Fractional calculus and BM Mishura 2021.pdf

Calculo fraccionario

1. Elements of fractional calculus. Fractional Brownian
motion. Wiener integration w.r.t. fBm. Some related
fractional processes
Yuliya Mishura
Taras Shevchenko National University of Kyiv
28 June 2021
Summer School "Stochastic Models and Complex Systems"
Yuliya Mishura Fractional calculus SMOCS 2021

2. Outline
1
The elements of fractional calculus
2
Fractional Brownian motion: denition and elementary properties
3
Mandelbrot-van-Ness representation of fBm
4
Wiener integration with respect to fBm
5
Representation of fBm via Wiener process on a nite interval.
6
Some related fractional processes
Sub-fractional Brownian motion
Bi-fractional Brownian motion
Mixed fractional Brownian motion
Yuliya Mishura Fractional calculus SMOCS 2021

3. Introduction
Fractional calculus is one of the best tools to characterize long-memory
processes and materials, anomalous diusion, long-range interactions,
long-term behaviors, power laws, allometric scaling laws, as well as the
respective short-memory eects, especially in nance. So the
corresponding mathematical models are, on the one hand, fractional
processes, and on the other hand, fractional dierential equations. Of
course, this is a very simplied situation, because the number of fractional
objects is much greater. There can be partial fractional dierential
equations etc. The evolutions of fractional processes behave in a much
more complicated way so to study the corresponding dynamics is much
more dicult.
Yuliya Mishura Fractional calculus SMOCS 2021

4. But the study of such processes and equations is absolutely necessary,
because fractality is inherent in almost all observed phenomena. It
manifests itself in particular in the fact that, for example, the transmission
of cellular signals cannot be described by dierential equations with
derivatives of an integer order, they are not smooth enough for this.
Yuliya Mishura Fractional calculus SMOCS 2021

5. It can be said that the derivatives of the integer order that have been used
to describe classical mechanical systems since Newton's, have become too
much luxury at the present time, when we observe changes that occur too
quickly in order to be residually smooth. So, derivatives and integrals with
integer indices are irreplaceable in describing suciently smooth
phenomena in nance, economics, modern technologies and natural
phenomena (moreover, almost all of the above evolve according to very
similar laws), and they are replaced by fractional order derivatives and
fractional order integrals that describe some quasi-smoothness. Therefore,
it is absolutely necessary to study and apply the elements of fractional
calculus.
Yuliya Mishura Fractional calculus SMOCS 2021

6. Remark 0.1
During the lectures, the technical details of the proofs and technical proofs
will be omitted. We shall discuss only some principal details.
Yuliya Mishura Fractional calculus SMOCS 2021

7. 1
The elements of fractional calculus
2
Fractional Brownian motion: denition and elementary properties
3
Mandelbrot-van-Ness representation of fBm
4
Wiener integration with respect to fBm
5
Representation of fBm via Wiener process on a nite interval.
6
Some related fractional processes
Sub-fractional Brownian motion
Bi-fractional Brownian motion
Mixed fractional Brownian motion
Yuliya Mishura Fractional calculus SMOCS 2021

8. Fractional integrals
Let >0 (in the most cases later <1 but it is not obligatory). Denote
the Riemann{Liouville left- and right- sided a;b) of
orderas the operatorsI

a+andI

b
of the form
(I

a+f)(x) :=
1
()
Z
x
a
f(t)(xt)
1
dt;
(I

b
f)(x) :=
1
()
Z
b
x
f(t)(tx)
1
dt:
We say that the functionf2 D(I

a+(b)
) (symbolD() denotes the domain
of corresponding operator) if the corresponding integrals converge for
almost all (a.a.)x2(a;b) (with respect to (w.r.t.) Lebesgue measure).
Yuliya Mishura Fractional calculus SMOCS 2021

9. Fractional integrals
The Riemann{Liouville left- and right- sided fractional integrals onRare
dened as
(I

+f)(x) :=
1
()
Z
x
1
f(t)(xt)
1
dt;
(I

f)(x) :=
1
()
Z
1
x
f(t)(tx)
1
dt:
The functionf2 D(I

) if the corresponding integrals converge for a.a.
x2R. According to [SKM93],Lp(R) D(I

