Drift parameter estimation in stochastic differential equation with multiplicative stochastic volatility

0.2 MB·

by  Meriem Bel Hadj Khlifa

ModernStochastics:TheoryandApplications3(2016)269–285
DOI:10.15559/16-VMSTA66
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a
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Drift parameter estimation in stochastic differential
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equation with multiplicative stochastic volatility
]
R
P
. MeriemBelHadjKhlifaa,YuliyaMishurab,KostiantynRalchenkob, ,
h ∗
at MounirZilia
m
[ aUniversityofMonastir,FacultyofSciencesofMonastir,Departmentof
Mathematics,Avenuedel’Environnement,5000,Monastir,Tunisia
1
bDepartmentofProbabilityTheory,StatisticsandActuarialMathematics,
v
8 TarasShevchenkoNationalUniversityofKyiv,
3 64Volodymyrska,01601Kyiv,Ukraine
2
1 [email protected](M.BelHadjKhlifa),[email protected](Yu.Mishura),
0 [email protected](K.Ralchenko),[email protected](M.Zili)
.
1
Received:9November2016,Revised:3December2016,Accepted:4December2016,
0
Publishedonline:13December2016
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: Abstract Weconsiderastochasticdifferentialequationoftheform
v
i
X dXt =θa(t,Xt)dt+σ1(t,Xt)σ2(t,Yt)dWt
r
a withmultiplicativestochasticvolatility,whereY issomeadaptedstochasticprocess.Weprove
existence–uniquenessresultsforweakandstrongsolutionsofthisequationundervariouscon-
ditionsontheprocessY andthecoefficientsa,σ1,andσ2.Also,westudythestrongconsis-
tencyofthemaximumlikelihoodestimatorfortheunknownparameterθ.WesupposethatY
isinturnasolutionofsomediffusionSDE.Severalexamplesofthemainequationandofthe
processY areprovidedsupplyingthestrongconsistency.
Keywords Stochasticdifferentialequation,weakandstrongsolutions,stochasticvolatility,
driftparameterestimation,maximumlikelihoodestimator,strongconsistency
2010MSC 60H10,62F10,62F12
∗Correspondingauthor.
©2016TheAuthor(s).PublishedbyVTeX.OpenaccessarticleundertheCCBYlicense.
www.i-journals.org/vmsta
270 M.BelHadjKhlifaetal.
1 Introduction
Thegoalofthepaperistostudythestochasticdifferentialequation(SDE),thediffu-
sioncoefficientofwhichincludesanadditionalstochasticprocess:
dX =θa(t,X )dt+σ(t,X ,Y )dW , (1)
t t t t t
whereσ(t,x,y) = σ (t,x)σ (t,y),andtoestimatethedriftparameterθ bytheob-
1 2
servationsofstochasticprocessesX andY.Suchequationsoftenariseasmodelsofa
financialmarketinmathematicalfinance.Forexample,oneofthefirstmodelsofsuch
atypewithσ(t,x,y) = xy wasproposedin[8],whereY wasthesquarerootofthe
geometricBrownianmotionprocess.AsimilarmodelwasconsideredbyHeston[6],
wherethevolatilitywasgovernedbytheOrnstein–Uhlenbeckprocess.Fouqueetal.
usedthemodelwithstochasticvolatilitydrivenbytheCox–Ingersoll–Rossprocess;
see [4, 5]. The case where σ(t,x,y) = xσ (y) and Y is the Ornstein–Uhlenbeck
2
processwasstudiedin[12,13].
