www.imstat.org/aihp 2011, Vol. 47, No. 2, 395–424
DOI:10.1214/10-AIHP357
© Association des Publications de l’Institut Henri Poincaré, 2011
Stochastic representations of derivatives of solutions of one-dimensional parabolic variational inequalities
with Neumann boundary conditions
Mireille Bossy
a, Mamadou Cissé
band Denis Talay
aaEPI TOSCA, INRIA, Sophia Antipolis, France. E-mails:Mireille.Bossy@sophia.inria.fr;Denis.Talay@sophia.inria.fr bENSAE-Sénégal, BP 45512 Dakar Fann, Dakar, Sénégal
Received 6 May 2009; revised 9 October 2009; accepted 7 January 2010
Abstract. In this paper we explicit the derivative of the flows of one-dimensional reflected diffusion processes. We then get sto- chastic representations for derivatives of viscosity solutions of one-dimensional semilinear parabolic partial differential equations and parabolic variational inequalities with Neumann boundary conditions.
Résumé. Dans cet article, nous explicitons la dérivée du flot d’un processus de diffusion réfléchi. Nous obtenons des représenta- tions stochastiques des dérivées des solutions de viscosité d’équations aux dérivées partielles paraboliques semi-linéaires. Nous en déduisons des représentations stochastiques des dérivées des solutions de viscosité d’inégalités variationnelles paraboliques avec conditions au bord de Neumann.
MSC:60H10; 60H30; 35K55
Keywords:Forward backward SDEs with refections; Feynman–Kac formulae; Derivatives of the flows of reflected SDEs and BSDEs
1. Introduction
Consider the parabolic variational inequality in the whole Euclidean space
⎧⎪
⎨
⎪⎩ min
V (t, x)−L(t, x); −∂V∂t (t, x)−AV (t, x)
−f
t, x, V (t, x), (∇V σ )(t, x) =0, (t, x)∈ [0, T )×Rd,
V (T , x)=g(x), x∈Rd,
(1)
whereAis the infinitesimal generator of a diffusion process. The numerical resolution of such a problem requires to introduce a boundary and artificial boundary conditions in order to allow the discretization of a PDE problem posed in a bounded domain. We thus localize the preceding variational inequality. If nonhomogeneous Neumann boundary conditions are chosen, one then has to solve
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩ min
v(t, x)−L(t, x); −∂v∂t(t, x)−Av(t, x)
−f
t, x, v(t, x), (∇vσ )(t, x) =0, (t, x)∈ [0, T )×O,
v(T , x)=g(x), x∈O,
∇v(t, x)+h(t, x);η(x)
=0, (t, x)∈ [0, T )×∂O,
(2)
where, for all x in∂O, η(x)denotes the inward unit normal vector at point x. From a numerical analysis point of view, one needs to estimate|V (t, x)−v(t, x)|. Berthelot, Bossy and Talay [3] have tackled this issue by using a stochastic approach based on Backward Stochastic Differential Equations (BSDE). Given the reflected forward Stochastic Differential Equation (SDE)
Xst,x=x+s t b
Xθt,x dθ+s
t σ Xt,xθ
dWθ+Kst,x, 0≤t≤s≤T, Kst,x=s
t η Xt,xθ
d|K|t,xθ with|K|t,xs =s
t I{Xt,x
θ ∈∂O}d|K|t,xθ , (3)
they have proven the following estimate: under smoothness conditions on the coefficients and on∂O, there exists C >0 such that, for all 0≤t≤T andx∈O,
V (t, x)−v(t, x)≤C E sup
t≤s≤T
∇V s, Xt,xs
+h s, Xst,x
;η
Xst,x4I{Xt,xs ∈∂O}1/4
.