), 1p<
1

.
Yuliya Mishura Fractional calculus SMOCS 2021

10. Hardy{Littlewood theorem
Moreover, the following Hardy{Littlewood theorem holds.
Theorem 1.1 ([SKM93])
Let1p;q<1,0< <1. Then the operatorsI

are bounded from
Lp(R)toLq(R)if and only if1<p<
1

andq=p(1p)
1
. It means,
in particular, that for any1<p<
1

andq=
p
1p
(or1=p1=q=)
there exists a constantCp;q;such that
Z
R

Z
R
jf(u)jjxuj
1
du

q
dx
1
q
Cp;q;kfk
Lp(R): (1)
Yuliya Mishura Fractional calculus SMOCS 2021

11. Properties of fractional integration
Fractional integration admits the following composition formula:
I

a+I

a+
f=I
+
a+
f;I

b
I

b
f=I
+
b
f
forf2L1[a;b]. If+1 we have these equalities at any point
x2(a;b) otherwise they hold for a. a.x. Also,
I

I


=I
+

f
forf2Lp(R),; >0+ <
1
p
. Forf2Lp[a;b],g2Lq[a;b],p;q1
and
1
p
+
1
q
1 +, wherep>1,q>1 for
1
p
+
1
q
= 1 +we have the
following integration-by-parts formula
Z
b
a
g(x)(I

a+f)(x)dx=
Z
b
a
f(x)(I

b
g)(x)dx:
Letf2Lp(R),g2Lq(R),p>1;q>1,
1
p
+
1
q
= 1 +. Then
Z
R
g(x)(I

+f)(x)dx=
Z
R
f(x)(I

g)(x)dx: (2)
Yuliya Mishura Fractional calculus SMOCS 2021

12. Holder continuous functions
LetC

(T) be set of Holder continuous functionsf:T!Rof order,
i.e.,
C

([a;b]) =
n
f: [a;b]!R


kfk
[a;b];:= sup
t2[a;b]
jf(t)j
+ sup
s;t2[a;b]
jf(s)f(t)j(ts)

<1
o
:
Forp1 denoteI

(Lp(R)) the class of functionsfthat can be presented
as the Riemann{Liouville integralsf=I

'for some'2Lp(R),p1.
Lemma 1.2
If >0,p>1, thenI

(Lp(R))C

([a;b])for any1<a<b<1
and0<
1
p
.
Yuliya Mishura Fractional calculus SMOCS 2021

13. The next result is evident.
Lemma 1.3
Let0< <1,f2Lp(R),1p<
1

andI

f= 0. Thenf(x) = 0for
a.a.x2R.
Yuliya Mishura Fractional calculus SMOCS 2021

14. Riemann{Liouville fractional derivatives
Lemma1.3supplies the uniqueness of such function'that its fractional
integral is some given functionf, and for 0< <1 this function'
coincides for a.a.x2Rwith the left- (right-) sided Riemann{Liouville
fractional derivative fof order:
(I

+
f)(x) = (D

+f)(x) :=
1
(1)
d
dx
Z
x
1
f(t)(xt)

dt;
(I


f)(x) = (D

f)(x) :=
1
(1)
d
dx
Z
1
x
f(t)(tx)

dt:
Yuliya Mishura Fractional calculus SMOCS 2021

15. Riemann{Liouville derivatives on the interval
The Riemann{Liouville fractional derivatives can be considered on any
interval [a;b]Rin the following way: we introduce the classI

(Lp[a;b])
of functionsfthat can be presented asf=I

a+'(f=I

b
') for
'2Lp[a;b],p1, and denote
(I

a+
f)(x) = (D

a+f)(x) =
1
(1)
d
dx
Z
x
a
f(t)(xt)

dt;
(I

b
f)(x) = (D

b
f)(x) =
1
(1)
d
dx
Z
b
x
f(t)(tx)

dt:
Yuliya Mishura Fractional calculus SMOCS 2021

16. Weyl representation
In this case Riemann{Liouville fractional derivativesD

a+fandD

b
fadmit
the following Weyl representation (we suppose thatf= 0 outside (a;b)):
(D

a+f)(x) =
1
(1)
(f(x)(xa)

