ELECTRONIC
COMMUNICATIONS in PROBABILITY
SHARP ESTIMATES FOR THE CONVERGENCE OF
THE DENSITY OF THE EULER SCHEME IN SMALL
TIME
EMMANUEL GOBET
Laboratoire Jean Kuntzmann, Universit´e de Grenoble and CNRS, BP 53, 38041 Grenoble Cedex 9, FRANCE
email: emmanuel.gobet@imag.fr
C´ELINE LABART
Centre de Math´ematiques Appliqu´ees, Ecole Polytechnique, Route de Saclay, 91128 Palaiseau, FRANCE
email: labart@cmap.polytechnique.fr
Submitted January 29, 2008, accepted in final form June 3, 2008 AMS 2000 Subject classification: 65C20 60H07 60H10 65G99 65M15 60J60
Keywords: stochastic differential equation, Euler scheme, rate of convergence, Malliavin cal-culus
Abstract
In this work, we approximate a diffusion process by its Euler scheme and we study the conver-gence of the density of the marginal laws. We improve previous estimates especially for small time.
1
Introduction
Let us consider a d-dimensional diffusion process (Xs)0≤s≤T and a q-dimensional Brownian motion (Ws)0≤s≤T. X satisfies the following SDE
dXsi =bi(s, Xs)ds+ q
X
j=1
σij(s, Xs)dWsj, X0i =xi,∀i∈ {1,· · ·, d}. (1.1)
We approximateX by its Euler scheme withN(N ≥1) time steps, sayXN, defined as follows. We consider the regular grid{0 =t0< t1<· · ·< tN =T} of the interval [0, T], i.e. tk =kNT. We putXN
0 =xand for alli∈ {1,· · · , d} we define
XuN,i=X N,i
tk +bi(tk, X
N tk)(u−t
N k) +
q
X
j=1
σij(tk, XtNk)(W
j u−W
j
tk), foru∈[tk, tk+1[. (1.2)
The continuous Euler scheme is an Itˆo process verifying
XuN =x+
Z u
0
b(ϕ(s), Xϕ(s)N )ds+
Z u
0
σ(ϕ(s), Xϕ(s)N )dWs
where ϕ(u) := sup{tk : tk ≤ u}. If σ is uniformly elliptic, the Markov process X admits a transition probability densityp(0, x;s, y). ConcerningXN (which is not Markovian except at times (tk)k), XN
s has a probability density pN(0, x;s, y), for any s > 0. We aim at proving sharp estimates of the differencep(0, x;s, y)−pN(0, x;s, y).
It is well known (see Bally and Talay [2], Konakov and Mammen [5], Guyon [4]) that this difference is of order 1
N. However, the known upper bounds of this difference are too rough for small values ofs. In this work, we provide tight upper bounds of|p(0, x;s, y)−pN(0, x;s, y)
|
ins(see Theorem 2.3), so that we can estimate quantities like
E[f(XTN)]−E[f(XT)] orE
" Z T
0
f(Xϕ(s)N )ds
#
−E
" Z T
0
f(Xs)ds
#
(1.3)
(without any regularity assumptions on f) more accurately than before (see Theorem 2.5). For other applications, see Labart [7]. Unlike previous references, we allow b and σ to be time-dependent and assume they are only C3 in space. Besides, we use Malliavin’s calculus tools.
Background results
The differencep(0, x;s, y)−pN(0, x;s, y) has been studied a lot. We can found several results in the literature on expansions w.r.t. N. First, we mention a result from Bally and Talay [2] (Corollary 2.7). The authors assume
Hypothesis 1.1. σ is elliptic (with σ only depending on x) and b, σ are C∞(Rd) functions whose derivatives of any order greater or equal to1 are bounded.
By using Malliavin’s calculus, they show that
p(0, x;T, y)−pN(0, x;T, y) = 1
NπT(x, y) +
1
N2R N
T(x, y), (1.4)
with |πT(x, y)|+|RNT(x, y)| ≤ K(T)
Tα exp(−c
|x−y|2
T ), where c > 0, α > 0 and K(·) is a non decreasing function. We point out that α is unknown, which doesn’t enable to deduce the behavior ofp−pN whenT
→0.
Besides that, Konakov and Mammen [5] have proposed an analytical approach based on the so-called parametrix method to boundp(0, x; 1, y)−pN(0, x; 1, y) from above. They assume
Hypothesis 1.2. σis elliptic and b, σ areC∞(Rd) functions whose derivatives of any order are bounded.
