Maximal inequalities for fractional brownian motion with variable drift

VNU Journal of Science: Mathematics – Physics, Vol. 36, No. 3 (2020) 1-9

Review Article

Maximal Inequalities for Fractional Brownian Motion
with Variable Drift
Trinh Nhu Quynh1, Tran Manh Cuong2,*
1

Military Information Technology Institute, 17 Hoang Sam, Cau Giay, Hanoi, Vietnam
Department of Mathematics, VNU University of Science, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam

2

Received 18 December 2019
Revised 06 March 2020; Accepted 15 June 2020

Abstract: Let BH be a fractional Brownian motion with H∈ (0, 1) and g be a deterministic
function. We study the asymptotic behaviour of the tail probability as for fixed x and as for
fixed T. Our results partially generalise those obtained by Prakasa Rao in [1].
Keywords: Fractional Brownian motion, Maximal inequalities, Variable drift.

1. Introduction
Let B H  ( BtH )t 0 be a standard fractional Brownian motion (fBm) with Hurst index , i.e. BH is a
centered Gaussian process with covariance function given by
1 2H
2H
RH  t , s  : E[ BtH BsH ] 
(t  s 2 H  t  s ), t , s  0.
2
1

We refer the readers to the monograph [2] for a short survey of properties of fBm. When H  ,
2
the following limit theorems were proved by Prakasa Rao in [1].
k
k 1
Theorem 1.1. Let g  t   ak t  ak 1t  ...  a1t be a polynomial of degree k with ak > 0. Then, for
any T > 0 and k ≥ 2 we have

________
Corresponding author.

Email address:
https//doi.org/ 10.25073/2588-1124/vnumap.4447

1


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T.N. Quynh, T.M. Cuong / VNU Journal of Science: Mathematics – Physics, Vol. 36, No. 3 (2020) 1-9

log P( sup ( BtH  g (t ))  x)
t[0, T]

lim
x 

Theorem 1.2. Let g  t   ak t  ak 1t
any x > 0 and k ≥ 1 we have
k

x
k 1

2



 ...  a1t be a polynomial of degree k with ak >0. Then, for
log P( sup ( BtH  g (t ))  x)

lim sup

t[0, T]

T 

It is known that when H 

1
,
2

1
.
2T 2 H

T 2k 2 H



ak2
.
2

BtH reduces to a standard Brownian motion. In this case, Prakasa Rao's

results reduce to those established previously by Jiao [3]. Naturally, one would like to ask the
following questions:
Q1: Are Theorems 1.1 and 1.2 still true when H <

1
?
2

Q2: Can we remove the polynomial structure of the drift g(t)?
The aim of this paper is to provide an affirmative answer to Q1 and Q2. Our method is different
from Prakasa Rao's where he mainly uses the classical Slepian's lemma. In the present paper, we
employ the techniques of Malliavin calculus which lead us to a shorter proof for more general results.
The rest of the paper is organized as follows. In Section 2, we recall some fundamental concepts of
Malliavin calculus. The main results of the paper are stated and proved in Section 3.
2. Preliminaries
It is well known that

BtH admits the so-called Volterra representation (see, e.g. [4])
t

B   K  t , s  dBs , t  0;T  ,
H
t

(2.1)

0

where (Bt)t≥0 is a standard Brownian motion, K(t, s) = 0 for s ≥ t and
3
H
 H  12

t
1
1
2
H
H
t
1
u

2
2
K (t , s )  CH
(t  s )
 (H  )
(u  s) du  , s  t
1
 H 1

H

2
s
s 2
 s 2


where CH 

 H (2 H  1)
1
1
(2  2 H )( H  ) 2 sin( ( H  ))
2
2

.

