PROBABILITY THEORY AND MATHEMATICAL STATISTICS 2017
International Kazan conference

Improved nonparametric estimation of the drift in
diffusion processes
1

Evgeny Pchelintsev,

arXiv:1710.03550v1 [] 10 Oct 2017

1

1

Svyatoslav Perelevskiy and

1

Irina Makarova

Tomsk State University, 36 Lenina avenue, Tomsk, 634050, Russian Federation;
evgen-pch@yandex.ru, slavaperelevskiy@mail.ru, star_irish@bk.ru
http://en.tsu.ru/

Abstract. In this paper, we consider the robust adaptive non parametric estimation problem for the drift
coefficient in diffusion processes. An adaptive model selection procedure, based on the improved weighted least
square estimates, is proposed. Sharp oracle inequalities for the robust risk have been obtained.
Keywords: Improved estimation, stochastic diffusion process, mean-square accuracy, model selection, sharp
oracle inequality.

1

Introduction

Let (Ω, F , (F )t≥0, P) be a filtred probability space on which the following stochastic differential
equation is defined:
y. = S(yt ) t. + w.t , 0 ≤ t ≤ T ,
(1)
t

where (wt )t≥0 is ascalar standard Wiener process, the initial value y0 is some given constant, and
S(·) is an unknown function.The problem is to estimate the function S(x), xin[a, b], from the observations (yt )0≤t≤T . The calibration problem for the model (1) is important in various applications.
In particular, it appears, when constructing optimal strategies for investor behavior in diffusion
financial markets. It is known that the optimal strategy depends on unknown market parameters,
in particular, on unknown drift coefficient S. Therefore, in practical financial calculations it is necessary to use statistical estimates for the function S which are reliable on some fixed time interval
[0, T ] [6]. Earlier, the problem of non-asymptotic estimation of the parameters of diffusion processes
was studied in [9]. Here it was shown that many difficulties of asymptotic estimation of parameters
for one-dimensional diffusion processes can be overcome by using a sequantial approach. It turns
out that the theoretical analysis of successive estimates is simpler than the analysis of classical
procedures. In particular, it is possible to calculate non-asymptotic bounds for quadratic risk. Owing to the use of a sequential approach, the problems of non-asymptotic estimation of parameters
were studied in [1] for multidimensional diffusion processes and recently in [2] for multidimensional
continuous and discrete semimartingales. In [7] a truncated sequential method for estimating the
parameters of diffusion processes was developed. Now about nonparametric estimation. A consistent approach to nonparametric criteria for minimax estimation of the drift coefficient in (ergodic)
diffusion processes was developed in [3]. In this article, sequential pointwise kernel estimates are
considered. For such estimates, non-asymptotic upper bounds of the root-mean-square risk are
obtained, and these estimates give the optimal convergence rate as T → ∞.
This paper deals with the estimating the unknown function S(x), a ≤ x ≤ b, in the sense of
the mean square risk
Z b
2
2
b
b
R(ST , S) = ES kST − Sk , kSk =
S 2 (x)x. ,
(2)
a

2

E. Pchelintsev, S. Perelevskiy, I. Makarova

where SbT is the estimate of S by observations (yt )0≤t≤T , a < b are some real numbers. Here ES
is the expectation with respect to the distribution PS of the random process (yt )0≤t≤T given the
drift function S.
The goal of this paper is to construct an adaptive estimate S ∗ of the drift coefficient S in (1)
and to show that the quadratic risk of this estimate is less then the one of the estimate proposed
in [3], i.e. we construct the improved estimate in the mean square acuracy sence. For this we use
the improved estimation approach proposed in [10] and [8] for parametric regression models and
recently developted in [11] for a nonparametric estimation problem. Moreover in this paper we
consider the estimation problem in adaptive setting, i.e. when the regulary of S is unknown. For
this we use a model selection method proposed in [4]. Such approach provides adaptive solution
for the nonparametric estimation through oracle inequalities which give the nonparametric upper
bound for the quadratic risk of estimate.
The rest of the paper is organized as follows. In section 2 we reduce the initial problem to an
estimation problem in a discrete time nonparametric regression model. In section 3 we construct
the improved weigted least square estimates. In section 4 the sharp nonasymptotic oracle inequality
for quadratic risk of model selection procedure is given.

