:20)Zti (cid:16) (cid:17) (cid:21)
(cid:12) (cid:12) 6 C(1+K2)δ .
g n
⊓⊔
4 Convergence of a modified time discretization scheme for the BSDE
4.1 Anapproximationofthe quadratic BSDE
In a first time we will approximate our quadratic BSDE (2.2) by another one. We set ε ]0,T[ and
N N. Let(YN,ε,ZN,ε)thesolution oftheBSDE ∈
∈ t t
T T
YN,ε = g (X )+ fε(s,X ,YN,ε,ZN,ε)ds ZN,εdW , (4.1)
t N T s s s − s s
Zt Zt
with
fε(s,x,y,z) := s6T−εf(s,x,y,z)+ s>T−εf(s,x,y,0),
1 1
andg aLipschitzapproximationofgwithLipschitzconstantN. fεverifiesassumption(HY0)withthe
N
sameconstants asf. Since g isaLipschitz function, ZN,ε has abounded version andthe BSDE(4.1)
N
isaBSDEwithalineargrowth. Moreover, wecanapplyTheorem3.2toobtain:
Proposition 4.1. Letusassume that (HX0),(HY0)and (HX1)or (HX1’)hold. Thereexists aversion of
ZN,ε and there exist three constants M ,M ,M R+ that do not depend on N and ε such that,
z,1 z,2 z,3
∈
s [0,T],
∀ ∈
M
ZN,ε 6 M + z,2 (M (N +1)).
s z,1 (T s)1/2 ∧ z,3
(cid:18) − (cid:19)
(cid:12) (cid:12)
ThankstoBMOtools(cid:12)weha(cid:12)veastability resultforquadratic BSDEs(see[2]and[14]):