For each pair (x, y) they get an expansion of arbitrary orderjofpN(0, x; 1, y). The coefficients of the expansion depend onN
p(0, x; 1, y)−pN(0, x; 1, y) = j−1
X
i=1 1
NiπN,i(0, x; 1, y) +O( 1
The coefficients have Gaussian tails : for each i they find constants c1 > 0,c2 >0 s.t. for allN ≥1 and all x, y∈Rd, |πN,i(0, x; 1, y)| ≤c1exp(−c2|x−y|2). To do so, they use upper bounds for the partial derivatives ofp(coming from Friedman [3]) and prove analogous results on the derivatives ofpN. Strong though this result may be, nothing is said when replacing 1 byt, fort→0. That’s why we present now the work of Guyon [4].
Guyon [4] improves (1.4) and (1.5) in the following way.
Definition 1.3. LetGl(Rd), l∈Zbe the set of all measurable functionsπ:Rd×(0,1]×Rd→R s.t.
• for allt∈(0,1], π(·;t,·)is infinitely differentiable,
• for all α, β∈Nd, there exist two constantsc1≥0 and c2 >0 s.t. for allt∈(0,1]and
x, y∈Rd,
|∂xα∂βyπ(x;t, y)| ≤c1t−(|α|+|β|+d+l)/2exp(−c2|x−y|2/t).
Under Hypothesis 1.2 and forT = 1, the author has proved the following expansions
pN
−p= π
N +
πN
N2, (1.6)
pN
−p= j−1
X
i=1
πN,i
Ni + j
X
i=2
µ
t−⌊N t⌋ N
¶i
π′
N,i+
π′′
N,j
Nj , (1.7)
where π ∈ G1(Rd) and (πN, N ≥ 1) is a bounded sequence in G4(Rd). For each i ≥ 1, (πN,i, N ≥ 1) is a bounded family in G2i−2(Rd), and (πN,i′ , N ≥ 1),(π′′N,i, N ≥ 1) are two bounded families inG2i(Rd). These expansions can be seen as improvements of (1.4) and (1.5) : it also allows infinite differentiations w.r.t. xandyand makes precise the way the coefficients explode whenttends to 0.
As a consequence (see Guyon [4], Corollary 22), one gets
|p(0, x;s, y)−pN(0, x;s, y)
| ≤ c1 N sd+22
e−c2|x−y| 2
s , (1.8)
for two positive constantsc1andc2, and for anyx, yands≤1. This result should be compared with the one of Theorem 2.3 (whenT = 1), in which the upper bound is tighter (shas a smaller power).
2
Main Results
Before stating the main result of the paper, we introduce the following notation
Definition 2.1. Cbk,l denotes the set of continuously differentiable bounded functions φ : (t, x) ∈ [0, T]×Rd with uniformly bounded derivatives w.r.t. t (resp. w.r.t. x) up to or-der k(resp. up to orderl).
Hypothesis 2.2. σ is uniformly elliptic,bandσ are inCb1,3 and∂tσis in Cb0,1.
Theorem 2.3. Assume Hypothesis 2.2. Then, there exist a constant c > 0 and a non de-creasing functionK, depending on the dimensiond and on the upper bounds ofσ, band their derivatives s.t. ∀(s, x, y)∈]0, T]×Rd×Rd, one has
Corollary 2.4. Assume Hypothesis 2.2. From the last inequality and Aronson’s inequality (A.1), we deduce
Theorem 2.3 enables to bound quantities like in (1.3) in the following way
Theorem 2.5. Assume Hypothesis 2.2. For any function f such that |f(x)| ≤ c1ec2|x|, it
Had we used the results stated by Guyon [4] (and more precisely the one recalled in (1.8)), we would have obtainedE[f(XTN)]−E[f(XT)] =O(1
N). Intuitively, this result is not optimal: the right hand side doesn’t tend to 0 whenT goes to 0 while it should. Analogously, regarding EhRT
and using Theorem 2.3 yield the first result. Concerning the second result, we split EhRT
0 (f(X
. First, using Theorem 2.3 leads to
Then, Proposition A.2 yields¯¯ ¯E
h RT
0 (f(Xϕ(s))−f(Xs))ds
i¯ ¯ ¯≤c1e
c2|x|
³
T N +C
RT T N ln(
s ϕ(s))ds
´
.
Moreover,RT T N ln(
s ϕ(s))ds=
PN−1
k=1
Rtk+1
tk ln(
s tk)ds=
T N
PN−1
k=1((k+ 1) ln( k+1
k )−1)≤C T N, using a second order Taylor expansion. This gives¯¯
¯E h
RT
0 (f(Xϕ(s))−f(Xs))ds
i¯ ¯ ¯≤c1e
c2|x|K(T)T
N.
