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УЧЕНЫЕ ЗАПИСКИ КАЗАНСКОГО УНИВЕРСИТЕТА. СЕРИЯ ФИЗИКО-МАТЕМАТИЧЕСКИЕ НАУКИ
2018, Т. 160, кн. 2 С. 364-372
ISSN 2541-7746 (Print) ISSN 2500-2198 (Online)
UDK 519.2
IMPROVED NONPARAMETRIC ESTIMATION OF THE DRIFT IN DIFFUSION PROCESSES
E.A. Pchelintsev, S.S. Perelevskiy, I.A. Makarova
National Research Tomsk State University, Tomsk, 634050 Russia
Abstract
In this paper, we have considered the robust adaptive nonparametric estimation problem for the drift coefficient in diffusion processes. It has been shown that the initial estimation problem can be reduced to the estimation problem in a discrete time nonparametric heteroscedastic regression model by using the sequential approach. We have developed a new sharp model selection method for estimating the unknown drift function using the improved estimation approach. An adaptive model selection procedure based on the improved weighted least square estimates has been proposed. It has been established that such estimate outperforms in non-asymptotic mean square accuracy the procedure based on the classical weighted least square estimates. Sharp oracle inequalities for the robust risk have been obtained.
Keywords: improved estimation, stochastic diffusion process, mean-square accuracy, model selection, sharp oracle inequality
Introduction
Let (Q, f, (f) t>o, P) be a filtered probability space on which the following stochastic differential equation is defined:
where (wt )t>o is a scalar standard Wiener process, the initial value yo is some given constant, and S(-) is an unknown function.The problem is to estimate the function S(x), x g [a,b], from the observations (yt)o<t<T. The calibration problem for the model (1) is important in various applications. In particular, it appears, when constructing optimal strategies for the 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] [1]. Earlier, the problem of non-asymptotic estimation of the parameters of diffusion processes was studied in [2]. 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 [3] for multidimensional diffusion processes and, recently, in [4] for multidimensional continuous and discrete semimartingales. In [5], a truncated sequential method for estimating the parameters of diffusion processes was developed. Nonparametric estimation has been covered in a number of publications. A consistent approach to nonparametric
dy t = S(y t) dt + dw t, 0 < t < T,
(1)
criteria for minimax estimation of the drift coefficient in (ergodic) diffusion processes was developed in [6]. In this paper, 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 ^ œ.
The present paper deals with estimation of the unknown function S(x), a < x < b, in the sense of the mean square risk
b
R(ST ,S) = Eg ISt - S||2, \\S\\2 =f S2(x)dx, (2)
where ST is the estimate of S by observations (y t) 0<t<T , a < b are some real numbers. Here E S is the expectation with respect to the distribution P g of the random process (y t )o<t<T given the drift function S.
The purpose 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 than the one of the estimate proposed in [6], i.e., we construct the improved estimate in the mean square accuracy sense. In order to fulfill this purpose, we use the improved estimation approach proposed in [7] and [8] for parametric regression models and recently developed in [9] for a nonparametric estimation problem. Moreover, we consider the estimation problem in adaptive setting, i.e., when the regularity of S is unknown, by using a model selection method proposed in [10]. This approach provides an adaptive solution for the nonparametric estimation through oracle inequalities, which give the nonparametric upper bound for the quadratic risk of estimate.
1. 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 [11]. One of the conditions sufficient for this purpose is the assumption that the process (y t) t>o in (1) returns to any neighborhood of each points x g [a, b]. As in [6], in order to get the ergodicity of the process (1), we define the following functional class:
£l,n = {S g LipL(R) : \S(N)| < L; v\x\ > N, 3 S(x) g C(R)
such that - L < inf S(x) < sup S(x) < -1/L}, (3)
\x\>N \x\>N
where L > 1, N > \a\ + \b\, S(x) is the derivative S(x),
Lipl (R) = (f g C(R) : sup \f ^x - y(y)\ < 4 •
[ x , y£R \x — y\ J
We note that if S g £ l n , then there exists an invariant density
{x 1 ( y 1
2 J S (z)dz\ J exp J 2 J S(z) dz > dy^ (4)
We note that the functions in £ l n are uniformly bounded on [a, b], i.e.