In the present paper, we investigate the existence and uniqueness of weak and
strong solutions to the equation (1). We adapt the approaches of Skorokhod [20],
StroockandVaradhan[21, 22],andKrylov[10,11] toestablishtheweakexistence
and weak uniqueness. Concerning the strong existence and uniqueness, we use the
well-knownapproachesofYamadaandWatanabe[23](see also[2]) forinhomoge-
neous coefficients and Lipschitz conditions. In the present paper, we consider only
thecaseofmultiplicativestochasticvolatility,where,asitwasmentioned,thediffu-
sioncoefficientisfactorizedasσ(t,x,y) = σ (t,x)σ (t,y).Thenweconstructthe
1 2
maximumlikelihoodestimatorfortheunknowndriftparameterandproveitsstrong
consistency. As an example, we consider a linear model with stochastic volatility
drivenbyasolutiontosomeItô’sSDE.Inparticular,westudyindetailsanSDEwith
constantcoefficients,theOrnstein–Uhlenbeckprocess,andthegeometricBrownian
motion,asthemodelforvolatility(notethatprocessY canbeinterpretednotonlyas
avolatility,butalsoasanadditionalsourceofrandomness).Notethatthemaximum
likelihoodestimationintheOrnstein–Uhlenbeckmodelwithstochasticvolatilitywas
studiedin[1].Similarstatisticalmethodsforthecaseofdeterministicvolatilitycan
befoundin[7,9,16,18].
Thepaperisorganizedasfollows.InSection2,weprovetheexistenceofweak
andstrongsolutionsunderdifferentconditions.InSection3,weestablishthestrong
consistencyofthemaximumlikelihoodestimatoroftheunknowndriftparameterθ.
Section 4 contains the illustrations of our results with some simulations. Auxiliary
statementsaregatheredinSection5.
2 Existenceanduniquenessresultsforweakandstrongsolutions
Let (Ω, , ,P) be a complete probability space with filtration = ,t 0
t
F F F {F ≥ }
satisfyingthestandardassumptions.Weassumethatallprocessesunderconsideration
areadaptedtothefiltration .
F
Driftparameterestimationinstochasticdifferentialequationwithmultiplicativestochasticvolatility 271
2.1 ExistenceofweaksolutionintermsofSkorokhodconditions
Considerthefollowingstochasticdifferentialequation:
dX =a(t,X )dt+σ (t,X )σ (t,Y )dW , (2)
t t 1 t 2 t t
whereX = X R,W isaWienerprocess,andY issomeadaptedstochastic
t=0 0
| ∈
processtobespecifiedlater.
Theorem1. LetY beameasurableandcontinuousprocess,a,σ ,andσ becon-
1 2
tinuousw.r.t.x R,y R,andt [0,T],σ bebounded,and
2
∈ ∈ ∈
σ (t,x) 2+ a(t,x) 2 K 1+ x2
1
≤ | |
forsomeconstantK >(cid:12)0.Then(cid:12)Eq.((cid:12)2)hasa(cid:12) weaks(cid:0)olution.(cid:1)
(cid:12) (cid:12) (cid:12) (cid:12)
Proof. Consider a sequenceof partitionsof [0,T]:0 = tn < tn < < tn = T
0 1 ··· n
suchthatlim max (tn tn)=0.Defineξnbyξn =X(0)and
n→∞ k k+1− k k 0
ξn =ξn+a tn,ξn ∆tn+σ tn,ξn σ tn,Y tn ∆Wn.
k+1 k k k k 1 k k 2 k k k
ItfollowsfromLemma1,(cid:0)Lemm(cid:1)a2,andPro(cid:0)positio(cid:1)n1(cid:0)inSec(cid:0)tion(cid:1)(cid:1)5thatitispossible
tochoosea subsequencen andconstructprocessesξ ,W ,andY suchthatthe
′ n n n
′ ′ ′
finite-dimensionaldistributionsofξn ,Wn ,andYn coincidewiththoseofξn′,W,
′ ′ ′
andY andξ ξ,W W andY Y inprobeabilfity,whereeξ,W,andY are
n n n
′ → ′ → ′ →
somestochasticprocesses(evidentley,WfisaWieneerprocess).Itsufficestoprovethat
ξisasolutieonofEeq.f(2)whenfW andeY areereplacedbyW andY. e f e
Wehavethatξ satisfiestheequaftion
n
′
e f e
ξn′(t)e=ξn′(0)+ a tnk′,ξn′ tnk′ ∆tnk′
tnkX+′1≤t (cid:0) (cid:0) (cid:1)(cid:1)
e e e
+ σ tn′,ξ tn′ σ tn′,Y tn′ ∆Wn′.
