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Vol. 8 (2003) Paper no. 3, pages 1–14.
Journal URL
http://www.math.washington.edu/˜ejpecp/
Paper URL
http://www.math.washington.edu/˜ejpecp/EjpVol8/paper3.abs.html
FRACTIONAL ORNSTEIN-UHLENBECK PROCESSES
Patrick Cheridito 1
Department of Mathematics, ETH Zürich
CH-8092 Zürich, Switzerland
dito@math.ethz.ch
Hideyuki Kawaguchi 2
Department of Mathematics, Keio University
Hiyoshi, Yokohama 223-8522, Japan
hide@1999.jukuin.keio.ac.jp
Makoto Maejima
Department of Mathematics, Keio University
Hiyoshi, Yokohama 223-8522, Japan
maejima@math.keio.ac.jp
Abstract The classical stationary Ornstein-Uhlenbeck process can be obtained in two different
ways. On the one hand, it is a stationary solution of the Langevin equation with Brownian
motion noise. On the other hand, it can be obtained from Brownian motion by the so called
Lamperti transformation. We show that the Langevin equation with fractional Brownian motion noise also has a stationary solution and that the decay of its auto-covariance function is
like that of a power function. Contrary to that, the stationary process obtained from fractional
Brownian motion by the Lamperti transformation has an auto-covariance function that decays
exponentially.
Keywords and phrases: Fractional Brownian motion, Langevin equation, long-range dependence, selfsimilar processes, Lamperti transformation.
AMS subject classification (2000): primary: 60H10; secondary: 60G15, 60G18, 45F05
Submitted to EJP on May 18, 2002. Final version accepted on January 9, 2003.
1
The first author would like to thank the Department of Mathematics, Keio University, for having made
possible his stay in Yokohama.
2
Current address: Risk Management Systems Department, Sumitomo Mitsui Banking Corporation, 1-3-2,
Marunouchi, Chiyoda, Tokyo 100-0005, Japan; kawaguchi hideyuki@dn.smbc.co.jp
1
1
Introduction
Let (Ω, F, P ) be a probability space.
Definition 1.1 A fractional Brownian motion with Hurst parameter H ∈ (0, 1], is an almost
surely continuous, centered Gaussian process (BtH )t∈R with
´
¢ 1 ³ 2H
¡
(1.1)
Cov BtH , BsH =
|t| + |s|2H − |t − s|2H , t, s ∈ R .
2
For an in-depth introduction to fractional Brownian motions we refer the reader to Section 7.2
of Samorodnitsky and Taqqu (1994) or Chapter 4 of Embrechts and Maejima (2002). It is clear
that for all H ∈ (0, 1], B0H = 0 almost surely. Moreover, it can be deduced from (1.1) that
for all H ∈ (0, 1], (BtH )t∈R has stationary increments and is H-selfsimilar, that is, for every
d
d
H)
H H
c > 0, (Bct
t∈R = (c Bt )t∈R , where = denotes equality of all finite-dimensional distributions.
1
(Bt2 )t∈R is a two-sided Brownian motion. In particular, it has independent increments. For
H ∈ (0, 12 ) ∪ ( 12 , 1), (BtH )t≥0 is not a semimartingale and it can be derived from (1.1) that for
all h ∈ R and 0 < t < s,
¢
¡
¡ H
¢
H
H
H
= Cov BtH , Bs+t
− BsH
Cov Bh+t
− BhH , Bh+s+t
− Bh+s
Ã2n−1
!
∞
X
Y
t2n
=
(2H − k) s2H−2n ,
(2n)!
n=1
k=0
in particular, for every N = 1, 2, . . . , for all h ∈ R and t > 0,
¡ H
¢
H
H
Cov Bh+t
− BhH , Bh+s+t
− Bh+s
!
Ã2n−1
N
Y
X
t2n
(2H − k) s2H−2n + O(s2H−2N −2 ) , as s → ∞ .
=
(2n)!
n=1
(1.2)
k=0
This shows that for H ∈ ( 12 , 1],
∞
X
n=0
Cov
³
H
BtH , B(n+1)t
−
H
Bnt
´
= ∞,
a phenomenon referred to as long-range dependence or long memory of the increments process
³
´∞
H
H
.