2

Passage to a discrete time regression model

To obtain a reliable estimate of the function S, it is necessary to impose on it certain conditions
that are analogous to the periodicity of the deterministic signal in the white noise model [5]. One
of the conditions sufficient for this purpose is the assumption that the process (yt )t≥0 in (1) returns
to any neighborhood of each points x ∈ [a, b]. As in [3] to get the ergodicity of the process (1) we
define the following functional class:
ΣL,N = {S ∈ LipL (R) : |S(N)| ≤ L ; ∀|x| ≥ N, ∃ Ṡ(x) ∈ C(R)
such that − L ≤ inf Ṡ(x) ≤ sup Ṡ(x) ≤ −1/L} ,
|x|≥N

(3)

|x|≥N

where L > 1, N > |a| + |b|, Ṡ(x)− derivative S(x),
(
LipL (R) =

)
|f (x) − f (y)|
f ∈ C(R) : sup
≤ L .
|x − y|
x,y∈R

We note that if S ∈ ΣL,N , then there exists an invariant density
Rx
exp{2 0 S(z)z.}
.
q(x) = qS (x) = R +∞
Ry
exp{2 0 S(z)z.}y.
−∞

(4)

We note that the functions in ΣL,N are uniformly bounded on [a, b], i.e.
s∗ = sup

sup S 2 (x) < ∞ .

a≤x≤b S∈ΣL,N

We start with the partition of the interva [a, b] by the points (xk )1≤k≤n , defined as
xk = a +

k
(b − a) ,
n

(5)

Improved nonparametric estimation of the drift in diffusion processes

3

where n = n(T ) is an integer-valued function of T such that
n(T ) ≤ T

n(T )
= 1.
T →∞ T

and

lim

(6)

Now at any point xk we estimate the function S by a sequential kernel estimation. We fix some
0 < t0 < T and put

Rt
y −x 
τk = inf{t ≥ t0 : t Q s h k s. ≥ Hk } ;


0

(7)
1 R τk

ys −xk 

 Sek =
Q
y. ,
h
s
Hk t0

where Q(z) = 1{|z|≤1}, 1A is an indicator of the set A, h = (b − a)/(2n) and Hk is a positive
threshold, which will be indicated below. From (1) it is easy to obtain that
Sek = S(xk ) + ζk .

The error ζk is represented as a sum of the approximating and stochastic parts, i.e.

Z τk 
ys − xk
1
1
Q
ξk , Bk =
∆S(ys , xk )s. ,
ζk = Bk + p
Hk t
h
Hk
0

where ∆S(y, x) = S(y) − S(x) and

1
ξk = p
Hk

Z

τk

Q

t0



ys − xk
h



w.s .

Taking into account that S is Lipshitz function , we obtain an upper bound for the approximating
part as
|Bk | ≤ Lh .
It is easy to see that random variables (ξk )1≤k≤n are independent identically distributed from
N (0, 1). In [3] it is established that an effective kernel estimate of the form (7) has a stochastic
part distributed as N (0, 2T hqS (xk )), where qS (xk ) is the ergodic density defined in (4). Therefore,
for an effective estimate at each point xk by the kernel estimate (7), we need to estimate the density
(4) from observations (yt )0≤t≤t0 .To this end, we establish that

where ǫT is positive, 0 < ǫT < 1,

qeT (xk ) = max{b
q(xk ) , ǫT } ,

1
qb(xk ) =
2t0 h

Z

0

t0

Q



ys − xk
h



s. .

Now choose the threshold Hk in (7):

Hk = (T − t0 )(2e
qT (xk ) − ǫ2T )h .
Suppose that the parameters t0 = t0 (T ) and ǫT satisfy the following conditions:

4

E. Pchelintsev, S. Perelevskiy, I. Makarova

H1 ) For any T ≥ 32,

16 ≤ t0 ≤ T /2 and

H2 )

lim t0 (T ) = ∞ ,

T →∞

√

1/8

2/t0

lim ǫT = 0 ,

T →∞

≤ ǫT ≤ 1 .

lim T ǫT /t0 (T ) = ∞ .

