Estimation of parameters of SDE driven by fractional Brownian motion with polynomial drift

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by  Kestutis Kubilius

Estimation of parameters of SDE driven by fractional
Brownian motion with polynomial drift
5 K. Kubilius *1, V. Skorniakov 2, D. Melichov 3
1
0
1 Vilnius University, Institute of Mathematics and Informatics, Akademijos 4,
2
LT-08663 Vilnius, Lithuania
y 2 Vilnius University, Faculty of Mathematics and Informatics, Naugarduko 24,
a
M LT-03225, Vilnius, Lithuania
3 Vilnius Gediminas Technical University, Saule˙tekio al. 11,

6 LT-10223, Vilnius, Lithuania
1
Abstract

] StronglyconsistentandasymptoticallynormalestimatorsoftheHurstindexandvolatility
R
parametersofsolutionsofstochasticdifferentialequationswithpolynomialdriftareproposed.
P
The estimators are based on discrete observations of the underlying processes.
.
h
t Keywords: fractionalBrownianmotion,Hurstindex,volatility,Black-Scholesmodel,Verhulst
a
equation, Landau-Ginzburg equation, consistent estimator
m
[
1 Introduction

2
v
Manyapplicationsuseprocessesthataredescribedbystochasticdifferentialequations(SDE).
0
Recently, much attention has been paid to SDEs driven by the fractional Brownian motion
5
(fBm) and to the problems of statistical estimation of model parameters. Statistical aspects
8
6 of the models including fBm were studied in many articles. Especially much attention was
0 paid to the estimations of the parameter of drift. We concentrate on estimation of the Hurst
. parameter and volatility of SDE with polynomial drift
1
0 (cid:90) t (cid:90) t
5 Xt =ξ+ (cid:0)aXsm+bXs(cid:1)ds+c XsdBsH, ξ(cid:62)0, t∈[0,T], (1)
1 0 0
: where1/2<H <1,m∈N,m(cid:62)2,andBH isafBmwithHurstindex1/2<H <1. Almost
v
i all sample paths of BH have bounded p-variation for each p>1/H on [0,T] for every T >0
X
(a more detailed definition of an fBm will be formulated in the next section). The second
r integral in (1) is the pathwise Riemann-Stieltjes integral with respect to the process having
a
finite p-variation.
It is well known that equation (1) for standard Brownian motion, i. e. for H = 1/2, is
nonlinear reducible SDE with polynomial drift of degree m. Under the substitution h(x) =
x1−m equation (1) reduces to a linear SDE with multiplicative noise. Some famous equations
can be obtained as particular cases of this equation. For example, we can obtain the Black-
Scholes model, the Verhulst and the Landau-Ginzburg equations.
Using an approximation approach in L2 Dung [7] obtained an explicit solution of the
equation (1) for H >1/2. His approximation is based on the fact that fBm of Liouville form
can be uniformly approximated by semimartingales.
Many authors (see [8], [9], [3], [5], [1], [2], [16], [14], [13]) considered an almost sure con-
vergence and an asymptotic normality of the generalized quadratic variations associated to
thefiltera(see[9])ofawideclassofprocesseswithGaussianincrements. Conditionsforthis
type of convergence are expressed in terms of covariances of a Gaussian process and depend
on some parameter γ ∈ (0,2). These results make it possible to obtain an estimator of the
parameter γ and to study its asymptotic properties. If the process under consideration is the
fBm then this parameter is the Hurst index 0<H <1.
∗Thisresearchwasfundedbyagrant(No. MIP-048/2014)fromtheResearchCouncilofLithuania.
1
There are a few results concerning estimation of the Hurst index if the observed process
is described by SDE driven by fBm. For the first time, to our knowledge, an estimator of the
HurstparameterofapathwisesolutionofalinearSDEdrivenbyafBmanditsasymptotical
behaviour were considered in [4]. A more general situation was considered in [10] using a
different approach. Estimators for the Hurst parameter were constructed making use of the
first and second order quadratic variations of the observed values of the solution. Only the
strong consistency of these estimators was proved. A comparison of different estimators of
the Hurst index for linear SDE driven by fBm was carried out in [11]. A rate of convergence
of these estimators with probability 1 to the true value of a parameter was obtained in [12],
when the maximum length of the interval partition tends to zero.
In this paper we will use a different approach to solving the equation (1) than suggested
by N. T. Dung. We apply a pathwise approach to stochastic integral equations and use a
chain rule for the composition of a smooth function and a function of bounded p-variation
with 1 < p < 2. This approach allows us to simplify the proof of existence and uniqueness
of the solution of the equation (1). The goal of the paper is to construct strongly consistent
and asymptotically normal estimators of the Hurst parameter and the volatility of SDE (1)
using discrete observations of the sample paths of the solution of the SDE. Estimators are
constructedusingquadraticvariationsofsecondorderincrements. Estimators’propertiesare
determined asymptotically from the properties of quadratic variations.
