Table Of ContentNumerical Schemes for Multivalued Backward Stochastic
Differential Systems
Lucian Maticiuc∗, Eduard Rotenstein∗
1
1
0
2 Faculty of Mathematics, “Alexandru Ioan Cuza” University,
n Carol I Blvd., no. 9, 700506, Ia¸si, Romˆania.
a
J
0
1
Abstract
] Wedefinesomeapproximationschemesfordifferentkindsofgeneralizedbackwardstochastic
R
differential systems, considered in the Markovian framework. We propose a mixed approxima-
P
tion scheme for the following backward stochastic variational inequality
.
h
t dY +F(t,X ,Y ,Z )dt+G(t,X ,Y )dA ∂ϕ(Y )dt+Z dW ,
a t t t t t t t t t t
∈
m
[ where(Xt,At)t∈[0,T] istheuniquesolutionofareflectedforwardstochasticdifferentialequation.
More precisely, we use an Euler scheme type for the system of decoupled forward-backward
1 variational inequality, combined with Yosida approximationtechniques.
v
1
Key words and phrases: Eulerscheme, Yosida approximation, error estimate, multivalued back-
3
ward SDEs, reflected SDEs
8
1
2010 Mathematics Subject Classification: 65C99, 60H30, 47H15
.
1
0
1
1 1 Introduction
:
v
i Thestochastic differential equations (SDE)withreflectingboundaryconditions,alsocalledreflected
X
stochastic differential equations (RSDE)appearsfromthemodelingofdifferentkindsofconstrained
r
a phenomenon. Theelliptic andparabolicPDEswithNeumanntypeandmixedboundaryconditions
leadustoprobabilisticinterpretations,viatheFeynman–Kac¸formula,intermsofreflecteddiffusion
processes, which are solutions of RSDEs. This type of equations were studied for the first time by
Skorohod in [22] and after this, considered in general domains (see [11], [14], [21], [23]...).
Since 1990s a lot of researchers focused their attention to numerical schemes, methods and
algorithms for the study of the behavior of the solution for RSDEs. In the recent years, some
new techniques consist in splitting-step algorithms and mixed penalization methods. The Euler
approximation was considered for the first time by Chitashvili and Lazrieva in [5], followed by
the Euler-Peano approximation, which was introduced by Saisho in [21]. Lepingle in [10] and
∗
Acknowledgements: Theworkofthefirstauthor wassupportedbyPOSDRU/89/1.5/S/49944 projectand,forthesecond
one,byIDEI395/2007 project.
E-mail addresses: [email protected] (LucianMaticiuc),[email protected](EduardRotenstein).
1
Slominski in [23] analyzed the corresponding numerical schemes and their rates of convergence. In
order to approximate the solution of RSDEs, the penalization method was also very useful (see
Menaldi, [14]). Approximation methods for a diffusion reflected and stopped at the boundary
appear in the literature in 1998, in the paper of Constantini, Pacchiarotti and Sartoretto [6]. They
defined a standard Euler projected approach to stopped reflected diffusions, approach which yields
a method with weak order of convergence (1/2 in particular) and they give an easy example where
this convergence rate is precise. Regarding adaptive approximations of one-dimensional reflected
Brownian motion, it can be used a simple method of two fixed step sizes chosen according to the
distance at the boundary.
In the paper [1], Asiminoaei and Ra˘¸scanu used a mixed method consisting in penalization and
splitting-up for the study of multivalued SDE with reflection at the boundary of the domain. The
penalization method was also used by Ra˘¸scanu in [19] for the study of the generalized Skorohod
problem and of its link to multivalued SDE governed by a general maximal monotone operator (of
sudifferential type). Recently, in [7], Ding and Zhang combined the penalization technique with
the splitting-step idea to propose some new schemes for the RSDE in the upper half space.
