Approximation for nonsmooth functionals of stochastic differential equations with irregular drift

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Approximation for non-smooth functionals of stochastic
differential equations with irregular drift
Hoang-Long Ngo∗

and

Dai Taguchi†

Abstract
We find upper bounds for the rate of convergence when the Euler-Maruyama approximation is used in order to compute the expectation of non-smooth functionals of some stochastic
differential equations whose diffusion coefficient is constant, whereas the drift coefficient may
be very irregular. As a byproduct of our method, we establish the weak order of the EulerMaruyama approximation for a diffusion processes killed when it leaves an open set. We also
apply our method to the study of the weak approximation of reflected stochastic differential
equations whose drift is H¨
older continuous.
2010 Mathematics Subject Classification: 60H35, 65C05, 65C30
Keywords: Euler-Maruyama approximation, Irregular drift, Monte Carlo method, Reflected
stochastic differential equation, Weak approximation

1

Introduction

Let (Xt )0≤t≤T be the solution to
dXt = b(Xt )dt + σ(Xt )dWt ,

X0 = x0 ∈ Rd , 0 ≤ t ≤ T,

where W is a d-dimensional Brownian motion. The diffusion (Xt )0≤t≤T is used to model many
random dynamical phenomena in many fields of applications. In practice, one often encounters the
problem of evaluating functionals of the type E[f (X)] for some given function f : C[0, T ] → R. For

example, in mathematical finance the function f is commonly referred as a payoff function. Since
they are rarely analytically tractable, these expectations are usually approximated using numerical
schemes. One of the most popular approximation methods is the Monte Carlo Euler-Maruyama
method which consists of two steps:
1. The diffusion process (Xt )0≤t≤T is approximated using the Euler-Maruyama scheme (Xth )0≤t≤T
with a small time step h > 0:
dXth = b(Xηhh (t) )dt + σ(Xηhh (t) )dWt ,

X0h = x0 , ηh (t) = kh,

∗ Hanoi National University of Education, 136 Xuan Thuy - Cau Giay - Hanoi - Vietnam, email:

† Ritsumeikan University, 1-1-1 Nojihigashi, Kusatsu, Shiga, 525-8577, Japan, email:

1

for t ∈ [kh, (k + 1)h), k ∈ N.
2. The expectation E[f (X)] is approximated using
independent copies of X h .

1
N

N
i=1

f (X h,i ) where (X h,i )1≤i≤N are N

This approximation procedure is influenced by two sources of errors: a discretization error and a

statistical error
Err(f, h) := Err(h) := E[f (X)] − E[f (X h )],

and E[f (X h )] −

1
N

N

f (X h,i ).
i=1

We say that the Euler-Maruyama approximation (X h ) is of weak order κ > 0 for a class H of
functions f if there exists a constant K(T ) such that for any f ∈ H,
|Err(f, h)| ≤ K(T )hκ .
The effect of the statistical error can be handled by the classical central limit theorem or large
deviation theory. Roughly speaking, if f (XTh ) has a bounded variance, the L2 -norm of the statistical
error is bounded by N −1/2 V ar(XTh )1/2 . Hence, if the Euler-Maruyama approximation is of weak
order κ, the optimal choice of the number of Monte Carlo iterations should be N = O(h−2κ )
in order to minimize the computational cost. Therefore, it is of both theoretical and practical
importance to understand the weak order of the Euler-Maruyama approximation.
It has been shown that under sufficient regularity on the coefficients b and σ as well as f ,
the weak order of the Euler-Maruyama approximation is 1. This fact is proven by writing the
discretization error Err(f, h) as a sum of terms involving the solution of a parabolic partial differential equation (see [1, 28, 24, 14, 7]). It should be noted here that besides the Monte Carlo
Euler-Maruyama method, there are many other related approximation schemes for E[f (XT )] which
have either higher weak order or lower computational cost. For example, one can use Romberg
extrapolation technique to obtain very high weak order as long as Err(h) can be expanded in terms
of powers of h (see [28]). When f is a Lipschitz function and the strong rate of approximation
is known, one can implement a Multi-level Monte Carlo simulation which can significantly reduce

the computation cost of approximating E[f (X)] in many cases (see [5]). It is also worth looking at
some algebraic schemes introduced in [18]. However, all the accelerated schemes mentioned above
require sufficient regularity condition on the coefficients b, σ and the test function f .
The stochastic differential equations with non-smooth drift appear in many applications, especially when one wants to model sudden changes in the trend of a certain random dynamical
phenomenon (see e.g., [14]). There are many papers studying the Euler-Maruyama approximations in this context. In [9] (see also [2]), it is shown that when the drift is only measurable,
the diffusion coefficient is non-degenerate and Lipschitz continuous then the Euler-Maruyama approximations converges to the solution of stochastic differential equation. The weak order of the
Euler-Maruyama scheme when both coefficients b and σ as well as payoff functions f are H¨older
continuous has been studied in [14, 23]. In the papers [15] and [25], the authors studied the
weak and strong convergent rates of the Euler-Maruyama scheme for specific classes of stochastic
differential equations with discontinuous drift.
The aim of the present paper is to investigate the weak order of the Euler-Maruyama approximation for stochastic differential equations whose diffusion coefficient σ is constant, whereas the
2