1 k n′ k 2 k n′ k k
tnkX+′1≤t (cid:0) (cid:0) (cid:1)(cid:1) (cid:0) (cid:0) (cid:1)(cid:1)
e e f
Sinceσ isboundedandσ isoflineargrowth,theirproductisoflineargrowth:
2 1
σ (t,x)σ (t,y) K (1+ x),
1 2 1
≤ | |
whereK >0isaconsta(cid:12)nt.Therefore, (cid:12)
1 (cid:12) (cid:12)
P sup σ t,ξ (t) σ t,Y (t) >C
1 n 2 n
′ ′
0 t T
(cid:16) ≤ ≤ (cid:12) (cid:0) (cid:1) (cid:0) (cid:1)(cid:12) (cid:17)
P (cid:12)sup Ke 1+ ξ (te) >C(cid:12) =P sup ξ (t) > C 1
1 n n
≤ ′ ′ K −
(cid:16)0≤t≤T (cid:0) (cid:12) (cid:12)(cid:1) (cid:17) (cid:18)0≤t≤T(cid:12) (cid:12) 1 (cid:19)
=P sup ξn′(t) (cid:12)>e C (cid:12) 1 . (cid:12)e (cid:12)
K −
(cid:18)0≤t≤T 1 (cid:19)
(cid:12) (cid:12)
UsingLemma1,weg(cid:12)etthat(cid:12)
P sup σ t,ξ (t) σ t,Y (t) >C 0 asC .
1 n 2 n
′ ′ → →∞
0 t T
(cid:16) ≤ ≤ (cid:12) (cid:0) (cid:1) (cid:0) (cid:1)(cid:12) (cid:17)
(cid:12) e e (cid:12)
272 M.BelHadjKhlifaetal.
Moreover, we have that σ (t,x)σ (t,y) is continuous w.r.t. t [0,T], x,y R.
1 2
∈ ∈
Then,foranyε>0,thereexistsδ >0suchthat
σ (t ,x )σ (t ,y ) σ (t ,x )σ (t ,y ) <ε
1 1 1 2 1 1 1 2 2 2 2 2
−
whenever t t(cid:12) <δ, x x <δ, y y <δ.Therefo(cid:12)re,
| 1− 2(cid:12)| | 1− 2| | 1− 2| (cid:12)
P σ t ,ξ (t ) σ t ,Y (t ) σ t ,ξ (t ) σ t ,Y (t ) >ε
1 1 n 1 2 1 n 1 1 2 n 2 2 2 n 2
′ ′ − ′ ′
(cid:0)(cid:12) P(cid:0) ξn (t1) (cid:1)ξn(cid:0)(t2) <δ, Y(cid:1)n (t1)(cid:0) Yn (t2)(cid:1)<(cid:0)δ, (cid:1)(cid:12) (cid:1)
≤(cid:12) e′ − ′ e ′ − e′ e (cid:12)
σ1(cid:0)(cid:12)t1,ξn (t1) σ2 t1(cid:12),Yn (t(cid:12)1) σ1 t2,ξn (t2(cid:12)) σ2 t2,Yn (t2) >ε
(cid:12)e ′ e (cid:12) ′ (cid:12)e − e ′ (cid:12) ′
(cid:12)(cid:12)+P(cid:0) ξne′(t1)−(cid:1) ξn(cid:0)′(t2)e ≥δ (cid:1)+P Y(cid:0)n′(te1)−Y(cid:1)n′(t(cid:0)2) ≥e δ (cid:1)(cid:12)(cid:12) (cid:1)
=P ξ(cid:0)n(cid:12) (t1) ξn (t2) (cid:12) δ +(cid:1)P Y(cid:0)(cid:12)(t1) Y(t2) δ(cid:12), (cid:1)
| (cid:12)′e − ′e |≥(cid:12) | (cid:12)e − e|≥ (cid:12)
andthelast(cid:0)relationimpliesthefoll(cid:1)owing(cid:0)one: (cid:1)
lim lim sup P σ t ,ξ (t ) σ t ,Y (t )
1 1 n 1 2 1 n 1
h→0n′→∞|t1−t2|≤h (cid:0)(cid:12) (cid:0) ′ (cid:1) (cid:0) ′ (cid:1)
(cid:12) σ t ,eξ (t ) σ t ,eY (t ) >ε =0.