Our proofs will be strongly based on techniques of Malliavin calculus. For the reader's convenience,
let us recall the definition of Malliavin derivative with respect to Brownian motion B, where B is used
to present BtH as in (2.1).
H

We suppose that ( Bt )t[0,T ] is defined on the complete probability space (, , P) , where


T.N. Quynh, T.M. Cuong / VNU Journal of Science: Mathematics – Physics, Vol. 36, No. 3 (2020) 1-9

(

t

)t[0,T ] is

a natural filter generated by the Brownian motion B. For

3

h  L2 0, T  we denote by

B(h) the Wiener integral
T

B  h  =  h  t  dBt .
0

Let S denote the dense subset of

L2  , , P  consisting of smooth random variables of the form
F  f ( B  h1  ,..., B(hn )),

where n  , f  Cb (

n

derivative as the process

(2.2)

) , h1, ..., hn ∈ L [0, T]. If F has the form (2.2), we define its Malliavin
2

DF : {Dt F ; t 0;T } given by
n

Dt F  
k 1

f
( B  h1  , ..., B(hn ))hk (t ).
xk

1,2
We will denote by
the space of Malliavin differentiable random variables, it is the closure of S
with respect to the norm

T

2
2
:

E
F

E
  Dt F dt  .
1,2
0


The next Proposition is a concrete version of Corollary 4.7.4 in [5].
F

Proposition 2.1. Let F be in

1,2

2

. Assume that

T

 ( D F ) d  
2

a.s.

(2.3)

0

for some   0 . Then, for all x > 0, we have

 x2 
P( F - E ( F )  x)  exp  
(2.4)
.
 2 
Remark 2.1. The random variable -F also satisfies the conditions of Proposition 2.1. We therefore

obtain the same bound for the left tail
 x2 
P( F - E ( F )   x)  P( F - E ( F )  x)  exp  
 , x  0.
 2 

(2.5)

3. The main results
We firstly establish the following technical result which plays a key role in this paper.
Proposition 3.1. Suppose that f is a continuously differentiable function on ℝ with bounded derivative
H

and g is a continuous function on [0, T]. Let Bt be a fBm with

H   0, 1 , it holds that


T.N. Quynh, T.M. Cuong / VNU Journal of Science: Mathematics – Physics, Vol. 36, No. 3 (2020) 1-9

4


( x  cT )2


H
P  sup ( Bt  g (t ))  x   exp  
2 2H
 t[0, T]


 2sup f '( x) T
x



, x  c ,
T




(3.1)

and


( x  cT )2


H
P  sup ( Bt  g (t ))  x   exp  
2 2H
 t[0, T]

 2sup f '( x) T
x




 ,0  x  c ,
T




(3.2)

where cT  E  sup  f ( BtH )  g  t    .
t[0,T ]

Proof. If sup f '( x)  0 , then the estimates (3.1) and (3.2) are trivial. Hence, we can and will assume
x

that sup f '( x)  0.
x

Consider a countable and dense subset S0  {tn , n  1} of [0, T]. Define
M n  sup{ X t1 , X t2 , ..., X tn },

where X t : f ( Bt ) +g  t  . Because f is continuous differential with bounded derivative, we know
from Proposition 1.2.3 in [4] that X t  D1,2 and
H

D X t  f '  BtH  D BtH  f '  BtH  K  t ,  ,   t.
It is known from Proposition 2.1.10 in [4] that Mn ∈ D1’2 and Mn converges in L2(Ω) to sup X t . In
t[0,T ]

order to evaluate the Malliavin derivative of Mn, we introduce the following sets:
A1  {M n =Xt1 },

........
Ak  {M n  Xt1 , ..., M n  X tk 1 , M n =X tk }, 2  k  n.

By the local property of the operator D; on the set Ak the derivatives of the random variables Mn and

X tk coincide. Hence, we can write
n

 

 

n

D M n   I Ak f ' BtHk D X tk  I Ak f ' BtHk K (tk , ) I Ak .
k 1

k 1

Consequently,
n

  

( D M n )2   f ' BtHk
k 1

2

K 2 (tk , ) I Ak .

And hence,



T

0

tk

( D M n )2 d   ( D M n )2 d
0

 sup f '( x)
x

2

tn n

 K
0

k 1

2

(tk , ) I Ak d

Denote by F(t, .) the antiderivative of K (t, .). Since K(t, θ) = 0 for θ ≥ t we can obtain
2

(3.3)

a.s.