B(n+1)t
− Bnt
n=0
The classical Ornstein-Uhlenbeck process with parameters λ > 0 and σ > 0 starting at
x ∈ R, is the unique strong solution of the Langevin equation with Brownian motion noise
Z t
1
Xs ds + σBt2 , t ≥ 0 ,
Xt = ξ − λ
(1.3)
0
with initial condition ξ = x. It is given by the almost surely continuous Gaussian Markov
process
µ
¶
Z t
1
1
,x
eλu dBu2 , t ≥ 0 .
Yt 2 := e−λt x + σ
0
2
The unique strong solution of (1.3) with initial condition
Z 0
1
ξ=σ
eλu dBu2 ,
−∞
is given by the restriction to non-negative t’s of the stationary, almost surely continuous,
centered Gaussian Markov process
Z t
1
1
2
Yt := σ
e−λ(t−u) dBu2 , t ∈ R .
−∞
It can easily be checked that
µ
1
2
1
2
Cov Yt , Yt+s
¶
=
σ 2 −λ|s|
e
,
2λ
t, s ∈ R .
1
This implies that (Yt 2 )t∈R is ergodic. Moreover, for all x ∈ R,
µ 1
¶
1
1
,x
−λt
2
2
2
Yt − Y t = e
Y0 − x → 0 , as t → ∞ ,
almost surely. From this it can be derived that if (Yt )t≥0 is a stationary process that solves
1
d
(1.3) with any initial condition ξ ∈ L0 (Ω), then (Yt )t≥0 = (Yt 2 )t≥0 .
Now let α > 0. Then,
1
1
2
Zt2 := e−λt Bαe
2λt ,
t ∈ R,
is also a stationary, almost surely continuous, centered Gaussian process, and
µ 1 1 ¶
2
Cov Zt2 , Zt+s
= αe−λ|s| , t, s ∈ R .
2
1
d
1
Hence, for α = σ2λ , (Yt 2 )t∈R = (Zt2 )t∈R .
It is shown in Lamperti (1962) that for every H > 0, a stochastic process (X t )t≥0 is Hselfsimilar if and only if for all λ, α > 0, the process
bt = e−λt X
X
α exp( λ t) ,
H
t ∈ R,
(1.4)
is stationary. We call (1.4) the Lamperti transformation from selfsimilar processes to stationary
bt )t∈R the Lamperti transform of (Xt )t≥0 .
processes and (X
For H = 1, fractional Brownian motion can be represented as follows:
Bt1 = tη ,
t ∈ R,
where η is a standard normal random variable. For every initial condition ξ ∈ L 0 (Ω), the
equation,
Z t
Xt = ξ − λ
Xs ds + σBt1 , t ≥ 0 ,
(1.5)
0
can path-wise be reduced to the ordinary differential equations,
Xt0 (ω) = −λXt (ω) + ση(ω) ,
3
ω ∈ Ω,
with initial conditions
X0 (ω) = ξ(ω) ,
ω ∈ Ω,
which have the unique solutions
o σ
n
σ
Yt1,ξ (ω) := e−λt ξ(ω) − η(ω) + η(ω) ,
λ
λ
t ≥ 0, ω ∈ Ω.
Equation (1.5) has only a stationary solution for the initial condition ξ = σλ η. It is given by
Yt1 :=
σ
η,
λ
t ≥ 0,
which, for all t ≥ 0, equals the Lamperti transform
Zt1 := e−λt Bα1 exp(λt) = αη ,
t ∈ R,
if α = σλ .
This leads us to the question whether for H ∈ (0, 12 ) ∪ ( 21 , 1), the Langevin equation with
noise process (σBtH )t≥0 has a stationary solution, if its distribution is unique and if it is equal
in some sense to the Lamperti transform
ZtH := e−λt BαHexp( λ t) ,
H
t ∈ R,
for an appropriately chosen α > 0.