+
Z
x
a
(f(x)f(t))(xt)
1
dt)1
(a;b)(x);
(D

b
f)(x) =
1
(1)
(f(x)(bx)

+
Z
b
x
(f(x)f(t))(tx)
1
dt)1
(a;b)(x);
where the convergence of the integrals holds pointwise for a.a.x2(a;b)
forp= 1 and inLp[a;b] forp>1.
Yuliya Mishura Fractional calculus SMOCS 2021

17. Properties of fractional derivatives
Composition formula for fractional derivatives has a form
D

a+D

a+
f=D
+
a+
f; (3)
for0,0,f2I
+
a+
(L1(R)):
Yuliya Mishura Fractional calculus SMOCS 2021

18. Integration-by-parts
Also, under the assumptions 0< <1,f2I

a+(Lp[a;b]),
g2I

b
(Lq[a;b]), 1=p+ 1=q1 +we have following
integration-by-parts formula
Z
b
a
(D

a+f)(x)g(x)dx=
Z
b
a
f(x)(D

b
g)(x)dx: (4)
Yuliya Mishura Fractional calculus SMOCS 2021

19. Caputo derivatives
For 0< <1 andf2C
1
[a;b] the derivativesD

a+fandD

b
fexist,
belong toLr[a;b] for 1r<1=and have a form
D

a+f=
1
(1)

f(a)(xa)

+
Z
x
a
f
0
(t)(xt)

dt

;
D

b
f=
1
(1)

f(b)(bx)


Z
b
x
f
0
(t)(tx)

dt

:
Let us consider only the integral in the latter formulas. We get Caputo
derivatives:
D
cap;
a+
f=
1
(1)
Z
x
a
f
0
(t)(xt)

dt
=D

a+f
1
(1)
f(a)(xa)

+;
D
cap;
b
f=
1
(1)
Z
b
x
f
0
(t)(tx)

dt
=D

b
f
1
(1)
f(b)(bx)

:
Yuliya Mishura Fractional calculus SMOCS 2021

20. Fractional integral of the indicator function
Let the general indicator function be given by
1
(a;b)(t) =
8
>
<
>
:
1;at<b;
1;bt<a;
0;otherwise:
Yuliya Mishura Fractional calculus SMOCS 2021

21. Lemma 1.4
LetH2(0;
1
2
)[(
1
2
;1),=H
1
2
. Then for allt2R
(I

1
(0;t))(x) =
1
(1 +)
((tx)

+(x)

+):
Proof.
LetH2(
1
2
;1), and, for example,x<0<t(other cases can be considered
similarly). Then
(I

1
(0;t))(x) =
1
()
Z
1
x
1
(0;t)(u)(ux)
1
du
=
1
()
Z
t
0
(ux)
1
du=
1
(+ 1)
((tx)

(x)

):(5)
Yuliya Mishura Fractional calculus SMOCS 2021

22. Remark 1.5
Obviously, (I

+1
(a;b)(x)) =
1
(1+)
((bx)

+(ax)

+),
1<a<b<1.
Yuliya Mishura Fractional calculus SMOCS 2021

23. 1
The elements of fractional calculus
2
Fractional Brownian motion: denition and elementary properties
3
Mandelbrot-van-Ness representation of fBm
4
Wiener integration with respect to fBm
5
Representation of fBm via Wiener process on a nite interval.
6
Some related fractional processes
Sub-fractional Brownian motion
Bi-fractional Brownian motion
Mixed fractional Brownian motion
Yuliya Mishura Fractional calculus SMOCS 2021

24. Let (;F;P) be a complete probability space.
Denition 2.1
(Two-sided, normalized)
indexH2(0;1) is a stochastic Gaussian processB
H
=fB
H
t;t2Rgon
(;F;P), having the properties
(i)B
H
0
= 0;
(ii)EB
H
t= 0;t2R,
(iii)EB
H
tB
H
s=
1
2
(jtj
2H
+jsj
2H
jtsj
2H
);s;t2R.
Yuliya Mishura Fractional calculus SMOCS 2021

25. Remark 2.2
SinceE(B
H
tB
H
s)
2
=jtsj
2H
and the process is Gaussian, it has a
continuous modication, according to Kolmogorov theorem, because for all
n1EjB
H
tB
H
sj
n
=
2
n
2