drift coefficient b may have a very low regularity, or could even be discontinuous. More precisely,
we consider a class of function A which contains not only smooth functions but also some discontinuous one such as indicator function. b will then be assumed to be either in A or α-H¨older
continuous. It should be noted that no smoothness assumption on the payoff function f is needed
in our framework. As a by product of our method, we establish the weak order of the EulerMaruyama approximation for a diffusion processes killed when it leaves an open set. We also
apply our method to study the weak approximation of reflected stochastic differential equation
whose drift is H¨
older continuous.
The remainder of this paper is organized as follows. In the next section we introduce some
notations and assumptions for our framework together with the main results. All proofs are
deferred to Section 3.

2
2.1

Main Results
Notations

A function ζ : Rd → R is called exponentially bounded or polynomially bounded if there exist positive
constants K, p such that |ζ(x)| ≤ KeK|x| or |ζ(x)| ≤ K(1 + |x|p ), respectively.
Let A be a class of exponentially bounded functions ζ : Rd → R such that there exists a
sequence of functions (ζN ) ⊂ C 1 (Rd ) satisfying:

ζN → ζ in L1loc (Rd ),

A(i) :
A(ii) : supN |ζN (x)| + |ζ(x)| ≤ KeK|x| ,

−|x|2 /u

A(iii) : supN,u>0; a∈Rd e−K|a|−Ku Rd |∇ζN (x + a)| ue (d−1)/2 dx < K,
for some positive constant K. We call (ζN ) an approximation sequence of ζ in A.
The following propositions shows that this class is quite large.
Proposition 2.1. i) If ξ, ζ ∈ A then ξζ ∈ A and a1 ξ + a2 ζ ∈ A for any a1 , a2 ∈ R.
ii) Suppose that A is a non-singular d × d-matrix, B ∈ Rd . Then ζ ∈ A iff ξ(x) := ζ(Ax + B) ∈ A.
It is easy to verify that the class A contains all C 1 (Rd ) functions which has all first order
derivatives polynomially bounded. Furthermore, the class A contains also some non-smooth functions of the type ζ(x) = (x1 − a)+ or ζ(x) = Iaa function ζ : Rd → R monotone in each variable separately if for each i = 1, . . . , d, the map
xi → ζ(x1 , . . . , xi , . . . , xn ) is monotone for all x1 , . . . , xi−1 , xi+1 , . . . , xd ∈ R.
Proposition 2.2. Class A contains all exponentially bounded functions which are monotone in
each variable separately.
The proofs of Propositions 2.1 and 2.2 and further properties of class A were presented in [16]
and [25].
We recall that a function ζ : Rd → Rd is called α-H¨older continuous for some α ∈ (0, 1] if there
exists a positive constant C such that |ζ(x) − ζ(y)| ≤ C|x − y|α for all x, y ∈ Rd . We denote by
B(α) the class of all measurable functions b : Rd → Rd such that b = bH + bA where bH is α-H¨older
continuous for some α > 0 and bA

j ∈ A for j = 1, . . . , d.
3

2.2

Weak approximation of stochastic differential equations

Let (Ω, G, {Gt }t≥0 , Q) be a filtered probability space and (Wt )t≥0 be a d-dimensional standard
Brownian motion. We consider a d-dimensional stochastic differential equation
t

Xt = x0 +

b(Xs )ds + σWt ,

x0 ∈ Rd , t ∈ [0, T ],

(1)

0

where σ is a d × d invertible deterministic matrix and b : Rd → Rd is a Borel measurable function.
Let X h , h > 0, denote the Euler-Maruyama approximation of X,
t

Xth = x0 +
0

b(Xηhh (s) )ds + σWt ,

t ∈ [0, T ],

(2)

where ηh (s) = kh if kh ≤ s < (k + 1)h for some nonnegative integer k. In this paper, we study the
convergent rates of the error
Err(h) = E[f (X)] − E[f (X h )]
as h → 0 for some payoff function f : C[0, T ] → R.
A Borel measurable function ζ : Rd → Rd is called sub-linear growth if ζ is bounded on
compact sets and ζ(y) = o(|y|) as y → ∞. ζ is called linear growth if |ζ(y)| < c1 |y| + c2 for
some positive constants c1 , c2 . It has been shown recently in [11] that when b is of super-linear
growth, i.e., there exist constants C > 0 and θ > 1 such that |ζ(y)| ≥ |y|θ for all |y| > C, then
the Euler-Maruyama approximation (2) converges neither in the strong mean square sense nor in
weak sense to the exact solution at a finite time point. It means that if E[|XT |p ] < ∞ for some
p ∈ [1, ∞) then
lim E |XT − XTh |p = ∞

h→0

lim E |XT |p − |XTh |p

and

h→0

= ∞.