In 1990, Pardoux and Peng introduced in [15] the notion of nonlinear backward stochastic
differential equation (for short, BSDE), and they obtained the existence and uniqueness result for
this kind of equation. Sincethen, theinterest in BSDEs has kept growing and therehave been alot
ofworksonthatsubject,bothinthedirectionofthegeneralization oftheequationsthatappearand
inconstructingschemesofapproximationforthem. Thebackward stochastic variational inequalities
(for short, BSVI) were analyzed by Pardoux and Ra˘¸scanu in [17] and [18] (the extension for Hilbert
spaces case) by a method that consists in a penalizing scheme, followed by its convergence.
Starting with the paper of Pardoux and Peng [16], have been given a stochastic approach for
the existence problem of a solution for many type of deterministic partial differential equations
(PDE for short). In [17] it is proved, using a probabilistic interpretation, the existence for the
viscosity solution for a multivalued PDE (with subdifferential operator) of parabolic and elliptic
type. More recently, Maticiuc andRa˘¸scanu in [12], prove an extended resultconcerning generalized
type of BSDE (including an integral with respect to an adapted continuous increasing function and
two subdifferential operators). These type of BSVI allows to prove Feynman-Kac type formula
for the representation of the solution of PVI with mixed nonlinear multivalued Neumann-Dirichlet
boundary conditions.
Even this type of the penalization approach is very useful when we deal with multivalued
backward stochastic dynamical systems governed by a subdifferential operator, it fails for the case
of a general maximal monotone operator. This motivated a new approach, via convex analysis, for
the study of both forward and backward multivalued differential systems. In [20], Ra˘¸scanu and
Rotenstein identified the solutions of those type of equations with the minimum points of some
proper, convex, lower semicontinuous functions, defined on well chosen Banach spaces.
Euler-type approximation schemes for BSDE, and for BSDE with exit time for the forwardpart
of the system, were introduced by Bouchard and Touzi in [4] and Bouchard and Menozzi in [3].
They considered the Markovian framework of a coupled forward-backward stochastic differential
system and they defined an adapted backward Euler scheme for the strong approximation of the
backwardSDEwithfinitestoppingtimehorizon,namelythefirstexittimeoftheforwardSDEfrom
a cylindrical domain. In [2], Bouchard and Chassagneux study the discrete-time approximation of
the solution of a BSDE with a reflecting barrier.
The paper is organized as follows. Section 2 presents some basic notations, hypothesis and
2
results that are used throughout this paper. Section 3 is dedicated to the analysis of the behavior
of an approximation scheme defined for a backward stochastic variational inequality. In Section
4 we present an existence and uniqueness result for a generalized BSVI and we propose a mixed
Euler type approximation scheme for its solution.
2 Notations. Hypothesis. Preliminaries
In all that follows we shall consider a finite horizon T > 0 and a complete probability space
(Ω, ,P) on which is defined a standard d-dimensional Brownian motion W = (W ) whose
F t t≤T
natural filtration is denoted F= , 0 t T . More precisely, F is the filtration generated by
t
{F ≤ ≤ }
the process W and augmented by P, the set of all P-null sets, i.e. t = σ Ws, s t P.
N F { ≤ }∨N
We denote by Lr (Ω;C([0,T];Rk)), r [1, ), the closed linear subspace of adapted stochastic
ad ∈ ∞
processes f Lr(Ω, ,P;C([0,T];Rk)), i.e. f(,t) : Ω Rk is -measurable for all t [0,T]
t
∈ F · → F ∈
and E sup f(t)r < . Also, we shall use the notation Lr (Ω;Lq([0,T];Rk)), r,q
t∈[0,T]| | ∞ ad ∈
[1, ) (cid:16)the Banach spac(cid:17)e of -measurable stochastic processes f : Ω [0,T] Rk such that
t
∞ F × →
r/q
E T f(t)qdt < .