The paper is organized in the following way. In Section 2 we present the main results of
the paper. Section 3 is devoted to several known results needed for the proofs. In Section 4
we prove the uniqueness of SDE (1) and give its explicit solution. Sections 5.1–5.3 contain
the proofs of the main results. Finally, in Section 6 some simulations are given in order to
illustrate the obtained results.
2 Main results
Let X be a stochastic process satisfying (1). Denote
∆(2)X =X −2X +X , ∆ X =X −X ,
n,k n,k+1 n,k n,k−1 n,k+1 n,k+1 n,k
whereX =X(tn),tn = kT. Forthesakeofsimplicity,weshallomittheindexnforpoints
n,k k k n
tn and for notions X , ∆(2), ∆ , where the sample size is equal to n. In case of different
k n,k n,k n,k
sample sizes the indices will be retained.
IntheformulationofTheorem2aswellasinthesequelwemakeuseofthesymbolsO ,o .
r r
Here is a brief explanation. Let (a ) be a sequence of real numbers. Y = O (a ), a ↓ 0,
n n r n n
means that there exists a.s. finite r.v. ς with the property |Y | (cid:54) ς·a ; Y = o (a ) means
n n n r n
that |Y | (cid:54) ς·b with b = o(a ). Particulary Y = o (1) denotes the sequence (Y ) which
n n n n n r n
tends to 0 a.s. as n→∞.
Theorem 1 Assume that X is a solution of SDE (1) with a (cid:54) 0. Let X > 0, c (cid:54)= 0, and
0
1/2<H <1. Then
H(cid:98)n(1) −→H a.s.,
√ n
2 n lnT (H(cid:98)n(1)−H)−−d→N(0,σ2) (2)
with a known variance σ2 =σ2(H) defined in Subsection 3.2, where for n>T
H(cid:98)n(1) =ϕ−n,1T(cid:32)n1 n(cid:88)−1(cid:32)∆c(kX2)X(cid:33)2(cid:33),
k
k=1
(cid:16)T(cid:17)2x
ϕ (x)= (4−22x) and ϕ−1 is the inverse function of ϕ , x∈(0,1).
n,T n n,T n,T
Theorem1givesanestimatorofH undertheassumptionthatcisknown. Thisisnotalways
the case. Therefore we present another estimator which is suitable when c is unknown.
2
Theorem 2 Let assumptions of Theorem 1 hold and
(cid:80)2n−1(cid:18)∆(22n),kX(cid:19)2
H(cid:98)n(2) = 12 − 2l1n2ln (cid:80)kn=−11(cid:18)∆X(n22,)knX,k(cid:19)2 .
k=1 Xn,k
Then
H(cid:98)n(2) −→H a.s.,
√
2ln2 n(H(cid:98)n(2)−H)−−d→N(0,σ∗2) (3)
with a known variance σ2 =σ2(H) defined in Subsection 3.2.
∗ ∗
To estimate c one can use the last theorem provided below.
Theorem 3 Assume that 1/2<H <1 and H(cid:98)n(3) =H+or(φ(n)) a.s. If φ(n)=o(cid:0)ln1n(cid:1) then
c2 = n2H(cid:98)n(3)−1 n(cid:88)−1(cid:32)∆(k2)X(cid:33)2 −→c2 a.s.;
(cid:98)n T2H(cid:98)n(3)(4−22H(cid:98)n(3)) k=1 Xk
(cid:16) (cid:17) √
if φ(n)=o √n1lnn then n((cid:98)cn2−c2)−−d→N(0,c4σ2).
Hence estimation of c and construction of confidence intervals is always possible. However
confidence intervals will shrink at the rate which is a bit worse than the usual n−1/2.
3 Preliminaries
3.1 Variation
In this subsection we remind several facts about p–variation. For details we refer the reader
to [6]. Recall that a function f :[a,b]→R has p–variation defined by
n
v (f;[a,b])=sup(cid:88)|f(x )−f(x )|p,
p k k−1
κ
k=1
where p∈(0,∞) is a fixed number and supremum is taken over all possible partitions
κ:a=x <x <···<x =b, n(cid:62)1.
0 1 n
If v (f;[a,b])<∞, f is said to have a bounded p-variation on [a,b]. Denote by W ([a,b]) the
p p
class of functions defined on the [a,b] with bounded p–variation. Let CW ([a,b]) denote the
p
class of functions that have bounded p–variation and are continuous. Define V (f;[a,b]) =
p
v1/p(f;[a,b]), which is a seminorm on W ([a,b]) provided p (cid:62) 1 and V (f;[a,b]) is 0 if and
p p p
only if f is a constant.