In the next section, we give results related to Malliavin’s calculus, that will be useful for the proof of Theorem 2.3.
3
Basic results on Malliavin’s calculus
We refer the reader to Nualart [8], for more details. Fix a filtered probability space (Ω,F,(Ft),P) and let (Wt)t≥0 be a q-dimensional Brownian motion. For h(·) ∈ H = L2([0, T],Rq), W(h) is the Wiener stochastic integral RT
0 h(t)dWt. Let S denote the class of random variables of the form F = f(W(h1),· · · , W(hn)) where f is a C∞ function with derivatives having a polynomial growth, (h1,· · ·, hn)∈Hn andn≥1. ForF ∈ S, we define its derivative DF = (DtF := (D1tF,· · ·,DtqF))t∈[0,T] as the H valued random variable given by
DtF = n
X
i=1
∂xif(W(h1),· · ·, W(hn))hi(t).
The operatorD is closable as an operator fromLp(Ω) to Lp(Ω;H), forp≥1. Its domain is denoted byD1,pw.r.t. the normkFk
1,p= [E|F|p+E(kDFk p
H)]1/p.We can define the iteration of the operatorD, in such a way that for a smooth random variableF, the derivativeDkF is a random variable with values onH⊗k. As in the casek= 1, the operator
Dk is closable from
S ⊂Lp(Ω) intoLp(Ω;H⊗k), p≥1. If we define the norm
kFkk,p= [E|F| p+
k
X
j=1
E(°°DjF ° °
p H⊗j)]
1/p,
we denote its domain byDk,p. Finally, setDk,∞=∩p≥1Dk,p, and D∞=∩k,p≥1Dk,p. One has the following chain rule property
Proposition 3.1. Fix p ≥1. For f ∈ C1
b(Rd,R), and F = (F1,· · ·, Fd)∗ a random vector whose components belong to D1,p,f(F)∈D1,p and for t≥0, one hasDt(f(F)) =f′(F)DtF, with the notation
DtF =
DtF1 .. .
DtFd
∈R
d
⊗Rq.
We now introduce the Skorohod integral δ, defined as the adjoint operator ofD.
Proposition 3.2. δ is a linear operator onL2([0, T]×Ω,Rq)with values inL2(Ω) s.t.
• the domain of δ (denoted by Dom(δ)) is the set of processesu∈L2([0, T]×Ω,Rq) s.t.
|E(RT
• If u belongs to Dom(δ), then δ(u) is the one element of L2(Ω) characterized by the integration by parts formula
∀F∈D1,2, E(F δ(u)) =E
à Z T
0 D
tF·utdt
!
.
Remark 3.3. If u is an adapted process belonging to L2([0, T]×Ω,Rq), then the Skorohod integral and the Itˆo integral coincide : δ(u) =RT
0 utdWt, and the preceding integration by parts formula becomes
∀F ∈D1,2, E
Ã
F
Z T
0
utdWt
!
=E
à Z T
0 D
tF·utdt
!
. (3.1)
This equality is also called the duality formula.
This duality formula is the corner stone to establish general integration by parts formula of the form
E[∂αg(F)G] =E[g(F)Hα(F, G)]
for any non degenerate random variablesF. We only give the formulation in the case of interest
F =XN t .
Proposition 3.4. We assume that σ is uniformly elliptic and b and σ are in Cb0,3. For all p > 1, for all multi-index α s.t. |α| ≤ 2, for all t ∈]0, T], all u, r, s ∈ [0, T] and for any functions f and g in Cb|α|, there exist a random variableHα∈Lp and a function K(T) (uniform inN, x, s, u, r, t, f andg) s.t.
E[∂α
xf(XtN)g(XuN, XrN, XsN)] =E[f(XtN)Hα], (3.2) with
|Hα|Lp≤
K(T)
t|α2|
kgkCb|α|. (3.3)
These results are given in the article of Kusuoka and Stroock [6]: (3.3) is owed to Theorem 1.20 and Corollary 3.7.
Another consequence of the duality formula is the derivation of an upper bound forpN.
Proposition 3.5. Assume σ is uniformly elliptic and b and σ are in Cb0,2. Then, for any
x, y∈Rd,s∈]0, T], one has
pN(0, x;s, y)≤K(T)
sd/2 e
−c|x−sy|2, (3.4)
for a positive constant c and a non decreasing function K, both depending on d and on the upper bounds forb, σ and their derivatives.