0 | | ∞
Co(cid:16)nRsider the fol(cid:17)lowing data:
the continuous coefficient functions b :[0,T] Rm Rm, σ : [0,T] Rm Rm×d, g :Rm
• × → × → →
Rn and F : [0,T] Rm Rn Rn×d R, which satisfies the following standard assumptions:
× × × →
for some constants α R, L, β, γ 0 and for all t [0,T], x, x˜ Rm, y, y˜ Rn and z,
∈ ≥ ∈ ∈ ∈
z˜ Rn×d :
∈
(i) b(t,x) b(t,x˜) + σ(t,x) σ(t,x˜) L x x˜ ,
| − | k − k ≤ | − |
(1) (ii) y y˜,F(t,x,y,z) F(t,x,y˜,z) α y y˜2,
h − − i ≤ | − |
(iii) F(t,x,y,z) F(t,x,y,z˜) β z z˜ ,
| − | ≤ k − k
and there exist some constants M > 0 and p, q N such that, for all t [0,T], x Rm and
∈ ∈ ∈
y Rn :
∈
(2) g(x) M(1+ x q) and F(t,x,y,0) M(1+ x p+ y ).
| | ≤ | | | | ≤ | | | |
the function ϕ : Rn ( ,+ ] which is a proper convex lower semicontinuous function
• → −∞ ∞
and satisfies that there exist M > 0 and r N such that, for all x Rm :
∈ ∈
(3) ϕ(g(x)) M(1+ x r).
| | ≤ | |
The following theorem summarizes some already well known results concerning forward and
backward SDE, considered in the Markovian framework (for the proof see Karatzas & Shreve [8],
for forward case, and Pardoux & Ra˘¸scanu [17] for the backward system).
Theorem 1 Let (t,x) [0,T] Rm be fixed. Under the assumptions (1), (2) and (3), the forward-
∈ ×
backward coupled system
t,x t,x t,x
dX = b(s,X )ds+σ(s,X )dW , s [0,T],
s s s s
∈
(4) dYt,x+F(s,Xt,x,Yt,x,Zt,x)ds ∂ϕ(Yt,x)ds+Zt,xdW , t [0,T],
s s s s s s s
∈ ∈
t,x t,x t,x
X = x, Y = g(X ),
t T T
3
has a unique solution, i.e. there exist a unique process Xt,x L2 (Ω;C([0,T];Rm)) such that
∈ ad
t∨s t∨s
(5) Xt,x =x+ b(r,Xt,x)dr+ σ(r,Xt,x)dW , s [0,T],
s r r r ∈
Zt Zt
and respectively
(Yt,x,Zt,x,Ut,x) L2 (Ω;C([0,T];Rn)) L2 (Ω;L2([0,T];Rn×d)) L2 (Ω;L2([0,T];Rn)),
∈ ad × ad × ad
such that
(6)
T T T
Yt,x+ Ut,xdr = g(Xt,x)+ 1 (r)F(r,Xt,x,Yt,x,Zt,x)dr Zt,xdW , s [0,T]
s r T [t,T] r r r − r r ∈
Zs Zs Zs
Ut,x ∂ϕ Yt,x , dP ds on Ω [0,T].
s s
∈ × ×
Moreover, for a(cid:16)ll p (cid:17)2, there exists some constant Cp > 0, q N∗, such that, for all t, t˜ [0,T],
≥ ∈ ∈
x, x˜ Rn :
∈
(j) E sup Xt,x p C (1+ x p),
s∈[0,T] s ≤ p | |
(jj) E(cid:0)sup (cid:12)Xt,x(cid:12) (cid:1)Xt˜,x˜ p C (1+ x p+ x˜ pq)(t t˜p/2+ x x˜ p),
s∈[0,T](cid:12) s (cid:12)− s ≤ p | | | | | − | | − |
(jjj) E(cid:0)sup (cid:12)Yt,x 2 C(cid:12)(1(cid:1)+ x 2),
s∈[0,T](cid:12) s ≤ (cid:12) | |
(jv) E(cid:0)sup (cid:12)Yt,x(cid:12) (cid:1)Yt˜,x˜ 2 C E g(Xt,x) g(Xt˜,x˜) 2 +
s∈[0,T](cid:12) s (cid:12)− s ≤ 2 T − T
(cid:0) +E(cid:12)(cid:12) T 1 (r)(cid:12)(cid:12)F(cid:1)(r,Xt,xh,Y(cid:12)(cid:12)t,x,Zt,x) 1 (r(cid:12)(cid:12))F(r,Xt˜,x˜,Yt,x,Zt,x) 2dr .
