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AdvancesinAppliedMathematicsA^êÆ?Ð,2023,12(5),2235-2254
PublishedOnlineMay2023inHans.https://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2023.125229
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Á‡
©‘3|^•¦{ïÄ‘‚5gü½¤£‘©êO-UL§ÚOíä"bB
H
=
{B
H
t
,t≥0}´Hurst•ê•
1
2
≤H<1©êÙK$ħ·‚•Äe•§§
dX
H
t
= dB
H
t
+σX
H
t
dt+νdt−θ

Z
t
0

X
H
s
−X
H
u

ds

dt
Ù¥,X
H
0
=0,θ<0Úσ,ν∈R´n‡ëê"ù‡L§´gáÚ*Ñ[(„Cranstonand
LeJan,Math.Ann.303(1995),87-93)§·‚̇8I´ïÄÙëê•¦O"
'…c
©êÙK$ħgü½*ѧ•¦O
StatisticalInferenceontheFractional
Ornstein-UhlenbeckProcesswiththe
LinearSelf-RepellingDrift
QingYang,LitanYan
CollegeofScience,DonghuaUniversity,Shanghai
Received:Apr.22
nd
,2023;accepted:May15
th
,2023;published:May24
th
,2023
©ÙÚ^:š,An".‘‚5gü½¤£‘©êO-UL§ÚOíä[J].A^êÆ?Ð,2023,12(5):
2235-2254.DOI:10.12677/aam.2023.125229
š§An"
Abstract
ThisdissertationaimistostudystatisticalinferenceonthefractionalOrnstein-
Uhlenbeckprocesswiththelinearself-attractingdriftbyleastsquaresestimation.
Let B
H
= {B
H
t
,t≥0}be a fractionalBrownian motion withHurstindex
1
2
≤H<1.We
considerthefollowingequation,
dX
H
t
= dB
H
t
+σX
H
t
dt+νdt−θ

Z
t
0

X
H
s
−X
H
u

ds

dt
with X
H
0
= 0, where θ<0and σ,ν∈Rarethree parameters.Theprocess is ananalogue
oftheself-attractingdiffusion(CranstonandLeJan,Math.Ann.303(1995),87-93).
Ourmainaimistostudytheleastsquaresestimationsofitsparameters.
Keywords
FractionalBrownianMotion,Self-RepellingDiffusions,LeastSquaresEstimation
Copyright
c
2023byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
1992c,DurrettÚRogers [1]é˜aO•àÜÔ.‰ïÄ"3,«^‡e§¦‚ïáX
e‘Ň©•§)ìC5Ÿµ
X
t
= B
t
+
Z
t
0
Z
s
0
f(X
s
−X
u
)duds,t≥0(1)
Ù¥B´˜‡d-‘IO ÙK$ħf´LipschitzëY"X
t
éAàÜÔ3žmt¤3 ˜"Š
ö‰Ñ3˜½^‡e3žmt→∞ž'u)X
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t
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t
†àÜÔ3tž•¤3 ˜k'"
DOI:10.12677/aam.2023.1252292236A^êÆ?Ð
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1995c§CranstonÚLeJan [3]*ÐT.§ïá¤¢gáÚ*ÑVg§¿…AOï
ÄXeü«˜‘œ/µ
(i)‚5gpœ/
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t
= B
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t
0
Z
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(ii)~gpœ/
X
t
= B
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+νt−σ
Z
t
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Z
s
0
sign(X
s
−X
u
)duds,t≥0(3)
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XJé¼êfØŠ?Û•½§@o•§(1)½Â˜‡gp*ÑL§"XJé?¿x∈R§
f÷vx·f(x) ≥0(†óƒ§§•–•ulÙƒcˆL ˜)§K¡•§(1))•gü½"
XJé?¿x∈R§f÷vx·f(x)≤0(†óƒ§§•–•u‚CÙƒcˆL ˜)§K¡•
§(1))•gáÚ"Š5¿´§ù«.Œ±'[•˜‡Ornstein-UhlenbeckL§§Ïd§
ïÄùa•§ìC1•†ëêO½N´ék¿Â"'ugü½ÚgáÚ*Ñ?˜ÚïÄ
Œë„©z[4]![5]![6]Ú[7]§'u˜„gp*ÑïČ넩z[8]![9]![10]Ú[11]"
2002c,Bena¨ım<[4]•Ä•6uòÈÿÝgp*Ñ"•§Xeµ
dX
t
=
√
2dB
t
−