Thus, in this paper we will consider the case that b is of at most linear growth.
Remark 2.3. In the one-dimensional case, d = 1, it is well-known that if σ = 0 and b is of linear

growth, then the strong existence and path-wise uniqueness hold for the equation (1) (see [3]).
In the multidimensional case, d > 1, it has been shown in [30] that if b is bounded then the
equation (1) has a strong solution and the solution of (1) is strongly unique. Moreover, if σ is
the identity matrix, then the equation (1) has a unique strong solution in the class of continuous
T
processes such that P 0 |b(Xs )|2 ds < ∞ = 1 provided that Rd |b(y)|p dy < ∞ for some p > d ∨ 2
(see [17]).
Throughout this paper, we suppose that equation (1) has a weak solution which is unique in
the sense of probability law (see Chapter 5 [13])).
Our main results requires no assumption on the smoothness of f .
Theorem 2.4. Suppose that b ∈ B(α) and b is of linear growth. Moreover, assume that f :
C[0, T ] → R is bounded. Then
lim E[f (X h )] = E[f (X)].
h→0

4

If b is of sub-linear growth, we can obtain the rate of convergence as follows.
Theorem 2.5. Suppose that b ∈ B(α) and b is of sub-linear growth. Moreover, assume that
f : C[0, T ] → R satisfies E[|f (x0 + σW )|r ] < ∞ for some r > 2. Then there exists a constant C
which does not depend of h such that
α

1

|E[f (X)] − E[f (X h )]| ≤ Ch 2 ∧ 4 .
For an integral type functional, we obtain the following corollary.
Corollary 2.6. Let h = T /n for some n ∈ N. If the drift coefficient b ∈ B(α) is bounded, then for
any Lipschitz continuous function f and g ∈ B(β) with β ∈ (0, 1], there exists a constant C which

does not depend of h such that
T

E f

T

g(Xs )ds

−E f

0

0

g(Xηhh (s) )ds

α

β

1

≤ Ch 2 ∧ 2 ∧ 4 .

Remark 2.7. In the paper [22], the author considered the weak rate of convergence of the EulerMaruyama scheme for equation (1) in the case of a one-dimensional diffusion. It was claimed that
if b was Lipschitz continuous, the weak rate of approximation is of order 1. However, we would
like to point out that the given proof contains several gaps (see for instance Lemma 2 of [22] and
Remark 3.3 below) which leave us unsure about the claim.
Remark 2.8. It has been shown in [14, 23] that for a stochastic differential equation with α-H¨older

continuous drift and diffusion coefficients with α ∈ (0, 1), one has
|E[f (XT )] − E[f (XTh )]| ≤ Chα/2 ,
where f ∈ Cb2 and the second derivative of f is α-H¨older continuous. On the other hand, in [10],
Gy¨
ongy and R´
asonyi have obtained the strong rate of convergence for a one-dimensional stochastic
differential equation whose drift is the sum of a Lipschitz continuous and a monotone decreasing
H¨
older continuous function, and its diffusion coefficient is H¨older continuous. In [25], the authors
improve the results in [10]. More precisely, we assume that the drift coefficient b is a bounded and
one-sided Lipschitz function, i.e., there exists a positive constant L such that for any x, y ∈ Rd ,
x − y, b(x) − b(y) Rd ≤ L|x − y|2 , bj ∈ A for any j = 1, ..., d and the diffusion coefficient σ is
bounded, uniformly elliptic and 1/2 + α-H¨older continuous with α ∈ [0, 1/2]. Then for h = T /n,
it holds that

−1
if α = 0 and d = 1,
 C(log 1/h)
α
h
Ch
if α ∈ (0, 1/2] and d = 1,
E[|XT − XT |] ≤

Ch1/2
if α = 1/2 and d ≥ 2.
Therefore, if the payoff function f is Lipschitz continuous, it is straightforward to verify that

−1
if α = 0 and d = 1,

 C(log 1/h)
α
h
Ch
if α ∈ (0, 1/2] and d = 1,
|E[f (XT ) − f (XT )]| ≤

Ch1/2
if α = 1/2 and d ≥ 2.

5

In the following we consider a special case of the functional f . More precise, we are interested
in the law at time T of the diffusion X killed when it leaves an open set. Let D be an open subset
of Rd and denote τD = inf{t > 0 : Xt ∈ D}. Quantities of the type E[g(XT )1(τD >T ) ] appear
in many domains, e.g. in financial mathematics when one computes the price of a barrier option
on a d-dimensional asset price random variable Xt with characteristics f, T and D (see [6, 8] and
h
h
the references therein for more detail). We approximate τD by τD
= inf{kh > 0 : Xkh
∈ D, k =
0, 1, . . .}.
Theorem 2.9. Assume the hypotheses of Theorem 2.5. Furthermore, we assume
(i) D is of class C ∞ and ∂D is a compact set (see [4] and [6]);
(ii) g : Rd → R is a measurable function, satisfying d(Supp(g), ∂D) ≥ 2 for some
g ∞ = supx∈Rd |g(x)| < ∞.

