Supplement A: Construction and properties of a reected
diusion
Jakub Chorowski
1
Construction
Assumption 1. For given constants 0 < d < D let the pair (σ, b) ∈ Θ, where
Θ := Θ(d, D) = {(σ, b) ∈ C 1 ([0, 1])×C 1 ([0, 1]) : kbk∞ ∨kσ 2 k∞ ∨kσ 0 k∞ < D, inf σ 2 (x) ≥ d}.
x∈[0,1]
For
(σ, b) ∈ Θ
consider the following Skorokhod type stochastic dierential equation:
dXt = b(Xt )dt + σ(Xt )dWt + dKt ,
X0 = x0 ∈ [0, 1]
where
(Wt , t ≥ 0)
and
Xt ∈ [0, 1]
for every
is a standard Brownian motion and
tinuous process with nite variation, starting form
´t
0 1(0,1) (Xs )dKs = 0.
0,
(1)
t ≥ 0,
(Kt , t ≥ 0)
is some adapted con-
and such that for every
t ≥ 0
holds
By the Engelbert-Schmidt theorem the SDE (1) has a weak solution,
see [13, Thm. 4.1]. In this section, we will present an explicit construction of a strong solution.
To that end we extend the coecients
Denition 2.
and
Dene
eb, σ
e:R→R
b, σ
to the whole real line.
f : R → [0, 1] by
(
x − 2n
: 2n ≤ x < 2n + 1
f (x) =
2(n + 1) − x : 2n + 1 ≤ x < 2n + 2
by
eb(x) = b(f (x))f 0 (x)
−
σ
e(x) = σ(f (x)).
Theorem 3. For every initial condition
motion W , the SDE
x0 ∈ [0, 1],
independent of the driving Brownian
dYt = eb(Yt )dt + σ
e(Yt )dWt ,
Y0 = x0 ,
has a non-exploding unique strong solution. Dene
Xt = f (Yt ).
The process (Xt , t ≥ 0) is a strong solution of the SDE (1).
1
(2)
Proof.
(Yt , t ≥ 0)
The existence of a non-exploding unique strong solution
follows from [8, Proposition 5.17]. Process
Y
is a continuous semimartingale, hence by [12,
Chapter VI Theorem 1.2] admits a local time process
([12, Chapter VI Theorem 1.5]) process
ˆ
t
Xt = x0 +
ˆ
0
ˆ
t
(LYt , t ≥ 0).
By the Itô-Tanaka formula
satises
σ
e(Ys )f−0 (Ys )dWs +
X
LYt (2n) −
n∈Z
X
LYt (2n + 1)
n∈Z
t
σ(Xs )dBs + Kt ,
b(Xs )ds +
= x0 +
t
eb(Ys )f 0 (Ys )ds +
−
0
ˆ
X
of the SDE (2)
0
0
´t
P
P
Bt = 0 f−0 (Ys )dWs and Kt = n∈Z LYt (2n)− n∈Z LYt (2n+1). Note that for any T > 0
the path (Xt , 0 ≤ t ≤ T ) is bounded, hence K is well dened. Using Lévy's characterization
theorem we nd that B is a standard Brownian motion. From the properties of the local time
LYt follows that K is anSadapted continuous process with nite variation, starting from zero
and varying on the set
n∈Z {Yt = 2n} ∪ {Yt = 2n + 1} ⊆ {Xt ∈ {0, 1}}. Consequently, X is
where
a strong solution of the SDE (1).
Notation.
We will write
C > 0. f ∼ g
f . g (resp. g & f )
f . g and g . f .
when
f ≤ C·g
for some universal constant
is equivalent to
From now on we take the Assumption (1) as granted. We denote by
Pσ,b
the law of the
X on the canonical space Ω of continuous functions over the positive axis with values
[0, 1], equipped with the topology of the uniform convergence on compact sets and endowed
with its σ−eld F . We denote by Eσ,b the corresponding expectation operator.
diusion
in
2
Modulus of continuity
In this Section we want to prove a uniform upper bound on the moments of the modulus of
continuity of the reected diusion
Denition 4.
Denote by
ωT
X.
the modulus of continuity of the path
(Xt , 0 ≤ t ≤ T ),
i.e.
sup |Xt − Xs |.
ωT (δ) =
0≤s,t≤T
|t−s|≤δ
Theorem 5. For every p ≥ 1 there exists constant Cp > 0 s.t.
sup Eσ,b [ωTp (∆)] ≤ Cp ∆p/2 (1 ∨ ln(2T /∆))p
(3)
(σ,b)∈Θ
Proof.
Fischer and Nappo [3] proved the above bound for the standard Brownian motion. We
will now generalize their result to diusions with boundary reection.
Step 1.
theorem
Consider a martingale
Mt = B´ t σ2 (Xu )du
M
with
dMt = σ(Xt )dWt . By Dambis,
B . Consequently
Dubins-Schwarz
for some Brownian motion
0
|Mt − Ms | = B´ t σ2 (Xu )du − B´ s σ2 (Xu )du ≤ ω B (|t − s|kσ 2 k∞ ),
0
0
where
ωB
is the modulus of continuity of
B.
Thus (3) holds for the martingale
constant that depends only on the upper bound on the volatility
2
σ.
M,
with a
Step 2.
Consider a semimartingale
ˆ
|Xt − Xs | ≤ ˆ
t
with
dXt = b(Xt )dt + dMt .