1
2
(
n+1
2
)jtsj
nH
. Moreover, the sucient
condition of Holder continuity of order%of the trajectories ofXis
EjXtXsj
n
Cjtsj
1+n%
for somen>0; % >0. In our case we can put
%=H1=n, and to get Holder property of the trajectories of fractional
Brownian motion up to orderH.
Yuliya Mishura Fractional calculus SMOCS 2021

26. Remark 2.3
ForH= 1 we setB
H
t=B
1
t=t, whereis standard normal random
variable.
Remark 2.4
It is possible to consider fBmB
H
tonly onR+(one-sided fBm) with
evident changes in Denition2.1.
The characteristic function has a form
'(t) :=Eexpfi
n
X
k=1
kB
H
tk
g= exp


1
2
(Ct; )

;
where the matrixCt= (EB
H
tk
B
H
ti
)i;k=1, (;) is inner product inR
n
.
Therefore, from (iii) of Denition 2.1, for any >0
'(t) = exp


1
2

2H
(Ct; )

: (6)
Yuliya Mishura Fractional calculus SMOCS 2021

27. Denition 2.5
Stochastic processX=fXt;t2Rgis called-self-similar
fXat;t2Rg
d
=fa

Xt;t2Rg
in the sense of nite-dimensional distributions.
It follows from Denition 2.5 and (6) thatB
H
isH-self-similar.
Note that
E(B
H
tB
H
s)(B
H
uB
H
v) =
1
2
(jsuj
2H
+jtvj
2H
jtuj
2H
jsvj
2H
):(7)
It follows from (7) that the processB
H
has stationary increments
(evidently, it is not stationary itself). LetH=
1
2
. Then the increments of
B
H
are non-correlated consequently independent, soB
H
=Wis a Wiener
process.
In what follows,=H1=2.
Yuliya Mishura Fractional calculus SMOCS 2021

28. ForH2(0;
1
2
)[(
1
2
;1) andt1<t2<t3<t4, from (7).
E(B
H
t4
B
H
t3
)(B
H
t2
B
H
t1
) =H(2H1)
Z
t2
t1
Z
t4
t3
(uv)
21
dudv:
Therefore, the increments are positively correlated forH2(
1
2
;1) and
negatively correlated forH2(0;
1
2
). Further, for anyn2Zn f0gthe
autocovariance function
r(n) :=EB
H
1(B
H
n+1B
H
n) =H(2H1)
Z
1
0
Z
n+1
n
(uv)
21
du dv
H(2H1)jnj
21
;jnj ! 1:
IfH2(0;
1
2
), then
P
n2Z
jr(n)j
P
n2Znf0g
jnj
21
<1:
IfH2(
1
2
;1), then
P
1
n=1
jr(n)j
P
n2Znf0g
jnj
21
=1. In this
connection we say that forH2(
1
2
;1) fBmB
H
has the property of
long-range dependence, while for H2(0;
1
2
) fBmB
H
has the property of
short-range dependence.
Yuliya Mishura Fractional calculus SMOCS 2021

29. 1
The elements of fractional calculus
2
Fractional Brownian motion: denition and elementary properties
3
Mandelbrot-van-Ness representation of fBm
4
Wiener integration with respect to fBm
5
Representation of fBm via Wiener process on a nite interval.
6
Some related fractional processes
Sub-fractional Brownian motion
Bi-fractional Brownian motion
Mixed fractional Brownian motion
Yuliya Mishura Fractional calculus SMOCS 2021

30. LetW=fWt;t2Rgbe the, i.e., Gaussian
process with independent increments,EWt= 0,EWtWs=s^t,s;t2R.
Evidently,W=B
1
2. Denote
kH(t;u) := (tu)

+(u)

+= (I

1
(0;t))(x);
=H
1
2
. The following representation belongs to Mandelbrot and van
Ness [MvN68].
Theorem 3.1
The processB
H
=fB
H
t;t2Rgwith
B
H
t:=C
(2)
H
Z
R
kH(t;u)dWu;H2

0;
1
2

[

1
2
;1

;
C
(2)
H
=

Z
R+

(1 +s)

s


2
ds+
1
2H


1
2
=

2HsinH(2H)