> 0 and

Then for any p > 1, there exist constants C and Cp independent of h such that
α

1

E[g(XT )1(τD >T ) ] − E[g(XTh )1(τDh >T ) ] ≤ Ch 2 ∧ 4 +

2.3

1
Cp g p ∞ 2p
h .
1 ∧ 4/p

(3)

Weak approximation of reflected stochastic differential equations

We first recall the Skorohod problem.
Lemma 2.10 ([13], Lemma III.6.14). Let z ≥ 0 be a given number and y : [0, ∞) → R be
a continuous function with y0 = 0. Then there exists unique continuous function = ( t )t≥0
satisfying the following conditions:
(i) xt := z + yt +
(ii)

t

≥ 0, 0 ≤ t < ∞;

is a non-decreasing function with

Moreover,

0

= 0 and

t

=

t
0

1(xs = 0)d s .

= ( t )t≥0 is given by
t

= max{0, max (−z − ys )} = max max(0,
0≤s≤t

0≤s≤t

s

− xs ).

Let us consider the following one-dimensional reflected stochastic Xt valued in [0, ∞) such

that
t

b(Xs )ds + σWt + L0t (X), x0 ∈ [0, ∞), t ∈ [0, T ],

Xt = x0 +

(4)

0
t

L0t (X) =

1(Xs =0) dL0s (X),
0

where (L0t (X))0≤t≤T is a non-decreasing continuous process stating at the origin and it is called
local time of X at the origin. In this paper, we assume that the SDE (4) has a weak solution and
the uniqueness in the sense of probability law holds (see [27, 29]). Using Lemma 2.10, we have
L0t (X) = sup max 0, L0s (X) − Xs .
0≤s≤t

6

Now we define the Euler-Maruyama scheme X h = (Xth )0≤t≤T for the reflected stochastic
differential equation (4). Let X0h := x0 and define
t

Xth = x0 +
0

b(Xηhh (s) )ds + σWt + L0t (X h ).

The existence of the pair (Xth , L0t (X h ))0≤t≤T is deduced from Lemma 2.10. Moreover
t

L0t (X h ) =
0

1(Xsh =0) dL0s (X h ).

By the definition of the Euler-Maruyama scheme, we have the following representation. For each
k = 0, 1, ...,
h
h
h
h
X(k+1)h
= Xkh
+ b(Xkh
)h + σ(W(k+1)h − Wkh ) + max(0, Ak − Xkh
),

where
Ak :=

sup

h
−b(Xkh
)(s − kh) − σ(Ws − Wkh ) .

kh≤s<(k+1)h

Though Ak is defined by the supremum of a stochastic process, it can be simulated by using the
following lemma.
Lemma 2.11 ([20], Theorem 1). Let t ∈ [0, T ] and a, c ∈ R. Define St := sup0≤s≤t (aWs + cs). Let
Ut be a centered Gaussian random variable with variance t and let Vt be an exponential random
variable with parameter 1/(2t) independent from Ut . Define
Yt :=

1
(aUt + ct + (a2 Vt + (aUt + ct)2 )1/2 ).
2

Then the processes (Wt , St )t∈[0,T ] and (Ut , Yt )t∈[0,T ] have the same law.
Under the Lipschitz condition for the coefficients of the reflected SDE (4), L´epingle [21] shows
that
E[ sup |Xt − Xth |2 ]1/2 ≤ Ch1/2 ,
0≤t≤T

for some constant C.
We obtain the following result on the weak convergence for the Euler-Maruyama scheme for
a reflected SDEs with non-Lipschitz coefficient.
Theorem 2.12. Suppose that the drift coefficient b is of sub-linear growth and α-H¨older continuous
with α ∈ (0, 1]. Moreover, assume that f : C[0, T ] → R is bounded. Then there exists a constant
C not depend of h such that
|E[f (X)] − E[f (X h )]| ≤ Chα/2 .

7

3

Proofs

From now on, we will repeatedly use without mentioning the following elementary estimate
2

sup |x|p ek|x|−x < ∞,

for any p ≥ 0, k ∈ R.

(5)

x∈R

Throughout this section, a symbol C stands for a positive generic constant independent of the
discretization parameter h, which nonetheless may depend on time T , coefficients b, σ and payoff
function f .

3.1

Change of Measures

From now on, we will use the following notations
t

Z t = e Yt ,

Yt =

Yth

Yth

(σ −1 b)j (x0 + σWs )dWsj −
0

Zth

=e

1
2

t

|σ −1 b(x0 + σWs )|2 ds,
0

t

,

=

(σ

−1

b)j (x0 +

σWηh (s) )dWsj

−

0

1
2

t

|σ −1 b(x0 + σWηh (s) )|2 ds,
0

where we use Einstein’s summation convention on repeated indices. We also use the following
auxiliary stopping times
W,h
W
τD
= inf{t ≥ 0 : x0 + σWt ∈ D}, and τD
= inf{kh ≥ 0 : x0 + σWkh ∈ D, k = 0, 1, . . .}.