Then
s
b(Xu )du + |Mt − Ms | ≤ |t − s|kbk∞ + ω M (|t − s|).
b(Xu )du −
0
X
0
Consequently (3) holds for semimartingales with a constant that depends only on the upper
bounds on
Step 3.
σ
b.
(σ, b) ∈ Θ
and
For
consider the reected diusion process
X
satisfying the SDE (1).
Let
dYt = eb(Yt )dt + σ
e(Yt )dWt ,
Xt = f (Yt ),
where
gale
Y
eb, σ
e
and
f
are as in Denition 2. From Step 2 follows that (3) holds for the semimartin-
with a uniform constant on
Since
ωX ≤ ωY ,
we conclude that the claim (3) holds
X.
for the reected diusion
3
Θ.
Local time
In this section we introduce some preliminary results regarding the local time of the reected
diusion
X.
A standard reference is [12, Chapter VI].
Denition 6.
of the path
Set t > 0. For any Borel set A ⊆ [0, 1] we dene the occupation measure Tt (A)
(Xs , 0 ≤ s ≤ t), with respect to the quadratic variation of X , by
ˆ t
Tt (A) =
1A (Xs )σ 2 (Xs )ds.
0
Tt (A) is absolutely continuous with respect to the Lebesgue measure dx on the interval
When
[0, 1],
we dene the local time by the Radon-Nikodym derivative:
dTt
.
dx
Lt (x) =
Theorem 7 (Itô-Tanaka formula). Let X be the solution of the SDE (1) with (σ, b) ∈ Θ.
Then, the local time L exists and has a continuous version in both t > 0 and x ∈ (0, 1). For
every x ∈ [0, 1] the process (Lt (x), t ≥ 0) is non-decreasing and increases only when Xt = x.
Furthermore, if f is the dierence of two convex functions, we have
ˆ
t
f (Xt ) = f (X0 ) +
ˆ
f−0 (Xs )σ(Xs )dWs
t
+
0
f−0 (Xs )b(Xs )ds
0
ˆ
+
t
1
+
2
ˆ
1
Lt (x)f 00 (dx)+
0
f−0 (Xs )dKs .
(4)
0
Proof.
Reected diusion
X
is a continuous semimartingale. By [12, Chapter VI, Theorem
(Lt (x) : x ∈ (0, 1), t ≥ 0), continuous and nonx and such that (4) holds. Furthermore, by [12, Chapter VI, Theorem
1.2 and Theorem 1.5] there exists a process
decreasing in t, cadlag in
1.7] for every
x ∈ (0, 1)
ˆ
Lt (x) − Lt (x− ) = 2
0
ˆ
t
1{Xs =x} b(Xs )ds + 2
The concentration of the associated measure
0
t
1{Xs =x} dKs = 0.
dLt on the set {Xt = x} follows from [12, Chapter
VI Proposition 1.3].
3
Lemma 8. For every T
>0
and p ≥ 1 we have
sup Eσ,b [sup Lpt (x)] < ∞.
sup
t≤T
(σ,b)∈Θ x∈(0,1)
Proof.
The usual way to bound the moments of the local time is to use the Itô-Tanaka formula
fx (y) = (y −x)+ , see e.g. [12, Chapter VI Theorem 1.7]. Because of the additional
0
term dKt , we make a less intuitive choice of the function f that guarantees f (0) =
for function
reection
f 0 (1) = 0.
Set T > 0, p ≥ 1
and
x ∈ (0, 1/2].
Let
fx (y) = 1(x ≤ y ≤ 3/4)(3y − 2y 2 ).
By (4)
ˆ
t
3 − 4x
Lt (x) = fx (Xt ) − fx (X0 ) −
1(x < Xs ≤ 43 )(3 − 4Xs )σ(Xs )dWs
2
0
ˆ t
ˆ t
3
1(x < Xs ≤ 4 )(3 − 4Xs )b(Xs )ds + 2
1(x < Xs ≤ 43 )σ 2 (Xs )ds.
−
0
Applying the uniform (on
0
Θ) bounds on b and σ , together with the Burkholder-Davies-Gundy
t≤T
inequality, we conclude that for
sup
Eσ,b [Lt (x)p ] ≤ Cp,T ,
sup
(σ,b)∈Θ x∈(0,1/2)
holds with some positive constant
y ≤ x)(y − 2y 2 )
Cp,T .
x ∈ (1/2, 1) we consider function fx (y) = 1(1/4 ≤
For
and proceed similarly.
Theorem 9. For any T
> 0, p ≥ 1 and x, y ∈ (0, 1) we have
sup Eσ,b sup |Lt (x) − Lt (y)|2p ≤ Cp,T |x − y|p .
(5)
t≤T
(σ,b)∈Θ
In particular, the family L of the local times can be chosen such that functions x 7−→ Lt (x)
are almost surely Hölder continuous of order α for every α < 1/2 and uniformly in t ≤ T .
Moreover, for every p ≥ 1 and t ≤ T we have
sup Eσ,b
(σ,b)∈Θ
Proof.
sup Lpt (x) < ∞.