1
t
Z
t
0
∇W(X
t
−X
s
)ds

dt,
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¤£‘رt"3Nõœ¹e§T *ÑL§Œ±†Ornstein-UhlenbeckL§ƒ'§ùŒ±•
ÄÙìC1•"
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K$Ä°Ä[µ
X
H
t
= B
H
t
−θ
Z
t
0
Z
s
0
(X
H
s
−X
H
u
)duds+νt(4)
Ù¥θ<0§B
H
´Hurst•ê÷v
1
2
≤H<1©êÙK$Ä"θ>0ž§Yan<[12]y²Ñ
tª•uឧþ㕧)Âñ53þ•ÚA7,^‡eÑ´¤á§¿…Ù)Âñ
˜‡‘ÅCþ"SunÚYan[13]q3dÄ:þéθÚν?1ëêO"
,˜•¡§32015c§Bena¨ım<[4]ïÄ±e/ªgü½p§
X
t
= B
t
+
Z
t
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s
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0
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s
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u
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+Feller5ŸÚØCÿÝ"
DOI:10.12677/aam.2023.1252292237A^êÆ?Ð
š§An"
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´˜‡Hurst•ê•
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H
t
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Z
t
0
Z
s
0
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H
s
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H
u
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HerrmannandScheutzow[7]§MountfordandP.Tarr´es[11]§SunandYan [13]±9ƒ'©z"
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ٵ
X
H
t
= B
H
t
+σ
Z
t
0
X
H
s
ds+νt−θ
Z
t
0
Z
s
0
(X
H
s
−X
H
u
)duds(5)
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H
´Hurst•ê÷v
1
2
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Mishura [19],Nualart [20],Nourdin [21],Tudor[22].3©¥·‚©ªb½H∈(0,1)´?¿‰
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H
,P)þ"þŠpd L§¡•Hurst•ê•H©êÙK$
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H
0
= 0±9
E

W
H
t
W
H
s

=
1
2

t
2H
+s
2H
−|t−s|
2H

,t,s≥0.
H´d«5¼ê{1
[0,t]
,t∈[0,T]}¤)¤‚5˜mE'uXeSÈzµ
h1
[0,s]
,1
[0,t]
i
H
=
1
2

t
2H
+s
2H
−|t−s|
2H

.

1
2
<H<1ž,§Œ¤
H= {ϕ: [0,T] →R|kϕk
H
<∞},
DOI:10.12677/aam.2023.1252292238A^êÆ?Ð
š§An"
Ù¥
kϕk
2
H
:= α
H
Z
T
0
Z
T
0
ϕ(s)ϕ(r)|s−r|
2H−2
dsdr
…α
H
= H(2H−1).½ÂNXeµ
1
[0,t]
7→W
H
(1
[0,t]
) :=
Z
T
0
1
[0,t]
dW
H
s
= W
H
t
,t∈[0,T]
ù‡NŒ±‚5*ÜEþµ
W
H
(ϕ) =
Z
T
0
ϕ(t)dW
H
t
.
KT‚5N´lEdW
H
)¤pd˜m˜‡åN¿…§Œ±òÿ Hþ.TN
¡•'uW
H
WienerÈ©.XJéz˜‡T>0,k
kϕk
2
H
:= α
H
Z
∞
0
Z
∞
0
ϕ(t)ϕ(s)|t−s|
2H−2
dsdt<∞,
K·‚Œ±½ÂÈ©µ
Z
∞
0
ϕ(t)dW
H
t
,
ùž,WienerÈ©
R
T
0
ϕ(t)dW
H
t
¡•Ø½È©.éuHurst•ê•H ∈(0,1)©êÙK$
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n
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i
∈H.é?¿F∈S,·‚½ÂSþêŽ
fD
H
(Malliavinê)Xeµ
D
H
F=
n
X
j=1
∂f
∂x
j
(W
H
(ϕ
1
),W
H
(ϕ
2
),...,W
H
(ϕ
n
))ϕ
j
.
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H
´lL
2
(Ω)L
2
(Ω;H)˜‡Œ4Žf.·‚^D
1,2
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1,2
:=
q
E|F|
2
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H
Fk
2
H
4•.Pδ
H
´êŽfD
H
ÝŽf,·‚r§¡•ÑÝŽf.•Ò´`·‚¡‘ÅC
þu∈L
2
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H
).eé?¿F∈Sk
E
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H
F,ui
H
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L
2
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,
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1,2
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(u)dXeéó'X½Â
E

Fδ
H
(u)

= EhD
H
F,ui
H
.(7)
DOI:10.12677/aam.2023.1252292239A^êÆ?Ð
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·‚kD
1,2
⊂Dom(δ
H
).¿…
1
2
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1,2
,k
E

δ
H
(u)
2

= Ekuk
2
H
+E
Z
[0,T]
4
D
H
ξ
u
r
D
H
η
u
s
φ(η,r)φ(ξ,s)dsdrdξdη.(8)
·‚ò¦^XePÒL«'uL§uSkorohodÈ©
δ
H
(u) =
Z
T
0
u
s
dW
H
s
,
…ؽȩ½Â•
R
t
0
u
s
dB
H
s
= δ
H
(u1
[0,t]
).·‚•Œ±½Âf
n
∈H
⊗n
'uW
H
-È©I
n
(f
n
),•
[SNë„Nualart9Ortiz-Latorre[23],Nualart-Peccati [24].
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K
H
(t,s) = Γ(H+
1
2
)
−1
(t−s)
H−
1
2
F(H−
1
2
,
1
2
−H,H+
1
2
,1−
t
s
),
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s,t≥0,•¼êR
H
(t,s)•:
R
H
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t∧s
0
K
H
(t,r)K
H
(s,r)dr.
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2
([0,T])‚5ŽfK
∗
H
:
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∗
H
ϕ)(s) = K
H
(T,s)ϕ(s)+
Z
T
s
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∂K
H
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∗
H
ϕ,K
∗
H
ψi
L
2
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H
.
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2
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H
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[0,t]
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
K
∗
H
1
[0,t]
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H
t
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Z
t
0
K
H
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s
.(10)
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H
ƒ'L
2
([0,T])þŽfK
H
´lL
2
([0,T])I
H+
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2
0
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2
([0,T]))ÓŽf¿
…0 ≤H≤
1
2
,§kXeLã/ª:
(K
H
h)(s) = I
2H
0
+
s
1
2
−H
I
1
2
−H
0
+
s
H−
1
2
h,h∈L
2
([0,T])
,,
1
2
≤H≤1,k
DOI:10.12677/aam.2023.1252292240A^êÆ?Ð
š§An"
(K
H
h)(s) = I
1
0
+
s
H−
1
2
I
H−
1
2
0
+
s
1
2
−H
h,h∈L
2
([0,T]),
Ù¥I
α
a+
L«†ý©êRiemann-LiouvilleÈ©…f∈L
1
([a,b])
êα>0,x∈(a,b),a,b∈Rk
I
α
a+
f(x) =
1
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Z
x
a
f(y)
(x−y)
1−α
dy
Ù¥Γ•Gamma¼ê.Ïdé?¿h∈I
H+
1
2
0
+
(L
2
[0,T]),0≤H≤
1
2
ž,_ŽfK
−1
H
äk±e
/ªµ
(K
−1
H
h)(s) = s
1
2
−H
D
1
2
−H
0
+
s
H−
1
2
D
2H
0+
h