Lemma 3.1. Suppose that b is a function with at most linear growth, then we have the following
representations
E[f (X)] − E[f (X h )] = E[f (x0 + σW )(ZT − ZTh )],

(6)

and
E[g(XT )1(τD >T ) ] − E[g(XTh )1(τDh >T ) ]
= E[g(x0 + σWT )(ZT 1(τDW >T ) − ZTh 1(τ W,h >T ) )],

(7)

D

for all measurable functions f : C[0, T ] → R and g : Rd → R provided that all the above expectations are integrable.
Proof. Let σ −1 be the inverse matrix of σ. Since b is of linear growth, so is σ −1 b. Thus, there exist
constants c1 , c2 > 0 such that |b(x)| < c1 |x| + c2 for any x ∈ Rd . For any 0 ≤ t ≤ t0 ≤ T ,
t

|Xt | ≤ |x0 | + |σWt | +

|b(Xs )|ds
0
t

≤ |x0 | + |σ| sup |Ws | + c2 t0 + c1
0≤s≤t0

|Xs |ds.
0

8

Applying Gronwall’s inequality for t ∈ [0, t0 ], one obtains
|Xt0 | ≤ (|x0 | + |σ| sup |Ws | + c2 t0 )ec1 t0
0≤s≤t0

≤ (|x0 | + c2 T )ec1 T + |σ|ec1 T sup |Ws |.

(8)

0≤s≤t0

On the other hand, for each integer k ≥ 1, one has
h
h
h
| + h|b(X(k−1)h
)| + 2|σ| sup |Wt |
|Xkh
| ≤ |X(k−1)h
0≤t≤kh

≤ (1 +

h
hc1 )|X(k−1)h
|

+ hc2 + 2|σ| sup |Wt |.
0≤t≤kh

Hence, a simple iteration yields that
h
|Xkh
| ≤ (1 + hc1 )k |x0 | + (hc2 + 2|σ| sup |Wt |)
0≤t≤kh

(1 + hc1 )k−1 − 1
.
hc1

Thus, for any t ∈ (0, T ],
|Xηhh (t) | ≤ (1 + hc1 )T /h |x0 | +

(1 + hc1 )T /h
c2 (1 + hc1 )T /h
+ 2|σ|
c1
hc1

sup

|Ws |.

0≤s≤ηh (t)

Moreover,
|Xth − Xηhh (t) | ≤ c1 h|Xηhh (t) | + c2 h + 2|σ| sup |Wt |.
0≤s≤t

Therefore, for any t ∈ (0, T ], we have

|Xth | ≤ (1 + c1 h)1+T /h

2|σ|(1 + hc1 )1+T /h + 2hc1
c1 |x0 | + c2
+ c2 h +
sup |Ws |.
c1
hc1
0≤s≤t

(9)

We define a new measure P and Ph as
dP
= exp −
dQ
dPh
= exp −
dQ

T

(σ −1 b)j (Xs )dWsj −
0

1
2

T

|σ −1 b(Xs )|2 ds ,
0

T
0

(σ −1 b)j (Xηhh (s) )dWsj −

1
2

T
0

|σ −1 b(Xηhh (s) )|2 ds .

It follows from Corollary 3.5.16 [13] together with estimates (8) and (9) that P and Ph are probability measures. Furthermore, it follows from Girsanov theorem that processes B = {(Bt1 , . . . , Btd ), 0 ≤
t ≤ T } and B h = {(Bth,1 , . . . , Bth,d ), 0 ≤ t ≤ T } defined by
t

t

(σ −1 b)j (Xs )ds, Bth,j = Wtj +

Btj = Wtj +
0

(σ −1 b)j (Xηh (s) )ds, 1 ≤ j ≤ d, 0 ≤ t ≤ T,
0

are d-dimensional Brownian motions with respect to P and Ph , respectively. Note that Xs =
x0 + σBs and Xsh = x0 + σBsh . Therefore,
E[f (X)] = EP f (X)

dQ
dP
9

T

(σ −1 b)j (x0 + σBs )dBsj −

= EP f (x0 + σB) exp
0

1
2

T

|σ −1 b(x0 + σBs )|2 ds
0

= E[f (x0 + σW )ZT ].
Repeating the previous argument leads to E[f (X h )] = E[f (x0 + σW )ZTh ], which implies (6). The
proof of (7) is similar and will be omitted.
From now on, we will use the representation formulas in Lemma 3.1 to analyze the weak rate
of convergence. We need the following estimates on the moments of Z and Z h .
Lemma 3.2. Suppose that b is of sub-linear growth. Then for any p > 0,

E[|ZT |p + |ZTh |p ] ≤ C < ∞,
for some constant C which is not depend on h.
Proof. For each p > 0,
T

(σ −1 b)j (x0 + σWs )dWsj −

E[epYT ] = E exp p
0

T

p
2

|σ −1 b(x + σWs )|2 ds
0

T

tn

(σ −1 b)j (x0 + σWs )dWsj − p2

= E exp p
0

p
+ (p2 − )
2

(δ|Ws |2 + M )ds ≤ T M + T δ sup |Ws |2
s≤T

0
d

((sup Wsj )2 + ( inf Wsj )2 ).