Motivated by applications in Finance, where the space derivative ofv(t, x)allows one to construct hedging strate- gies of American options, we aim in this paper to estimate|∂xV (t, x)−∂xv(t, x)|, where the derivatives are understood in the sense of the distributions. We thus have to check that the probabilistic interpretations, in terms of BSDEs, of V (t, x)and ofv(t, x), are differentiable in the sense of the distributions, and to exhibit formulae which are suitable to estimate|∂xV (t, x)−∂xv(t, x)|. Unfortunately, so far we are able to deal with one-dimensional problems only. which means thatOis reduced to a bounded interval(d, d). Two main reasons explain the limitation to one-dimensional problems: first, we need to prove an explicit representation of the derivative ∂xXt,xt , where Xt,x is as in (3); this representation appears to be simple and of exponential type; exhibiting such an explicit formula seems difficult for general multi-dimensional flows1(Malliavin derivatives were also explicited by Lépingle, Nualart and Sanz [10] in the one-dimensional case only); second, in order to get stochastic representations for∂xv(t, x)whenh=0, that is, in the case of nonhomogeneous Neumann boundary conditions, we use an integration by parts technique which seems limited to the one-dimensional case (see Lemma3.7).
We aim to provide a stochastic representation for∂xv(t, x)in terms of the derivative of the solution(Yt,x,Zt,x, Rt,x)of the reflected BSDE with the reflected forward diffusionXt,x
⎧⎪
⎪⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎩
Yt,xs =g XTt,x
+T s f
r, Xrt,x,Yt,xr ,Zt,xr
dr+T s h
r, Xt,xr dKrt,x
+Rt,xT −Rt,xs −T
s Zt,xr dWr, Yt,xs ≥L
s, Xst,x
for all 0≤t≤s≤T , Rt,xs ,0≤t≤s≤T
is a continuous increasing process such that T
t
Yt,xr −L
s, Xrt,x
dRt,xr =0.
As we suppose that the coefficientsbandσ are only Lipschitz (and not necessarily differentiable), we need to extend various approaches developed to solve problems without or with reflexion: Bouleau and Hirsch [6] have explicited the derivatives w.r.t. the initial data of the solutions of nonreflected forward SDEs with Lipschitz coefficients; Lépingle et al. [10] have explicited the Malliavin derivatives of the solutions of one-dimensional reflected forwards SDEs.
Pardoux and Zhang [19] have established stochastic representations, in terms of BSDEs driven by forward reflected SDEs, for viscosity solutions of semilinear partial differential equations with Neumann boundary conditions. In [12]
and [13] Ma and Zhang have represented, without differentiating the coefficientsgandf, derivatives of solutions of BSDEs and reflected BSDEs driven by nonreflected forward SDEs with differentiable coefficients. N’Zi, Ouknine and Sulem [16] have extended Ma and Zhang’s results for nonreflected BSDEs to the case where the coefficients of the nonreflected forward SDEs are supposed Lipschitz only.
The paper is organized as follows. In Section2we explicit the derivative of the flow of the reflected flow(Xt,x) defined in (3). In Section3we get two stochastic representations for derivatives of solutions of semilinear parabolic partial differential equations (which corresponds to the case whereL(t, x)≡ −∞andR≡0): the first representation involves the gradient off, the second one does not involve it. We distinguish the homogeneous Neumann boundary
1The differentiability, in the sense of the distributions, seems easy to get by localization procedures when the boundary of the domain is smooth.
condition case, that is, the case whereh(t, x)≡0, and the inhomogeneous case. In Section4we get stochastic repre- sentations for derivatives of parabolic variational inequalities. We conclude by using our representations to estimate
|∂xV (t, x)−∂xv(T , x)|.
Notation. In all the paper we denote byC,C1,C2positive constants which may vary from line to line but only depend ond,d,T,theL∞-norms and the Lipschitz constants of the functionsb,σ,g,f andh,and the strong ellipticity constantα∗which appears in the inequality(6)below.
2. Derivative of the flow of the reflected diffusionX 2.1. Main result and examples
From now on we consider a one-dimensional stochastic differential equation in the interval[d, d], with reflection at pointsdandd:
Xt,xs =x+s t b
Xt,xr dr+s
t σ Xrt,x
dWr+Kst,x, Kst,x=s
t η Xrt
d|K|t,xr with|K|t,xs =s
t I{Xrt,x∈{d,d}}d|K|t,xr , (4) whereη(d)=1 andη(d)= −1. Our objective in this section is to explicit the derivative w.r.t.x of the stochastic flowXt,x.