Iff ∈W ([a,b]),thenfisboundedandf ∈W ([a,b])foreveryp >p(cid:62)1. Let0<p<∞
p p1 1
and f,g∈W ([a,b]). Then fg∈W ([a,b]).
p p
Let f ∈ W ([a,b]) and h ∈ W ([a,b]) with 0 < p < ∞, q > 0, 1/p+1/q > 1. Then
q p
the integral (cid:82)bfdh exists as the Riemann–Stieltjes integral if f and h have no common
a
discontinuities. If the integral exists, the Love–Young inequality
(cid:12) b (cid:12)
(cid:12)(cid:90) (cid:12)
(cid:12) fdh−f(y)(cid:2)h(b)−h(a)(cid:3)(cid:12)(cid:54)C V (cid:0)f;[a,b](cid:1)V (cid:0)h;[a,b](cid:1) (4)
(cid:12) (cid:12) p,q q p
(cid:12) (cid:12)
a
holds for all y∈[a,b], where C =ζ(p−1+q−1) and ζ(s)=(cid:80) n−s.
p,q n(cid:62)1
Let f ∈W ([a,b]) and h∈W ([a,b]). From (4) it follows that
q p
(cid:32) · (cid:33)
(cid:90)
(cid:0) (cid:1) (cid:0) (cid:1)
V fdh;[a,b] ≤C V f;[a,b] V h;[a,b] ,
p p,q q,∞ p
a
3
where V (f;[a,b]) = V (f;[a,b])+sup |f(x)|. Note that V (f;[a,b]) is a norm on
q,∞ q a(cid:54)x(cid:54)b q,∞
W ([a,b]), q(cid:62)1.
q
If h∈CW ([a,b]), then the indefinite integral (cid:82)yfdh, y∈[a,b], is a continuous function.
p a
Toverifythatsomeprocesssolvestheequation(1)weuseachainruleforthecomposition
of a smooth function and a function of bounded p-variation with 0 < p < 2. The chain rule
is based on the Riemann–Stieltjes integrals. As usual, the composition g◦h on [a,b] of two
functions g and h is defined by (g◦h)(x)=g(h(x)) whenever h is defined on [a,b] and g on
the range of h.
Theorem 4 (Chain rule (see [6])) Let f = (f ,...,f ) : [a,b] → Rd be a function such
1 d
that,foreachk=1,...,d,f ∈CW ([a,b])with1(cid:54)p<2. Letg:Rd →Rbeadifferentiable
k p
functionwithlocallyLipschitzpartialderivativesg(cid:48),k=1,...,d. Theneachg(cid:48)◦f isRiemann-
k l
Stieltjes integrable with respect to f and
k
(g◦f)(b)−(g◦f)(a)=(cid:88)d (cid:90) b(g(cid:48) ◦f)df .
k k
k=1 a
Proposition 5 (Substitution rule (see [6])) Let f, g, and h be functions in CW ([a,b]),
p
1(cid:54)p<2. Then
(cid:90) b (cid:18)(cid:90) x (cid:19) (cid:90) b
f(x)d g(y)dh(y) = f(x)g(x)dh(x).
a a a
3.2 Several results on fBm
Recall that fBm BH ={BH: t(cid:62)0} with Hurst index H ∈(0,1) is a real-valued continuous
t
centered Gaussian process with covariance given by
E(BHBH)= 1(cid:0)s2H+t2H−|t−s|2H(cid:1).
t s 2
For consideration of strong consistency and asymptotic normality of the given estimators we
need several facts about BH.
Limit results. Let
V = n2H−1 n(cid:88)−1(cid:0)T−H∆(2)BH(cid:1)2, H (cid:54)= 1.
n,T 4−22H n,k 2
k=1
Then (see [3], [9], [2])
V −−−−→1 a.s.;
n,T
n→∞
√n(cid:18) Vn,T −1 (cid:19)−−d→N(cid:18)(cid:18) 0 (cid:19),(cid:18) σ2(H) σ1,2(H) (cid:19)(cid:19)
V −1 0 σ (H) σ2(H)/2
2n,T 1,2
with
∞
σ2(H)=2+c (H)+c (H)(cid:88)(cid:0)ρ (l)(cid:1)2, σ (H)=2−2H(cid:0)3σ2+σ2+4σ2(cid:1),
2 1 2−2H 1,2 1 2
l=2
where
(cid:18)2H(2H−1)(2H−2)(2H−3)(cid:19)2 (cid:18)22H+2−7−32H(cid:19)2
c (H)= , c (H)= ,
1 4−22H 2 4−22H
|l−2|2H−4|l−1|2H+6|l|2H−4|l+1|2H+|l+2|2H
ρ (l)= , l∈Z,
2−2H 2H(1−2H)(2−2H)(3−2H)
σ2 =c2(H) +c (H)(cid:88)∞ ρ (l)ρ (l−2),
1 2 1 2−2H 2−2H
l=2
∞
σ2 =2(cid:112)c (H)+c (H)(cid:88)ρ (l)ρ (l−1).