Proof. The inequality (1.32) of Kusuoka and Stroock [6], Theorem 1.31 givespN(0, x;s, y)
≤
K(T)
sd/2 for any xand y. This implies the required upper bound when |x−y| ≤
√s. Let us
now consider the case |x−y|>√s. Using the same notations as in Kusuoka and Stroock [6], we denote ψ(y) = ρ(|y−rx|) wherer > 0 and ρ is a C∞
b function such that 1{[3/4,∞[} ≤ρ≤
1{[1/2,∞[}. Then, combining inequality (1.33) of Kusuoka and Stroock [6], Theorem 1.31 and Corollary 3.7 leads to
sup
|y−x|≥r
pN(0, x;s, y)≤K(T)e
−cr2 s
sd/2
µ
1 +
r
s r2
¶
,
where we use ° °ψ(XsN)
°
°1,q ≤K(T)e−c r2
s ¡1 +ps
r2
¢
. This easily completes the proof in the case|x−y| ≥√s.
4
Proof of Theorem 2.3
In the following, K(·) denotes a generic non decreasing function (which may depend ond, b
andσ). To prove Theorem 2.3, we take advantage of Propositions 3.4 and 3.5. The scheme of the proof is the following
• Use a PDE and Itˆo’s calculus to write the differencepN(0, x;s, y)
−p(0, x;s, y)
=
Z s
0 E
" d X
i=1
(bi(ϕ(r), XN
ϕ(r))−bi(r, XrN))∂xip(r, X
N r ;s, y)
+1 2
d
X
i,j=1
(aij(ϕ(r), Xϕ(r)N )−aij(r, XrN))∂2xixjp(r, X
N r ;s, y)
dr:=E1+E2. (4.1)
• Prove the intermediate result∀(r, x, y)∈[0, s[×Rd×Rd andc >0
E
·
exp
µ
−c|y−X
N r |2
s−r
¶¸
≤K(T)
µ
s−r s
¶d2
exp
µ
−c′|x−y|
2
s
¶
, (4.2)
wherec′ >0.
• Use Malliavin’s calculus, Proposition 3.5 and the intermediate result, to show that each termE1 andE2 (see (4.1)) is bounded by K(TN)T
1 sd+12
exp(−c|x−sy|2).
Definition 4.1. We say that a termE(x, s, y)satisfies propertyPif∀(x, s, y)∈Rd×]0, T]×Rd
|E(x, s, y)| ≤ K(T)T
N
1
sd+12
exp
µ
−c|x−y|
2
s
¶
. (P)
4.1
Proof of equality (4.1)
First, the transition density function (r, x)7−→p(r, x;s, y) satisfies the PDE
where L(r,x) is defined by L(r,x) =
its first derivatives, are uniformly bounded by a constant depending onǫ for|s−r| ≥ǫ (see Appendix A).
Second, sincepN(0, x;s, y) is a continuous function insandy (convolution of Gaussian den-sities), we observe that
pN(0, x;s, y)−p(0, x;s, y) = lim
From the PDE, the above equality becomes
E[p(s−ǫ, XsN−ǫ;s, y)−p(0, x;s, y)] =
4.2
Proof of the intermediate result (4.2)
We prove inequality (4.2). E[exp(−c|y−X Proposition 3.5, we get
E
r )dzi is the convolution product
of the density of two independant Gaussian random variables N(−xi,2cr′) and N(yi, s−r
computed at 0. Hence, the integral is equal to q 1
≤k≤q are uniformly bounded. Using (4.3) and the duality formula (3.1) yield
Using the intermediate result (4.2) yields
|E11| ≤K(T)
and thus,E11 satisfies propertyP (see Definition 4.1).
To rewrite E12, we use the expression of βi
u and Proposition 3.1, which gives
Du(∂xip(r, X
Using the integration by parts formula (3.2), we get that
E12=−
whereek is a vector whosek-th component is 1 and other components are 0. From (3.3), we deduceE[|He
the H¨older inequality, it follows that
|E12| ≤K(T) this inequality with the intermediate result (4.2) yields
E[|∂xp(r, XrN;s, y)|d+1d ]d/(d+1)≤ K(T)
Hence,E12is bounded by
K(T)
The above integral equals s12−
d+1
2(d+1)) where B is the function Beta. Thus
|E12| ≤ K(Tsd/2)NT exp(−c Then, the duality formula (3.1) leads to
Bound for E21=Pdij=1
Rs
0 E[
Rr
ϕ(r)∂ 2
xixjp(r, X
N
r ;s, y)γuijdu]dr.
As σ, b, ∂tσ, ∂xσ, ∂x2σ are Cb1 in space, γuij has the same smoothness properties as the term [(σσ∗)(ϕ(r), XN
ϕ(r))(∇xbi(u, XuN))∗]k appearing in (4.5). Thus,E21can be treated as E12and satisfies to the same estimate.