1
2
≤H≤1ž,_Žf/ªXe:
(K
−1
H
h)(s) = s
H−
1
2
D
H−
1
2
0
+
s
1
2
−H
h
0
,
Ù¥D
α
a
+
´†ýRiemannian-LiouvilleêŽf…f∈I
α
a+
(L
2
)ê•α∈(0,1),½ÂXe:
D
α
a+
f(x) =
1
Γ(1−α)
d
dx
Z
x
a
f(y)
(x−y)
α
dy.
•õ'u©ê‡È©•£Œë„Samko<[26].3‰ÑëêO(؃c,·‚k£'u
Xe•§®²‰ÑÜ©)5Ÿ
X
H
t
= B
H
t
+σ
Z
t
0
X
H
s
ds+νt−θ
Z
t
0
Z
s
0
(X
H
s
−X
H
u
)duds,
Ún1.bθ<0¿…
1
2
≤H<1"½ÂL§
ξ
H
t
:=
Z
t
0
(s−
σ
θ
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1
2
θs
2
−σs
dB
H
s
,t≥0.
tª•uឧkξ
H
t
→ξ
H
∞
:=
R
∞
0
(s−
σ
θ
)e
1
2
θs
2
−σs
dB
H
s
3L
2
ÚA??¿Âe¤á"?˜Ú
/§½Â
Ψ
H
t
(θ,σ) :=
Z
t
0
e
−
1
2
θu
2
+σu

ξ
H
∞
−ξ
H
u

du,t≥0.
Ké?¿γ≥0§tª•uឧk
t
γ
e
1
2
θt
2
−σt
Ψ
H
t
(θ,σ) −→0(11)
3L
2
ÚA??¿Âe¤á"
3.eZO
3!¥§·‚ò‰Ñ˜O"•{ü儧bθ<0§C´˜‡ŒU•6†H!θ!
νÚσ~꧿…§Š3ØÓœ/eŒUØÓ§é~þc•ŠÓb"
DOI:10.12677/aam.2023.1252292241A^êÆ?Ð
š§An"
Ún2.bX
H
••§7)…θ<0,
1
2
≤H<1,s,t∈[0,T]§Kk
c|t−s|
2H
≤E[(X
H
t
−X
H
s
)
2
] ≤C|t−s|
2H
(12)
y².·‚Œ±uy§évϑ∈(0,H)§L§t7→X
H
t
´k.
1
H−ϑ
C"Ïd§X
Ju´k.p−C§…1 ≤p<
1
1−H+ϑ
§@oYoungÈ©
Z
t
0
u
s
dX
H
s
= u
t
X
H
t
−u
0
X
H
0
−
Z
t
0
X
H
s
du
s
•3§qÏ•
Y
H
t
=
Z
t
0
(u−
σ
θ
)dX
H
u
,t≥0
9
X
H
t
= B
H
t
−θ
Z
t
0
Y
H
s
ds+νt,t≥0.(13)
Ïd§d©ÜÈ©úª§é?¿t≥0§k
Y
H
t
= (t−
σ
θ
)X
H
t
−
Z
t
0
X
H
s
ds=
Z
t
0
(u−
σ
θ
)dX
H
u
,t≥0.
(Ü13§
dY
H
t
= −θ(t−
σ
θ
)Y
H
t
dt+(t−
σ
θ
)dB
H
t
+ν(t−
σ
θ
)dt,t≥0.(14)
ÏL~êC´{§·‚Œ±b½L§
Y
H
t
= C
H
t
e
−
1
2
θt
2
+σt
12œ)
Ù¥C
H
0
= Y
H
0
= 0§KŠâ12§·‚k
e
−
1
2
θt
2
+σt
dC
H
t
= (t−
σ
θ
)dB
H
t
+ν(t−
σ
θ
)dt,t≥0.
k
C
H
t
=
Z
t
0
(s−
σ
θ
)e
1
2
θs
2
−σs
dB
H
s
+ν
Z
t
0
(s−
σ
θ
)e
1
2
θs
2
−σs
ds
=
Z
t
0
(s−
σ
θ
)e
1
2
θs
2
−σs
dB
H
s
+
ν
θ
(e
−
1
2
θt
2
+σt
−1),t≥0
Kk
Y
H
t
= e
−
1
2
θt
2
+σt
Z
t
0
(s−
σ
θ
)e
1
2
θs
2
−σs
dB
H
s
+
ν
θ
(1−e
−
1
2
θt
2
+σt
),t≥0(15)
•ÄL§ξ
t
:=
R
t
0
(s−
σ
θ
)e
1
2
θs
2
−σs
dB
H
s
,t≥0§K
Y
H
t
= e
−
1
2
θt
2
+σt
ξ
t
+
ν
θ
(1−e
−
1
2
θt
2
+σt
),t≥0(16)
DOI:10.12677/aam.2023.1252292242A^êÆ?Ð
š§An"
…kXeOµ
E(ξ
t
−ξ
s
)
2
≤C
H,θ,σ
(t−s)
2
H,t>s≤0.(17)
¯¢þ§H=
1
2
ž§w,k
E(ξ
t
−ξ
s
)
2
=
Z
t
s
(r−
σ
θ
)
2
e
θr
2
−2σr
dr≤θ
−1
(t−s),t>s≥0.