2 2 1 2−2H 2−2H
l=2
4
These relationships imply that
(cid:18) (cid:19)
1 1 V
H(cid:101)n = 2 − 2ln2ln 22H−2n1,VT →H a.s.,
n,T
√ 3
2ln2 n(H(cid:101)n−H)−−d→N(0;σ∗2(H)), σ∗2(H)= 2σ2(H)−2σ1,2(H).
Variation of BH. ItisknownthatalmostallsamplepathsofanfBmBH arelocallyHo¨lder
of order strictly less than H, 0 < H < 1. To be more precise, for all 0 < ε < H and T > 0,
there exists a nonnegative random variable G such that E(|G |p)<∞ for all p(cid:62)1, and
ε,T ε,T
|BH−BH|(cid:54)G |t−s|H−ε a.s. (5)
t s ε,T
for all s,t∈[0,T]. Thus BH ∈CW ([0,T]), H = 1 .
Hε ε H−ε
SDEdrivenbyfBmBH. Asolutionofthestochasticintegralequation(1),onagivenfiltered
probability space (Ω, F, P, F) and with respect to the fixed fBm (BH, F), 1/2<H <1, and
initialconditionξ,isanadaptedtothefiltrationFcontinuousprocessX ={X : 0≤t≤T}
t
such that X0 =ξ a.s., P((cid:82)0t|f(Xs)|ds+(cid:12)(cid:12)(cid:82)0tg(Xs)dBsH(cid:12)(cid:12)<∞)=1 for every 0≤t≤T, and
its almost all sample paths satisfy (1).
4 Solution of SDE with polynomial drift
First, we consider a non-random integral equation
(cid:90) t (cid:90) t
x =x + (axm+bx )ds+c x dh , x (cid:62)0, a(cid:54)0, t∈[0,T], (6)
t 0 s s s s 0
0 0
where m∈N, m(cid:62)2, h∈CW ([0,T]), 1<p<2.
p
4.1 Uniqueness of the solution of equation (6)
Theorem 6 Theintegralequation(6)hasauniquesolutionintheclassoffunctionsCW ([0,T]),
p
1<p<2.
Proof. It is well-known that
m−1
am−bm =(a−b)(cid:88)akbm−1−k.
k=0
Letusassumethatwehavetwosolutionsx andy ofequation(6). Sincethesolutionsbelong
t t
to the class of functions CW ([0,T]), 1<p<2, then
p
|xm−ym|(cid:54)mL |x −y |,
t t x,y,m,T t t
where
(cid:16) (cid:110) (cid:111)(cid:17)m−1
L = max max |x |, max |y | .
x,y,m,T t t
0(cid:54)t(cid:54)T 0(cid:54)t(cid:54)T
Further, one can find a subdivision τ ={0<τ <τ <···<τ =T} such that
1 2 n
V (h;[τ ,τ ])(cid:54)(cid:0)4|c|C (cid:1)−1
p k−1 k p,p
for all k. Assume we have proved that x =y . Then
τk−1 τk−1
V (x−y;[τ ,τ ])=V (x−y−(x −y );[τ ,τ ])
p,∞ k−1 k p,∞ τk−1 τk−1 k−1 k
(cid:90) τk (cid:90) τk
(cid:54)2|a| |xm−ym|dt+2|b| |x −y |dt
t t t t
τk−1 τk−1
+2|c|C V (x−y;[τ ,τ ])V (h;[τ ,τ ])
p,p p,∞ k−1 k p k−1 k
(cid:54)(cid:0)2L |a|+2|b|(cid:1)(cid:90) τk |x −y |dt
x,y,m,T t t
τk−1
+2|c|C V (x−y;[τ ,τ ])V (h;[τ ,τ ])
p,p p,∞ k−1 k p k−1 k
5
and
V (x−y;[τ ,τ ])(cid:54)4(cid:0)L |a|+|b|(cid:1)(cid:90) τk |x −y |dt
p,∞ k−1 k x,y,m,T t t
τk−1
(cid:54)4(cid:0)L |a|+|b|(cid:1)(cid:90) τk V (x−y;[τ ,t])dt.
x,y,m,T p,∞ k−1
τk−1
ByGronwall’sinequalitywethenhaveV (x−y;[τ ,τ ])=0. Thusx =y on[τ ,τ ].
p,∞ k−1 k t t k−1 k
Sincex =x =y theclaimofthetheoremfollowsfromrepetitiveapplicationoftheabove
τ0 0 τ0
reasoning.