s* = sup sup S2(x) <
a<x<b Se'S L N
We start with the partition of the interval [a, b] by the points (x k) k k<n , defined as
k
x k = a +--(b — a), (5)
n
where n = n(T) is an integer-valued function of T, such that
n(T) < T and lim n(T) = 1. (6)
t^W T
Now, at any point x k , we estimate the function S by sequential kernel estimation. We fix some 0 <t 0 < T and put
t
Tk =inf 11 > to :J ds > Hkj;
t (7)
•"O
Tk
S=htJ d%
tO
where Q(z) = 1 {\z\<iy , 1 a 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
Sk = S(x k ) + Z k ■
The error Zk is represented as a sum of the approximating and stochastic parts, i.e.,
T k / \
Zk = Bk +-H= tk, Bk = Hrk j AS(ys ,x k )ds,
where AS(y, x) = S(y) — S(x) and
T k
1 ) dWs.
vH
tO
Taking into account that S is the Lipshitz function, we obtain an upper bound for the approximating part as
\Bk \< Lh.
It is easy to see that random variables (tk) Kk<n are independent identically distributed from n(0,1). In [6], it is established that an effective kernel estimate of the form (7) has a stochastic part distributed as n(0, 2ThqS (x k)), where qS (x k) is the ergodic density defined in (4). Therefore, for an effective estimate at each point x k by the kernel estimate (7), we need to estimate the density (4) from observations (vt) 0<t<t 0 .To this end, we establish that
q_T(x k ) = max{q(x k ) , e t k where et is positive, 0 < et < 1,
t 0
q(xk) = ^ [q("y^)ds.
2t 0 h
t
k
Now, we choose the threshold Hk in (7):
Hk = (T — t o) (2qT(x k ) — c^t ) h. Let us suppose that the parameters t o = t o (T) and ct satisfy the following conditions: H!) For any T > 32,
16 < t o < T/2 and V2/tl/8 < ct < 1.
H 2 )
lim t o (T) = to, lim ct = 0, lim Tct /t o (T) = to.
T^w T^w T^w
H 3) For any v > 0 and m > 0,
lim Tcm = to and lim Tm = 0.
T^w 1 T^w
For example, for T > 32 ,
to = max{min{ln4 T, T/2} , 16} and ct = V21-1/8.
Let
r = { max tl < T} and Yk = Sk 1 r . (8)
1<l<n
Then, there exists a temporary heteroscedastic regression model on the set r
Yk = S(x k) + ck, zk = *k ek + sk (9)
with S k = B k and
a! =
k (T - t o )(qr (x k ) - 4/2)(b - a)"
^ 0^ W'T Kx k) — e TI
It should be noted that from (6) and H x ), we get the following upper bound
ima< —k < (b 4a)e =—* (10) i<k<n (b — a)e t
for which, by condition H 3 ),
lim —— = 0 for any m > 0 .
t^^ Tm
To estimate the function S from the observations of (9), we should study some properties of the set r in (8).
Proposition 1. Let us suppose that the parameters 10 and et satisfy the following conditions: H ) - H 3 ). Then
sup P S (rc) < nT,
L,N
where lim Tm n T = 0 for any m > 0.
TT
n
2. 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 Yi,... ,Yn.
The accuracy of any estimator S will be measured by the empirical squared error of the form
b _ n
- s\\n = (S - s,s - s ) n = (S(x i ) - S(x i ))2.
i=i
Now, we fix a basis (&j ) i<j<n , which is orthonormal for the empirical inner product:
b — a n
& j) n = -^Yl i)& j(x i) = Kr ij ' i=i
where Kr j is Kronecker's symbol. By making use of this basis, we apply the discrete Fourier transformation to (9) and obtain the Fourier coefficients and their least square estimates
b n b n
0 j,n = (x i) & j(x i)' ôj,n = & j(x i).