[t,T] r r r − [t˜,T] r r r
Z0 (cid:21)
(cid:12) (cid:12)
(cid:12) (cid:12)
3 Approximations schemes for BSVI
We will consider a partition of [0,T],
π = t = ih :0 i n , with h:= T/n, n N∗,
i
{ ≤ ≤ } ∈
onwhichweapproximatethesolution ofthebackwardstochastic variational inequality (6). For the
numerical simulations of the forward part, the most standard approach consists in approximating
the SDE in a proper way on each interval [t ,t ] by the classical Euler scheme (see, e.g. Kloeden
i i+1
& Platen [9]):
Xh = Xh +b Xh h+σ Xh W W , i = 0,n 1
ti+1 ti ti ti ti+1 − ti −
( Xh = X . (cid:0) (cid:1) (cid:0) (cid:1)(cid:0) (cid:1)
0 0
We remark that the above numerical scheme is easy to implement since it requires only the simu-
lation of d-independent Gaussian variables for the Brownian increments, providing a weak error of
h order.
For t [t ,t ] let
i i+1
∈
Xh = Xh +b(Xh)(t t )+σ(Xh)(W W ).
t ti ti − i ti t− ti
We have the following estimation of the error given by the Euler scheme (see [9]).
4
Proposition 2 Under the assumptions (1) on the coefficients b and σ, for all p 1, there exists
≥
C > 0 such that
p
1/p
maxE sup X Xh p+ sup X X p C √h.
t− t t− ti ≤ p
0,n−1 (cid:18)t∈[0,T] t∈[ti,ti+1] (cid:19)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
Here and subsequently we will consider the one-dimensional BSDE case.
Using the Yosida approximation ϕ of the multivalued operator ∂ϕ, with ε = ha and a
ε
∇ ∈
(0,1/2) (the way of choosing this constant will be detailed later), we deduce that the following
approximate equation
T T T
(7) Yth+ ∇ϕha(Yrh)dr = g(XT)+ F(r,Xr,Yrh,Zrh)dr− ZrhdWr, ∀t ∈ [0,T], P−a.s.,
Zt Zt Zt
admits a unique solution Yh,Zh L2 (Ω;C([0,T];R)) L2 (Ω;L2([0,T];Rd)).
t t ∈ ad × ad
Further, inspired by the paper of Bouchard and Touzi, [4], let us define an Euler type approx-
(cid:0) (cid:1)
imation for the Yosida approximation process Yε. For an intuitive introduction, let Yh := g(Xπ)
T T
be the initial condition, and, for i = n 1,0, remark that
−
(8) Ythi ∼ Ythi+1 +h F(ti,Xthi,Ythi,Zthi)−∇ϕha(Ythi) −Zthi(Wti+1 −Wti);
h i
taking the conditional expectation Ei( ):= E( ), we obtain
· · |Fti
Ythi ∼ Ei(Ythi+1)+h F(ti,Xthi,Ythi,Zthi)−∇ϕha(Ythi) .
h i
If we multiply (8) by W W it follows
ti+1 − ti
1
Zh Ei(Yh (W W )).
ti ∼ h ti+1 ti+1 − ti
Therefore,weproposethefollowingimplicitdiscretizationprocedure,whichdefinethepair Yh,Zh
inductively, for i= n 1,0 :
− (cid:0) (cid:1)
Y˜h := g(Xh), Z˜h = 0,
T T T
Y˜thi := Ei,h(Y˜thi+1)+h F(ti,Xthi,Y˜thi,Z˜thi)−∇ϕha(Y˜thi) ,
(9) 1 h i
Z˜h := Ei,h(Y˜h (W W )),
ti h ti+1 ti+1 − ti
U˜thi := ∇ϕha(Ei,h(Y˜thi+1)),
where Ei,h() := E h and h := σ(Xh :0 j i).