4.2 Explicit solution of equation (6)
Theorem 7 Function
(cid:18) (cid:90) t (cid:19)1/(1−m)
x =ebt+cht x1−m+(1−m)a e(m−1)(bs+chs)ds , t∈[0,T], (7)
t 0
0
is an element of the class of functions CW ([0,T]), 1<p<2, and satisfies equation (6).
p
Proof. We firstshow thatx∈CW ([0,T]), 1<p<2. Letus denote z =exp{bt+ch }and
p t t
z = z , where {0 = t < t < ··· < t = T} is a subdivision of the interval [0,T]. By the
k tk 0 1 n
mean value theorem for the function ex, x∈R, we obtain
n−1 n−1
(cid:88)|zk+1−zk|p (cid:54)exp(cid:8)p(cid:2)|b|T +2|c|Vp(cid:0)h;[0,T](cid:1)(cid:3)(cid:9)(cid:88)(cid:12)(cid:12)b∆k+1+c∆hk+1(cid:12)(cid:12)p
k=0 k=0
(cid:54)2exp(cid:8)p(cid:2)|b|T +2|c|V (cid:0)h;[0,T](cid:1)(cid:3)(cid:9)(cid:2)|b|pTp+|c|pv (cid:0)h;[0,T](cid:1)(cid:3),
p p
where ∆ =t −t and ∆h =h −h . Let
k+1 k+1 k k+1 tk+1 tk
(cid:90) t
f =x1−m+(1−m)a e(m−1)(bs+chs)ds.
t 0
0
Repeatedly using the mean value theorem for the function u (cid:55)→ u1/(1−m), u > 0, m (cid:62) 2, we
obtain
n−1 n−1
(cid:88)|f1/(1−m)−f1/(1−m)|(cid:54)xm(cid:88)|f −f |(cid:54)xmv (cid:0)f;[0,T](cid:1).
k+1 k 0 k+1 k 0 1
k=0 k=0
Since z∈CW ([0,T]) and f ∈CW ([0,T]), we have x∈CW ([0,T]).
p 1 p
Now we verify that function (7) satisfies the equation (6) by using the chain rule. Indeed,
let F(t,u,v)=ebt+cuv1/(1−m). Then by Theorem 4 and Proposition 5 we get
(cid:90) t (cid:90) t
x =F(t,h ,f )=F(0,h ,f )+b x ds+c x dh
t t t 0 0 s s s
0 0
1 (cid:90) t
+ ebs+chsf1/(1−m)−1df
1−m s s
0
(cid:90) t (cid:90) t 1 (cid:90) t
=x +b x ds+c x dh + ebs+chsfm/(1−m)df
0 s s s 1−m s s
0 0 0
(cid:90) t (cid:90) t 1 (cid:90) t
=x +b x ds+c x dh + (1−m)aem(bs+chs)fm/(1−m)ds
0 s s s 1−m s
0 0 0
(cid:90) t (cid:90) t (cid:90) t
=x +b x ds+c x dh +a xmds.
0 s s s s
0 0 0
4.3 Solution of SDE
Since almost all sample paths of BH, 1/2<H < 1, are continuous and have bounded H =
ε
1 -variation,1/2<H−ε<1,thepathwiseRiemann-Stieltjesintegral(cid:82)tX dBH existsfor
H−ε 0 s s
X ∈CW ([0,T]). So SDE (1) makes sense for almost all ω. Thus the obtained result for a
Hε
non-random integral equation can be applied to an equation driven by a fBm.
6
Theorem 8 Suppose that X >0 and m(cid:62)2. The stochastic process
0
(cid:18) (cid:90) t (cid:19)1/(1−m)
Xt =ebt+cBtH X01−m+(1−m)a e(m−1)(bs+cBsH)ds , t∈[0,T], (8)
0
for almost all ω belongs to CW ([0,T]) and is the unique solution of (1).
Hε
5 Proof of the main Theorems
5.1 Proof of Theorem 1
Beforepresentingtheproofofthistheorem,wegiveanauxiliarylemma. Toavoidcumbersome
expressionsweextendnotionsofo ,O tocontinuoustimeprocessesasfollows: Y =O (q(t))
r r t r
meansthatthereexistssomealmostsurelyfinitenon-negativer.v. ζ andq(t)−−−−→0+,with
t→0+
the property |Y |(cid:54)ζq(t), t→0, whereas Y =o (q(t)) means that |Y |(cid:54)ζw(t), t→0, with
t t r t
w(t)=o(q(t)), t→0.
For t∈[0,T) and h(cid:62)0 we also denote
∆Y =Y −Y , ∆(2)Y =∆Y −∆Y .
t,t+h t+h t t,t+h t+h,t+2h t,t+h
Lemma 9 Assume that H ∈ (1/2,1),a (cid:54) 0,m (cid:62) 2 and X is the solution of equation (1).