(6)
x∈[0,1]
The proof goes along the same lines as [12, Chapter VI Theorem 1.7]. We will rst
show the inequality (5). For
x ∈ (0, 1),
by the Itô-Tanaka formula (4)
ˆ t
1
Lt (x) = (Xt − x)+ − (X0 − x)+ −
1(Xs > x)σ(Xs )dWs +
2
0
ˆ t
ˆ t
−
1(Xs > x)b(Xs )ds −
1(Xs = 1)dKs .
0
0
´t
x 7−→ (Xt − x)+ − (X0 − x)+ − 0 1(Xs = 1)dKs is uniformly Lipschitz on
´t
Θ, we need only to consider the martingale term Mtx = 0 1(Xs > x)σ(Xs )dWs and the nite
´t
x
variation term Dt =
0 1(Xs > x)b(Xs )ds. For x, y ∈ (0, 1) Hölder's inequality and Lemma 8
Since the function
yield
Eσ,b sup |Dtx
t≤T
−
Dty |2p
. Eσ,b
h ˆ
y
LT (z)dz
x
4
2p i
ˆ
. |y − x|2p−1 y
Eσ,b [L2p
(z)]dz
. Cp,T |y − x|2p ,
T
x
for some constant
ep,T > 0.
C
To bound the increments of the martingale
Mx
we use the
Burkholder-Davies-Gundy inequality together with Hölder's inequality, obtaining
Eσ,b sup |Mtx
t≤T
−
Mty |2p
. Cp Eσ,b
h ˆ
y
LT (z)dz
p i
ep,T |y − x|p .
≤C
x
We nished the proof of the bound (5). From the Kolmogorov continuity criterion (see [12,
e of the family of local times
L
surely Hölder continuous of order α for every
for any α < 1/2 and p ≥ 2 we have
Chapter I, Theorem 2.1]) follows that there exists a modication
L, such that
α < 1/2 and
e t (x) are almost
functions x 7−→ L
uniformly in t ≤ T . Furthermore,
sup Eσ,b
h
x6=y
(σ,b)∈Θ
Fix
x0 ∈ (0, 1).
sup
By the bound above and Lemma 8 we conclude that
e p (x)] ≤ sup Eσ,b
sup Eσ,b [ sup L
t
(σ,b)∈Θ
e t (x) − L
e t (y)| p i
|L
< ∞.
|x − y|α
(σ,b)∈Θ
x∈[0,1]
Theorem 10. Set T
>0
h
p i
e t (x) − L
e t (x0 )|
|L
e t (x0 )
+
L
< ∞.
|x − x0 |α
sup
x6=x0
and dene the (chronological) occupation density µT by
LT (x)
.
T σ 2 (x)
µT (x) =
Then, for any bounded Borel measurable function f , the following occupation formula holds:
1
T
ˆ
ˆ
T
1
f (Xs )ds =
0
f (x)µT (x)dx.
0
Furthermore, the occupation density µT inherits the regularity properties of the local time LT .
In particular, for every p ≥ 1 and T > 0 we have
sup Eσ,b
sup µpT (x)
< ∞.
(7)
x∈[0,1]
(σ,b)∈Θ
sup Eσ,b |µT (x) − µT (y)|2p ≤ Cp,T |x − y|p .
(8)
(σ,b)∈Θ
Proof.
The existence and form of the occupation density follow from Theorem 7 and Denition
σ 2 & 1 inequality (7)
2
property of σ imply (8).
6. Given that
Lipschitz
4
follows directly from (6). Finally, (5) and the uniform
Estimation of the occupation time of an interval
Estimation of the local time from nite data observations has been extensively studied and
is nowadays well established, see e.g.
[6, 7].
Nevertheless, until recently, much less was
known about the estimation of the occupation time of a given Borel set. In the breakthrough
paper Ngo and Ogawa [10] authors considered Riemann sum approximations of the occupation
5
time of a half-line.
[10, Theorem 2.2] stated a convergence rate
3
∆4
for diusion processes
with bounded coecients, which was dened as normalization required for tightness of the
estimation errors. It is important to note that
3
∆4
is a better rate than could be obtained
using only the regularity properties of the local time.
3
∆4
occupation time of the positive half-line,
In the special case of the Brownian
was shown to be the upper bound of the root
mean squared error (see [10, Theorem 2.3]) and to be optimal. Further development was done
in Kohatsu-Higa et al. [9]. By means of the Malliavin calculus [9, Theorem 2.3] proved that for
any suciently regular scalar diusion
X
and an exponentially bounded function
h inequality
N −1
p i
h 1 ˆ T
1 X
Eσ,b h(Xs )ds −
h(Xn∆ ) ≤ CT,X,h ∆p+1/2
T 0
N
(9)
n=0
holds with constant
CT,X,h > 0
depending on the time horizon
In what follows, we extend (9), for
coecients in
h
T,
diusion
X
and function
h.
being characteristic function, to reected diusions with
Θ.
Theorem 11. For any T
>0
and α ∈ (0, 1) we have
ˆ
−1
2 i 1
h 1 NX
2
1 T
2
sup Eσ,b ≤ CT ∆ 3 ,
1(Xn∆ < α) −
1(Xs < α)ds
N
T 0
(σ,b)∈Θ
n=0
with some positive constant CT .
Remark
.
12
As the right hand side of (9) does not scale linearly in
p, it is not optimal to use the
Girsanov theorem to generalize the Brownian bound to diusions with bounded coecients.
Nevertheless, we proceed with this approach, as the suboptimal rate
∆2/3
is sucient for our
purposes.