≤ TM + Tδ
j=1

s≤T

Hence,
T

|σ −1 b(x0 + σWs )|2 ds

E exp (2p2 − p)
0

10

s≤T

d

≤ e(2p

2

−p)M T

E exp T δ(2p2 − p)

((sup Wsj )2 + ( inf Wsj )2 )
j=1

≤ e(2p

2

−p)M T

s≤T

s≤T

E exp 2T δ(2p2 − p)|WT1 |2

d/2

,

where the last inequality follows from H¨older inequality and the fact that
law

law

sup Wsj = − inf Wsj = |WT1 |.
s≤T

s≤T

< ∞ if one chooses δ < (4T 2 (2p2 −p))−1 , we obtain E[|ZT |p ] <

Because E exp 2T δ(2p2 −p)WT2

∞. Furthermore, since equation (10) still holds if one replaces b(x0 + σWs ) with b(x0 + σWηh (s) ),
a similar argument yields E[|ZTh |p ] < ∞.
Remark 3.3. In general, the conclusion of Lemma 3.2 is no longer correct if we only suppose that
b is of linear growth or even Lipschitz.
Indeed, consider the particular case that d = 1, σ = 1 and b(x) = x, which is a Lipschitz
function. It follows from H¨
older inequality that
T

p
2

E exp

T

Ws2 ds

Ws dWs −

E exp p

0

0

= e−pT /2 E exp

p
2

T

Ws2 ds

E exp

0

p
2

p 2 p
W −
2 T
2

T

Ws2 ds

0
T

Ws2 ds
0

2

2

≥ e−pT /2 E epWT /4

.
2

Furthermore, for any p, T > 0 such that pT ≥ 2 and pT 2 < 1/2, we have E epWT /4 = ∞, whereas
E exp

p
2

T

Ws2 ds
0

pT
sup |Ws |2
2 s≤T
pT

pT
≤ E exp
(sup Ws )2 +
( inf Ws )2 )
2 s≤T
2 s≤T

≤ E exp

2

≤ E epT |WT |

2

< ∞.

Therefore,
T

Ws dWs −

E exp p
0

3.2

p
2

T

Ws2 ds

= ∞,

0

Some auxiliary estimates

The following result plays a crucial role in our argument.

11

if pT ≥ 2, pT 2 <

1
.
2

Lemma 3.4. For any ζ ∈ A, any p ≥ 1, t > s > 0,
√
t−s
E[|ζ(Wt ) − ζ(Ws )|p ] ≤ Cp √ ,
s

(11)

for some constant Cp not depending on neither t nor s. On the other hand, if ζ is α-H¨older

=
Rd

Rd

e−|x| /2s e−|y| /2(t−s)
dxdy
(2πs)d/2 (2π(t − s))d/2
2

|ζN (x + y) − ζN (x)|(eK(|x+y| + eK|y| )p−1

≤C
Rd
d

Rd

i=1

Rd

1

≤C
√

Rd

0

1

≤C t − s
Rd

2

2

∂ζN (y + θx) K(p−1)(|x|+|y|) e−|x| /2s e−|y| /2(t−s)
dθdxdy
yi
e
∂xi
(2πs)d/2 (2π(t − s))d/2

d

i=1

2

e−|x| /2s e−|y| /2(t−s)
dxdy
(2πs)d/2 (2π(t − s))d/2

Rd

0

2

2

∂ζN (y + θx) e−|x| /4s e−|y| /4(t−s)
dθdxdy.
∂xi
(2πs)d/2 (2π(t − s))d/2

It follows from A(iii) that
d

2

sup
N

i=1

Rd

∂ζN (y + θx)
e−|y| /4(t−s)
dy ≤ CeK|θx| ,
∂xi
(2π(t − s))(d−1)/2

thus
√

t−s
E[|ζN (Wt ) − ζN (Ws )| ] ≤ C √
s
√
t−s
≤C √ .
s

2

1

CeK|θx|

p

Rd

0

e−|x| /4s
dθdx
(2πs)d/2

From (13) and (14) we get (11). The proof of (12) is straightforward.

12

Lemma 3.5. Suppose that b ∈ B(α) and b is of linear growth. Then there exists a constant C > 0

such that
YT − YTh 2 ≤ Ch1/4 + Chα/2 .
Proof. Using Minkowski’s inequality, we obtain E[|YT − YTh |2 ] ≤ C(S1 + S2 ) where
T

[(σ −1 b)j (x0 + σWs )dWsj − (σ −1 b)j (x0 + σWηh (s) )]dWsj

S1 = E

2

,

0
T

[|σ −1 b(x0 + σWs )|2 − |σ −1 b(x0 + σWηh (s) )|2 ]ds

S2 = E

2

.

0

It follows from Doob’s inequality (see [13]) that,
d

T

S1 ≤ C

[(σ −1 b)j (x0 + σWs ) − (σ −1 b)j (x0 + σWηh (s) )]2 ds

E
0

j=1
d

T

E[(σ −1 b)j (x0 + σWs ) − (σ −1 b)j (x0 + σWηh (s) )]2 ds.

≤C
0

j=1

Since b is of linear growth,
d

h

E[(σ −1 b)j (x0 + σWs ) − (σ −1 b)j (x0 + σWηh (s) )]2 ds ≤ Ch.
0

j=1

Furthermore, it follows from Proposition 2.1 ii) and Lemma 3.4 that
d

T

E[(σ −1 b)j (x0 + σWs ) − (σ −1 b)j (x0 + σWηh (s) )]2 ds
h

j=1

T

s − ηh (s)

|s − ηn (s)|α +

≤C

ηh (s)

h

ds ≤ C(hα + h1/2 ).