Then for every ε>0 such that H >1/2+ε
∆X =X (cid:0)bh+c∆BH +O (h)(cid:1), h→0+, t∈[0,T),
t,t+h t t,t+h r
and
∆(2)X =X (cid:0)c∆(2)BH +O (h2(H−ε))(cid:1), h→0+, t∈[0,T).
t,t+h t t,t+h r
Proof. We rewrite the solution of equation (1) in the following form
X =X Z A ,
t 0 t t
where
(cid:18) (cid:90) t (cid:19)1/(1−m)
Z =exp{bt+cBH} and A = 1+(1−m)aXm−1 Zm−1ds .
t t t 0 s
0
Let h→0+. By (5) and Maclaurin’s expansion of ex
Zt+h =eb(t+h)ecBtH+h =ebt(1+bh+o(h2(H−ε)))ecBtH+c∆BtH,t+h
(cid:16) (cid:17)(cid:16) (cid:17)
=Z 1+bh+o(h2(H−ε)) 1+c∆BH +O (h2(H−ε))
t t,t+h r
(cid:16) (cid:17)
=Z 1+bh+c∆BH +O (h2(H−ε)) . (9)
t t,t+h r
Next note that t(cid:55)→A (cid:62)1 is non-decreasing whereas
t
(cid:90) t+h
Zm−1ds=hZm−1 =O (h), θ∈(0;1), (10)
s t+θh r
t
vanishes. Therefore Maclaurin’s expansion of x(cid:55)→(1+x)α gives
(cid:18) (cid:90) t+h (cid:19)1/(1−m)
A = A1−m+(1−m)aXm−1 Zm−1ds
t+h t 0 s
t
(cid:16) (cid:17)
=A 1+I +o (h2(H−ε)) , (11)
t t,t+h r
where
aXm−1 (cid:90) t+h
I = 0 Zm−1ds.
t,t+h A1−m s
t t
Hence
(cid:16) (cid:17)(cid:16) (cid:17)
X =X 1+bh+c∆BH +O (h2(H−ε)) 1+I +o (h2(H−ε))
t+h t t,t+h r t,t+h r
(cid:16) (cid:17)
=X 1+bh+c∆BH +I +O (h2(H−ε)) . (12)
t t,t+h t,t+h r
7
Consequently,
(cid:16) (cid:17)
∆X =X bh+c∆BH +I +O (h2(H−ε))
t,t+h t t,t+h t,t+h r
and taking into account (12) it follows that
(cid:16) (cid:17)
∆(2)X =X bh+c∆BH +I +O (h2(H−ε))
t,t+h t+h t+h,t+2h t+h,t+2h r
−X (cid:0)bh+c∆BH +I +O (h2(H−ε))(cid:1)
t t,t+h t,t+h r
=X (cid:0)1+bh+c∆BH +I +O (h2(H−ε))(cid:1)
t t,t+h t,t+h r
×(cid:0)bh+c∆BH +I +O (h2(H−ε))(cid:1)
t+h,t+2h t+h,t+2h r
(cid:16) (cid:17)
−X bh+c∆BH +I +O (h2(H−ε))
t t,t+h t,t+h r
(cid:16) (cid:17)
=X c∆(2)BH +I −I +O (h2(H−ε)) .
t t,t+h t+h,t+2h t,t+h r
It remains to show that I −I = O (h2(H−ε)). To see this observe that by
t+h,t+2h t,t+h r
(9)–(11) and the mean value theorem, for some θ ∈(0;1),i=1,2,
i
aXm−1 (cid:32)(cid:90) t+2h (cid:18)A (cid:19)m−1(cid:90) t+h (cid:33)
I −I = 0 Zm−1ds− t+h Zm−1ds
t+h,t+2h t,t+h Amt+−h1 t+h s At t s
aXm−1 (cid:18)(cid:90) t+2h (cid:90) t+h (cid:19)
= 0 Zm−1ds−(1+O (h))m−1 Zm−1ds
Am−1 s r s
t+h t+h t
aXm−1 (cid:18)(cid:90) t+2h (cid:90) t+h (cid:19)
= 0 Zm−1ds− Zm−1ds+O (h2)
Am−1 s s r
t+h t+h t
=aX0m−1 (cid:0)hZm−1 −hZm−1 +O (h2)(cid:1)
Am−1 t+h+θ1h t+θ2h r
t+h
aXm−1 (cid:32) (cid:32)(cid:18)Z (cid:19)m−1 (cid:33) (cid:33)
= 0 hZm−1 s+τ −1 +O (h2)
Amt+−h1 s Zτ r
aXm−1 (cid:18) (cid:18)(cid:16) (cid:17)m−1 (cid:19) (cid:19)
= 0 hZm−1 1+O (hH−ε) −1 +O (h2)
Am−1 s r r
t+h
aXm−1
= 0 O (h1+H−ε)=O (h2(H−ε)),
Am−1 r r
t+h
where s=t+θ h and τ =h(1+θ −θ ).