Proof.
Fix
T > 0.
The proof is divided in several steps, generalizing the result from a standard
Brownian motion process to reected diusions.
Step 1.
CT > 0
Let
Wt
be a standard Brownian motion. We will show that there exists a constant
such that for any
α∈R
we have
ˆ
−1
2p i
h 1 NX
1 T
E 1(Wn∆ < α) −
1(Ws < α)ds
≤ CT ∆p+1/2 .
N
T 0
(10)
n=0
Set
α∈R
and
hα (x) = 1(x < α).
Following [9, proof of Proposition 2.1], for
M ∈ N,
denote
1
ρM (x) = cρ M e (M x)2 −1 1(−1,1) (x),
ˆ
hα,M (x) =
hα (x − y)ρM (y)dy,
R
where the constant
cρ =
´
1
1
y 2 −1 dy
−1 e
−1
is such that
ρM
integrates to
1.
Direct calculations
show that



A(i) :
A(ii) :


A(iii) :
hα,M → hα in L1
supM supx∈R |hα (x)| + |hα,M (x)| ≤ 2
1
´
´1
(α+y/M )2
x2
u
supM supu≥0 |h0α,M (x)|e− u dx = cρ −1 e y2 −1 e−
dy ≤ 1.
6
Let
ˆ
N −1
1 X
1 T
hα (Wn∆ ) −
hα (Ws )ds,
N
T 0
n=0
ˆ
N
−1
1 X
1 T
hα,M (Wn∆ ) −
hα,M (Ws )ds.
N
T 0
SN,α =
SN,α,M
=
n=0
Arguing as in [9, proof of Eq. 3.10], we obtain that for any
α
2p
2p
lim Eσ,b [SN,α,M
] = Eσ,b [SN,α
].
M →∞
Hence, it is sucient to show that
2p
] ≤ CT N −(p+1/2) ,
Eσ,b [SN,α,M
for some constant
CT
M
independent of
and
α.
(11)
But, given uniform bounds
A(ii)
and
A(iii),
inequality (11) follows by the same calculations as in [9, proof of Theorem 3.2].
Step 2.
h.
Consider diusion
Y
satisfying
dYt = h(Yt )dt + dWt , with uniformly bounded drift
We will show that
ˆ
−1
2 i 1
h 1 NX
1 T
2
E 1(Ys < α)ds
≤ CT,khk∞ ∆17/24 .
1(Yn∆ < α) −
N
T 0
(12)
n=0
ˆ
Denote
t
Zt = exp −
0
Since
Q
h
1
h(Ys )dWs −
2
is bounded, by Novikov's condition
Z
ˆ
t
h2 (Ys )ds .
0
is a martingale. Dene the probability measure
by the Radon-Nikodym derivative:
dQ
|F = Zt .
dP t
By Girsanov's theorem the process
measure
Q.
Y
is a standard Brownian motion under the probability
From Hölder's inequality, for
1
p
+
1
q
= 1,
follows that
ˆ
−1
2 i 1
h 1 NX
1 T
2
E 1(Yn∆ < α) −
1(Ys < α)ds
=
N
T 0
n=0
ˆ
−1
2
i1
h 1 NX
1 T
2
= EQ 1(Yn∆ < α) −
1(Ys < α)ds ZT−1 .
N
T 0
n=0
ˆ
−1
2p i 1
h 1 NX
1 T
2p −q 2q1
≤ EQ 1(Yn∆ < α) −
EQ ZT
.
1(Ys < α)ds
N
T 0
n=0
Since the drift function
h
is uniformly bounded,
ˆ
h
−q −(q−1) EQ ZT = E ZT
= E exp (q − 1)
0
7
T
q−1
h(Ys )dWs +
2
ˆ
T
h2 (Ys )ds
0
i
≤ exp
Hence, by (10), with
Step 3.
q(q − 1)
2
p = 6/5,
2q
T khk2∞ = Cq,khk
.
∞
inequality (12) holds.
Consider diusion
Y
dYt =´eb(Yt )dt + σ
e(Yt )dWt , with bounded drift eb
x −1
S(x) = 0 σ
e (y)dy , dZt = S(Yt ). It follows from
satisfying
and positive, Lipschitz continuous
σ
e.
Let
Itô's formula that
dZt = g(Yt )dt + dWt ,
e
b(x) 1 0
where g(x) =
e (x). Since σ
e is a strictly positive function,
σ
e(x) − 2 σ
−1
Denote h(x) = g(S
(x)). We have
S is increasing and invertible.
dZt = h(Zt )dt + dWt ,
with
kebk∞ 1 0
+ ke
σ k∞ .
inf σ
e
2
constant CT , depending
khk∞ ≤
From (12) follows, that there exists a
only on the bounds on the
diusion coecients, such that
ˆ
−1
2 i 1
h 1 NX
1 T
2
1(Yn∆ < α) −
1(Ys < α)ds
=
E N
T 0
n=0
ˆ
−1
2 i 1
h 1 NX
17
1 T
2
=E 1(Zn∆ < S(α)) −
1(Zs < S(α))ds
≤ CT ∆ 24 .
N
T 0
(13)
n=0
Step 4.
Fix
(σ, b) ∈ Θ.