T →∞

H3 ) For any ν > 0 and m > 0,
lim T ǫm
= ∞ and
T

T →∞

lim T m e−ν

√

t0

For example, for T ≥ 32,
t0 = max{min{ln4 T , T /2} , 16} and ǫT =
Let

= 0.

T →∞

√

−1/8

2 t0

.

Γ = { max τl ≤ T } and Yk = Sek 1Γ .

(8)

1≤l≤n

Then on the set Γ there exists a temporary heteroscedastic regression model
Yk = S(xk ) + ζk ,
with δk = Bk and
σk2 =

ζk = σk ξk + δk

(9)

n
.
(T − t0 )(e
qT (xk ) − ǫ2T /2)(b − a)

It should be noted that from (6) and H1 ), we get the following upper bound
max σk2 ≤

1≤k≤n

4
= σ∗
(b − a)ǫT

(10)

for which, by condition H3 ),
σ∗
= 0 for any m > 0 .
T →∞ T m
lim

To estimate the S function from the observations of (9) should study some properties of the set Γ
in (8).
Proposition 1. Suppose that the parameters t0 and ǫT satisfy the following conditions: H1 ) –
H3 ). Then
sup PS (Γ c ) ≤ ΠT ,
S∈ΣL,N

where limT →∞ T m ΠT = 0 for any m > 0.

3

Improved estimates

In this section we consider the estimation problem for the model (9). The function S(·) is unknown
and has to be estimated from observations Y1 , . . . , Yn .
The accuracy of any estimator Sb will be measured by the empirical squared error of the form
n

b−aX b
kSb − Sk2n = (Sb − S, Sb − S)n =
(S(xl ) − S(xl ))2 .
n
l=1

Improved nonparametric estimation of the drift in diffusion processes

5

Now we fix a basis (φj )1≤j≤n which is orthonormal for the empirical inner product:
n

b−aX
φi (xl )φj (xl ) = Krij ,
(φi , φj )n =
n
l=1

where Krij is Kronecker’s symbol. By making use of this basis we apply the discrete Fourier
transformation to (9) and we obtain the Fourier coefficients
n

b−aX
θbj,n =
Yl φj (xl ) ,
n

n

θj,n

l=1

b−aX
=
S(xl ) φj (xl ) .
n
l=1

From (9) it follows directly that these Fourier coefficients satisfy the following equation
r
b−a
θbj,n = θj,n + ζj,n with ζj,n =
ξ + δj,n ,
n j,n

where

ξj,n =

r

n

n

l=1

l=1

b−aX
b−aX
σl ξl φj (xl ) and δj,n =
δl φj (xl ) .
n
n

Note that the upper bound (10) and the Bounyakovskii-Cauchy-Schwarz inequality imply that
|δj,n | ≤ kδkn kφj kn = kδkn .
We estimate the function S in (9) on the sieve (5) by the weighted least squares estimator
Sbλ (xl ) =

n
X
j=1

λ(j) θbj,n φj (xl ) 1Γ ,

1 ≤ l ≤ n,

where the weight vector λ = (λ(1), . . . , λ(n)) belongs to some finite set Λ ⊂ [0, 1]n . We set for any
a≤x≤b
n
X
b
b
Sλ (x) = Sλ (x1 )1{a≤x≤x1 } +
Sbλ (xl )1{xl−1 <x≤xl } .
(11)
l=2

Further we suppose that the first d ≤ n components of the weight vector λ are equal to 1, i.e.
λ(j) = 1 for any 1 ≤ j ≤ d.
We consider a new estimate for the function S in (9) of the form
Sλ∗ (xl )

=

n
X

∗
λ(j) θj,n
φj (xl ) 1Γ ,

j=1

where
∗
θj,n
=

c(d)
1−
1{1≤j≤d}
kθe k
n

where

(d − 1)σ∗2 L(b − a)1/2
p
,
c(d) =
n(s∗ + dσ∗ /n)

!