We start with introducing some notation coming from [6]. We equip the spaceΩ:=(d, d)×Ω with its natural σ-field and the measure dP:=dx⊗dP. LetD1be the space of functionsγ (x, ω)satisfying: there exists a measur- able functionγ:Ω→Rsuch thatγ=γ,P-a.s, and, for all(x, ω), the mapy→γ (x+y, ω)is locally absolutely continuous.
Forγ∈D1, set
∂xγ (x, ω):=lim inf
y→0
γ (x+y, ω)−γ (x, ω)
y .
Bouleau and Hirsch [5] have shown that this definition is proper in the sense that,P-a.s.,∂xγ (x, ω)is well defined and does not depend on the choice ofγ. Finally, set
D:=
γ∈L2(P)∩D1;∂xγ∈L2(P) and
γD=
(d,d)×Ω
γ2dP+
(d,d)×Ω
(∂xγ )2dP 1/2
.
As in Lépingle et al. [10] we introduce the random set Est,x:=
ω∈Ω: d < inf
r∈[t,s]Xt,xr (ω)≤ sup
r∈[t,s]Xrt,x(ω) < d
. (5)
Our main result in this section is the following statement.
Theorem 2.1. Suppose thatbandσ are bounded Lipschitz functions,and that
∃α∗>0,∀x∈ d, d
σ (x) > α∗. (6)
Denote bybandσversions of the a.e.derivatives ofbandσ.Then the flowXt,xbelongs toDand2
∂xXt,xs =Jst,xIEst,x, P-a.s., (7)
2We recall that dP:=dx⊗dP.
where
Jst,x=exp s
t
σ Xt,xr
dWr+ t
s
b
Xt,xr
−1 2σ2
Xrt,x dr
. (8)
Before proceeding to the proof of this theorem we illustrate it with two examples.
Example 2.2. Brownian motion reflected at0.Letx >0.The resolution of the Skorokhod problem(see,e.g.,Karatzas and Shreve[8]),shows that the adapted increasing process
kxs(ω):=sup
0,−x+ sup
0≤r≤s
Wr(ω)
(9) is such that the processXxs :=x−Ws+ksxis positive and satisfies
T
0
I(0,∞)
Xxs
dksx=0.
We obviously have
∂xXxs =1+ ∂
∂xksx=Iinf0≤r≤sXxr>0.
Example 2.3. Brownian motion reflected at pointsdandd.Letx∈(d, d).Kruk et al. [9]have solved explicitly the Skorokhod problem corresponding to a two-sided reflection.To simplify the notation we suppose here thatd=0.We therefore consider the processXxs :=x−Ws+ ˜ksx,where we define the increasing processk˜ssby
−˜ksx:=
0∧ inf
0≤r≤s(x−Wr) ∨ sup
0≤r≤s
x−Wr−d
∧ inf
r≤θ≤s(x−Wθ)
.
Notice that,on the eventEs0,x the process(k˜xr, r≤s) is null and thus ∂x∂ Xxs =1, whereas on Ω−Es0,x one has
−˜ksx=x+Gs for some random variableGsindependent ofx,and thus∂x∂ Xxs =0.
We start the proof of Theorem2.1 with checking that the right-hand side of equality (7), that we will denote byΦt,x(s), is properly defined.
Proposition 2.4. The process(Φt,x(s), t≤s≤T ) is well defined in the sense that it does not depend on the Borel versions of the a.e.derivatives ofbandσ.
Proof. Observe thatΦt,x(s)=0 on the eventΩ−Est,x. To prove the desired result on the eventEst,x, consider two Borel versionsb1andb2(respectively,σ1 andσ2) of the a.e. derivative ofb(respectively,σ). Fori=1,2 set
Φi(s):=exp s
t
σi Xrt,x
dWr+ s
t
bi
Xrt,x
−1 2
σi2 Xt,xr
dr
,
andΦi(s)=Φi(s)IEst,x, so that E sup
t≤s≤T
Φ1(s)−Φ2(s)2
≤E sup
t≤s≤T
Φ1(s)−Φ2(s)2
≤C
E T
t
b1 Xrt,x
−b2 Xt,xr
− σ12
Xt,xr
− σ22
Xt,xr 4dr 1/2
+C
E T
t
σ1 Xt,xr
−σ2
Xt,xr 4dr 1/2
.