Let
dYt = eb(Yt )dt + σ
e(Yt )dWt ,
Xt = f (Yt ),
where
eb, σ
e
and
f
X is the reected
α ∈ (0, 1) and s > 0,
are as in Denition 2. Recall that by Theorem 3 process
diusion with coecients
(σ, b).
By denition of the function
f,
for any
we have
{Xs < α} =
[
{Ys ∈ (2m − α, 2m + α)}.
(14)
m∈Z
Denote
ΓN (m) =
ˆ
N −1
1 T
1 X
1(Yn∆ ∈ (2m − α, 2m + α)) −
1(Ys ∈ (2m − α, 2m + α))ds.
N
T 0
n=0
By (13) there exists a uniform on
Θ
constant
CT > 0,
such that for any
m ∈ Z,
we have
h
i1
17
2
Eσ,b Γ2N (m) ≤ CT ∆ 24 .
Let
Yt∗ = sups≤t |Ys |.
From (14) follows that
ˆ T
N −1
X
X
1 X
1(Xn∆ < α) −
1(Xs < α)ds =
ΓN (m) =
ΓN (m)1(YT∗ ≥ 2|m| − 1).
N
0
n=0
m∈Z
8
m∈Z
Since
ΓN (m) ≤ 2,
for any
M,
using (14) we obtain
2 i
h X
Eσ,b ΓN (m)1(YT∗ ≥ 2|m| + 1) .
m∈Z
. M Eσ,b
2 i
i
h X
Γ2N (m) + Eσ,b 1(YT∗ ≥ 2m − 1)
h X
m>M
|m|≤M
. CT M 2 ∆
17
12
X
+
Pσ,b (YT∗ ≥ 2(m ∨ k) − 1).
m,k>M
By the Burkholder-Davies-Gundy inequality, together with uniform on
coecients, for any
Θ
bounds on diusion
p ≥ 1,
h
Eσ,b (YT∗ )p . Eσ,b
ˆ
0
T
σ
e2 (Ys )ds
p i
2
+ (T kebk∞ )p . Cp,T .
Consequently, by the Markov inequality
X
X
Pσ,b (YT∗ ≥ 2(m ∨ k) − 1) ≤ 2
m,k>M
Cp,T (2m − 1)−p
M <k≤m
≤ 2Cp,T
X
(2m − 1)−(p−1) . M −(p−2) .
m>M
We conclude that
2 i
h X
17
Eσ,b ΓN (m)1(Ys∗ ≥ 2|m| + 1) . M 2 ∆ 12 + M −(p−2) .
m∈Z
Hence the claim follows for
5
3
M ∼ ∆− 8
and any
p ≥ 4.
Upper bounds on the transition kernel
In this section we prove a uniform Gaussian upper bound on the transition kernel of the
reected diusion
X
with coecients in
Θ.
Under the assumption of smooth coecients, ex-
istence of Gaussian o-diagonal bounds follows from the general theory of partial dierential
equations, see [4, 9] c.f. [5, Chapter 9]. Sharp upper bounds are also established for diusion
processes in the divergence form [2, Chapter VII] and more recently were derived for multivariate diusions with the innitesimal generator satisfying Neumann boundary conditions,
see [14]. Nevertheless, as demonstrated by [11, Theorem 2] Gaussian upper bounds do not
hold in general for scalar diusions with bounded measurable drift.
Theorem 13. The transition kernel pt of the reected diusion X satises
CT (x−y)2
sup pt (x, y) . √ e− ct ,
t
(σ,b)∈Θ
for all x, y ∈ [0, 1], 0 ≤ t ≤ T and c, CT > 0.
9
Proof.
We will generalize the bound on the transition kernel from diusions with bounded
(σ, b) ∈ Θ.
dZt = g(Zt )dt + dWt , where g is a bounded
Z of the process Z
Then by [11, Theorem 1] the transition kernel p
ˆ ∞
√
1
Z
−(z−kgk∞ t)2 /2
pt (x, y) . √
ze
dz.
t |x−y|/√t
drift and unit volatility to reected processes with coecients
Step 1.
Consider diusion
measurable function.
satises
Z
satisfying
Using the inequality [1, Formula 7.1.13]:
ˆ
∞
−z 2
e
x
we obtain that
ˆ
∞
√ ze
a/ t
Thus
Step 2.
and
√
−(z−b t)2 /2
ˆ
√
2
π −x2
e−x
p
dz ≤
≤
e ,
2
2
x + x + 4/π
(15)
√ √ −w2 /2
(a−bt)2
πt
− 2t
dz =
dw ≤ e
1+b √ .
√ (w + b t)e
a
2
√ −b t
t
∞
1 − (x−y)2 +kgk∞ |x−y|
2t
pZ
.
(16)
t (x, y) ≤ CT,kgk∞ √ e
t
´ x −1
σ
e (y)dy
Consider diusion Y satisfying dYt = e
b(Yt )dt + σ
e(Yt )dWt . Let S(x) =
0
Zt = S(Yt ).
It follows from Itô's formula that
dZt = g(Yt )dt + dWt ,
where
g(x) =
e
b(x)
σ
e(x)
e0 (x).
− 12 σ
For any
x, y
we have
−1
−1
ke
σ −1 k−1
σ k∞ |x − y|.
∞ |x − y| ≤ |S (x) − S (y)| ≤ ke
Hence, from (16) follows that
(S −1 (x) − S −1 (y))2
CT,kgk∞
√
exp −
+ kgk∞ |S −1 (x) − S −1 (y)|
2t
t
2
(x − y)
exp −
+ kgk∞ ke
σ k∞ |x − y| .