Therefore, S1 ≤ C(hα + h1/2 ).
Now we estimate S2 . Using H¨
older’s inequality, we obtain
T

2

Thus, YT −

YTh 2

1/4

≤ Ch

s − ηh (s)
ηh (s)

1/2

ds + Chα ≤ C(h1/2 + hα ).

+ Chα/2 .
13

3.3

Proof of Theorem 2.4

It follows from Lemma 3.5 that YTh converges in probability to YT as h → 0. Thus ZTh also converges
in probability to ZT as h → 0. Moreover E[ZTh ] = E[ZT ] = 1 for all h > 0. Therefore, it follows
from Proposition 4.12 [12] that
lim E[|ZTh − ZT |] = 0.
(15)
h→0

2/r

E |ZT + ZTh |2r/(r−2)

(r−2)/r

≤ C < ∞.

This together with Lemma 3.5 implies the desired result.

3.5

Proof of Corollary 2.6

We first note that if b is bounded, then it holds from Theorem 2.1 in [19] (see also Corollary 3.2
in [25]) that there exists a density function pht of Xth for t ∈ (0, T ] and it satisfies the following
Gaussian upper bound, i.e.,
pht (x)

≤C

e−

|x−x0 |2
2ct

td/2

.

for some positive constants C and c.
r

T

Now we prove that E f 0 g(x0 + Ws )ds
is finite for any r > 2. Since |g(x)| ≤ KeK|x| ,
it follows from Jensen’s inequality that for any r > 2,
r

T

E

f

g(x0 + Ws )ds
0

T

T

d
i

E[|g(x + Ws )|r ]ds ≤ C + C

≤C +C

0

E[erKWs ]ds.
0

14

i=1

Since x2 /(4s) + K 2 r2 s ≥ Krx, we have
r

T

T

edKrs ds < ∞.

≤C +C

g(x0 + Ws )ds

f

E

0

0

Thanks to Theorem 2.5, it remains to prove that
T

T

g(Xsh )ds

E f
0

≤ Chβ/2 .

g(Xηhh (s) )ds

−E f
0

Since f is a Lipschitz continuous function, we have
T

T

T

g(Xsh )ds

E f

−E f

0

0

g(Xηhh (s) )ds

E g(Xsh ) − g(Xηhh (s) ) ds

≤C
0

If s ∈ (0, h], then by using the Gaussian upper bound for phs (x), we have
h
0

h

E g(Xsh ) − g(Xηhh (s) ) ds ≤

E[|g(Xsh )|]ds + |g(x0 )|h
0
h

≤C

|g(x)|

ds
0

|x−x0 |2
2cs

e−

sd/2

Rd

+ |g(x0 )|h

≤ Ch.
On the other hand, for s ∈ [h, T ], using the Gaussian upper bound for phηh (s) and following the
proof of Lemma 3.4 (see also Lemma 3.5 of [25]), we have
E[|g(Xsh ) − g(Xηhh (s) )|] ≤

C

s − ηh (s)
ηh (s)

+ Chβ/2 .

Therefore, we conclude the proof of the statement.

3.6

Proof of Theorem 2.9

It suffices to proof the statement for the case that g is positive. It follows from (7) that E[g(XT )1(τD >T ) ]−
E[g(XTh )1(τDh >T ) ] = E1 + E2 where
E1 = E[g(x0 + σWT )(ZT − ZTh )1(τ W,h >T ) ],
D

E2 = E[g(x0 + σWT )ZT (1(τDW >T ) − 1(τ W,h >T ) )].
D

It follows from the proof of Theorem 2.5 that
α

1

|E1 | ≤ E[|g(x0 + σWT )(ZT − ZTh )|] ≤ Ch 2 ∧ 4 .
Applying H¨
older’s inequality, we have
|E2 | ≤ ZT

q

g(x0 + σWT )(1(τDW >T ) − 1(τ W,h >T ) ) p ,
D

15

(16)

W,h
W

where q is the conjugate of p. Thanks to Lemma 3.2 and the fact τD
≥ τD
, we have

|E2 | ≤ Cp E g p (x0 + σWT )1(τ W,h ≥T ) − E g p (x0 + σWT )1(τDW ≥T )

1/p

.

D

It follows from Theorem 2.4 in [6] that there exists a constant K(T ) such that
K(T ) g p
1∧ 4

|E2 | ≤ Cp

1/p

∞

1

h 2p .

Combining this estimate with (16) completes the proof.