2 1 2
Proof of Theorem 1. Observe first that the function
(cid:16)T(cid:17)2x
ϕ (x)= (4−22x), x∈(0,1),
n,T n
is continuous and strictly decreasing for n > T. Thus it has the inverse function ϕ−1 for
n,T
n>T. By Lemma 9 we obtain
ϕn,T(H(cid:98)n(1)) =(cid:104)(cid:16)T(cid:17)2H(4−22H)(cid:105)−1ϕ (cid:32)ϕ−1 (cid:32)1 n(cid:88)−1(cid:32)∆(k2)X(cid:33)2(cid:33)(cid:33)
ϕ (H) n n,T n,T n cX
n,T k
k=1
=(cid:104)(cid:16)T(cid:17)2H(4−22H)(cid:105)−1(cid:32)1 n(cid:88)−1(cid:32)∆(k2)X(cid:33)2(cid:33)
n n cX
k
k=1
= n2H−1 (cid:18)n(cid:88)−1(cid:16)∆(2)BH(cid:17)2+O (cid:16) 1 (cid:17)(cid:19)
T2H(4−22H) k r n3(H−ε)−1
k=1
= n2H−1 n(cid:88)−1(cid:16)T−H∆(2)BH(cid:17)2+O (cid:16) 1 (cid:17)=V +O (cid:16) 1 (cid:17)
4−22H k r nH−3ε n,T r nH−3ε
k=1
for 3ε<H.
8
Fixδ>0. Letβ = ϕn,T(H(cid:98)n(1)). SinceV →1a.s.,thesameappliestoβ . Consequently,
n ϕn,T(H) n,T n
there exists n (ω) such that
0
(cid:18)T(cid:19)2(H+δ) (cid:18)T(cid:19)2H
ϕ (H+δ)=(4−22(H+δ)) <β (4−22H)
n,T n n n
(cid:18)T(cid:19)2(H−δ)
<(4−22(H−δ)) =ϕ (H−δ)
n n,T
for all n (cid:62) n0, i.e. ϕn,T(H(cid:98)n(1)) ∈ (cid:0)ϕn,T(H +δ),ϕn,T(H −δ)(cid:1) for all n (cid:62) n0. Since the
function ϕn,T is strictly decreasing for all n>T we conclude that H(cid:98)n(1) ∈(H−δ,H+δ) for
all n(cid:62)max(n0,T). Thus H(cid:98)n(1) is strongly consistent.
Now we prove the asymptotic normality of H(cid:98)n(1). Note that
ln(cid:18)ϕn,T(H(cid:98)n(1))(cid:19)=ln(cid:16)n(cid:17)−2(H(cid:98)n(1)−H)+ln4−22H(cid:98)n(1)
ϕ (H) T 4−22H
n,T
=−2(H(cid:98)n(1)−H)ln(cid:16)Tn(cid:17)+ln(cid:0)4−22H(cid:98)n(1)(cid:1)−ln(cid:0)4−22H(cid:1). (13)
By the Lagrange theorem for the function h(x)=ln(4−22x), x∈(0,1), we obtain
ln(cid:0)4−22H(cid:98)n(1)(cid:1)−ln(cid:0)4−22H(cid:1)=(H(cid:98)n(1)−H)h(cid:48)(cid:0)H+θn(H(cid:98)n(1)−H)(cid:1)
for some θn ∈(0,1). Since H(cid:98)n(1) →H the sequence (h(cid:48)(H+θn(H(cid:98)n(1)−H)))n(cid:62)1 is bounded.
The equality (13) can be rewritten in the following way:
−2√n ln(cid:16)Tn(cid:17)(H(cid:98)n(1)−H)=√(cid:18)n1(cid:16)−lnh(cid:16)(cid:48)(cid:0)ϕHϕn,n+T,Tθ(nH(cid:98)(H(n(H1(cid:98)))n()1(cid:17))−−H)l(cid:1)n(cid:19)1(cid:17)
2ln(n)
T
√n(cid:16)ln(cid:16)ϕn,T(H(cid:98)n(1))(cid:17)−ln1(cid:17)
= ϕn,T(H) . (14)
1+o (1)
r
NextnotethattheDeltamethodtogetherwithSlutsky’stheoremandlimitresultsofsection
3.2 imply
√n(cid:16)ϕn,T(H(cid:98)n(1)) −1(cid:17)=√n(cid:18)V −1(cid:19)+O (cid:16) 1 (cid:17)−−d→N(0;σ2(H))
ϕ (H) n,T r nH−3ε−1/2
n,T
once ε>0 is chosen to satisfy 3ε<H−1/2. It remains to observe that assertion (2) imme-
diately follows from (14) and repeated application of the Delta method along with Slutsky’s
theorem.