1 2 n 2 2 2 n 2
− ′ ′
ApplyingLemma1,wegetthat (cid:0) (cid:1) (cid:0) (cid:1)(cid:12) (cid:1)
e e (cid:12)
T
σ tn′,ξ tn′ σ tn′,Y tn′ ∆Wn′ σ s,ξ(s) σ s,Y(s) dW(s)
1 k n′ k 2 k n′ k k → 1 2
tnkX+′1≤t (cid:0) (cid:0) (cid:1)(cid:1) (cid:0) (cid:0) (cid:1)(cid:1) Z0 (cid:0) (cid:1) (cid:0) (cid:1)
e e e e f
inprobabilityasn ,andwealsohavethat
′
→∞
T
a tn′,ξ tn′ ∆tn′ a s,ξ(s) ds,
k n′ k k →
tnkX+′1≤t (cid:0) (cid:0) (cid:1)(cid:1) Z0 (cid:0) (cid:1)
e e
whencetheprooffollows.
2.2 ExistenceanduniquenessofweaksolutionintermsofStroock–Varadhancon-
ditions
Inthisapproach,weassumeadditionallythattheprocessY alsoisasolutionofsome
diffusionstochasticdifferentialequation.LetW1 andW2 betwoWienerprocesses,
possiblycorrelated,sothatdW1dW2 = ρdt forsome ρ 1.Inthiscase,wecan
t t | | ≤
representW2 =ρW1+ 1 ρ2W3,whereW3isaWienerprocessindependentof
t t − t
W1.
p
Theorem2. Considerthesystemofstochasticdifferentialequations
dX =a(t,X )dt+σ (t,X )σ (t,Y )dW1, (3)
t t 1 t 2 t t
dY =α(t,Y )dt+β(t,Y )dW2, (4)
(cid:26) t t t t
where all coefficientsa, α, σ , σ , and β are nonrandommeasurableand bounded
1 2
functions, σ , σ , and β are continuous in all arguments. Let ρ < 1, and let
1 2
| |
σ (t,x) > 0, σ (t,y) > 0, β(t,y) > 0 for all t,x,y. Then the weak existence
1 2
anduniquenessinlawholdforsystem(3)–(4),andinparticular,theweakexistence
anduniquenessinlawholdforEq.(3)withY beingaweaksolutionofEq.(4).
Driftparameterestimationinstochasticdifferentialequationwithmultiplicativestochasticvolatility 273
Proof. Equations(3)and(4)areequivalenttothetwo-dimensionalstochasticdiffer-
entialequation
dZ(t)=A(t,Z )dt+B(t,Z )dW(t),
t t
whereZ(t)=(X(t)),W(t)=(W1(t))isatwodimensionalWienerprocess,
Y(t) W3(t)
a(t,x) σ (t,x)σ (t,y) 0
A(t,x,y)= , and B(t,x,y)= 1 2 .
α(t,y) ρβ(t,y) 1 ρ2β(t,y)
(cid:18) (cid:19) (cid:18) − (cid:19)
It follows from the measurability and boundednessof a and α apnd from the conti-
nuity and boundedness of σ , σ , and β that the coefficients of matrices A and B
1 2
arenonrandom,measurable,andbounded,andadditionallythecoefficientsofB are
continuousinallarguments.ThenwecanapplyTheorems4.2and5.6from[21](see
also Prop. 1.14in [3]) anddeducethatwe haveto provethe followingrelation:for
any(t,x,y) R+ R2,thereexistsε(t,x,y)>0suchthat,forallλ R2,
∈ × ∈
B(t,x,y)λ ε(t,x,y) λ . (5)
≥ k k
Relation(5)isequivalentto(cid:13)thefollowin(cid:13)gone(weomitarguments):
(cid:13) (cid:13)
σ2σ2λ2+β2 ρλ + 1 ρ2λ 2 ε2 λ2+λ2
1 2 1 1 − 2 ≥ 1 2
or (cid:0) p (cid:1) (cid:0) (cid:1)
σ2σ2+β2ρ2 λ2+β2 1 ρ2 λ2+2ρ 1 ρ2β2λ λ ε2 λ2+λ2 . (6)
1 2 1 − 2 − 1 2 ≥ 1 2
The(cid:0)quadraticform(cid:1) (cid:0) (cid:1) p (cid:0) (cid:1)
Q(λ ,λ )= σ2σ2+β2ρ2 λ2+β2 1 ρ2 λ2+2ρ 1 ρ2β2λ λ
1 2 1 2 1 − 2 − 1 2
intheleft-handside(cid:0)of(6)ispositi(cid:1)vedefinite(cid:0)sincei(cid:1)tsdiscrimpinant
D =ρ2 1 ρ2 β4 β2 1 ρ2 σ2σ2+β2ρ2 = β2 1 ρ2 σ2σ2 <0.