1
2
<H<1§é?¿t>s≥0§
CaseI:σ≥0¿…
σ
θ
≤s<t.
E

Z
t
s
(r−
σ
θ
)e
θr
2
−2σr
dB
H
r

2
= α
H
·
Z
t
s
Z
t
s
(u−
σ
θ
)(v−
σ
θ
)|u−v|
2H−2
e
1
2
θ(u
2
+v
2
)−σ(u+v)
dudv
≤(t−
σ
θ
)e
θt
2
−2σt
·α
H
Z
t
s
Z
t
s
|u−v|
2H−2
dudv
≤C
θ,σ
(t−s)
2H
dž¤á¶
CaseII:σ<0¿…0 <s<t≤
σ
θ
.
E

Z
t
s
(r−
σ
θ
)e
θr
2
−2σr
dB
H
r

2
= α
H
·
Z
t
s
Z
t
s
(u−
σ
θ
)(v−
σ
θ
)|u−v|
2H−2
e
1
2
θ(u
2
+v
2
)−σ(u+v)
dudv
≤
σ
2
θ
e
σ
2
θ
·α
H
Z
t
s
Z
t
s
|u−v|
2H−2
dudv
≤C
θ,σ
(t−s)
2H
dž¤á¶
CaseIII:σ<0¿…0 <s<
σ
θ
≤t.
E

Z
t
s
(r−
σ
θ
)e
θr
2
−2σr
dB
H
r

2
= α
H
·
Z
t
s
Z
t
s
(u−
σ
θ
)(v−
σ
θ
)|u−v|
2H−2
e
1
2
θ(u
2
+v
2
)−σ(u+v)
dvdu
= α
H
Z
t−
σ
θ
s−
σ
θ
s−
σ
θ
t−
σ
θ
uv|u−v|
2H−2
e
1
2
θ(u
2
+v
2
)
dvdu
≤α
H
Z
t−
σ
θ
0
Z
t−
σ
θ
0
uv|u−v|
2H−2
e
1
2
θ(u
2
+v
2
)
dvdu
+
Z
0
s−
σ
θ
Z
0
s−
σ
θ
uv|u−v|
2H−2
e
1
2
θ(u
2
+v
2
)
dvdu
!
≤C
θ,σ