5.2 Proof of Theorem 2
(cid:16) (cid:17)
Fix ε∈ 0,H−1/2 . By Lemma 9
3
n(cid:88)−1(cid:32)∆(n2,)kX(cid:33)2 =n(cid:88)−1(cid:0)c∆(2)BH+O (cid:0)n−2(H−ε)(cid:1)(cid:1)2
X n,k r
n,k
k=1 k=1
n−1
=c2(cid:88)(cid:0)∆(2)BH(cid:1)2+O (cid:0)n1−3(H−ε)(cid:1)
n,k r
k=1
=c2T2H(4−22H)V +O (cid:0)n1−3(H−ε)(cid:1)
n2H−1 n,T r
and in the same way
2(cid:88)n−1(cid:32)∆(22n),kX(cid:33)2 = c2T2H(4−22H)V +O (cid:0)n1−3(H−ε)(cid:1).
X (2n)2H−1 2n,T r
2n,k
k=1
9
Since V ,V →1 a.s. (see section 3.2),
n,T 2n,T
H(cid:98)n(2) =21 − 2l1n2lncc22TT(222nHH)(2(4H4−−−22122HH))VV2n,T++OOr(cid:0)(cid:0)nn11−−33((HH−−εε))(cid:1)(cid:1)
n2H−1 n,T r
=1 − 1 ln(cid:32) V2n,T (cid:32)1+Or(cid:0)n−H+3ε(cid:1)(cid:33)(cid:33)
2 2ln2 22H−1Vn,T 1+Or(cid:0)n−H+3ε(cid:1)
=H(cid:101)n− 2l1n2ln(cid:16)1+Or(cid:0)n−H+3ε(cid:1)(cid:17)=H(cid:101)n+Or(cid:0)n−H+3ε(cid:1).
NowtofinishtheproofapplySlutsky’stheoremandlimitresultsofsection3.2. Notethatthe
limit variance σ∗2(H) of H(cid:98)n(2) equals to that of H(cid:101)n.
5.3 Proof of Theorem 3
Let 0<ε<1/2 be such that 3ε<H−1/2 and
n2(H−H(cid:98)n(3))(4−22H(cid:98)n(3)))
c2 = c2.
(cid:101)n T2(H−H(cid:98)n(3))(4−22H) (cid:98)n
By Maclaurin’s expansion and the mean value theorem we obtain
n2(H−H(cid:98)n(3))(4−22H(cid:98)n(3))) =exp(cid:26)o (φ(n))ln(cid:16)n(cid:17)2+ln(cid:0)4−22H(cid:98)n(3)(cid:1)−ln(4−22H)(cid:27)
T2(H−H(cid:98)n(3))(4−22H) r T
=(1+or(φ(n)lnn))exp(cid:8)or(φ(n))h(cid:48)(cid:0)H+θn(H(cid:98)n(3)−H)(cid:1)(cid:9)
=(1+o (φ(n)lnn))(1+o (φ(n)))=(1+o (φ(n)lnn))
r r r
for some θ ∈ (0,1), where h(x) = ln(4−22x), x ∈ (0,1). Hence it suffices to show that c2
n (cid:101)n
satisfies
√
c2 →c2 a.s., n(c2−c2)−−d→N(0;c4σ2)
(cid:101)n (cid:101)n
and the claim will be proved. By Lemma 9
n(cid:88)−1(cid:32)∆(k2)X(cid:33)2 =c2n(cid:88)−1(cid:0)∆(2)BH(cid:1)2+O (cid:0)n1−3(H−ε)(cid:1).
X k r
k
k=1 k=1
Thus
c2−c2 =c2(cid:18) n2H−1 n(cid:88)−1(cid:0)∆(2)BH(cid:1)2−1(cid:19)+O (cid:0)n−H+3ε(cid:1)
(cid:101)n T2H(4−22H) k r
k=1
=c2(cid:0)V −1(cid:1)+O (cid:0)n−H+3ε(cid:1).
n,T r
It remains to apply Slutsky’s theorem and limit results stated in section 3.2.
6 Simulations
ThesimulationsoftheobtainedestimatorspresentedbelowwereperformedusingtheRsoft-
wareenvironment[15]. Theperformanceoftheseestimatorswasassessedusingsamplepaths
of the Black-Scholes model, the Verhulst equation and the Landau-Ginzburg equation as de-
fined below. In Equation 1, if we set a = 0, b = λ, c = σ we obtain the Black-Scholes
model
dX =λX dt+σX dBH.
t t t t
Letusnotethattheresults,obtainedinSection2areapplicablefortheBlack-Scholesmodel.
If we set a=−1, b=λ, c=σ, m=2 we obtain the Verhulst equation
dX =(cid:0)−X2+λX (cid:1)dt+σX dBH
t t t t t
with the explicit solution
ξexp{λt+σBH}
X = t .
t 1+ξ(cid:82)texp{λs+σBH}ds
0 s
10