− − − 1 2 − − 1 2
The continu(cid:0)ity of Q(cid:1)(λ ,λ )(cid:0)implies(cid:1)(cid:0)the existence o(cid:1)f min (cid:0) Q(cid:1)(λ ,λ ) > 0.
1 2 λ2+λ2=1 1 2
1 2
Then,puttingε=min Q(λ ,λ )andusinghomogeneity,weget(6).
λ2+λ2=1 1 2
1 2
2.3 ExistenceofstrongsolutionintermsofYamada–Watanabeconditions
Now we consider strong existence–uniquenessconditions for Eq. (2), adapting the
Yamada–Watanabeconditionsforinhomogeneouscoefficientsfrom[2].
Theorem 3. Let a, σ , and σ be nonrandommeasurable bounded functions such
1 2
that
(i) Thereexistsapositiveincreasingfunctionρ(u),u (0, ),satisfyingρ(0)=0
∈ ∞
suchthat
σ (t,x) σ (t,y) ρ(x y ), t 0, x,y R,
1 1
− ≤ | − | ≥ ∈
and 0∞ρ−2(cid:12)(cid:12)(u)du=+∞. (cid:12)(cid:12)
(ii) TherReexistsapositiveincreasingconcavefunctionk(u),u (0, ),satisfying
∈ ∞
k(0)=0suchthat
274 M.BelHadjKhlifaetal.
a(t,x) a(t,y) k(x y ), t 0, x,y R,
− ≤ | − | ≥ ∈
(cid:12) (cid:12)
and 0∞k−1((cid:12)u)du = +∞. A(cid:12)lso, let Y be an adapted continuous stochastic
process.ThenthepathwiseuniquenessofsolutionholdsforEq.(2),andhence
R
ithasauniquestrongsolution.
Proof. Let1>a >a > >a > >0bedefinedby
1 2 n
··· ···
1 a1 an 1
ρ 2(u)du=1, ρ 2(u)du=2,..., − ρ 2(u)du=n,....
− − −
Za1 Za2 Zan
We have that a 0 as n . Let φ (u), n = 1,2,..., be a continuousfunc-
n n
→ → ∞
tion with support contained in (a ,a ) such that 0 φ (u) 2ρ−2(u) and
aann−1φn(u)du=1.Suchafunctinononb−v1iouslyexists.Se≤t n ≤ n
R
x y
ϕ (x)= | | φ (u)dudy, x R.
n n
∈
Z0 Z0
Clearly,ϕ C2(R), ϕ (x) 1,andϕ (x) x asn .
n ∈ | ′n |≤ n ր| | →∞
Let X and X be two solutions of Eq. (2) on the same probability space with
1 2
the same Wienerprocessandsuch thatX (0) = X (0). Thenwe can presenttheir
1 2
differenceas
t
X (t) X (t)= σ s,Y(s) σ s,X (s) σ s,X (s) dW(s)
1 2 2 1 1 1 2
− −
Z0
t (cid:0) (cid:1)(cid:0) (cid:0) (cid:1) (cid:0) (cid:1)(cid:1)
+ a s,X (s) a s,X (s) ds.