· · |Fti Fti tj ≤ ≤
(cid:0) (cid:1)
Remark 3 Observe that Y˜h is defined implicitly as the solution of a fixed point problem. Since
ti
the involved functions are Lipschitz, it is well defined. Moreover, for small values of h > 0 it can
be estimated numerically in an accurate way.
5
Remark 4 We can also use an explicit scheme to define
Y˜thi := Ei,h(Y˜thi+1)+hEi,h F(ti,Xthi,Y˜thi+1,Z˜thi)−∇ϕha(Y˜thi+1) .
h i
The advantage of this scheme is that it does not require a fixed point procedure but, from a numerical
point of view, adding a term in the conditional expectation makes it more difficult to estimate.
Therefore the implicit scheme can be more tractable in practice.
Remark 5 We have that the filtration generated by the Brownian motion coincides with the
t
F
filtration generated by the diffusion process X, i.e. = X, and, from the Markov property of the
Ft Ft
process Xh, it follows that
Ei(Y˜h ) = Ei,h(Y˜h ) = E(Y˜h Xh),
ti+1 ti+1 ti+1 | ti
Ei(Y˜h (W W )) =Ei,h(Y˜h (W W )) = E(Y˜h (W W ) Xh).
ti+1 ti+1 − ti ti+1 ti+1 − ti ti+1 ti+1 − ti | ti
Consider now a continuous version of (9). From the martingale representation theorem there
exists a square integrable process Z˜h such that
ti+1
(10) Y˜h = Ei(Y˜h )+ Z˜hdW ,
ti+1 ti+1 s s
Zti
and, therefore, we define, for t (t ,t ],
i i+1
∈
t
(11) Y¯th := Y˜thi −(t−ti) f(ti,Xthi,Y˜thi,Z˜thi)−∇ϕha(Y˜thi) + Z˜shdWs.
h i Zti
Obviously, we obtain that Y¯h = Y˜h, and, for the simplicity of the notation, we will continue to
ti ti
write Y˜h for Y¯h.
t t
Remark 6 From (9), (10) and the isometry property, we notice that, for i= 0,n 1,
−
ti+1 ti+1
(12) h Z˜h = Ei(Y˜h (W W )) = Ei (W W ) Z˜hdW = Ei Z˜hds .
ti ti+1 ti+1 − ti ti+1 − ti s s s
(cid:20) Zti (cid:21) (cid:20)Zti (cid:21)
To approximate Zh we use
t
1 ti+1
Z¯h := Ei Zhds , t [t ,t )
t h s ∈ i i+1
(cid:20)Zti (cid:21)
rather than Zh, which is the best approximation in L2(Ω [0,T]) of Zh by adapted processes
ti ×
which are constant on each interval [t ,t ) (see Lemma 3.4.2 from Zhang [24]):
i i+1
ti+1 ti+1
E Zh Z¯h 2ds E Zh η 2ds ,
| s − ti| ≤ | s − |
(cid:20)Zti (cid:21) (cid:20)Zti (cid:21)
for all -measurable stochastic process η.