The structure of the paper is as follows. In Section 2 we show that for all H ∈ (0, 1],
the Langevin equation with fractional Brownian motion noise has for all initial conditions
ξ ∈ L0 (Ω), a unique strong solution (YtH,ξ )t≥0 . Moreover, there exists a stationary, almost
surely continuous, centered Gaussian process (YtH )t∈R such that (YtH )t≥0 solves the Langevin
equation with fractional Brownian motion noise, and every other stationary solution is equal
to (YtH )t≥0 in distribution. The decay of the auto-covariance function of (YtH )t∈R is for all
H ∈ (0, 21 ) ∪ ( 12 , 1) similar to that of the increments of (BtH )t∈R (see (1.2)). In particular,
(YtH )t∈R is ergodic, and for H ∈ ( 12 , 1], it exhibits long-range dependence. In Section 3 we
show that for all H ∈ (0, 1) the auto-covariance function of (ZtH )t∈R decays exponentially,
which implies that for H ∈ (0, 12 ) ∪ ( 12 , 1), (YtH )t∈R cannot have the same distribution as
(ZtH )t∈R .
2
Fractional Ornstein-Uhlenbeck processes
Let λ, σ > 0 and ξ ∈ L0 (Ω). Since the Langevin equation,
Z t
Xs ds + Nt , t ≥ 0 ,
Xt = ξ − λ
0
only involves an integral with respect to t, it can be solved path-wise for much more general
noise processes (Nt )t≥0 than Brownian motion. For example, it follows from Proposition A.1
that for each H ∈ (0, 1] and for every a ∈ [−∞, ∞),
Z t
eλu dBuH , t > a ,
a
4
exists as a path-wise Riemann-Stieltjes integral, which is almost surely continuous in t, and
µ
¶
Z t
H,ξ
−λt
λu
H
Yt
:= e
e dBu , t ≥ 0 ,
ξ+σ
0
is the unique almost surely continuous process that solves the equation,
Z t
Xt = ξ − λ
Xs ds + σBtH , t ≥ 0 .
(2.1)
0
In particular, the restriction to positive t’s of the almost surely continuous process
Z t
YtH := σ
e−λ(t−u) dBuH , t ∈ R ,
−∞
solves (2.1) with initial condition ξ = Y0H . It is clear that (YtH )t∈R is a Gaussian process, and
it follows immediately from the stationarity of the increments of fractional Brownian motion
that it is stationary. Furthermore, as in the Brownian motion case, for every ξ ∈ L 0 (Ω),
¡
¢
YtH − YtH,ξ = e−λt Y0H − ξ → 0 , as t → ∞ , almost surely,
which implies that every stationary solution of (2.1) has the same distribution as (Y tH )t≥0 . We
call (YtH,ξ )t≥0 a fractional Ornstein-Uhlenbeck process with initial condition ξ and (Y tH )t∈R a
stationary fractional Ornstein-Uhlenbeck process.
In Pipiras and Taqqu (2000) it is shown that for H ∈ ( 21 , 1) and two real-valued measurable
functions
¾
½
Z ∞Z ∞
2H−2
|f (u)| |f (v)| |u − v|
dudv < ∞ ,
f, g ∈ f :
−∞
R∞
−∞
R∞
the two integrals −∞ f (u)dBuH , −∞ g(u)dBuH can in a consistent way be defined as limits of
integrals of elementary functions, and
·Z ∞
¸
Z ∞
Z ∞Z ∞
E
f (u)dBuH
g(u)dBuH = H(2H − 1)
f (u)g(v) |u − v|2H−2 dudv .
−∞
−∞
−∞
−∞
For H ∈ (0, 12 ), the kernel |u − v|2H−2 cannot be integrated over the diagonal. However, if f
and g are regular enough and the intersection of their supports is of Lebesgue measure zero,
the same holds true. We will only need this result for the case where f and g are given by
f (u) = g(u) = eλu and their supports are disjoint intervals. However, the following lemma can
easily be generalized.
Lemma 2.1 Let H ∈ (0, 12 ) ∪ ( 21 , 1], λ > 0 and −∞ ≤ a < b ≤ c < d < ∞. Then
E
·Z
b
e
a
λu
dBuH
Z
d
e
c
λv
dBvH
¸
= H(2H − 1)
5
Z
b
e
a
λu
µZ
d
λv
e (v − u)
c
2H−2
¶
dv du .