The hypotheses made in Theorem2.1allow us to apply the Proposition 4.1 in Lépingle et al. [10]: for alls > tandx, the probability distribution ofXt,xr has a densitypt,x(r,·)w.r.t. Lebesgue’s measure. To conclude, it then remains to
useb1≡b2 andσ1≡σ2 a.e.
2.2. On approximations by penalization
The proof of Theorem2.1essentially consists in approximatingXby the solution of a penalized stochastic differential equation. We use the following proposition which precises the convergence rate ofEsupt≤s≤T|Xst,x−Xt,x,ns |p for p >2 and is easily derived from the inequality (3.23) in Menaldi [14]:
Proposition 2.5 ([14]). Forn≥1define the functionβnby
βn(y):=
⎧⎨
⎩
−n y−d
ify≥d,
0 ifd≤y≤d,
n(d−y) ify≤d. Then the solutionXt,x,nto
Xt,x,ns =x+ s
t
b Xrt,x,n
dr+ s
t
σ Xt,x,nr
dWr+ s
t
βn Xrt,x,n
dr (10)
satisfies,for allp≥1,
∀t≤T lim
n→∞ sup
x∈(d,d)
E sup
t≤s≤T
Xt,xs −Xst,x,np=0. (11)
In order to explicit the limit of∂xXsx,nwe use the following convergence criterion used in Bouleau and Hirsch [6], p. 49.
Proposition 2.6. Let(Hsx,n, s∈ [0, T], n≥1)be a sequence of random fields which are time continuous from[0, T] toD. Suppose that
sup
n≥1
sup
s∈[0,T]
d
d
EHsx,n2dx+ d
d
E∂xHsx,n2dx
<+∞. (12)
Suppose that there exists a stochastic flowHsxcontinuous in(s, x)such that d
d
E sup
s∈[0,T]
Hsx,n−Hsx2dx−−−→
n→+∞0. (13)
Then,for alls∈ [0, T],∂xHsxis well definedP-a.s.,Hsxis inDand d
d
EHsx2dx+ d
d
E∂xHsx2dx <+∞. (14)
In addition,Hsx,n converges weakly toHsx in the following sense:for all stochastic flowUsx such that∂xUsxis well definedP-a.s.and
d
d
EUsx2dx+ d
d
E
∂xUsx2
dx <+∞, then
d
d
E Usx
Hsx,n−Hsx
+∂xUsx
∂xHsx,n−∂xHsx
dx−−−→
n→+∞0.
The next lemma states that the processXt,x,nsatisfies (12).
Lemma 2.7. For allp≥1we have sup
n≥1
sup
x∈(d,d)
E sup
s∈[t,T]
Xt,x,ns p+E sup
s∈[t,T]
∂xXt,x,ns p
<+∞.
Proof. In view of (11) we only need to estimateEsups∈[t,T]|∂xXst,x,n|p. Setbn:=b+βn. From the Theorem 1 and the discussion in [6], p. 56, we deduce that,P-a.s., the derivative∂sXt,x,ns in the sense of the distributions is well defined and satisfies
∂xXt,x,ns =exp s
t
σ Xt,x,nr
dWr+ s
t
bn
Xrt,x,n
−1 2σ
Xrt,x,n2 dr
.
It then suffices to use the one-side bound from abovebn(y)≤ b∞for all integernand ally∈Rto get sup
n≥1
sup
x∈(d,d)
E sup
s∈[t,T]
∂xXt,x,ns p<+∞. (15)
Our next step consists in identifying the process∂xXt,xs . 2.3. Proof of Theorem2.1:The one-sided reflection case
We are now in a position to explicit the derivative ofXst,x. We start with the case of the reflection at the sole pointd. Proposition 2.8. Letx∈(d, d)andXt,xbe the solution to
Xt,xs =x+ s
t
bXt,xr dr+
s
t
σXrt,x
dWr+ΛdsXt,x ,
whereΛd(Xt,x)is the local time at pointd of the semi-martingaleXt,x.The flowXt,xbelongs toDand,setting Est,x:=
ω∈Ω, inf
t≤r≤sXrt,x(ω) > d
,
we have:for allt≤s≤T,P-a.s.,
∂xXt,xs =exp s
t
σXt,xr dWr+
s
t
bXt,xr (ω)
−1
2σ2Xt,xr dr
IEst,x. (16)
Proof. For alln≥1 consider the solutions(Xt,x,n)to Xt,x,ns =x+
s
t
bXt,x,nr dr+
s
t
σXrt,x,n dWr+
s
t
n
d−Xt,x,nr + dr.