Fti
6
Remark 7 From (12), the definition of Z¯h and Jensen inequality we obtain
ti
1 ti+1 2 1 ti+1 2
E Z¯h Z˜h 2 = E Ei ∆hZ ds E Ei∆hZ ds
| ti − ti| h2 s ≤ h s ≤
(13) (cid:20) Zti (cid:21) Zti h i
1 ti+1 1 ti+1
E Ei ∆hZ 2ds = E ∆hZ 2ds.
s s
≤ h | | h | |
Zti Zti
In order to prove an error estimate of the scheme first we use the solution Yh,Zh of the
t t t∈[0,T]
approximating equation (7). The next result is a straightforward consequence of Proposition 2.3
(cid:0) (cid:1)
from Pardoux & Ra˘¸scanu [17].
Proposition 8 Under the assumptions (1)-(3), there exists C > 0 such that
T
(14) sup E Y Yh 2+E Z Zh 2dt CΓ(T)ha,
| t− t | | t− t | ≤
t∈[0,T] Z0
where Γ(T):= E 1+ g(X ) 2+ X r + T F 0,Xh,0,0 ds
| T | | T| 0 s
h i
R (cid:0) (cid:1)
We recall Theorem 3.4.3 from [24], applied for the solution Yh,Zh of (7). To otain a similar
t t
conclusion we have to impose more restrictive assumptions than (1-3):
(cid:0) (cid:1)
there exists some constant K > 0, such that
•
(i) b(x) b(x˜) + σ(x) σ(x˜) K x x˜ , x,x˜ Rm,
| − | k − k ≤ | − | ∀ ∈
(15) (ii) F(ξ) F(ξ˜) K ξ ξ˜, ξ,ξ˜ [0,T] Rm R Rd,
| − | ≤ | − | ∀ ∈ × × ×
(iii) g(y) g(y˜) K y y˜ , y,y˜ R;
| − | ≤ | − | ∀ ∈
the function ϕ :Rn ( ,+ ] is a proper convex lower semicontinuous function and there
• → −∞ ∞
exist M > 0 and r N such that
∈
(16) ϕ(g(x)) M(1+ x r), x Rm.
| | ≤ | | ∀ ∈
Proposition 9 Let the assumptions (15) and (16) be satisfied. We have the following estimate,
for some C > 0,
n ti+1
max sup E Yh Yh 2+ E Zh Z¯h 2ds Ch,
| t − ti+1| | s − ti| ≤
i=0,n−1t∈[ti,ti+1] i=1 Zti
X
where Z¯h := 1Ei ti+1Zhds .
ti h ti s
h i
R
Proof. The inequality
max sup E Yh Yh 2 Ch
| t − ti+1| ≤
i=0,n−1t∈[ti,ti+1]
can be obtained by classical calculus, using Itˆo’s formula, Lipschitz property of the coefficient
functions and the bounds of the approximate solution Yh,Zh of (7) (see Proposition 2.1
t t t∈[0,T]
and 2.2 from [17]).
(cid:0) (cid:1)
7
For the proof of the inequality
n ti+1
E Zh Z¯h 2ds Ch
| s − ti| ≤
i=1 Zti
X
is sufficient to recall the proof of Theorem 3.4.3 from [24].
Using the estimates from the above Propositions we can prove the following:
Proposition 10 Let the assumptions (15) and (16) be satisfied. Then there exists C > 0 such that
T
sup E Yh Y˜h 2+E Zh Z˜h 2dt Ch1−2a.
| t − t | | t − t | ≤
t∈[0,T] Z0
Proof. From (7) and (11) we deduce that, for i = 0,n 1 and t [t ,t ],
i i+1
− ∈
ti+1 ti+1
Yth = Ythi+1 + F(s,Xs,Ysh,Zsh)−∇ϕha(Ysh) ds− ZshdWs,
Zt h i Zt
ti+1 ti+1
Y˜th = Y˜thi+1 + F(ti,Xthi,Y˜thi,Z˜thi)−∇ϕha(Y˜thi) ds− Z˜shdWs.