(17)
2tke
σ −1 k∞
−1
−1
pYt (x, y) = pZ
t (S (x), S (y)) .
.
Step 3.
CT,kgk∞
√
t
For
(σ, b) ∈ Θ
let
eb, σ
e
and
that by (17) there exist uniform on
Θ
f
be as in Denition 2 and
constants
cT , c1 , c2 > 0
Y
as in Step 2. Note rst,
such that
(x − y)2
cT
pYt (x, y) ≤ √ exp −
+ c2 |x − y| .
(18)
c1 t
t
By Theorem 3 Xt = f (Yt ) is the reected diusion process corresponding to the coecients
(σ, b). For y ∈ [0, 1] let (ym )m∈Z be such that ym ∈ [m, m + 1] and f (ym ) = y . Then
X
pX
(x,
y)
=
pYt (x, ym ).
t
m∈Z
Since for any
m∈Z
|x − y| + (|m| − 2)+ ≤ |x − ym | ≤ |m| + 2, from (18)
(x − y )2
cT X
m
pX
exp −
+ c2 |x − ym |
t (x, y) ≤ √
c1 t
t m∈Z
2
[(|m| − 2) ]2
cT − (x−y) X
+
≤ √ e c1 t
exp −
+ c2 (|m| + 2) .
c1 T
t
m∈Z
we have
10
follows that
6
Mean crossings bounds
Throughout this section we consider xed time horizon, for simplicity we assume
Denition 14.
For
α ∈ (0, 1)
n = 0, ..., N − 1
and
T = 1.
denote
χ(n, α) = 1[0,α) (X(n+1)∆ ) − 1[0,α) (Xn∆ ).
Theorem 15. For every α ∈ (0, 1) we have
Eσ,b
−1
h NX
|χ(n, α)|(X(n+1)∆ − Xn∆ )2
2 i 1
2
. ∆1/2 .
n=0
Proof.
α ∈ (0, 1).
Fix
α,
the level
Since|χ(n, α)|
=1
if and only if the increment
(Xn∆ , X(n+1)∆ )
crosses
the claim is equivalent to the inequalities:
Eσ,b
Eσ,b
−1
h NX
n=0
N
−1
h X
1(Xn∆ < α)1(X(n+1)∆ > α)(X(n+1)∆ − Xn∆ )2
2
1(Xn∆ > α)1(X(n+1)∆ < α)(X(n+1)∆ − Xn∆ )
2 i 1
2
2 i 1
2
. ∆1/2 ,
. ∆1/2 .
n=0
Below, we only prove the rst inequality. The second one can be obtained in a similar way
or by a time reversal argument. Denote
ηn = 1(Xn∆ < α)1(X(n+1)∆ > α)(X(n+1)∆ − Xn∆ )2 .
We have
Eσ,b
−1
h NX
2
1(Xn∆ < α)1(X(n+1)∆ > α)(X(n+1)∆ −Xn∆ )
2 i
=
n=0
Denote by
N
−1
X
n=0
pt
the transition kernel of the diusion
X.
Eσ,b [ηn2 ]+2
N
−1
X
Eσ,b [ηn ηm ].
0≤n<m
From Theorem 13 and the inequality
(15) follows that
ˆ
αˆ 1
ˆ
αˆ 1
ˆ α ˆ √1−x
∆c −z 2 4
1 − (y−x)2
4
2
√ e c∆ (y − x) dydx . ∆
p∆ (x, y)(y − x) dydx .
e z dzdx
α−x
∆
√
0
α
0
∆c
ˆ α ˆ √1−x
ˆ α
2
2
(α−x)
∆c − z
. ∆2
e 2 dzdx . ∆2
e− 2c∆ dx . ∆5/2 .
(19)
4
0
α
0
Similarly
ˆ
α−x
√
∆c
0
αˆ 1
p∆ (x, y)(y − x)2 dydx . ∆3/2 .
0
For simplicity we will use the stationarity of
µ.Using
(20)
α
X,
which is granted by the assumption
d
x0 =
more elaborated arguments the result could be obtained for an arbitrary initial con-
dition. By stationarity, for any
t,
the one dimensional margin
11
Xt
is distributed with respect
to the invariant measure
density
µ
µ(x)dx.
Conditioning on
Xn∆ , from (19) and uniform bounds on the
follows
ˆ
αˆ 1
Eσ,b ηn2 =
p∆ (x, y)(y − x)4 dyµ(x)dx . ∆5/2 .
0
Hence
α
N
−1
X
5
3
Eσ,b [ηn2 ] . N ∆ 2 = ∆ 2 .
n=0
The Cauchy-Schwarz inequality implies
N
−2
X
Eσ,b [ηn ηn+1 ] .
n=0
N
−2
X
1
1
5
3
2
]2 . N∆2 . ∆2 .