In view of Theorem 1 in [6], the stochastic flow Xt,x,n is differentiable in the sense of the distributions, and its derivative, denoted by∂xXt,x,ns , satisfiesP-a.s.,
∂xXt,x,ns =exp s
t
σXt,x,nr dWr+
s
t
bXrt,x,n
−1
2σ2Xrt,x,n dr
×exp
−n s
t
IXrt,x,n<ddr
.
We can easily get a result similar to Lemma2.7, that is, sup
n≥1
sup
x∈(d,d)
E sup
s∈[t,T]
Xst,x,n2+E sup
s∈[t,T]
∂xXt,x,ns 2
<+∞, (17)
which establishes (12) withHsx,n≡Xst,x,n. To obtain (13) we observe that we may substituteXtoXinto (11): indeed, in [14] the diffusion process is reflected at the boundary of a bounded domain whereas, here, the domain is the infinite interval(d,+∞); however, it is easy to see that Menaldi’s proof of inequality (3.23) also applies in this latter case.3 Therefore, in view of Proposition2.6, for allt≤s≤T,P-a.s.,∂xXt,x,ns converges weakly into some process that we denote by∂xXst,xandXst,x∈D. Suppose now that we have proven, for all xin(d, d):
At,x,ns :=exp
−n s
t IXt,x,nr (ω)<d
IEst,x−−−→
n→+∞IEst,x, P-a.s., (18)
and
EBst,x,n−−−→
n→+∞0, (19)
where
Bst,x,n:=exp
−n s
t
IXt,x,nr (ω)<d
IΩ−Est,x.
Let us check that we then could deduce (16). Indeed, denoting byGt,xs the r.h.s. of (16), it suffices to prove that, for all stochastic fieldUsxas in Proposition2.6,
E d
d
UsxXst,x,n−Xst,x dx+E
d
d
∂xUsx
∂xXst,x,n−Gt,xs IEst,xdx +E
d
d
∂xUsx∂xXt,x,ns IΩ−Est,xdx
tends to 0 asntends to infinity. Now, it is easy to check that each one of the three terms in the right-hand side tends to 0: for example, one has
E d
d
∂xUsx∂xXst,x,nIΩ−Est,xdx 2≤C
d
d
E
∂xUsx2
dx d
d
E Bst,x,n2
dx, and the right-hand side tends to 0 in view of (19).
Therefore it now remains to prove (18) and (19).
We start with (18). It suffices to prove that, on the event {inft≤r≤sXrt,x> d}, for alln large enough,P-a.s., inft≤r≤sXt,x,nr > d. A sufficient condition is
sup
t≤r≤s
Xrt,x,n−Xrt,x≤1 2
t≤infr≤sXt,xr −d
.
In view of Menaldi [14], Remark 3.1, p. 742, for all 2<2q < pthere existsC >0 such that, for alln, E sup
t≤s≤T
Xst,x,n−Xst,xp≤ C nq.
Thus Borel–Cantelli’s lemma implies that supt≤s≤T|Xst,x,n−Xt,xs |tends to 0 almost surely. We thus have proven (18).
3The properties (3.1) and (3.2) of the penalization function in [14] are clearly satisfied byβn.
Let us now prove thatEBst,x,nconverges to 0. The comparison theorem for stochastic differential equations shows that, for allm < nandt < r < T,Xrt,x,m≤Xt,x,nr ; therefore, for allnandt < r < T,Xrt,x,n≤Xt,xr . Thus
EBst,x,n≤E
exp
−n s
t
IXt,x,nr <ddr
Iinft≤r≤sXrt,x,n≤d
.