1 2
−
Z0
(cid:0) (cid:0) (cid:1) (cid:0) (cid:1)(cid:1)
BytheItôformula,
ϕ X (t) X (t)
n 1 2
−
t
(cid:0) (cid:1)
= ϕ X (s) X (s) σ s,Y(s) σ s,X (s) σ s,X (s) dW(s)
′n 1 − 2 2 1 1 − 1 2
Z0
t (cid:0) (cid:1) (cid:0) (cid:1)(cid:0) (cid:0) (cid:1) (cid:0) (cid:1)(cid:1)
+ ϕ X (s) X (s) a s,X (s) a s,X (s) ds
′n 1 − 2 1 − 2
Z0
1 t (cid:0) (cid:1)(cid:0) (cid:0) (cid:1) 2 (cid:0) (cid:1)(cid:1) 2
+ ϕ X (s) X (s) σ s,Y(s) σ s,X (s) σ s,X (s) ds
2 ′n′ 1 − 2 2 1 1 − 1 2
Z0
=J +J +J(cid:0) . (cid:1) (cid:0) (cid:1) (cid:0) (cid:0) (cid:1) (cid:0) (cid:1)(cid:1)
1 2 3
WehavethatE(J )=0,
1
t
E(J ) E a s,X (s) a s,X (s) ds
2 1 2
| |≤ −
Z0
t (cid:12) (cid:0) (cid:1) (cid:0) (cid:1)(cid:12) t
E((cid:12)k X (s) X (s) ds (cid:12) k(E X (s) X (s) ds
1 2 1 2
≤ | − | ≤ | − |
Z0 Z0
(cid:0) (cid:1) (cid:0) (cid:1)
Driftparameterestimationinstochasticdifferentialequationwithmultiplicativestochasticvolatility 275
byJensen’sinequality,and
C2 t 2
E(J3) E ρ−2 X1(s) X2(s) ρ2 X1(s) X2(s) ds
| |≤ 2 n − −
Z0 (cid:18) (cid:19)
t (cid:0)(cid:12) (cid:12)(cid:1) (cid:0)(cid:12) (cid:12)(cid:1)
0 asn (cid:12) . (cid:12) (cid:12) (cid:12)
≤ n → →∞
Sobylettingn weget
→∞
t
E X (s) X (s) k(E X (s) X (s) ds.
1 2 1 2
− ≤ −
Z0
(cid:0)(cid:12) (cid:12)(cid:1) (cid:0)(cid:12) (cid:12)(cid:1)
We have that 0∞(cid:12)k−1(u)du = +(cid:12)∞. Then we(cid:12)get E(|X1(s)−(cid:12)X2(s)|) = 0, and
henceX (s)=X (s)a.s.
1 2
R
2.4 ExistenceanduniquenessforstrongsolutionintermsofLipschitzconditions
Theorem4. Leta,σ ,andσ benonrandommeasurablefunctions,andletY bean
1 2
adaptedcontinuousstochasticprocess.Considerthefollowingassumptions:
(i) ThereexistsK >0suchthat,forallt 0andx R,
≥ ∈
σ (t,x) 2+ a(t,x) 2 K2 1+ x2 ;
1
≤ | |
(cid:12) (cid:12) (cid:12) (cid:12) (cid:0) (cid:1)
(ii) Foranyn N,the(cid:12)reexists(cid:12)K (cid:12)> 0su(cid:12)chthat,forallt 0andforall(x,y)
N
∈ ≥
satisfying x N and y N,
| |≤ | |≤
a(t,x) a(t,y) + σ (t,x) σ (t,y) K x y ;
1 1 N
− − ≤ | − |
(cid:12) (cid:12) (cid:12) (cid:12)
(iii) sup sup(cid:12) σ (s,x) (cid:12) C(cid:12) . (cid:12)
s≥0 |x|≤N| 2 |≤ N
ThenEq.(2)hasauniquestrongsolution.
Thisresultcanbeprovedbyusingthesuccessiveapproximationmethod;see,for
example,[19,Thm.1.2].