Bound for E22=Pdi,j=1
Rs
0 E[
Rr
ϕ(r)Du(∂ 2
xixjp(r, X
N
r ;s, y))·δijudu]dr.
To rewrite E22, we use the expression of δij
u and Proposition 3.1, which asserts
Du(∂2
xixjp(r, X
N
r ;s, y)) =∇x(∂x2ixjp(r, X
N
r ;s, y))σ(ϕ(r), Xϕ(r)N ). Thus,
E22=− d
X
i,j,k=1
Z s
0
dr
Z r
ϕ(r) E[∂x3
ixjxkp(r, X
N
r ;s, y)[(σσ∗)(ϕ(r), Xϕ(r)N )(∇xaij(u, XuN))∗]k]du.
To complete this proof, we splitE22in two terms : E221 (respE222) corresponds to the integral in rfrom 0 to s2 (resp. from s2 tos).
• On [0,s 2], E
1
22 is bounded byCNT
R2s
0 E[|∂ 3
xixjxkp(r, X
N
r ;s, y)|]dr. Using Proposition A.2 and (4.2), it gives
|E221 | ≤
K(T)T N
1
sd/2exp
µ
−c|x−y|
2
s
¶ Z s2
0 1 (s−r)3/2dr.
Hence,E22 satisfiesP.
• On [s2, s], we use the integration by parts formula (3.2) of Proposition 3.4, with|α|= 2.
E2 22=−
d
X
i,j,k=1
Z s s
2
dr
Z r
ϕ(r) E[∂x
ip(r, X
N
r ;s, y)Hejk(i)]du,
where ejk is a vector full of zeros except the j-th and the k-th components. Using H¨older’s inequality and (3.3) (remember thatσ∈Cb1,3), we obtain
|E2
22| ≤K(T)
T N
Z s s
2
1
rE[|∂xp(r, X
N r ;s, y)|
d+1
d ]d+1d dr. (4.7)
By applying (4.6), we get
|E2
22| ≤K(T)
T N
1
s1+ d
2 2(d+1)
exp
µ
−c|x−y|
2
s
¶ Z s s
2
1
(s−r)22dd+1+2
dr,
and the result follows.
A
Bounds for the transition density function and its
derivatives
We bring together classical results related to bounds for the transition probability density of
Proposition A.1(Aronson [1]). Assume that the coefficientsσandbare bounded measurable functions and that σis uniformly elliptic. There exist positive constantsK, α0, α1 s.t. for any
x, yin Rd and any0≤t < s≤T, one has
K−1 (2πα1(s−t))d2
e−
|x−y|2
2α1(s−t) ≤p(t, x;s, y)≤K 1
(2πα2(s−t))d2
e−
|x−y|2
2α2(s−t). (A.1)
Proposition A.2(Friedman [3]). Assume that the coefficientsb andσare H¨older continuous in time, C2
b in space and thatσ is uniformly elliptic. Then, ∂m+ax ∂ybp(t, x;s, y)exist and are continuous functions for all 0 ≤ |a|+|b| ≤2,|m| = 0,1. Moreover, there exist two positive constants candK s.t. for anyx, yin Rd and any0≤t < s≤T, one has
|∂xm+a∂byp(t, x;s, y)| ≤
K
(s−t)(|m|+|a|+|b|+d)/2exp
µ
−c|y−x|
2
s−t
¶
.
References
[1] Aronson D.G., 1967. Bounds for the fundamental solution of a parabolic equation. Bulletin of the American Mathematical Society, 73, 890-903. MR0217444
[2] Bally, V., Talay, D., 1996. The law of Euler scheme for stochastic differential equations: convergence rate of the density. Monte Carlo Methods Appl., 2(2), 93-128. MR1401964
[3] Friedman A., 1964. Partial differential equations of parabolic type. Prentice-Hall Inc., En-glewood Cliffs, N.J. MR0181836
[4] Guyon J., 2006. Euler schemes and tempered distributions. Stochastic Processes and their Applications, 116, 877-904. MR2254663
[5] Konakov V., Mammen E., 2002. Edgeworth type expansions for Euler schemes for stochas-tic differential equations. Monte Carlo Methods Appl, 8(3), 271-285. MR1931967
[6] Kusuoka S. Stroock D., 1984. Applications to the Malliavin calculus I. Stochastic Anal-ysis, Proceeding of the Taniguchi International Symposium on Katata and Kyoto 1982. Kinokuniya, Tokyo, Itˆo K. Ed, 271-306.
[7] Labart C., 2007. PhD Thesis, Ecole Polytechnique, Paris. Available at http://pastel.paristech.org/3086/