Proof. We first assume b = 0 = c. By Proposition A.1 a) we get
·Z 0
¸
Z d
λu
H
λv
H
e dBu
e dBv
E
a
0
¶µ
·µ
Z
Z 0
λu H
λd H
λa H
e Bu du
e Bd − λ
= E −e Ba − λ
a
d
e
λv
0
BvH dv
¶¸
£
¤
1
= − eλa eλd (−a)2H + d2H − (d − a)2H
2
Z
¤
1 λa d λv £
e (−a)2H + v 2H − (v − a)2H dv
+ λe
2
0
Z
¤
1 λd 0 λu £
− λe
e
(−u)2H + d2H − (d − u)2H du
2
a
µZ 0
¶
Z d
£
¤
1 2
λv
λu
2H
2H
2H
+ λ
e
e
(−u) + v − (v − u)
du dv .
2
0
a
After partial integration with respect to u, this becomes
Z 0
£
¤
λd
−He
eλu (−u)2H−1 − (d − u)2H−1 du
a
µZ 0
¶
Z d
£
¤
λv
λu
2H−1
2H−1
e
e
(−u)
− (v − u)
+Hλ
du dv ,
0
a
which, by partial integration with respect to v, is equal to
¶
µZ d
Z 0
λv
2H−2
λu
e (v − u)
dv du .
e
H(2H − 1)
0
a
Now we assume b = 0 < c. It follows from above that
·Z 0
¸
·Z 0
¸
Z d
Z d
Z 0
Z c
λu
H
λv
H
λu
H
λv
H
λu
H
λv
H
E
e dBu
e dBv = E
e dBu
e dBv −
e dBu
e dBv
a
c
a
0
a
0
¶ ¸
µZ c
¶
µZ d
·Z 0
Z 0
λv
2H−2
λu
λv
2H−2
λu
e (v − u)
dv du
e
e (v − u)
dv du −
e
= H(2H − 1)
0
a
0
a
µZ d
¶
Z 0
λu
λv
2H−2
= H(2H − 1)
e
e (v − u)
dv du .
a
c
H − B H , t ∈ R, is again a
For general −∞ ≤ a < b ≤ c < d < ∞, the process B̃tH = Bt+b
b
fractional Brownian motion. Therefore,
¸
¸
·Z 0
·Z b
Z d−b
Z d
λ(x+b)
H
λ(w+b)
H
λv
H
λu
H
e
dB̃x
e
dB̃w
e dBv = E
e dBu
E
a
= H(2H − 1)
= H(2H − 1)
c
Z
0
e
a−b
Z b
e
a
λ(w+b)
λu
µZ
d
µZ
c−b
a−b
d−b
e
λ(x+b)
(x − w)
c−b
λv
e (v − u)
c
and the proof is complete.
2H−2
¶
2H−2
¶
dx dw
dv du ,
¤
6
Lemma 2.2 Let β < 0. Then for each N = 0, 1, 2, . . . ,
!
Ãn−1
Z ∞
N
Y
X
(β − k) xβ−n + O(xβ−N −1 ) , as x → ∞ ,
ex
e−y y β dy = xβ +
x
n=1
k=0
and
e
where
−x
P0
Z
x
y β
β
e y dy = x +
1
n=1
N
X
(−1)
n=1
n
Ãn−1
Y
!
(β − k) xβ−n + O(xβ−N −1 ) , as x → ∞ ,
k=0
means 0.
Proof. We have
Z
∞
e−y y β dy
µx
Z
x
−x β
= e e x +β
ex
∞
e
−y β−1
y
dy
x
¶
= ...
= xβ + βxβ−1 + β(β − 1)xβ−2 + . . . + β(β − 1) . . . (β − N + 1)xβ−N
Z ∞
e−y y β−N −1 dy ,
+ex β(β − 1) . . . (β − N )
x
and
ex
Z
∞
e−y y β−N −1 dy ≤ ex
x
Z
∞
e−y xβ−N −1 dy = xβ−N −1 ,
x
which proves the first assertion. On the other hand,
Z x
−x
ey y β dy
e
µ1
¶
Z x
x β
y β−1
−x
e x −e−β
e y
dy = . . .
= e
1
β
β−1
= x − βx
+ . . . + (−1)N β(β − 1) . . . (β − N + 1)xβ−N
©
ª
−e−x e 1 − β + . . . + (−1)N β(β − 1) . . . (β − N + 1)
Z x
−x
N
ey y β−N −1 dy ,
−e (−1) β(β − 1) . . . (β − N )
1
and
e
−x
Z
x
y β−N −1
e y
1
dy ≤ e
−x
ÃZ
x
2
y
e dy +
1
Z
x
e
x
2
y
³ x ´β−N −1
2
dy
!
x
≤ e− 2 +
³ x ´β−N −1
2
.