(t−
σ
θ
)
2H
+(
σ
θ
−s)
2H

≤C
θ,σ
(t−s)
2H
y."
DOI:10.12677/aam.2023.1252292243A^êÆ?Ð
š§An"
Ún3.b
1
2
≤H<1§é?Ûk•êp≥1§Tª•uጞ§A??k
te
θt
2
−2σt
Z
t
0
e
−θs
2
+2σs
ξ
p
s
ds→
1
2θ
ξ
p
∞
.
y².ŠâÚn1Œ•§é?¿
1
2
≤H<1§‘ÅCþξ
∞
Ñl"þŠ©Ù§Ïd§
P(ξ
∞
6= 0) = 1
dL§{ξ
t
,t≥0}ëY5§Œ
lim
t→∞
inf
1
2
t≤s≤t
ξ
s
= ξ
∞
a.s.
Ïd§tª•uጞ,
Z
t
0
e
−θs
2
+2σs
ξ
p
s
ds→∞.
ÏL$^â7ˆ£L’Hopital¤{K§tª•uጞ§k
lim
t→∞
R
t
0
e
−θs
2
+2σs
ξ
p
s
ds
t
−1
e
−θt
2
+2σt
=lim
t→∞
e
−θt
2
+2σt
ξ
p
t
2θ(t−
σ
θ
)t
−1
e
−θt
2
+2σt
=
1
2θ
ξ
p
∞
y."
Ún4.-
1
2
<H<1"tª•uឧ
(t−
σ
θ
)
2H
e
θt
2
−2σt
Z
t
0
Z
t
0
e
−
1
2
θ(s
2
+r
2
)+σ(s+r)
|s−r|
2H−2
dsdr→θ
−2H
Γ(2H−1).
y².´•§é¤kšKëY¼êf§4•
lim
t→∞
Z
t
0
f(x)dx
•3…k•§K
lim
t→∞
Z
t
0
f(x)
dx
p
1−
x
t
=lim
t→∞
Z
t
0
f(x)dx(18)
âd§dâ7ˆ{K§$^CþO†−
1
2
θ(t
2
−r
2
)+σ(t−r) = xÚ››Âñ½n§Œ
DOI:10.12677/aam.2023.1252292244A^êÆ?Ð
š§An"
lim
t→∞
(t−
σ
θ
)
2H
e
θt
2
−2σt
Z
t
0
Z
t
0
e
−
1
2
θ(s
2
+r
2
)+σ(s+r)
|s−r|
2H−2
dsdr
= 2lim
t→∞
1
(t−
σ
θ
)
−2H
e
−θt
2
+2σt
Z
t
0
e
−
1
2
θs
2
+σs
Z
s
0
e
−
1
2
θr
2
+σr
|s−r|
2H−2
dsdr
=lim
t→∞
1
θ(t−
σ
θ
)
1−2H
e
−
1
2
θt
2
+σt
Z
t
0
e
−
1
2
θr
2
+σr
(t−r)
2H−2
dr
= θ
−1
lim
t→∞
(t−
σ
θ
)
2H−1
e
1
2
θ(t
2
−r
2
)−σ(t−r)
(t−r)
2H−2
dr
= θ
−2
lim
t→∞
(t−
σ
θ
)
2H−1
Z
−
1
2
θt
2
+σt
0
e
−x
(t−
σ
θ
)−
r
(t−
σ
θ
)
2
−
2x
θ
!
2H−2
dx
q
(t−
σ
θ
)
2
−
2x
θ
= 2
2H−2
θ
−2H
lim
t→∞
(t−
σ
θ
)
2H−1
Z
−
1
2
θt
2
+σt
0
e
−x
(t−
σ
θ
)+
r
(t−
σ
θ
)
2
−
2x
θ
!
2−2H
x
2H−2
dx
q
(t−
σ
θ
)
2
−
2x
θ
= 2
2H−2
θ
−2H
lim
t→∞
Z
−
1
2
θt
2
+σt
0
e
−x
1+
s
1+
2x
θ(t−
σ
θ
)
2
!
2−2H
x
2H−2
dx
q
1−
2x
θ(t−
σ
θ
)
2
= θ
−2H
Γ(2H−1)
y."
Ún5.b
1
2
≤H<1§Ké?¿χB
H
t
,t≥0Œÿ¿÷vP(F<∞)=1‘ÅCþF§
tª•uጞ§•©Ùkµ
(F,(t−
σ
θ
)
H
e
1
2
θt
2
−σt
Z
t
0
e
−
1
2
θs
2
+σs
dB
H
s
) →(F,
p
HΓ(2H)θ
−H
N),(19)
Ù¥N´ÕáuB
H
IO‘ÅCþ"
y².w,§é?¿t>0§k
(t−
σ
θ
)
H
e
1
2
θt
2
−σt
Z
t
0
e
−
1
2
θs
2
+σs
dB
H
s
= Nχ
H,σ,θ
(t)
ùpÒ/=0L«•©Ùƒ§N´˜‡IO‘ÅCþ…
χ
2
H,σ,θ
(t) = (t−
σ
θ
)
2H
e
θt
2
−2σt
E(
Z
t
0
e
−
1
2
θs
2
+σs
dB
H
s
)
2
= α
H
(t−
σ
θ
)
2H
e
θt
2
−2σt
Z
t
0
Z
t
0
e
−
1
2
θ(s
2
+r
2
)+σ(s+r)
|s−r|
2H−2
dsdr,t>0.
2dÚn4§
(t−
σ
θ
)
H
e
1
2
θt
2
−σt
Z
t
0
e
−
1
2
θs
2
+σs
dB
H
s
→
p
HΓ(2H)θ
−H
N,t→∞.
du19ü>þÑl‘©Ù§Šâ©z[27]§=Iy²é?¿d≥1,s
1
,...,s
d
∈
DOI:10.12677/aam.2023.1252292245A^êÆ?Ð
š§An"
[0,∞)§tª•uጞ§•©Ùk
(B
H
s
1
,...,B
H
s
d
,(t−
σ
θ
)
H
e
1
2
θt
2
−σt
Z
t
0
e
−
1
2
θs
2
+σs
dB
H
s
) →(B
H
s
1
,...,B
H
s
d
,θ
−H
p
HΓ(2H)N).(20)
•20§=Iy²Ù•ÝÂñuƒA(J=Œ"
H=
1
2
ž§y²'{ü§•‡•Ä
1
2
<H<1"é?¿½s>0§k
E

B
H
s
·(t−
σ
θ
)
H
e
1
2
θt
2
−σt
Z
t
0
e
−
1
2
θr
2
+σr
dB
H
r

= α
H
(t−
σ
θ
)
H
e
1
2
θt
2
−σt
Z
t
0
e
−
1
2
θv
2
+σv
dv
Z
s
0
|u−v|
2H−2
du
= α
H
(t−
σ
θ
)
H
e
1
2
θt
2
−σt
Z
s
0
e
−
1
2
θv
2
+σv
dv
Z
s
0
|u−v|
2H−2
du
+α
H
(t−
σ
θ
)
H
e
1
2
θt
2
−σt
Z
t
s
e
−
1
2
θv
2
+σv
dv
Z
s
0
|u−v|
2H−2
du
=: η
1
(t)+η
2
(t)
w,§t→∞ž§η
1
(t) →0"$^â7ˆ{K§t→∞ž§é?¿s>0§k
0 <η
2
(t) = H(t−
σ
θ
)
H
e
1
2
θt
2
−σt
Z
t
s
e
−
1
2
θv
2
+σv
[v
2H−1
−(v−s)
2H−1
]dv
≤Hs
2H−1