Zt h i Zt
Throughout the proof let ∆hF := F(t,X ,Yh,Zh) F(t ,Xh,Y˜h,Z˜h), ∆hY := Yh Y˜h and
t t t t − i ti ti ti t t − t
∆hZ := Zh Z˜h, t [t ,t ].
t t − t ∈ i i+1
Applying Energy equality we obtain that
ti+1 ti+1
(17) E ∆hY 2+ E ∆hZ 2ds = E ∆hY 2+2E ∆hY ∆hF ds
| t| | s| | ti+1| s s
Zt Zt
ti+1
−2E ∆hYs ∇ϕha(Ysh)−∇ϕha(Y˜thi) ds,
Zt (cid:16) (cid:17)
We first compute ∆hYs · ∆hFs−(∇ϕha(Ysh)−∇ϕha(Y˜thi)) for which we use Lipschitz property
of F and ϕha : h i
∇
(18)
ti+1
2E ∆hYs· ∆hFs −(∇ϕha(Ysh)−∇ϕha(Y˜thi)) ds ≤
Zt h i
ti+1 1
2KE ∆hY s t + X Xh + Yh Y˜h + Zh Z˜h + Yh Y˜h ds
≤ s · | − i| | s− ti| | s − ti| | s − ti| ha| s − ti| ≤
Zt (cid:12) (cid:12) (cid:20) (cid:21)
(cid:12) ti+(cid:12)1 4 ti+1 4
K2α+β (cid:12)E (cid:12) ∆hY 2ds+ E X Xh 2ds+ h3+
≤ | s| α | s − ti| α
Zt Zt
4(cid:0) ti+1 (cid:1) 4 1 ti+1
+ E Zh Z˜h 2ds+ + E Yh Y˜h 2ds,
α | s − ti| α βh2a | s − ti|
Zt (cid:16) (cid:17) Zt
where α,β > 0 will be chosen later.
From now on, let C > 0 be a constant independent of h, constant which can take different
values from one line to another.
From Proposition 2 we have that there exists C > 0 such that
E X Xh 2 2E X X 2+2E X Xh 2 Ch,
| s− ti| ≤ | s− ti| | ti − ti| ≤
8
and, from Proposition 9,
E Yh Y˜h 2 2E Yh Yh 2+2E Yh Y˜h 2 Ch+2E ∆hY 2.
| s − ti| ≤ | s − ti| | ti − ti| ≤ | ti|
Using (13)
2 ti+1
E Zh Z˜h 2 2E Zh Z¯h 2+2E Z¯h Z˜h 2 = 2E Zh Z¯h 2+ E ∆hZ 2ds.
| s − ti| ≤ | s − ti| | ti − ti| | s − ti| h | s|
Zti
Then (18) yields
ti+1 ti+1
(19) A (t) := E ∆hY 2+ E ∆hZ 2ds K2α+β E ∆hY 2ds+B ,
i t s s i
| | | | ≤ | |
Zt Zt
(cid:0) (cid:1)
where
8 ti+1 8 ti+1 4 1
B := E ∆hY 2+ E Zh Z¯h 2ds+ E ∆hZ 2ds+ + Ch2+
i | ti+1| α | s − ti| α | s| α βh2a
(20) Zti Zti (cid:16) (cid:17)
+2h 4 + 1 E ∆hY 2.
α βh2a ti
(cid:16) (cid:17) (cid:12) (cid:12)
Using a backward Gronwall(cid:12)type in(cid:12)equality we deduce
ti+1
(K2α+β)ds
E ∆hYt 2 BieZt Bie(K2α+β)h CBi,
| | ≤ ≤ ≤
and, therefore,
ti+1
A (t) B + K2α+β CB ds = B 1+C K2α+β h B [1+Ch], h (0,1).
i i i i i
≤ ≤ ∈
Zt
(cid:0) (cid:1) (cid:2) (cid:0) (cid:1) (cid:3)
The above inequality and the definition of B implies
i
4 1 8 ti+1
1 (1+Ch) + 2h E ∆hY 2+ 1 (1+Ch) E ∆hZ 2ds
− α βh2a | ti| − α | s| ≤
(cid:20) (cid:16) (cid:17) (cid:21) (cid:20) (cid:21)Zti
8 ti+1
(1+Ch) E ∆hY 2+ E Zh Z¯h 2ds+Ch2−2a .