1=2
(H+ 1=2)
;
has a continuous modication that is a normalized two-sided fBm.
Yuliya Mishura Fractional calculus SMOCS 2021

31. 1
The elements of fractional calculus
2
Fractional Brownian motion: denition and elementary properties
3
Mandelbrot-van-Ness representation of fBm
4
Wiener integration with respect to fBm
5
Representation of fBm via Wiener process on a nite interval.
6
Some related fractional processes
Sub-fractional Brownian motion
Bi-fractional Brownian motion
Mixed fractional Brownian motion
Yuliya Mishura Fractional calculus SMOCS 2021

32. Dene the operator
M
H
f:=
(
C
(3)
H
I

f;H2(0;
1
2
)[(
1
2
;1);
f; H=
1
2
;
(8)
whereC
(3)
H
=C
(2)
H
(H+
1
2
).
Consider the spaceL
H
2
(R) :=ff:M
H
f2L2(R)gequipped with the norm
jjfjj
L
H
2
(R)
=jjM
H
fjj
L2(R).
Denition 4.1
Letf2L
H
2
(R). Then
IH(f) :=
Z
R
f(s)dB
H
s:=
Z
R
(M
H
f)(s)dWs: (9)
HereB
H
sandWsare connected as in Theorem 3.1. As a particular case,
consider stepwise functionf:R!Rthat has a form
f(t) =
n
X
k=1
ak1
[tk1;tk)(t);
wheret0<t1< : : : <tn2Randak2R;1kn.
Yuliya Mishura Fractional calculus SMOCS 2021

33. Then, from the linearity of the operatorM
H
, the integralIH(f) equals
IH(f) =
n
X
k=1
ak
Z
R
M
H
1
[tk1;tk)(s)dWs=
n
X
k=1
ak(B
H
tk
B
H
tk1
) (10)
and it coincides with usual Riemann{Stieltjes sum. So, the question arises:
in what sense can we consider formula (9) as the extension of the sum
(10)? Note that for stepwise function
kIH(f)k
2
L2()
=
n
X
i;k=1
aiak
Z
R
M
H
1
[tk1;tk)(x)M
H
1
[ti1;ti)(x)dx
=


M
H
f



2
L2(R)
=H(2H1)
Z
R
2
f(u)f(v)juvj
21
du dv;
(11)
where the last equality holds forH2(1=2;1). One can see that the
situation is very dierent forH2(0;1=2) andH2(1=2;1).
Yuliya Mishura Fractional calculus SMOCS 2021

34. Nevertheless, there is a fact that holds for any 0<H<1.
Lemma 4.2 ([Ben03a])
For any0<H<1the linear span of the setfM
H
1
(u;v);u;v2Rgis
dense inL2(R).
Yuliya Mishura Fractional calculus SMOCS 2021

35. 1
The elements of fractional calculus
2
Fractional Brownian motion: denition and elementary properties
3
Mandelbrot-van-Ness representation of fBm
4
Wiener integration with respect to fBm
5
Representation of fBm via Wiener process on a nite interval.
6
Some related fractional processes
Sub-fractional Brownian motion
Bi-fractional Brownian motion
Mixed fractional Brownian motion
Yuliya Mishura Fractional calculus SMOCS 2021

36. Sometimes it is convenient to consider \one-sided" fBm
B
H
=fB
H
t;t0gand to represent it as a functional of some Wiener
processB=fBt;t0gof the formB
H
t='(Bs;0st). For this
purpose, consider the following kernels
lH(t;s) =C
(5)
H
s

(ts)

I
f0<s<tg;
mH(t;s) =C
(6)
H
h
t
s


(ts)

s

Z
t
s
u
1
(us)

du
i
;
with=H
1
2
;H2(0;1) and with the constants
C
(5)
H
=

(22)
(1)
3
()2H
1
2
;C
(6)
H
=

2H(1)
(12)(+ 1)
1
2
:
Yuliya Mishura Fractional calculus SMOCS 2021

37. (i) LetH2(
1
2
;1). Then for anyt>0
Z
t
0
Z
t
0
(tu)