(t−
σ
θ
)
H
e
1
2
θt
2
−σt
Z
t
s
e
−
1
2
θv
2
+σv
dv

→0
l§é?¿s>0§k
lim
t→∞
E

B
H
s
·(t−
σ
θ
)
H
e
1
2
θt
2
−σt
Z
t
0
e
−
1
2
θr
2
+σr
dB
H
r

= 0
y."
Ún6.é
1
2
<H<1§tª•uጞ§
(t−
σ
θ
)
H
e
1
2
θt
2
−σt
Z
t
0
(s−
σ
θ
)e
1
2
θs
2
−σs
δB
H
s
Z
s
0
e
−
1
2
θr
2
+σr
δB
H
r
L
2
→0(21)
(t−
σ
θ
)
H
e
1
2
θt
2
−σt
Z
t
0
(s−
σ
θ
)e
1
2
θs
2
−σs
ds
Z
s
0
e
−
1
2
θr
2
+σr
|s−r|
2H−2
dr→0(22)
y².Âñ522´w,§e¡y²Âñ521"d-È©åúªŒ§é?¿t>0§
k
DOI:10.12677/aam.2023.1252292246A^êÆ?Ð
š§An"
E

(t−
σ
θ
)
H
e
1
2
θt
2
−σt
Z
t
0
(s−
σ
θ
)e
1
2
θs
2
−σs
(
Z
s
0
e
−
1
2
θr
2
+σr
δB
H
r
)δB
H
s

2
= (t−
σ
θ
)
2H
e
θt
2
−2σt
E

Z
t
0
(s−
σ
θ
)e
1
2
θs
2
−σs
(
Z
s
0
e
−
1
2
θr
2
+σr
δB
H
r
)δB
H
s

2
= (α
H
)
2
(t−
σ
θ
)
2H
e
θt
2
−2σt
Z
t
0
(s−
σ
θ
)e
1
2
θs
2
−σs
ds
Z
s
0
e
−
1
2
θx
2
+σx
dx
·
Z
t
0
(r−
σ
θ
)e
1
2
θr
2
−σr
dr
Z
r
0
dye
−
1
2
θy
2
+σy
·(|s−y|
2H−2
|r−x|
2H−2
+|s−r|
2H−2
|x−y|
2H−2
).
2dØª
Z
s
0
dξ
Z
r
0
|r−ξ|
2H−2
|s−η|
2H−2
dy≤
2
(2H−1)
2
r
2H−1
s
2H−1
Œ§tª•uឧ
E

(t−
σ
θ
)
H
e
1
2
θt
2
−σt
Z
t
0
(s−
σ
θ
)e
1
2
θs
2
−σs
(
Z
s
0
e
−
1
2
θr
2
+σr
δB
H
r
)δB
H
s