i=1 i=1
From (9), it follows directly that these Fourier coefficients satisfy the following equation
9j,n = 0 j,n + Cj,n with Cj,n = \/ j,n + S j,n ,
b
where
t j,n =J l11 4 j (x l ) and 5i,n = ^-^Y,61 4 j (x l )■
l=1 l=1
Note that the upper bound (10) and the Bounyakovskii-Cauchy-Schwarz inequality imply that
\6j,n I < M6Mn \\4j M n = 6 n ■
We estimate the function S in (9) on the sieve (5) by the weighted least squares estimator
n
S\ (x i) = y1 A(j) ®j,n 4j (x i) 1 r > 1 < l < n
j=1
where the weight vector A = (A(1),.. ■, A(n)) belongs to some finite set A C [0,1]n. We set for any < x < b
n
S\ (x) = S\ (x 1 )1 {a<x<x , } +Y1 (x l )1 {x _ <x<x l } ■ (11)
l = 2
Hereafter, we suppose that the first d < n components of the weight vector A are equal to 1, i.e., A(j) = 1 for any 1 < j < d.
We consider a new estimate for the function S in (9) of the form
n
S*\ (x l) = J2 A(j) °*j,n 4 j (x l) 1 r , 1 < l < n, j=1
where
- = (' — K\\1 {1<3<d}) 3
where
T2r" M/2 d
(d) = (d - ^^1/2, k f = £
n(s* + y da*/n)
Now, we define the estimate for S in (1). We set for any a < x < b
n
S\ (x) = S A (x 1 )X {a<x<x , } + £ S*X (x l {x _ <x<x t } . (12)
l = 2
We denote the difference of quadratic risks of the estimates (12) and (11) as
An(S) := Eg \iS*a — S||2n — EgS — S||2n. The choice of estimate (12) is motivated by the desire to control the quadratic risk.
Theorem 1. The estimate (12) outperforms in the mean square accuracy the estimate (11), i.e.,
sup An(S) < —c2(d).
se'Eln
3. Oracle inequalities
In order to obtain a good estimator, we have to write a rule to choose a weight vector A G A in (12). It is obvious that the best way is to minimize the empirical squared error with respect to A :
Errn (A) = \\S*a — S||2n ^ min . Making use of (12) and the Fourier transformation of S implies
n n n
Err n (A) = £ ^ )j — 2 £ A(j j 9 3,n + £ jn .
3=1 3=1 3=1
Since the coefficient 93 n is unknown, we need to replace the term 9* 93 n by some
3 n ' i j n 3 , n -J
estimator, which we choose as
~ b — a b — a 22
93 , n = 93 , n 9 3 , n ——s 3, n with s 3 ,n = — l 4 3 (x l ).
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
Jn (A) = £ A2(j)9*ln — 2 £ A(j)~3n + pPn (A),
3=1 3=1
where the penalty term is defined as
Pn (A) = b-^£A2(j> j n
n
j=1
and 0 < p < 1 is some positive constant which will be chosen later. We set
A = argmin Jn(A)
ma
and define an estimator of S of the form (11):
S*(x) = S*~ (x) for a < x < b. (13)
Now, we obtain the non asymptotic upper bound for the quadratical risk of the estimator (13).
Theorem 2. Let A c [0, 1]n be any finite set such that the first -d < n components of the weight vector A are equal to 1. Then, for any n > 3 and 0 < p < 1/6, the estimator (13) satisfies the following oracle inequality
^ - .,,2 < 1±«£ mn E \\A s\\2 , * n p s" 11 n " 1 — 6p \ea
es \\&> — s\\n < t^t, min es \\six — s\\n +
where lim * n (p)/n = 0.
Now, we consider the estimation problem (1) via model (9). We apply the estimating procedure (13) with special weight set introduced in [6] to the regression scheme (9). Denoting S*a = S* we set
S* = S1 with a = argmin Jn (Aa )■
a aiAe
We obtain through theorem 2 the following oracle inequality.