T.N. Quynh, T.M. Cuong / VNU Journal of Science: Mathematics – Physics, Vol. 36, No. 3 (2020) 1-9

tn n

 K
0

k 1

2

(tk , ) I Ak d

t1 n

tn n

tn

0

t1

tn1



5

 K 2 (tk , ) I Ak d  
k 1

 K 2 (tk , ) I Ak d  ...  
k 1

  F (t1 , t1 )  F (t1 ,0)  I A1  ...   F (tn , tn )  F (tn ,0)  I An
2

n

K

2

k 1

(tn , ) I An d

2

 E BtH1 I A1  ...  E BtHn I An

 3.4 

 t12 H I A1  ...  tn2 H I An  T 2 H .
Combining (3.3) and (3.4) yields



T

0

( D M n )2 d  sup f '( x) T 2 H , a.s.
2

(3.5)

x

The inequalitiy (3.5) shows that the random variable Mn satisfies the condition (2.3) of Proposition 2.1.
Consequently, we can get


x2
P  M n  E[M n ]  x   exp  
2 2H
 2sup f '( x) T
x

Then, by Fatou's lemma we deduce


 , x  0.






P  sup ( BtH  g (t ))  cT  x   lim inf P  M n  E[M n ]  x 
 t[0, T]
 n


( x  cT ) 2

 , x  0,
 exp  
2 2H 
 2sup f '( x) T 
x


which gives us (3.1). Similarly, we can obtain (3.2) by using the estimate (2.5).
So the proof of Proposition is complete.
Remark 3.1. We state Proposition 3.1 in a general form because it can be useful for the other
researches. Let us give here an example. Consider the fractional stochastic differential equation
xt  x0    ( xs )dBsH , t [0, T ].
t

0

Under suitable assumptions on σ and H, the Doss-Sussmann representation of xt is given by (see, e.g.
[6, 7])

xt  f  BtH  ,

where f (x) solves the ordinary differential equation:

f '  x     f  x  , f  0  x 0 .
Thus f (x) will satisfy the condition of Proposition 3.1 if σ(x) is continuous and bounded on ℝ.
We now are in a position to formulate and prove the first main result which generalises and improves
Theorem 1.1.


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T.N. Quynh, T.M. Cuong / VNU Journal of Science: Mathematics – Physics, Vol. 36, No. 3 (2020) 1-9

H

Theorem 3.1. Let Bt be a fBm with H ∈ (0, 1). Fixed T > 0, let g be a continuous function on [0, T].
It holds that
log P( sup ( BtH  g (t ))  x)
t[0, T]

lim
x 

x

2



1
.
2T 2 H

Proof. Obviously, we have


P  sup ( BtH  g (t ))  x   P  ( BTH  g (T ))  x   P  BTH  x  g (T )  .
 t[0, T]

H

Since Bt is a normal random variable with mean zero and variance T2H, we can obtain
x  g (T ) 



P  sup ( BtH  g (t ))  x   P  Z 
,
TH
t

[0,
T]






(3.7)

x  g (T )
 1 for sufficiently large x, we can apply
TH

where Z has a standard normal distribution. Since
Lemma 1 in [2] to get


( x  g (T )) 2

e 2T


for sufficiently large x.
P  sup ( BtH  g (t ))  x  
 t[0, T]
 6( x  g (T ))
TH
2H

As a consequence,
log P( sup ( BtH  g (t ))  x)  
t[0, T]

( x  g (T ))2

 log(6 x  6 g (T ))  log T H ,
2T 2 H

and hence,
log P( sup ( BtH  g (t ))  x)
t[0, T]

lim inf
x 

x

2



1
.
2T 2 H

(3.8)

On the other hand, we obtain from Proposition 3.1 that

 ( x  g (T ))2 


P  sup ( BtH  g (t ))  x   exp  
,
2T 2 H 

 t[0, T]


which gives us
log P( sup ( BtH  g (t ))  x)
t[0, T]

x2

Notice



( x  cT )2
.
2 x 2T 2 H

cT  E[ sup ( BtH  g  t )]

that

is

finite

because

t[0,T ]

cT  E[ sup BtH ]  sup [g  t ]  E[ sup BtH ]T H  sup [g  t ].

t[0,T ]

t[0,T ]

t[0,1]

t[0,T ]

Taking the limit x→∞ we get
log P( sup ( BtH  g (t ))  x)
lim sup
x 

t[0, T]

x

2

So we can finish the proof by combining (3.8) and (3.9).