≤ | ti+1| α | s − ti|
(cid:20) Zti (cid:21)
Taking a (0,1/2), we can chose h > 0 sufficiently small and α,β > 0 large enough such that
∈
C := 1 (1+Ch) 4 + 1 2h > 0 and C := 1 (1+Ch) 8 > 0
1 − α βh2a 2 − α
(cid:16) (cid:17)
and, therefore,
ti+1
C E ∆hY 2+C E ∆hZ 2ds
1 | ti| 2 | s| ≤
(21) Zti
8 ti+1
(1+Ch) E ∆hY 2+ E Zh Z¯h 2ds+Ch2−2a .
≤ | ti+1| α | s − ti|
(cid:20) Zti (cid:21)
9
Writing the above inequality for each i= 0,n 1, we can deduce
−
n ti+1
E ∆hY 2 (1+Ch)n h2−2a+E ∆hY 2+ E Zh Z¯h 2ds , i = 0,n 1.
| ti| ≤ | T| | s − ti| −
" i=1Zti #
X
From the Lipschitz property of g and Proposition 9 we obtain, for each i = 0,n 1, since a
− ∈
(0,1/2), that
(22) E ∆hY 2 Ch, h (0,1) small enough.
| ti| ≤ ∀ ∈
For the proof of the inequality concerning ∆hZ we act in the following manner
s L2(Ω×[0,T])
(see, e.g. Bouchard & Touzi [4]). From (21) it follows that
(cid:13) (cid:13)
(cid:13) (cid:13)
T n−1 ti+1
E ∆hZ 2ds = E ∆hZ 2ds
s s
| | | | ≤
Z0 i=0Zti
X
n−1 8n−1 ti+1 n−1
(1+Ch) E ∆hY 2+Ch2−2a n+ E Zh Z¯h 2ds C E ∆hY 2 =
≤ | ti+1| α | s − ti| − 1 | ti|
"i=0 i=0Zti # i=0
X X X
8n−1 ti+1
= (1+Ch) E ∆hY 2+Ch1−2a+ E Zh Z¯h 2ds +
| T| α | s − ti|
" i=0Zti #
X
n−1
+ (1+Ch)+(1+Ch) 4 + 1 2h 1 E ∆hY 2 C E ∆hY 2.
α βh2a − | ti| − 1 | 0|
(cid:16) (cid:0) (cid:1) (cid:17)Xi=1
and, therefore, from (22),
T n−1 ti+1
E ∆hZ 2ds C E ∆hY 2+ E Zh Z¯h 2ds+h1−2a +
| s| ≤ | T| | s − ti|
Z0 " i=0Zti #
X
n−1
8 2 8 2
+ Ch+ h+ h1−2a+ h2+ h2−2a E ∆hY 2
α β α β | ti| ≤
(cid:18) (cid:19) i=1
X
n−1 ti+1
C E ∆hY 2+ E Zh Z¯h 2ds+h1−2a+h1−2aCh n =
≤ | T| | s − ti|
" i=0Zti #
X
= C Ch+Ch+Ch1−2a Ch1−2a.
≤
Using the definitio(cid:2)n (20) of B we ded(cid:3)uce that B Ch and, respectively, maxE ∆hY 2 Ch,
i i ≤ | ti| ≤
which completes the proof.
Consequently we have proved our main result:
Theorem 11 There exists the constant C > 0 which depends only on the Lipschitz constants of
the coefficients, such that:
T
(23) sup E Y Y˜h 2+E Y Y˜h 2+ Z Z˜h 2 dt Cha∧(1−2a).
| t− t | | t− t | | t− t | ≤
t∈[0,T] Z0 h i
10