Theorem 3. Let us assume that S G £L 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) < (1 + ,)2(1 + 6P) mm U(S* , S) + bT (,)
1 — 6p aeA e a n
where lim BT (p)/n(T) = 0^
T—>oo
Acknowledgements. This work was supported by the Russian Science Foundation
(results of Section 2, project no. 17-11-01049) and by the Ministry of Education and
Science of the Russian Federation (results of Section 3, project no. 2.3208.2017/4.6).
References
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6. Galtchouk L.I., Pergamenshchikov S.M. Asymptotically efficient sequential kernel estimates of the drift coefficient in ergodic diffusion processes. Stat. Inference Stoch. Processes, 2006, vol. 9, no. 1, pp. 1-16. doi: 10.1007/s11203-005-3248-4.
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Recieved October 24, 2017
Pchelintsev Evgeniy Anatolevich, Candidate of Physics and Mathematics, Associate Professor of the Department of Mathematical Analysis and Theory of Functions National Research Tomsk State University
pr. Lenin, 36, Tomsk, 634050 Russia E-mail: evgen-pch@yandex.ru
Perelevsky Svyatoslav Sergeevich, PhD Student of the Department of Mathematical Analysis and Theory of Functions
National Research Tomsk State University
pr. Lenin, 36, Tomsk, 634050 Russia E-mail: slavaperelevskiy@mail.ru
Makarova Irina Alekseevna, Student of the Department of Mathematical Analysis and Theory of Functions
National Research Tomsk State University
pr. Lenin, 36, Tomsk, 634050 Russia E-mail: starirish@bk.ru
УДК 519.2
Об улучшенном оценивании функции сноса в диффузионных процессах
Е.А. Пчелинцев, С.С. Перелевский, И.А. Макарова
Национальным исследовательский Томский государственным университет, г. Томск, 634050, Россия
Аннотация
В работе рассмотрена задача робастного адаптивного непараметрического оценивания коэффициента сноса в диффузионных процессах. На основе последовательного подхода показано, что исходную задачу оценивания можно свести к задаче оценивания функции в дискретной непараметрической гетероскедастичной регрессионной модели. Предложена адаптивная процедура выбора модели на основе улучшенных взвешенных оценок по методу наименьших квадратов (МНК). Установлено, что такая оценка имеет более высокую неасимптотическую среднеквадратическую точность, чем процедура, построенная на основе классических взвешенных оценках МНК. Получено точное оракульное неравенство для квадратического риска предложенной процедуры оценивания, которое дает неасимптотическую верхнюю границу для риска.
Ключевые слова: улучшенное оценивание, стохастический диффузионный процесс, среднеквадратическая точность, выбор модели, оракульное неравенство
Поступила в редакцию 24.10.17
Пчелинцев Евгений Анатольевич, кандидат физико-математических наук, доцент кафедры математического анализа и теории функций
Национальный исследовательский Томский государственный университет
пр. Ленина, д. 36, г. Томск, 634050, Россия E-mail: evgen-pch@yandex.ru
Перелевский Святослав Сергеевич, аспирант кафедры математического анализа и теории функций
Национальный исследовательский Томский государственный университет
пр. Ленина, д. 36, г. Томск, 634050, Россия E-mail: slavaperelevskiy@mail.ru
Макарова Ирина Алексеевна, студент кафедры математического анализа и теории функций
Национальный исследовательский Томский государственный университет
пр. Ленина, д. 36, г. Томск, 634050, Россия E-mail: starirish@bk.ru
I For citation: Pchelintsev E.A., Perelevskiy S.S., Makarova I.A. Improved nonparametric ( estimation of the drift in diffusion processes. Uchenye Zapiski Kazanskogo Universiteta. \ Seriya Fiziko-Matematicheskie Nauki, 2018, vol. 160, no. 2, pp. 364-372.
/ Для цитирования: Pchelintsev E.A., Perelevskiy S.S., Makarova I.A. Improved ( nonparametric estimation of the drift in diffusion processes // Учен. зап. Казан. ун-та. \ Сер. Физ.-матем. науки. - 2018. - Т. 160, кн. 2. - С. 364-372.