This proves the second part of the lemma.
¤
Theorem 2.3 Let H ∈ (0, 12 ) ∪ ( 12 , 1] and N = 1, 2, . . . . Then for fixed t ∈ R and s → ∞,
!
Ã2n−1
N
Y
¡ H H ¢ 1 2X
Cov Yt , Yt+s = σ
(2H − k) s2H−2n + O(s2H−2N −2 ) .
λ−2n
2
n=1
k=0
7
Proof. By Lemma 2.1,
¢
¢
¡
¡
H
= Cov Y0H , YsH
Cov YtH , Yt+s
· Z 0
¸
Z s
λu
H
−λ(s−v)
H
= E σ
e dBu σ
e
dBv
−∞
−∞
" Z
#
Z 1
= e−λs E σ
0
−∞
eλu dBuH σ
+σ 2 H(2H − 1)e−λs
Z
0
−∞
λ
−∞
eλu
eλv dBvH
ÃZ
s
1
λ
!
eλv (v − u)2H−2 dv du
(by the change of variables: w = λu , x = λv)
¶
µZ λs
Z 0
σ2
x
2H−2
−λs
w
e (x − w)
dx dw + O(e−λs )
H(2H − 1)e
e
=
λ2H
1
−∞
(by the change of variables: y = x − w , z = x + w)
½Z λs
¶
µZ y
σ2
−λs
2H−2
z
=
H(2H − 1)e
y
e dz dy
2λ2H
1
2−y
¶ ¾
µZ 2λs−y
Z ∞
z
2H−2
e dz dy + O(e−λs )
y
+
λs
2−y
σ2
=
H(2H − 1)e−λs
2λ2H
¾
½Z λs
Z ∞
Z ∞
y 2H−2
2λs−y 2H−2
2−y 2H−2
e y
dy +
e
y
dy −
e y
dy + O(e−λs )
×
1
λs
1
¾
½
Z ∞
Z λs
σ2
−y 2H−2
y 2H−2
λs
−λs
e y
dy + O(e−λs ) .
e y
dy + e
=
H(2H − 1) e
2λ2H
λs
1
The proof can now be concluded by applying Lemma 2.2.
¤
Theorem 2.3 shows that for H ∈ (0, 21 ) ∪ ( 12 , 1], the decay of
¡
¢
H
Cov YtH , Yt+s
, for s → ∞ ,
is very similar to the decay of
¡ H
¢
H
H
Cov Bh+t
− BhH , Bh+s+t
− Bh+s
, for s → ∞
(see (1.2)). In particular, (YtH )t∈R is ergodic, and for H ∈ ( 12 , 1], it exhibits long-range dependence.
Remark 2.4 Let s ∈ R. For all H ∈ (0, 1), the functions f (x) = 1{x≤0} eλx and g(x) =
1{x≤s} eλx belong to the inner product space Λ̃H defined on page 289 of Pipiras and Taqqu
¡
¢
H
(2000). Hence, for all t, s ∈ R, Cov YtH , Yt+s
is equal to
Z
Γ(2H + 1) sin(πH) ∞ isx |x|1−2H
σ 2 e−λs (f, g)Λ̃H = σ 2
e
dx .
(2.2)
2π
λ2 + x 2
−∞
Therefore, the expression given in the the statement of Theorem 2.3 is an asymptotic expansion
of the right hand side in (2.2) as s → ∞.
8
The next corollary shows that for the solution (YtH,x )t≥0 of (2.1) with deterministic initial value
Y0H,x = x ∈ R,
³
´
H,x
Cov YtH,x , Yt+s
, for s → ∞ ,
decays like a power function of the order 2H − 2 as well.
Corollary 2.5 Let H ∈ (0, 12 ) ∪ ( 12 , 1], x ∈ R and N = 1, 2, . . . . Then for fixed t ≥ 0 and
s → ∞,
³
´
H,x
Cov YtH,x , Yt+s
Ã2n−1
!
N
n
o
1 2 X −2n Y
λ
(2H − k)
s2H−2n − e−λt (t + s)2H−2n + O(s2H−2N −2 ) .