2
≤C
H
(t−
σ
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H−1
e
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2
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T
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p
HΓ(2H)
N
ξ
∞
−
ν
θ
(25)
T
1+H
(ˆν
T
−ν−
B
H
T
T
) →2(
p
HΓ(2H)θ
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)N(26)
T
1−H
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T
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DOI:10.12677/aam.2023.1252292247A^êÆ?Ð
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Φ
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t
0
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H
t
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Y
H
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T
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T
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t
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H
T
Z
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0
Y
H
t
dt)
1
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T
0
Y
H
t
dB
H
t
−B
H
T
Z
T
0
Y
H
t
dt)
1
Φ
T
(28)
¿…
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1
T
B
H
T
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θ
T
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Z
T
0
Y
H
t
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θ
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2
(T→∞)(30)
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T
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e
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2
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Z
T
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Y
H
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H
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Z
T
0
Y
H
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dt) →0a.s.(31)
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H
T
T
→0a.s.(32)
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T
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e
1
2
θT
2
−σT
(B
H
T
Z
T
0
Y
H
t
dt) =
B
H
T
T
·(T
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e
1
2
θT
2
−σT
Z
T
0
Y
H
t
dt) →0a.s.(33)
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T
Z
T
0
Y
H
t
dB
H
t
= TY
T
B
H
T
−T
Z
H
0
B
H
t
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H
t
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T
B
H
T
−T
Z
H
0
B
H
t
d
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e
−
1
2
θt
2
+σt
ξ
t
+
ν
θ
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−
1
2
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2
+σt
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T
B
H
T
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Z
T
0
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θ
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2
θt
2
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B
H
t
ξ
t
dt
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Z
T
0
e
−
1
2
θt
2
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B
H
t
dξ
t
−νT
Z
T
0
(t−
σ
θ
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−
1
2
θt
2
+σt
B
H
t
dt
= TY
T
B
H
T
−θT
Z
T
0
(t−
σ
θ
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−
1
2
θt
2
+σt
B
H
t
ξ
t
dt
−T
Z
T
0
tB
H
t
dB
H
t
−νT
Z
T
0
(t−
σ
θ
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−
1
2
θt
2
+σt
B
H
t
dt
DOI:10.12677/aam.2023.1252292248A^êÆ?Ð
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d16ÚÚn3¿$^â7ˆ{K§Œ±y²Tª•uጞ§eÂñ5±VÇ1¤áµ
T
−2
e
1
2
θt
2
−σt
(TY
T
B
H
T
) =
B
H
T
T
(e
1
2
θt
2
−σt
Y
T
) →0,
T
−2
e
1
2
θt
2
−σt
(T
Z
T
0
(t−
σ
θ
)e
−
1
2
θt
2
+σt
B
H
t
ξ
t
dt) = T
−1
e
1
2
θt
2
−σt
Z
T
0
(t−
σ
θ
)B
H
t
ξ
t
dt→0,
T
−2
e
1
2
θt
2
−σt
(T
Z
T
0
(t−
σ
θ
)e
−
1
2
θt
2
+σt
B
H
t
dt) = T
−1
e
1
2
θt
2
−σt
Z
T
0
(t−
σ
θ
)e
−
1
2
θt
2
+σt
B
H
t
dt→0.
d§Tª•uጞ§•k
T
−1
e
1
2
θt
2
−σt
Z
T
0
tB
H
t
dB
H
t
) =
1
2
T
−1
e
1
2
θt
2
−σt
Z
T
0
td(B
H
t
)
2
=
1
2
T
−1
e
1
2
θt
2
−σt
(T(B
H
T
)
2
−
Z
T
0
(B
H
t
)
2
dt)
→0a.s.
l
lim
T→∞
e
θt
2
−2σt
(T
Z
T
0
Y
H
t
dB
H
t
) =lim
T→∞
(T
2
e
1
2
θT
2
−σT
)T
−2
e
1
2
θt
2
−σt
(T
Z
T
0
Y
H
t
dB
H
t
)
= 0.
(34)
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θ
T
−θ=
1
e
θT
2
−2σT
Φ
T
e
θT
2
−2σT
(T
Z
T
0
Y
H
t
dB
H
t
−B
H
T
Z
T
0
Y
H
t
dt)
→0a.s.
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Te
1
2
θt
2
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Z
T
0
Y
H
t
dt= Te
1
2
θt
2
−σt

Z
T
0
e
−
1
2
θt
2
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ξ
t
dt−
ν
θ
Z
T
0
e
−
1
2
θt
2
+σt
dt+
ν
θ
T

→
1
θ
(ξ
∞
−
ν
θ
)a.s.
(35)
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DOI:10.12677/aam.2023.1252292249A^êÆ?Ð
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ˆν
T
−ν=
1
T
(X
H
T
+
ˆ
θ
T
Z
T
0
Y
H
t
dt−Tν)
=
1
T
B
H
t
−(θ−
ˆ
θ
T
)
1
T
Z
T
0
Y
H
t
dt
=
1
T
B
H
t
−
1
e
θT
2
−2σT
Φ
T
T
−2
e
1
2
θT
2
−σT
·

T
Z
T
0
Y
H
t
dB
H
t
−B
H
T
Z
T
0
Y
H
t
dt

Te
θT
2
−2σT
Z
T
0
Y
t
dt

→0a.s.
(36)
y."
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T
H−1
e
−
1
2
θT
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θ
T
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1
Φ
T
T
H
e
−
1
2
θT
2
+σT
Z
T
0
Y
H
t
dt−T
H−1
e
−
1
2
θT
2
+σT
1
Φ
T
B
H
T
Z
T
0
Y
H
t
dt
≡
1
Φ
T
(Υ
1
(T)−Υ
2
(T)).
(37)
w,§dL«16ÚÚn3Œ
e
θT
2
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e
1
2
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H
T
Z
T
0
Y
H
t
dt
=
B
H
T
T
2−H

Te
1
2
θT
2
−σT
Z
T
0
e
−
1
2
θT
2
+σT
ξ
t
dt+Te
1
2
θT
2
−σT
Z
T
0
ν
θ
(1−e
−
1
2
θt
2
+σt
)dt

→0(T→∞)
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1
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Z
T
0
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θ
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1
2
θs
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Z
s
0
e
−
1
2
θt
2
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t

dB
H
s
=
Z
T
0
(s−
σ
θ
)e
1
2
θs
2
−σs

Z
s
0
e
−
1
2
θt
2
+σt
δB
H
t

dB
H
s
=
Z
T
0
(s−
σ
θ
)e
1
2
θs
2
−σs

Z
s
0
e
−
1
2
θt
2
+σt
δB
H
t

δB
H
s
·
Z
T
0
(s−
σ
θ
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1
2
θs
2
−σs
Z
T
0
D
H
r
(
Z
s
0
e
−
1
2
θt
2
+σt
δB
H
t
)|s−r|
2H−2
dsdr
=
Z
T
0
(s−
σ
θ
)e
1
2
θs
2
−σs

Z
s
0
e
−
1
2
θt
2
+σt
δB
H
t

δB
H
s
+α
H
Z
T
0
(s−
σ
θ
)e
1
2
θs
2
−σs
(s−r)
2H−2
dsdr.
DOI:10.12677/aam.2023.1252292250A^êÆ?Ð
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dd§é?¿T≥0§k
Z
T
0
e
−
1
2
θt
2
+σt
ξ
t
dB
H
t
=
Z
T
0
e
−
1
2
θt
2
+σt