Letϕbe the increasing one-to-one mapϕ(z):=z
0 1
σ (y)dy. SetX¯st,x,n:=ϕ(Xt,x,ns ). We have:
X¯t,x,ns =ϕ(x)+ s
t
b(ϕ−1(X¯t,x,nr ))+n(d−ϕ−1(X¯t,x,nr ))+ σ (ϕ−1(X¯t,x,nr )) −1
2σ
ϕ−1X¯t,x,nr dr +Ws−Wt.
Using the Girsanov transformation removing the drift coefficient of(X¯t,x,ns )and denoting byEt,ϕ(x)the conditional expectation knowing thatWt=ϕ(x)we get
EBst,x,n≤Et,ϕ(x)
Msnexp
−n s
t
IWr<ϕ(d)dr
Iinft≤r≤sWr≤ϕ(d)
,
where
Msn=exp s
t
b(ϕ−1(Wr))+n(d−ϕ−1(Wr))+ σ (ϕ−1(Wr)) −1
2σ
ϕ−1(Wr) dWr
×exp
−1 2
s
t
b(ϕ−1(Wr))+n(d−ϕ−1(Wr))+ σ (ϕ−1(Wr)) −1
2σ
ϕ−1(Wr)2 dr
.
SetΦσ(z):=ϕ(d) z
(d−ϕ−1(y))+
σ (ϕ−1(y)) dy. Then Φσ(Ws)=Φσ(Wt)−
s
t
(d−ϕ−1(Wr))+
σ (ϕ−1(Wr)) dWr+1 2
s
t
Iϕ−1(Wr)<ddr +1
2 s
t
σ(ϕ−1(Wr)) σ (ϕ−1(Wr))
d−ϕ−1(Wr)+ dr.
In addition, observe that, for allx∈(d, d),Φσ(ϕ(x))=0, and thatΦσ(z)≥0 for allzinR. Therefore,Pt,ϕ(x)-a.s., n
s
t
(d−ϕ−1(Wr))+
σ (ϕ−1(Wr)) dWr ≤n 2
s
t
IWr<ϕ(d)dr+n 2
s
t
σ(ϕ−1(Wr)) σ (ϕ−1(Wr))
d−ϕ−1(Wr)+ dr
≤n 2
s
t
IWr<ϕ(d)dr+K n 2α∗
s
t
d−ϕ−1(Wr)+
dr, (20)
whereKis the Lipschitz constant ofσ, andα∗is as in (6). We deduce that, for some positive constantsC1andC2
and bounded continuous functionsρ1andρ2, all of them independent ofn,Pt,ϕ(x)-a.s., Msn≤exp
n 2
s
t IWr<ϕ(d)dr+C1n s
t
d−ϕ−1(Wr)+ dr
−C2n2 s
t
d−ϕ−1(Wr)+2
dr+ s
t
ρ1(Wr)dWr+ s
t
ρ2(Wr)dr
. (21)
As there existsC0>0 such thatC1nY−C2n2Y2< C0for all integernand allY ≥0, Cauchy–Schwartz inequality implies
EBst,x,n≤C√
Υn, (22)
where
Υn:=Et,ϕ(x)
exp
−n s
t
IWr<ϕ(d)dr
Iinft≤r≤sWr≤ϕ(d)
.
Now setx0:=ϕ(d)−ϕ(x)and letτx0:=inf{r≥0, Wr=x0}be the first passage time of the Brownian motionW at pointx0. The strong Markov property and the definition ofx0imply that
Υn≤ s−t
0
Ex0exp
−n s−t−θ
0
IWr≤x0dr
dPWτx0(θ ), where (see, e.g., Borodin and Salminen [4], p. 198)
dPWτx0(θ )= |x0|
√2πθ3exp
−x02 2θ
dθ.
Using formula (1.5.3) in Borodin and Salminen [4], p. 160, we deduce Υn≤
s−t
0
I0
n(s−t−θ ) 2
exp
−n
2(s−t−θ ) |x0|
√2πθ3exp
−x02 2θ
dθ, (23)
whereI0is a Bessel function whose definition can be found in, e.g., Abramowitz and Stegun [1], p. 375. We split the integral in the right-hand side of(23)into the two following terms:
Υ1n:=
s−t−1/√n
0
I0
n(s−t−θ ) 2
exp
−n
2(s−t−θ ) |x0|
√2πθ3exp
−x02 2θ
dθ,
Υ2n:=
s−t
s−t−1/√ n
I0
n(s−t−θ ) 2
exp
−n
2(s−t−θ ) |x0|
√2πθ3exp
−x02 2θ
dθ.