σ
=
2
n=1
k=0
Proof.
=
=
=
=
³
H,x
YtH,x , Yt+s
´
Cov
¸
· Z t
Z t+s
e−λ(t+s−v) dBvH
e−λ(t−u) dBuH σ
E σ
0
0
¶¸
µZ t+s
· Z t
Z 0
−λ(t−v)
H
−λ(t+s−v)
H
−λs
−λ(t−u)
H
e
dBv
e
dBv − e
e
dBu σ
E σ
−∞
−∞
0
¶¸
¶ µZ t+s
· µZ t
Z 0
−λ(t+s−v)
H
λu
H
−λ(t−u)
H
−λt
e
dBv
e dBu
e
dBu − e
E σ
−∞
−∞
−∞
· Z t
¸
Z 0
−λs
−λ(t−u)
H
−λ(t−v)
H
−e E σ
e
dBu σ
e
dBv
0
−∞
¡
¢
¡
¢
H
H
Cov YtH , Yt+s
− e−λt Cov Y0H , Yt+s
+ O(e−λs ) .
Now, the corollary follows from Theorem 2.3.
3
¤
The Lamperti transform of fractional Brownian motion
Let λ > 0 and α > 0. For each H ∈ (0, 1], we set
ZtH := e−λt BαHexp( λ t) ,
H
t ∈ R.
Theorem 3.1 Let H ∈ (0, 1] and t, s ∈ R. Then
)
(
µ ¶
∞
X
¡ H H ¢ α2H
n
2H
e−λ|s| +
(−1)n−1
Cov Zt , Zt+s =
e−λ( H −1)|s| .
2
n
n=1
Proof. Without loss of generality we can assume that s ≥ 0. Then,
½
³ λ
´2H ¾
2H
¡ H H ¢
λ
−λt −λ(t+s) α
2λ(t+s)
2λt
(t+s)
t
Cov Zt , Zt+s = e e
e
+ e − eH
− eH
2
9
(3.1)
=
=
=
½
³
´2H ¾
λ
α2H λs
−2λs
−H
s
e
1+e
− 1−e
2
)
(
¶
∞ µ
X
2H ³ − λ s ´n
α2H λs
−2λs
e
1+e
−
−e H
2
n
n=0
(
)
µ ¶
∞
2H
X
n
α
−1)s
−λ( H
n−1 2H
−λs
(−1)
e
+
,
e
n
2
n=1
which proves the theorem.
¤
It follows from Theorem 3.1 that for every N = 1, 2, . . ., for each H ∈ (0, 1) and all t ∈ R,
(
)
µ ¶
N
´
³
X
¡ H H ¢ α2H
N +1
n
−λ( H
−1)|s|
n−1 2H
−λ|s|
e
(−1)
e
+
+ O e−λ( H −1)|s| ,
Cov Zt , Zt+s =
n
2
n=1
as s → ∞. This shows that for all H ∈ (0, 1), the auto-covariance function of (Z tH )t∈R decays
exponentially. It follows that for H ∈ (0, 21 )∪( 12 , 1), (ZtH )t∈R cannot have the same distribution
as (YtH )t∈R . For H ∈ (0, 12 ), the leading term in (3.1) for s → ∞, is
α2H −λ|s|
e
,
2
whereas for H ∈ ( 12 , 1), it is
1
α2H He−λ( H −1)|s| .
¡
¢
H
Note that for H ∈ (0, 21 ),¡ the leading
term of Cov ZtH , Zt+s
for s → ∞, is positive, whereas
¢
H
H
the leading term of Cov Yt , Yt+s for s → ∞, is negative (see Theorem 2.3).