3 Driftparameterestimation
3.1 Generalresults
Let (Ω, , ,P) be a complete probability space with filtration = ,t 0
t
F F F {F ≥ }
satisfyingthestandardassumptions.Weassumethatallprocessesunderconsideration
areadaptedtothefiltration .ConsideraparameterizedversionofEq.(2)
F
dX =θa(t,X )dt+σ (t,X )σ (t,Y )dW , (7)
t t 1 t 2 t t
where W is a Wiener process. Assume that Eq. (7) has a unique strong solution
X = X ,t [0,T] .Ourmainproblemistoestimatetheunknownparameterθby
t
{ ∈ }
continuousobservationsofX andY.
276 M.BelHadjKhlifaetal.
Denote
a(t,x) a(t,x)
f(t,x,y)= , g(t,x,y)= .
σ2(t,x)σ2(t,y) σ (t,x)σ (t,y)
1 2 1 2
Assumethat,forallt>0,
σ (t,X )σ (t,Y )=0 a.s., (8)
1 t 2 t
6
t
g2(s,X ,Y )ds< a.s., (9)
s s
∞
Z0
∞g2(s,X ,Y )ds= a.s. (10)
s s
∞
Z0
ThenalikelihoodfunctionforEq.(7)hastheform
dP (T) T θ2 T
θ =exp θ f(t,X ,Y )dX g2(t,X ,Y )dt ;
t t t t t
dP0(T) ( Z0 − 2 Z0 )
see[15,Ch.7].Hence,themaximumlikelihoodestimatorofparameterθconstructed
byobservationsofX andY ontheinterval[0,T]hastheform
T T
f(t,X ,Y )dX g(t,X ,Y )dW
θˆ = 0 t t t =θ+ 0 t t t. (11)
T R0T g2(t,Xt,Yt)dt R0T g2(t,Xt,Yt)dt
R R
Theorem5. Underassumptions(8)–(10),theestimatorθˆ isstronglyconsistentas
T
T .
→∞
t
Proof. Note that, under condition (9) the process M = g(s,X ,Y )dW is a
t 0 s s s
square-integrable local martingale with quadratic variation M =
t
tg2(s,X ,Y )ds. According to the strong law of large nRumbers forhmairtingales
0 s s
[14,Ch.2,§6,Thm.10,Cor.1],underthecondition M a.s.asT ,
T
wRehavethat MT 0a.s.asT .Therefore,itfolhlowisfr→om∞representatio→n(1∞1)
thatθˆ isstrohnMgilTyc→onsistent. →∞
T
3.2 Linearequationwithstochasticvolatility
Asanexample,letusconsiderthemodel
dX =θX dt+X σ (Y )dW , X =x R, (12)
t t t 2 t t 0 0
∈
whereW isaWienerprocess,andY isacontinuousstochasticprocesswithvalues
t t
from an open intervalJ = (l,r) (further,in examples,we will considerJ = R or
J =(0,+ )).ByTheorem4,undertheassumption
∞
(A1) σ (y)islocallyboundedonJ,
2
thereexistsauniquestrongsolutionof(12).
Driftparameterestimationinstochasticdifferentialequationwithmultiplicativestochasticvolatility 277
LetY beaJ-valuedsolutionoftheequation
dY =α(Y )dt+β(Y )dW1, Y =y J, (13)
t t t t 0 0 ∈
whereW1isaWienerprocess,possiblycorrelatedwithW.
ByL1 (J)wedenotethesetofBorelfunctionsJ [ , ]thatarelocally
loc → −∞ ∞
integrableonJ,thatis,integrableoncompactsubsetsofJ.ByL1 (l+)wedenote
loc
z
thesetofBorelfunctionsf: J [ , ]suchthat f(y) dy < forsome
→ −∞ ∞ l | | ∞
z J.ThenotationL1 (r )isintroducedsimilarly.
∈ loc − R
Assumethatcoefficientsαandβ satisfytheEngelbert–Schmidtconditions
(A2) β(y)=0forally J,and
6 ∈
(A3) β 2,αβ 2 L1 (J).
− − ∈ loc
Letusintroducethefollowingnotation:
y α(u)
ρ(y)=exp 2 du , y J,
− β2(u) ∈
(cid:26) Zc (cid:27)
y
s(y)= ρ(u)du, y J¯=[l,r],
∈
Zc
forsomec J.Assumeadditionallythat
∈
(A4) s(r)=∞or s(ρrβ)−2s ∈/ L1loc(r−),
(A5) s(l)=−∞or s−ρβs(2l) ∈/ L1loc(l+).