3.7

Proof of Theorem 2.12

In the same way as in subsection 3.1, we have the following Lemma.
Lemma 3.6. If b is a measurable function with sub-linear growth then
E[f (X)] − E[f (X h )] = E[f (U )(ZˆT − ZˆTh )]
for all measurable functions f : C[0, T ] → R provided that the above expectations are integrable.
Here the process U = (Ut )0≤t≤T is the unique solution of the equation Ut = x0 + σWt + L0t (U ) and
ˆ
Zˆt := eYt ,

t

Yˆt :=

b(Us )dWs −
0

t

1
2

b2 (Us )ds
0

t

ˆh
Zˆth := eYt ,

Yˆth :=

b(Uηh (s) )dWs −
0

1
2

t

b2 (Uηh (s) )ds,
0

ˆ and P
ˆ h as
Proof. We define new measures P
ˆ
dP
= exp −
dQ
ˆh
dP
= exp −
dQ

T

σ −1 b(Xs )dWs −
0

ˆt )0≤t≤T and B
ˆ h = (B
ˆth )0≤t≤T defined by
from the Girsanov theorem that the processes B
t

ˆ t = Wt +
B

t

ˆth = Wt +
σ −1 b(Xs )ds, B
0

0

σ −1 b(Xηhh (s) )ds, 0 ≤ t ≤ T,

ˆ and P
ˆ h respectively. Note that Xs = x0 + σ B
ˆs + L0s (X)
are Brownian motions with respect to P
h
h
0
h
ˆ
and Xs = x0 + σ Bs + Ls (X ). Therefore,
E[f (X)] = EPˆ f (X)

dQ
ˆ
dP
16

T

= EPˆ f (X) exp

0

T

ˆs − 1
σ −1 b(Xs )dB
2
T

ˆ + L0 (X)) exp
= EPˆ f (x0 + σ B

0

|σ −1 b(Xs )|2 ds
0

ˆs + L0s (X))dB
ˆs − 1

σ −1 b(x0 + σ B
2

T

ˆs + L0s (X))|2 ds
|σ −1 b(x0 + σ B
0

d

ˆ ˆ = (U, W )|Q , the above term equals to
Since (X, B)|
P
T

σ −1 b(x0 + σWs + L0s (U ))dWs −

E f (x0 + σW + L0 (U )) exp
0
T

σ −1 b(Us )dWs −

= E f (U ) exp
0

1
2

1
2

T

|σ −1 b(x0 + σWs + L0s (U ))|2 ds
0

T

|σ −1 b(Us )|2 ds
0

= E[f (U )ZˆT ].
Repeating the previous argument leads to E[f (X h )] = E[f (U )ZˆTh ], which concludes the statement.
In the same way as Lemma 3.2, we have the following estimate of the moments of Zˆ and Zˆ h .
Lemma 3.7. Suppose that b is of sub-linear growth. Then for any p > 0,
E[|ZˆT |p + |ZˆTh |p ] ≤ C < ∞,
for some constant C which is not depend on h.
Finally, we introduce the following auxiliary estimate.
Lemma 3.8. Let U as in Lemma 3.6. Suppose that ζ is α-H¨older continuous with α ∈ (0, 1], then
for any t > s > 0,
E[|ζ(Ut ) − ζ(Us )|p ] ≤ Cp (t − s)pα/2 .
Proof. By H¨
older continuity of ζ, we have
E[|ζ(Ut ) − ζ(Us )|p )] ≤ E[|Ut − Us |αp ] ≤ CE[|Wt − Ws |pα ] + CE[|L0t (U ) − L0s (U )|pα ].
Hence it is sufficient to prove that
E[|L0t (U ) − L0s (U )|p α] ≤ Cp (t − s)pα/2 .
Using Lemma 2.10, we have
L0s (U ) ≤ L0t (U ) ≤ L0s (U ) + sup max (0, −σ(Wu − Ws )) .

s≤u≤t

Therefore, |L0t (U )−L0s (U )| ≤ |σ| sups≤u≤t |Wu −Ws |. Hence applying the Burkholder-Davis-Gundy
inequality, we have
E[|L0t (U ) − L0s (U )|pα ] ≤ CE[ sup |Wu − Ws |pα ] ≤ C(t − s)pα/2 .
s≤u≤t

This concludes the proof.

17

.

Proof of Theorem 2.12 Using Lemmas 3.6 and the elementary estimate |ex − ey | ≤ (ex + ey )|x − y|,
we have
|E[f (X)] − E[f (X h )]| ≤ E f (U )(ZˆT + ZˆTh )(YˆT − YˆTh ) .
Thanks to Lemma 3.7 and the H¨
older inequality, for some r > 2, we have
|E[f (X)] − E[f (X h )]| ≤ CE[|f (U )|r ]2/r ||YˆT − YˆTh ||2 .
By a similar argument as the proof of Lemma 3.5, we can show that
||YˆT − YˆTh ||2 ≤ Chα/2 ,
which concludes the proof.
Remark 3.9. The conclusion of Theorem 2.12 still holds if we relax the condition f bounded to
E[|f (U )|r ] < ∞ for some r > 2.
Acknowledgment. This work was completed while the H.N. was staying at Vietnam Institute
for Advanced Study in Mathematics (VIASM). He would like to thank the institute for support.
This work was also partly supported by the Vietnamese National Foundation for Science and
Technology Development (NAFOSTED) under Grant Number 101.03-2014.14. The authors thank
Prof. Arturo Kohatsu-Higa for his helpful comments.

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20

Approximation for nonsmooth functionals of stochastic differential equations with irregular drift

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