Appendix: The Langevin equation
Langevin (1908) pioneered the following approach to the movement of a free particle immersed in a liquid: He described the particle’s velocity v by the equation of motion
dv(t)
f
F (t)
= − v(t) +
dt
m
m
(A.1)
where m > 0 is the mass of the particle, f > 0 a friction coefficient and F (t) a fluctuating force
resulting from impacts of the molecules of the surrounding medium. Uhlenbeck and Ornstein
(1930) imposed probability hypotheses on F (t) and then derived that for v(0) = x ∈ R, v(t) is
¢
2 ¡
f
normally distributed with mean xe−λt and variance σ2λ 1 − e−2λt , for λ = m
and σ 2 = 2fmkT
2 ,
where k is the Boltzmann constant and T the temperature. Doob (1942) noticed that if v(0) is
a random variable which is independent of (F (t))t≥0 and normally distributed with mean zero
2
and variance σ2λ , then the solution (v(t))t≥0 of (A.1) is stationary and
µ
¶
1
1
1{t>0} t 2 v
ln t , t ≥ 0 ,
2λ
is a Brownian motion, from which he concluded that every solution of (A.1) has almost surely
continuous paths which are nowhere differentiable. To avoid the “embarrassing situation” that
10
the equation (A.1) involves the derivative of v but leads to solutions v that do not have a
derivative, he gave a rigorous meaning to stochastic differential equations of the form
dXt = −λXt dt + dNt ,
(A.2)
for the case that N is a Lévy process and showed that for all x ∈ R, the equation (A.2) with
initial condition X0 = x ∈ R, has the unique solution
µ
¶
Z t
x
−λt
λu
Xt = e
x+
e dNu , t ≥ 0 .
0
In the modern theory of stochastic differential equations (see e.g. Protter (1990)) the equation
(A.2) with initial condition X0 = ξ ∈ L0 (Ω) is understood as the integral equation
Xt = ξ − λ
Z
t
Xs ds + Nt ,
t ≥ 0,
(A.3)
0
and it can be shown that the unique strong solution of (A.3) is given by
µ
¶
Z t
Xtξ := e−λt ξ +
eλu dNu , t ≥ 0 ,
0
whenever (Nt )t≥0 is a semimartingale with respect to the filtration generated by (Nt )t≥0 and
ξ.
Proposition A.1 Let (BtH )t∈R be a fractional Brownian motion with Hurst parameter H ∈
(0, 1] and ξ ∈ L0 (Ω). Let −∞ ≤ a < ∞ and λ, σ > 0. Then for almost all ω ∈ Ω, we have the
following:
a) For all t > a,
Z t
eλu dBuH (ω)
a
exists as a Riemann-Stieltjes integral and is equal to
Z t
λt H
λa H
BuH (ω)eλu du .
e Bt (ω) − e Ba (ω) − λ
a
b) The function
Z
t
a
eλu dBuH (ω) ,
t > a,
is continuous in t.
c) The unique continuous function y that solves the equation,
y(t) = ξ(ω) − λ
is given by
y(t) = e
−λt
Z
t
0
y(s)ds + σBtH (ω) ,
½
¾
Z t
λu
H
ξ(ω) + σ
e dBu (ω) ,
0
11
t ≥ 0.
t ≥ 0.
(A.4)
(A.5)
In particular, the unique continuous solution of the equation,
Z t
Z 0
λu
H
y(s)ds + σBtH (ω) ,
e dBu (ω) − λ
y(t) = σ
t ≥ 0,
0
−∞
is given by
y(t) = σ
Z
t
−∞
e−λ(t−u) dBuH (ω) ,
t ≥ 0.
Proof. It can easily be checked that
2H H
B̃sH := 1{s<0} (−s)2H B H
B1 ,
1 + 1{s>0} s
s
s
s ∈ R,
is again a fractional Brownian motion. It follows from the Kolmogorov- Čentsov theorem (see
e.g. Theorem 2.2.8 of Karatzas and Shreve (1991)) that there exists a measurable null set
N ⊂ Ω, such that for every ω ∈ Ω \ N , BsH (ω) and B̃sH (ω) are continuous in s, and for all
β < H,
B̃ H (ω)
lim s β = 0 .
s→0 |s|
This implies that for all γ > H,
BsH (ω)
= 0.
γ
|s|→∞ |s|
lim
Hence, for all t > a,
Z
t
a
BuH (ω)eλu du
exists as a Riemann integral, which, by Theorem 2.21 of Wheeden and Zygmund (1977), implies
that the Riemann-Stieltjes integral
Z
t
a
eλu dBuH (ω)
exists too and is equal to
e
λt
BtH (ω)
−e
λa
BaH (ω)
−λ
Z
t
a
BuH (ω)eλu du .
This proves a).
b) follows from a) and the fact that the function
Z t
eλt BtH (ω) − λ
BuH (ω)eλu du ,
t > a,
a
is continuous in t.