For allθin(0, s−t−√1n)one has n(s−2t−θ ) ≥√2n; in addition (see, e.g., Borodin and Salminen [4], p. 638), I0(y)≈ ey
√2πy asy→ +∞.
Therefore, there existsC >0, uniformly bounded inx0∈(ϕ(d)−ϕ(d),0)such that, for allnlarge enough, Υ1n≤ C
n1/4.
Now, we use thatI0(y)e−y≤1 for ally≥0 (see, e.g., Abramowitz and Stegun [1], p. 375) and deduce that Υ2n≤
s−t
s−t−1/√ n
|x0|
√2πθ3exp
−x02 2θ
dθ≤ C
√n,
from which Υn≤ C
n1/4, (24)
whereCis uniformly bounded inx∈(d, d). In view of (22) we thus have obtained EBst,x,n≤ C
n1/8,
which ends the proof.
2.4. Proof of Theorem2.1:The two-sided reflection case We now consider the penalized system (10).
With the arguments used at the beginning of the proof of Proposition2.8one can deduce that,P-a.s., the map x→Xst,xbelongs to the Sobolev space
H1 d, d
= f∈L
d, d
;f∈L d, d .
We now aim to prove the representation formula (7). We first consider the eventEst,x. On this eventXt,x,ns satisfies Xt,x,ns =x+
s
t
b Xt,x,nr
dr+ s
t
σ Xrt,x,n
dWr.
Pathwise uniqueness holds for both the above stochastic differential equation and Eq. (4). Therefore(Xt,x,nr , r∈ [t, s]) and(Xrt,x, r∈ [t, s])coincide onEst,x. We deduce the equality (7) onEst,x.
We next consider the eventΩ−Est,x. We are inspired by Lépingle et al. [10] to reduce our study to the one-sided reflection case. For all rational numbersc1ands1such thatd < c1< dandt < s1< sset
Ad,cs1 1:=
ω∈Ω: d= inf
r∈[t,s1]Xt,xr , sup
r∈[t,s1]Xrt,x=c1
,
Acs11,d:=
ω∈Ω: inf
r∈[t,s1]Xrt,x=c1, sup
r∈[t,s1]Xt,xr =d
.
Set also Ad:=
ω∈Ω: ∀r∈ [t, s), d < Xt,xr < d, Xt,xs =d , Ad:=
ω∈Ω: ∀r∈ [t, s), d < Xt,xr < d, Xt,xs =d .
We have
Ω−Est,x=Ad∪Ad
d<c1<d t <s1<s
Ad,cs1 1∪Acs11,d .
LetXt,x,nbe defined as in (10). As observed in the proof of Lemma2.7, settingbn:=b+βnwe have,P-a.s.,
∂xXt,x,ns =exp s
t
σ Xt,x,nr
dWr+ s
t
bn
Xrt,x,n
−1 2σ
Xrt,x,n2 dr
.
LetXt,x be the one-sided reflected diffusion process defined in Proposition2.8, and, as in the proof of Proposi- tion2.8, letXt,x,n be the corresponding penalized process. On the eventAd,cs1 1 we haveXt,x=Xt,xand, as already noticed, we also haveXt,x,n≤Xt,x; therefore, onAd,cs1 1 the paths ofXt,x,ndo not hit the pointd, which implies that Xt,x,n=Xt,x,non this event, from which, by a classical local property of Brownian stochastic integrals,
∂xXt,x,ns
1 IAd,c1 s1
=exp s1
t
σXt,x,nr dWr+
s1
t
bnXrt,x,n
−1
2σXrt,x,n2 dr
IAd,c1
s1
.
Moreover, the arguments used to prove (19) imply that E
exp
s1 t
σXt,x,nr dWr+
s1
t
bnXt,x,nr
−1
2σXt,x,nr 2 dr
IAd,c1
s1
−−−→n→+∞0.