(ts)

u

s

jusj
21
du dst
12
<1:
(12)
Yuliya Mishura Fractional calculus SMOCS 2021

38. Therefore, we can consider the integral
I
H
t(lH) =
Z
t
0
lH(t;s)dB
H
s:=
Z
R
lH(t;s)dB
H
s
=
Z
R
(M
H
lH)(t;)(x)dWx;
(13)
whereWis the underlying Wiener processfWx;x2Rg. Similarly to (12),
for any 0<t<t
0
the scalar product equals
EI
H
t(lH)I
H
t
0(lH) =

lH(t;);lH(t
0
;)

jRHj;2
= (C
(5)
H
)
2
2H
Z
t
0
(tu)

u


Z
t
0
0
(t
0
s)

s

jusj
21
ds
!
du
= (C
(5)
H
)
2
2Ht
12
B(;1)B(1;1) =t
12
:
(14)
Yuliya Mishura Fractional calculus SMOCS 2021

39. From (13),

I
H
t;t0

is a centered Gaussian process, and from (14), for
any 0<s<ts
0
<t
0
E

I
H
t
0(lH)I
H
s
0(lH)

I
H
t(lH)I
H
s(lH)

=t
12
t
12
s
12
+s
12
= 0;
i.e., the increments ofI
H
t(lH) are non-correlated, hence, independent.
Therefore,I
H
t(lH) is a martingale w.r.t. its natural ltration
F
H
t:=
n
I
H
s(lH);0st
o
;
having independent increments and with quadratic characteristic

I
H
t(lH)

=t
12
;I
H
0
(lH) = 0.
Yuliya Mishura Fractional calculus SMOCS 2021

40. By Levy theorem, there exists some Wiener processB=fBt;t0gsuch
that
M
H
t:=I
H
t(lH) = (12)
1=2
Z
t
0
s

dBs: (15)
The processM
H
is called Molchan martingale, since it was considered
originally in the papers [Mol69, MG69], see also [NVV99].
(ii) Now, letH2(0;
1
2
).
Let the functionfbelong toBV[0;T], the class of functions of bounded
variation on [0;T], andf= 0 outside [0;T]:Then the integral
R
T
0
f(s)dB
H
s
exists for anyf2BV[0;T] if we dene it via integration by parts:
Z
T
0
f(s)dB
H
s=f(t)B
H
t
Z
T
0
B
H
sdf(s):
Yuliya Mishura Fractional calculus SMOCS 2021

41. Evidently, for any xedt>0 the kernellH(t;)2BV[0;t]\C[0;t], if
H2(0;
1
2
). Therefore,
I
H
t(lH) =
Z
t
0
lH(t;s)dB
H
s=
Z
t
0
B
H
sdlH(t;s) =
Z
t
0
B
H
sl
0
H
(t;s)ds
=C
(5)
H
Z
t
0
B
H
ss

(ts)
1
(t2s)ds;
and this integral is obviously a Gaussian random variable. We can easily
calculateEI
H
t(lH)I
H
t
0(lH) =t
12
=t
22H
for any 0<t<t
0
, taking into
the account the fact thatlHvanishes at the endpoints:
EI
H
t(lH)I
H
t
0(lH)
=
1
2
Z
t
0
Z
t
0
0
(u
2H
+s
2H
jusj
2H
)l
0
H
(t;s)l
0
H
(t
0
;u)du ds
=
1
2
Z
t
0
Z
t
0
0
jusj
2H
l
0
H
(t;s)l
0
H
(t
0
;u)du ds
Yuliya Mishura Fractional calculus SMOCS 2021

42. =
1
2
Z
t
0
l
0
H
(t;s)
Z
s
0
(su)
2H
l
0
H
(t
0
;u)du

ds

1
2
Z
t
0
l
0
H
(t;s)

Z
t
0
s
(us)
2H
l
0
H
(t
0
;u)du
!
ds
=H
Z
t
0
lH(t;s)
Z
s
0
(su)
2
l
0
H
(t
0
;u)du

ds
+H
Z
t
0
lH(t;s)

Z
t
0
s
(us)
2
l
0
H
(t
0
;u)du
!
ds
=HC
(5)
H
Z
t
0
lH(t;s)

Z
t
0
0
jusj
2
sign(us)u
1
(t
0
u)
1
(t
0
2u)du ds(16)
Yuliya Mishura Fractional calculus SMOCS 2021