1
.
2T 2 H

□

(3.9)

T.N. Quynh, T.M. Cuong / VNU Journal of Science: Mathematics – Physics, Vol. 36, No. 3 (2020) 1-9

7

The second main result of this paper is the following theorem.
H

Theorem 3.2. Let Bt be a fBm with H ∈ (0, 1) and g be a continuous function on [0, 1). Assume
that there exists a positive constant k > H such that
g (T )
 k : lim k  0.
T  T
Then, for any x > 0 we have
log P( sup ( BtH  g (t ))  x)
t[0, T]

lim sup

T 2k 2 H

T 



 k2
2

(3.10)

.

and
log P( sup ( BtH  g (t ))  x)
t[0, T]

lim inf

T 2k 2 H

T 



 k2
2

(3.11)

.

Proof. It is clear that cT  E ( sup BtH  g  t )  E ( BTH  g T )  g (T )  . as T→∞.
t[0,T ]

Hence x < cT for sufficiently large T. Once again, we apply Proposition 3.1 to get
log P( sup ( BtH  g (t ))  x)
t[0, T]

T

2k 2 H



( x  cT )2
for sufficiently large T,
2T 2 k

which leads us to the following

log P( sup ( BtH  g (t ))  x)
lim sup

T 

Since lim

T 

t[0, T]

T

2k 2 H

c
1
   lim Tk
2  T  T


.


(3.12)

cT
g (T )
 lim k   k  0. This, together with (3.12), yields
k
T

T
T
log P( sup ( BtH  g (t ))  x)
lim sup

t[0, T]

T 2k 2 H

T 



 k2
2

.

Thus the estimate (3.10) was proved.
The remaining of the proof is to show (3.11). Because αk > 0 and k > H, we have lim

T 

for any x>0. Hence,
x  g (T )  1
g (T )  x 


P Z 
   P Z 

TH
TH

 2



for sufficiently large T. Recalling (3.7) and using Lemma 1 in [3], we have


( g (T )  x ) 2

e 2T


for sufficiently large T.

P  sup ( BtH  g (t ))  x  
 t[0, T]
 6( g (T )  x)
TH

We therefore obtain

2H

x  g (T )
 
TH


T.N. Quynh, T.M. Cuong / VNU Journal of Science: Mathematics – Physics, Vol. 36, No. 3 (2020) 1-9

8

log P( sup ( BtH  g (t ))  x)  
t[0, T]

( g (T )  x)2
 log(6 g (T )  6 x)  log T H
2T 2 H

and

log P( sup ( BtH  g (t ))  x)
t[0, T]

lim inf

T 2k 2 H

T 


1
g (T ) 
   lim k    k .
2  T  T 
2
2

2

The proof of Theorem is complete.
We end up this paper with a remark.
Remark 3.2. The method used in the paper can be applied to a larger class of Gaussian processes of
form
Yt   k (t , s)dBs , t [0, T ],
t

0

where the Volterra kernel k(t, s) is continuous and satisfies the function t

E Yt

2

  k 2  t , s  ds is
t

0

2

non-decreasing. Here we note that the non-decreasing property of E Yt is used to prove the inequality
(3.5).
T

For example, when Yt   e
0

 (t  s )

dBs is an Ornstein-Uhlenbeck process we have

log P( sup (Yt  g (t ))  x)
limsup
x 

t[0, T]

x

2



1
T

2 k 2 (T , s)ds




.
1  e2T

0

3. Conclusion
Thus, we have generalized Rao's studies of fractional Brownian motion with continuous drift, H ∈
(0, 1). And we got the answers to question 1 one and question 2 who are the two issues raised in the
introduction. In these proofs we also use images of the Malliavin’s calculus, which are quite different
from Rao's.
Acknowledgments
This work was partially supported by Vietnam National University, Hanoi (grant no. QG.20.26).
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