1 ≤ l ≤ n,

θbj,n ,

kθen k2 =

d
X
j=1

2
θbj,n
.

6

E. Pchelintsev, S. Perelevskiy, I. Makarova

Now we define the estimate for S in (1). We set for any a ≤ x ≤ b
Sλ∗ (x)

=

Sλ∗ (x1 )1{a≤x≤x1 }

+

n
X

Sλ∗ (xl )1{xl−1 <x≤xl } .

(12)

l=2

We denote the difference of quadratic risks of the estimates (12) and (11) as
∆n (S) := ES kSλ∗ − Sk2n − ES kSbλ − Sk2n .

The choice of estimate (12) is motivated by the desire to control the quadratic risk.
Theorem 1. The estimate (12) outperforms in mean square accuracy the estimate (11), i.e.
sup ∆n (S) < −c2 (d).

S∈ΣL,N

4

Oracle inequalities

In order to obtain a good estimator, we have to write a rule to choose a weight vector λ ∈ Λ in
(12). It is obvious, that the best way is to minimize the empirical squared error with respect to λ:
Errn (λ) = kSλ∗ − Sk2n → min .
Making use of (12) and the Fourier transformation of S imply
Errn (λ) =

n
X

λ

2

∗2
(j)θj,n

j=1

−2

n
X

∗
λ(j)θj,n
θj,n

+

j=1

n
X

2
θj,n
.

j=1

∗
θj,n by some its estimator
Since the coefficient θj,n is unknown, we need to replace the term θj,n
which we choose as
n
b−a
b−a X 2 2
∗
θej,n = θbj,n θj,n
−
σl φj (xl ) .
sj,n with sj,n =
n
n
l=1

One has to pay a penalty for this substitution in the empirical squared error. Finally, we define
the cost function of the form
n
n
X
X
2
∗2
Jn (λ) =
λ (j)θj,n − 2
λ(j) θej,n + ρPn (λ) ,
j=1

j=1

where the penalty term is defined as

n

Pn (λ) =

b−aX 2
λ (j)sj,n
n
j=1

and 0 < ρ < 1 is some positive constant which will be chosen later. We set
b = argmin
λ
J (λ)
λ∈Λ n

and define an estimator of S of the form (11):

S ∗ (x) = Sλb∗ (x) for a ≤ x ≤ b .

(13)

Now we obtain the non asymptotic upper bound for the quadratical risk of the estimator (13).

Improved nonparametric estimation of the drift in diffusion processes

7

Theorem 2. Let Λ ⊂ [0, 1]n be any finite set such that the first d ≤ n components of the weight
vector λ are equal to 1. Then, for any n ≥ 3 and 0 < ρ < 1/6, the estimator (13) satisfies the
following oracle inequality
ES kS ∗ − Sk2n ≤
where limn→∞ Ψn (ρ)/n = 0.

1 + 6ρ
Ψ (ρ)
min ES kSbλ − Sk2n + n
,
1 − 6ρ λ∈Λ
n

Now we consider the estimation problem (1) via model (9). We apply the estimating procedure
(13) with special weight set introduced in [3] to the regression scheme (9). Denoting Sα∗ = Sλ∗ we
α
set
S ∗ = Sαb∗ with α
b = argminα∈A Jn (λα ) .
ε

We obtain through Theorem 2 the following oracle inequality.

Theorem 3. Assume that S ∈ ΣL,N and the number of the points n = n(T ) in the model (9)
satisfies (6). Then the procedure S ∗ satisfies, for any T ≥ 32, the following inequality
R(S ∗ , S) ≤

B (ρ)
(1 + ρ)2 (1 + 6ρ)
min R(Sα∗ , S) + T
,
1 − 6ρ
n
α∈Aε

where limT →∞ BT (ρ)/n(T ) = 0.

Acknowledgement
The results of Section 3 of this work are supported by the RSF grant number 17-11-01049. The results of section 4 are supported by the Ministry of Education and Science of the Russian Federation
in the framework of the research project no. 2.3208.2017/4.6.

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