Eσ,b [ηn2 ] 2 Eσ,b [ηn+1
n=0
m > n + 1, we obtain
ˆ αˆ 1ˆ αˆ 1
Eσ,b [ηn ηm ] =
p∆ (x, y)(y − x)2 p(m−n−1)∆ (z, x)(z − w)2 p∆ (w, z)µ(w)dydxdzdw
0
α
0
α
ˆ αˆ 1
ˆ αˆ 1
1
.
p∆ (x, y)(y − x)2 dydx p
(z − w)2 p∆ (w, z)dzdw
(m − n − 1)∆ 0 α
0
α
1
. ∆5/2 √
.
m−n−1
Finally, using (20), for
Consequently
N
−3 N
−1
X
X
Eσ,b [ηn ηm ] . ∆
5/2
n=0 m=n+2
N
−3 N X
−n−2
X
n=0
= ∆5/2
k=1
N
−2
X
√
N
−3
X
√
1
5/2
√ .∆
N −n−2
k
n=0
n . ∆5/2 N 3/2 = ∆.
n=1
Denition 16.
Dene the event
R1 := {ω1 (∆)kµ1 k∞ ≤ ∆5/11 v}.
Using Markov's inequality together with Theorem 5 and (6) we obtain that
Pσ,b (Ω \ R1 ) . ∆2/3 .
Theorem 17. For any α ∈ [ J1 , 1 − J1 ] holds
−1
NX
h
2 i 21
Eσ,b 1R1 · 1[0,α) (X(n+1)∆ ) − 1[0,α) (Xn∆ ) (X(n+1)∆ − α)2 − (Xn∆ − α)2 . ∆2/3 .
n=0
Proof.
α ∈ [ J1 , 1 − J1 ]. On the event R1 , whenever 1[0,α) (X(n+1)∆ ) − 1[0,α) (Xn∆ ) 6= 0 we
4/9 . Consider function d : [0, 1] → R given by
must have |Xn∆ − α|, |X(n+1)∆ − α| ≤ ω(∆) < ∆
Fix
d(x) = (x − α)2 1(|x − α| ≤ ∆4/9 ).
12
We have
1[0,α) (X(n+1)∆ ) − 1[0,α) (Xn∆ ) (X(n+1)∆ − α)2 − (Xn∆ − α)2 =
= 1[0,α) (X(n+1)∆ ) − 1[0,α) (Xn∆ ) d(X(n+1)∆ ) − d(Xn∆ ) .
Step 1.
We will rst show that
h
Eσ,b 1R1
−1
NX
2 i 12
·
1[0,α) (Xn∆ ) d(X(n+1)∆ ) − d(Xn∆ ) . ∆2/3 .
(21)
n=0
Note that
d0 (x) = 2(x − α)1(|x − α| ≤ ∆4/9 ),
1 00
d (x) = −∆4/9 δ{α−∆4/9 } + 1(|x − α| ≤ ∆4/9 ) − ∆4/9 δ{α+∆4/9 } ,
2
where the second derivative must be understood in the distributional sense. Since we xed
0
separated from the boundaries, d (0)
=
d0 (1)
=0
for
∆
α
small enough. Denote by
Ls,t (x) := Lt (x) − Ls (x),
the local time of the path fragment
(Xu , s ≤ u ≤ t).
From the Itô-Tanaka formula (4) follows
that
ˆ
(n+1)∆
d(X(n+1)∆ ) − d(Xn∆ ) =
ˆ
ˆ
0
(n+1)∆
d (Xs )σ(Xs )dWt +
n∆
d0 (Xs )b(Xs )ds+
n∆
(n+1)∆
σ 2 (Xs )1(|Xs − α| ≤ ∆4/9 )ds − ∆4/9 Ln∆,(n+1)∆ (α − ∆4/9 )
+
n∆
ˆ
(n+1)∆
− ∆4/9 Ln∆,(n+1)∆ (α + ∆4/9 ) :=
d0 (Xs )σ(Xs )dWt + Dn .
n∆
First, we will bound the sum of the martingale terms. Since martingale increments are uncorrelated, using Itô isometry, we obtain that
ˆ
−1
h NX
Eσ,b 1[0,α) (Xn∆ )
=
(n+1)∆
n∆
n=0
N
−1
X
ˆ
h
Eσ,b 1[0,α) (Xn∆ )
n=0
8
= ∆9
2 i
d0 (Xs )σ(Xs )dWt =
(n+1)∆
i
hˆ
8
(d0 (Xs ))2 σ 2 (Xs )ds . ∆ 9 Eσ,b
n∆
ˆ
4
α+∆ 9
4
α−∆ 9
1
i
4
1(|Xs − α| ≤ ∆ 9 )ds
0
4
Eσ,b [µ1 (x)]dx . ∆ 3 ,
where the last inequality follows from Theorem 10. Now, we will bound the sum of the nite
variation terms:
N
−1
X
n=0
PN −1
n=0
ˆ
1[0,α) (Xn∆ )Dn .
(n+1)∆
Note rst, that since
ˆ
0
d (Xs )b(Xs )ds . ∆4/9
1[0,α) (Xn∆ )
n∆
is uniformly bounded, we have
1
1(|x−α| ≤ ∆4/9 )µ1 (x)dx . ∆8/9 kµ1 k∞ .