Z
t
0
(s−
σ
θ
)e
1
2
θs
2
−σs
dB
H
s

dB
H
t
=
Z
T
0
e
−
1
2
θt
2
+σt

Z
T
0
(s−
σ
θ
)e
1
2
θs
2
−σs
dB
H
s

dB
H
t
−
Z
T
0
e
−
1
2
θt
2
+σt

Z
T
t
(s−
σ
θ
)e
1
2
θs
2
−σs
dB
H
s

dB
H
t
= ξ
T
Z
T
0
e
−
1
2
θt
2
+σt
dB
H
t
−
Z
T
0
(s−
σ
θ
)e
1
2
θs
2
−σs

Z
s
0
e
−
1
2
θt
2
+σt
dB
H
t

dB
H
s
= ξ
T
Z
T
0
e
−
1
2
θt
2
+σt
dB
H
t
−
Z
T
0
(s−
σ
θ
)e
1
2
θs
2
−σs

Z
s
0
e
−
1
2
θt
2
+σt
δB
H
t

δB
H
s
−α
H
Z
T
0
(s−
σ
θ
)e
1
2
θs
2
−σs
Z
s
0
e
−
1
2
θr
2
+σr
(s−r)
2H−2
dsdr.
(ÜÚn1!5!6!Âñ530ÚSlutsky½nŒ§Tª•uጞ§
Υ
1
(T)
1
Φ
T
= T
H
e
1
2
θT
2
−σT

Z
T
0
e
−
1
2
θt
2
+σt
ξ
t
dB
H
t
+
ν
θ
B
H
T
−
ν
θ
Z
T
0
e
−
1
2
θt
2
+σt
dB
H
t

1
e
θT
2
−2σT
Φ
T
= (ξ
T
−
ν
θ
)

T
H
e
1
2
θT
2
−σT
Z
T
0
e
−
1
2
θt
2
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dB
H
t

1
e
θT
2
−2σT
Φ
T
−T
H
e
1
2
θT
2
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
−
ν
θ
B
H
T
+
Z
T
0
(s−
σ
θ
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1
2
θs
2
−σs

Z
s
0
e
−
1
2
θt
2
+σt
δB
H
t

δB
H
s
+α
H
T
H
e
−
1
2
θT
2
+σT
Z
T
0
(s−
σ
θ
)e
1
2
θs
2
−σs
Z
s
0
e
−
1
2
θr
2
+σr
(s−r)
2H−2
dsdr

1
e
θT
2
−2σT
Φ
T
→2θ
1−H
p
HΓ(2H)
N
ξ
∞
−
ν
θ
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T
1+H
( ˆν
T
−ν−
1
T
B
H
T
) = T
1+H
(X
H
T
−
ˆ
θ
T
Z
T
0
Y
H
t
dt−Tν)
=

T
H−1
e
−
1
2
θT
2
+σT
(
ˆ
θ
T
−θ)


Te
1
2
θT
2
−σT
Z
T
0
Y
t
dt

→2
p
HΓ(2H)θ
−H
N,T→∞.
(Ü35-37ŒyÂñ526¤á"
DOI:10.12677/aam.2023.1252292251A^êÆ?Ð
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•§Šâ29!35Ú37Œ±y²Tª•uጞ§
T
1−H
( ˆν
T
−ν) =
T
H
B
H
T
+{T
H−1
e
−
1
2
θT
2
+σT
(
ˆ
θ
T
−θ)}·{T
1−2H
e
1
2
θT
2
−σT
Z
T
0
Y
H
t
dt}
→M∼N(0,1)
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[1]Durrett,R. andRogers,L.C.G.(1991)Asymptotic Behavior of Brownian Polymer. Probability
TheoryandRelatedFields,92,337-349.https://doi.org/10.1007/BF01300560
[2]Coppersmith,D.andDiaconis,P.(1986)RandomWalkswithReinforcement.Unpublished
manuscript.
[3]Cranston,M.andLeJan,Y.(1995)Self-AttractingDiffusions:TwoCaseStudies.Mathema-
tischeAnnalen,303,87-93.https://doi.org/10.1007/BF01460980
[4]Bena¨ım,M.,Ciotir,I.andGauthier,C.-E.(2015)Self-RepellingDiffusionsviaanInfinite
Dimensional Approach. StochasticPartialDifferentialEquations:AnalysisandComputations,
3,506-530.https://doi.org/10.1007/s40072-015-0059-5
[5]Gauthier,C.-E.(2016)SelfAttractingDiffusionsonaSphereandApplicationtoaPeriodic
Case.ElectronicCommunicationsinProbability,21,1-12.
https://doi.org/10.1214/16-ECP4547
[6]Herrmann,S. andRoynette, B.(2003) BoundednessandConvergence ofSome Self-Attracting
Diffusions.MathematischeAnnalen,325,81-96.https://doi.org/10.1007/s00208-002-0370-0
[7]Herrmann,S.andScheutzow,M.(2004)RateofConvergenceofSomeSelf-AttractingDiffu-
sions.StochasticProcessesandTheirApplications,111,41-55.
https://doi.org/10.1016/j.spa.2003.10.012
DOI:10.12677/aam.2023.1252292252A^êÆ?Ð
š§An"
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