43. According to [NVV99, formula (2.5)], the interior integral in the
right-hand side of (16) equals fors<t
0
Z
t
0
0
jusj
2
sign(us)u
1
(t
0
u)
1
(t
0
2u)du
=

H(C
(5)
H
)
2
B(1;1)

1
;
whenceEI
H
t(lH)I
H
t
0(lH) =t
22H
. We can conclude, similarly to part (i),
thatI
H
t(lH) is a martingale according to its natural ltration, and
I
H
t(lH) = (12)
1=2
Z
t
0
s

dBs
for some Wiener processB. Thus, we have proved the following result.
Yuliya Mishura Fractional calculus SMOCS 2021

44. Theorem 5.1
LetB
H
be an fBm withH2(0;1),
M
H
t=I
H
t(lH) =
Z
t
0
lH(t;s)dB
H
s: (17)
Then there exists Wiener processBsuch that
M
H
t= (12)
1=2
R
t
0
s

dBs. It is clear that


B
H
s;0st

=fBs;0stg.
Yuliya Mishura Fractional calculus SMOCS 2021

45. The converse relation can be obtained for anyH2(0;1) by the similar
way and has a form:
B
H
t=
Z
t
0
m(t;s)dW(s);
where
mH(t;s) =C
(6)
H
h
t
s


(ts)

s

Z
t
s
u
1
(us)

du
i
:
In the caseH>1=2 the kernelmH(t;s) can be simplied to
mH(t;s) =C
(6)
H
s

R
t
s
u

(us)
1
du.
Yuliya Mishura Fractional calculus SMOCS 2021

46. 1
The elements of fractional calculus
2
Fractional Brownian motion: denition and elementary properties
3
Mandelbrot-van-Ness representation of fBm
4
Wiener integration with respect to fBm
5
Representation of fBm via Wiener process on a nite interval.
6
Some related fractional processes
Sub-fractional Brownian motion
Bi-fractional Brownian motion
Mixed fractional Brownian motion
Yuliya Mishura Fractional calculus SMOCS 2021

47. Sub-fractional Brownian motion
Consider some related Gaussian processes.
Sub-fractional Brownian motion is a zero-mean Gaussian process
C
H
= (C
H
t)t0with parameterH2(0;1);such that its covariance
function equals
EC
H
tC
H
s=t
2H
+s
2H

1
2
(jt+sj
2H
+jtsj
2H
);t;s0:
This process was introduced in [BGT04] in connection with the occupation
time uctuations of branching particle systems. In the caseH= 1=2, it
coincides with the standard Brownian motion:C
1=2
=B
1=2
. ForH6= 1=2;
C
H
is, in a sense, a process intermediate between the standard Brownian
motionB
1=2
and the fBmB
H
. Sub-fractional Brownian motionC
H
has
non-stationary increments, and its incremental covariance satises the
inequalities (see [BGT04] for details)
jtsj
2H
EjC
H
tC
H
sj
2
(22
H1
)jtsj
2H
;t;s0:(18)
Yuliya Mishura Fractional calculus SMOCS 2021

48. Bi-fractional Brownian motion
LetB
H;K
= (B
H;K
t)t0;whereH2(0;1),K2(0;1] are parameters, be a
zero-mean Gaussian process with covariance function
EB
H;K
tB
H;K
s= 2
K

(t
2H
+s
2H
)
K
jtsj
2HK

;t;s0:
This process can be considered as an extension of the fBm, the latter
being a special case whenK= 1 (see [HV03, RT006, LN09]). The process
B
H;K
satises the following version of (18):
2
K
jtsj
2HK
EjB
H;K
tB
H;K
sj
2
2
1K
jtsj
2HK
;t;s0:
Yuliya Mishura Fractional calculus SMOCS 2021

49. Mixed fractional Brownian motion
Consider a zero-mean Gaussian process of the form
M
H
t=Wt+B
H
t;t0;
whereWandB
H
are independent stochastic processes,Wis a standard
Brownian motion, andB
H
is a fractional Brownian motion. Such process
is called a mixed fractional Brownian motion. It was considered in detail
by Cheridito in [Cher01].
Yuliya Mishura Fractional calculus SMOCS 2021

50. References
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51. References
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53. References
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54. References
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55. References
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Yuliya Mishura Fractional calculus SMOCS 2021