0
13
b
kµ1 k∞
Since by the inequality (7)
has all moments nite, the root mean squared value of
∆2/3 . Now, note that since on the event R1 ω(∆) <
Xn∆ < α implies Ln∆,(n+1)∆ (α + ∆4/9 ) = 0. On the other hand, whenever
Ln∆,(n+1)∆ (α − ∆4/9 ) =
6 0 we must have Xn∆ < α . Hence
this sum is of smaller order than
∆4/9 , condition
N
−1
X
1[0,α) (Xn∆ )(∆4/9 Ln∆,(n+1)∆ (α − ∆4/9 ) + ∆4/9 Ln∆,(n+1)∆ (α + ∆4/9 )) = ∆4/9 L1 (α − ∆4/9 ).
n=0
Using rst the Cauchy-Schwarz inequality and then Theorem 9 we obtain
ˆ
h
Eσ,b ∆4/9 L1 (α − ∆4/9 ) −
α
α−∆4/9
2 i
L1 (x)dx . ∆4/9
ˆ
α
α−∆4/9
Eσ,b [|L1 (x) − L1 (α − ∆4/9 )|2 ]dx
. ∆4/3 .
Consequently, to prove (21) we just have to argue that the root mean squared error of
ˆ
N
−1
X
α
L1 (x)dx −
α−∆4/9
=
=
n=0
n=0 n∆
N
−1 ˆ (n+1)∆
X
.
(1(Xs < α) − 1(Xn∆ < α))σ 2 (Xs )ds
(22)
n∆
From the Lipschitz property of
(n+1)∆
n∆
σ 2 (Xs )1(|Xs − α| ≤ ∆4/9 )ds
n∆
(1(Xs < α) − 1(Xn∆ < α))σ 2 (Xs )1(|Xs − α| ≤ ∆4/9 )ds
∆2/3 .
−1 ˆ
NX
(n+1)∆
1[0,α) (Xn∆ )
n=0
ˆ
N
−1
X (n+1)∆
n=0
is of order
ˆ
σ2
follows that
(1(Xs < α) − 1(Xn∆ < α))(σ 2 (Xs ) − σ 2 (α))ds .
N
−1 ˆ (n+1)∆
X
n=0
4
9
4
9
1(|Xs − α| ≤ ∆ )∆ ds . ∆
n∆
4
9
ˆ
α+∆4/9
α−∆4/9
8
µ1 (dx) . ∆ 9 kµ1 k∞ .
Thus, by (7), we reduced (22) to
ˆ
0
1
N −1
1 X
1(Xs < α)ds −
(1(Xn∆ < α),
N
n=0
which is of the right order by Theorem 11. We conclude that (21) holds.
Step 2.
Yt = X1−t . Since X is reversible, the process
Y , under the measure Pσ,b , has the same law as X . Furthermore, the occupation density and
the modulus of continuity of processes Y and X are identical, hence R1 is a good event also
for Y . Inequality (21) is equivalent to
Consider the time reversed process
−1
NX
2 i 1
h
2
2
Eσ,b 1R1 · 1[0,α) (Ym∆ )(d(Y(m+1)∆ ) − d(Ym∆ ))
. ∆3 .
m=0
Substituting
N
−1
X
m=0
n=N −m
we obtain
1[0,α) (Ym∆ )(d(Y(m+1)∆ ) − d(Ym∆ )) = −
N
−1
X
n=0
14
1[0,α) (X(n+1)∆ )(d(X(n+1)∆ ) − d(Xn∆ )).
References
[1] Abramowitz, M. and Stegun, I. A. (1972).
Formulas, Graphs, and Mathematical Tables.
[2] Bass, R. (1997).
Handbook of Mathematical Functions with
Dover, New York, 9th edition.
Diusions and Elliptic Operators.
Springer-Verlag, New York.
[3] Fischer, M. and Nappo, G. (2010). On the moments of the modulus of continuity of Itô
processes.
Stochastic Analysis and Applications, 28(1).
[4] Florens-Zmirou, D. (1993). On estimating the diusion coecient from discrete observations.
J. Appl. Prob., 30.
[5] Friedman, A. (1964).
Partial dierential equations of parabolic type.
Prentice-Hall Inc.,
Englewood Clis, N.J., 2nd edition.
[6] J. M. Azaïs (1989). Approximation des trajectoires et temps local des diusions.
Inst. Henri Poincaré Probab. Statist., 25(2):175194.
[7] Jacod, J. (1998). Rates of convergence of the local time of a diusion.
Poincaré Probab. Statist., 34(4):505544.
[8] Karatzas, I. and Shreve, S. E. (1991).
Ann.
Anns. Inst. Henri
Brownian motion and stochastic calculus.
Springer-
Verlag, New York.
[9] Kohatsu-Higa, A., Makhlouf, A., and Ngo, H. L. (2014). Approximations of non-smooth
integral type functionals of one dimensional diusion processes.
their Applications, 124:1881 1909.
Stochastic Processes and
[10] Ngo, H.-L. and Ogawa, S. (2011). On the discrete approximation of occupation time of
diusion processes.
Electronic Journal of Statistics, 5:13741393.
[11] Qian, Z. and Zheng, W. (2002). Sharp bounds for transition probability densities of a
class of diusions.
C. R. Acad. Sci. Paris, I 335:953957.
[12] Revuz, D. and Yor, M. (1999).
Continuous Martingales and Brownian Motion, volume I,
II. Springer, New York, 3 edition.
[13] Rozkosz, A. and Sªomi«ski, L. (1997). On stability and existence of solutions of SDEs
with reection at the boundary.
Stochastic processes and their applications, 68:285302.
[14] Yang, X. and Zhang, T. (2013).
Conditions.
Estimates of Heat Kernels with Neumann Boundary
Potential Analysis, 38:549572.
15