Under(A2)–(A3),theSDE(13)hasaweaksolution,uniqueinlaw,whichpossi-
blyexitsJ atsometimeζ.Moreover,ζ = a.s.ifandonlyifconditions(A4)–(A5)
∞
aresatisfied, see,forexample,[17,Prop.2.6].
Assumealsothat
(A6) β−2σ2−2 ∈L1loc(J),
(A7) oneofthefollowingfourconditionsholds:
(i) s(r)= ,s(l)= ,
∞ −∞
(ii) s(r)<∞,s(l)=−∞, sρ(βr2)−σ22s ∈/ L1loc(r−),
(iii) s(r)=∞,s(l)>−∞, sρ−β2sσ(l22) ∈/ L1loc(l+),
(iv) s(r)<∞,s(l)>−∞, sρ(βr2)−σ22s ∈/ L1loc(r−), sρ−β2sσ(l22) ∈/ L1loc(l+),
(A8) X σ(Y )=0a.s.,t 0.
t t
6 ≥
Themaximumlikelihoodestimator(11)formodel(12)equals
θˆ = 0T Xt−1σ2−2(Yt)dXt. (14)
T R 0T σ2−2(Yt)dt
R
278 M.BelHadjKhlifaetal.
Theorem6. Underassumptions(A1)–(A8),theestimatorθˆ isstronglyconsistent
T
asT .
→∞
Proof. We need to verify conditions (8)–(10) of Theorem 5. For model (12), they
readasfollows:
X σ (Y )=0, t 0, a.s., (15)
t 2 t
6 ≥
t
σ2−2(Ys)ds<∞, t>0, a.s., (16)
Z0
∞σ2−2(Ys)ds=∞ a.s. (17)
Z0
Note that (15) is assumption (A8). By [17, Thm. 2.7] the local integrability con-
dition (A6), together with (A2)–(A5), implies (16). Further, if assumption (A7)(i)
holds,then(17)issatisfied by[17,Thm.2.11].Intheremainingcase s(r) < or
∞
s(l)> ,wehavethat
−∞
Ω = limY =r limY =l ;
t t
t ∪ t
(cid:26)↑∞ (cid:27) (cid:26) ↑∞ (cid:27)
see[17].Moreover,ifs(r) = ,thenP(lim Y = r) = 0by[17,Prop.2.4].If
t t
s(r) < ∞and sρ(βr2)σ−22s ∈/ L1loc(r∞−),then 0∞σ↑2−∞2(Ys)ds = ∞a.s.on{limt↑∞Yt =
r by[17,Thm.2.12].Thesimilarstatementsholdfor lim Y =l .Thisimplies
} R { t↑∞ t }
that(17)issatisfiedundereachofconditions(ii)–(iv)ofassumption(A7).
NowweconsiderseveralexamplesoftheprocessY,namelytheBacheliermodel,
the Ornstein–Uhlenbeck model, the geometric Brownian motion, and the
Cox–Ingersoll–Rossmodel. We concentrate on verification of assumption (A7) for
thesemodels,assumingthatotherconditionsofTheorem6aresatisfied.
Example1(Bacheliermodel). LetY beasolutionoftheSDE
dY =αdt+βdW1, Y =y R,
t t 0 0 ∈
whereα∈Randβ 6=0aresomeconstants.Assumethatσ2−2(y)∈L1loc(R)andone
ofthefollowingassumptionsholds:
(i) α=0,
(ii) α>0andσ2−2(y)∈/ L1loc(+∞),
(iii) α<0andσ2−2(y)∈/ L1loc(−∞).
Thenestimator(14)isstronglyconsistent.
Indeed,inthiscase,J =R,
2α y 2α
ρ(y)=exp y , and s(y)= exp u du.
−β2 −β2
(cid:26) (cid:27) Z0 (cid:26) (cid:27)
Ifα = 0,thens(y) = y,s(+ ) = ,s( ) = ,andassumption(A7)(i)is
∞ ∞ −∞ −∞
satisfied.Otherwise,wehave