A continuous function y solves (A.4) if and only if the function
Z t
z(t) =
y(s)ds , t ≥ 0 ,
0
solves the linear differential equation:
z 0 (t) = −λz(t) + ξ(ω) + σBtH (ω) ,
12
z(0) = 0 .
(A.6)
Since the unique solution of (A.6) is given by
Z t
¡
¢
−λt
z(t) = e
eλu ξ(ω) + σBuH (ω) du ,
t ≥ 0,
0
the unique continuous function y that solves (A.4) is given by
Z t
¡
¢
−λt
eλu ξ(ω) + σBuH (ω) du + ξ(ω) + σBtH (ω) ,
−λe
t ≥ 0,
0
which, by a), is equal to (A.5). This shows c).
¤
Remark A.2 Equation (A.3) can be solved path-wise for all stochastic processes (N t )t≥0 that
have almost all paths in
½
¾
Z t
1
Lloc (R+ ) := h : R+ → R : h is measurable and ∀t ≥ 0 ,
|h(s)| ds < ∞ ,
0
and even when the constant λ is replaced by a stochastic process with almost all paths in
½
¾
∞
Lloc (R+ ) := g : R+ → R : g is measurable and ∀t ≥ 0 , sup |g(s)| < ∞ .
0≤s≤t
Indeed, if h ∈ L1loc (R+ ) and g ∈ L∞
loc (R+ ), then it can easily be checked that the function
Z t
Rt
g(s)e s g(u)du h(s)ds , t ≥ 0 ,
(A.7)
f (t) := h(t) +
0
is in
L1loc (R+ )
and solves the integral equation
Z t
f (t) =
g(s)f (s)ds + h(t) ,
t ≥ 0.
(A.8)
0
On the other hand, if f˜ ∈ L1loc (R+ ) is a solution of (A.8), then
Z t
h
i
˜
g(s) f (t) − f˜(t) ds , t ≥ 0 ,
f (t) − f (t) =
0
and it follows from a variant of Gronwall’s lemma that
f (t) − f˜(t) = 0 ,
t ≥ 0.
Hence, (A.7) is the only function in L1loc (R+ ) that solves (A.8).
If the functions g and h are both in L∞
loc (R+ ) and continuous on R+ \ C, where C is of
Lebesgue measure zero, then it can be deduced from Theorems 5.54 and 2.21 of Wheeden and
Zygmund (1977) that f can be written as follows:
µ
¶
Z t R
Rt
s
g(u)du
−
g(u)du
h(0) +
e 0
dh(s) , t ≥ 0 ,
f (t) = e 0
0
Rt
Rs
where 0 e− 0 g(u)du dh(s) is a Riemann-Stieltjes integral.
Note that almost all paths of a semimartingale are right-continuous and have left limits, in
particular, they are in L∞
loc (R+ ) and have at most countably many discontinuities.
13
References
Doob, J.L. (1942), The Brownian movement and stochastic equations, Ann. of Math. (2) 43,
351-369.
Embrechts, P. and Maejima, M. (2002), Selfsimilar Processes, Princeton Series in Applied
Mathematics, Princeton University Press.
Karatzas, I. and Shreve, S.E. (1991), Brownian Motion and Stochastic Calculus, Graduate
Texts in Mathematics, Springer-Verlag, New York.
Lamperti, J.W. (1962), Semi-stable stochastic processes, Trans. Amer. Math. Soc. 104, 62-78.
Langevin, P. (1908), Sur la théorie du mouvement brownien, C.R. Acad. Sci. Paris 146, 530533.
Pipiras, V. and Taqqu, M. (2000), Integration questions related to fractional Brownian motion,
Prob. Th. Rel. Fields 118, 121-291.
Protter, P. (1990), Stochastic Integration and Differential Equations, Springer-Verlag, Berlin.
Samorodnitsky, G. and Taqqu, M.S. (1994), Stable Non-Gaussian Random Processes, Chapman
& Hall, New York.
Uhlenbeck, G.E. and Ornstein, L.S. (1930), On the theory of the Brownian motion, Physical
Review 36, 823-841.
Wheeden, R.L. and Zygmund, A. (1977), Measure and Integral, Marcel Dekker, New YorkBasel.
14