On differentiability of solution to stochastic differential equation with fractional Brownian motion
Stochastic differential equation with pathwise integral with respect to fractional Brownian motion is considered. For solution of such equation, under different conditions, the Malliavin differentiability is proved. Under infinite differentiability and boundedness of derivatives of the cofficients i...
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| Цитувати: | On differentiability of solution to stochastic differential equation with fractional Brownian motion / Yu.S. Mishura, G.M. Shevchenko // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 243-250. — Бібліогр.: 10 назв.— англ. |
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irk-123456789-44932009-11-20T12:00:43Z On differentiability of solution to stochastic differential equation with fractional Brownian motion Mishura, Yu.S. Shevchenko, G.M. Stochastic differential equation with pathwise integral with respect to fractional Brownian motion is considered. For solution of such equation, under different conditions, the Malliavin differentiability is proved. Under infinite differentiability and boundedness of derivatives of the cofficients it is proved that the solution is infinitely differentiable in the Malliavin sense with all derivatives bounded. 2007 Article On differentiability of solution to stochastic differential equation with fractional Brownian motion / Yu.S. Mishura, G.M. Shevchenko // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 243-250. — Бібліогр.: 10 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4493 en Інститут математики НАН України |
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Stochastic differential equation with pathwise integral with respect to fractional Brownian motion is considered. For solution of such equation, under different conditions, the Malliavin differentiability is proved. Under infinite differentiability and boundedness of derivatives of the cofficients it is proved that the solution is infinitely differentiable in the Malliavin sense with all derivatives bounded. |
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Article |
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Mishura, Yu.S. Shevchenko, G.M. |
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Mishura, Yu.S. Shevchenko, G.M. On differentiability of solution to stochastic differential equation with fractional Brownian motion |
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Mishura, Yu.S. Shevchenko, G.M. |
| author_sort |
Mishura, Yu.S. |
| title |
On differentiability of solution to stochastic differential equation with fractional Brownian motion |
| title_short |
On differentiability of solution to stochastic differential equation with fractional Brownian motion |
| title_full |
On differentiability of solution to stochastic differential equation with fractional Brownian motion |
| title_fullStr |
On differentiability of solution to stochastic differential equation with fractional Brownian motion |
| title_full_unstemmed |
On differentiability of solution to stochastic differential equation with fractional Brownian motion |
| title_sort |
on differentiability of solution to stochastic differential equation with fractional brownian motion |
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Інститут математики НАН України |
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2007 |
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http://dspace.nbuv.gov.ua/handle/123456789/4493 |
| citation_txt |
On differentiability of solution to stochastic differential equation with fractional Brownian motion / Yu.S. Mishura, G.M. Shevchenko // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 243-250. — Бібліогр.: 10 назв.— англ. |
| work_keys_str_mv |
AT mishurayus ondifferentiabilityofsolutiontostochasticdifferentialequationwithfractionalbrownianmotion AT shevchenkogm ondifferentiabilityofsolutiontostochasticdifferentialequationwithfractionalbrownianmotion |
| first_indexed |
2025-07-02T07:43:29Z |
| last_indexed |
2025-07-02T07:43:29Z |
| _version_ |
1836520266602119168 |
| fulltext |
Theory of Stochastic Processes
Vol.13 (29), no.1-2, 2007, pp.243-250
YU. S. MISHURA AND G. M. SHEVCHENKO
ON DIFFERENTIABILITY OF SOLUTION TO
STOCHASTIC DIFFERENTIAL EQUATION WITH
FRACTIONAL BROWNIAN MOTION
Stochastic differential equation with pathwise integral with respect
to fractional Brownian motion is considered. For solution of such
equation, under different conditions, the Malliavin differentiability is
proved. Under infinite differentiability and boundedness of deriva-
tives of the coefficients it is proved that the solution is infinitely
differentiable in the Malliavin sense with all derivatives bounded.
1. Introduction
Let
{
BH
t = (B1,H
t , . . . , Bm,H
t ), t ≥ 0
}
be m-dimensional fractional Brow-
nian motion (fBm in short) of Hurst parameter H > 1
2
on a filtered proba-
bility space (Ω, F , {Ft, t ≥ 0}, P ). That is, BH
t is an Ft-adapted Gaussian
process, whose components are independent and have the covariance func-
tion
E
[
Bi,H
t Bi,H
s
]
= RH(t, s) :=
1
2
(|t|2H + |s|2H − |t − s|2H
)
.
There are different ways to define stochastic integrals with respect to fBm.
We choose in this paper the approach of Zähle [10], that is, the Riemann–
Stiltjes integral defined in pathwise sense.
In this paper we consider the following equation
Xt = X0 +
∫ t
0
b(Xs)ds +
∫ t
0
σ(Xs)dBH
s
= X0 +
∫ t
0
b(Xs)ds +
m∑
j=1
∫ t
0
σj(Xs)dBj,H
s , t ≥ 0,
(1)
or
X i
t = X i
0 +
∫ t
0
bi(Xs)ds +
m∑
j=1
∫ t
0
σj
i (Xs)dBj,H
s , i = 1, ..., d.
2000 Mathematics Subject Classifications. Primary: 60H05, Secondary: 60H07
Key words and phrases. Fractional Brownian motion, pathwise integral, stochastic
differential equation, Malliavin derivative.
243
244 YU. S. MISHURA AND G. M. SHEVCHENKO
Many authors studied existence and uniqueness of solution of (1), see
[3], [4], [5], [9], [7].
In this paper we investigate stochastic differentiability of the solution
of (1). This question was already studied in the paper of Nualart and
Saussereau [8], where they have proved that under the conditions that
the coefficients are infinitely differentiable and bounded together with their
derivatives, the solution will be infinitely differentiable (in a local sense).
From a point of view of stochastic derivatives of Nelson’s type this problem
was studied by Darses and Nourdin [2]. In this paper we establish strong
(non-local) differentiability results in two cases: for the diffusion coefficient
is linear and for one-dimensional equation with infinitely differentiable co-
efficients. In the latter case we also prove the uniform boundedness of
stochastic derivatives.
1. Stochastic derivative w.r.t. fBm
We briefly recall the notion of the stochastic derivative with respect to
the fBm, the detailed description can be found in [1]. First we define the
Hilbert space H associated to the fBm as the closure of the space R
m-valued
step function with respect to the scalar product
〈
(111[0,t1], . . . ,111[0,tm]), ((111[0,s1], . . . ,111[0,sm])
〉
H :=
m∑
i=1
RH(ti, si).
The space H contains not only usual functions, but also distributions. For
ϕ, ψ ∈ L
1
H ([0, T ]; Rm) one has
〈ϕ, ψ〉H = H(2H − 1)
∫ T
0
∫ T
0
φ(r)ψ(u) |r − u|2H−2 dr du.
The mapping
B : (111[0,t1], . . . ,111[0,tm]) �−→
m∑
i=1
Bi,H
ti
can be extended to the isometry between H and the Hilbert space H1(B
H)
associated with BH .
For a smooth variable of the form F = f(B(ϕ1), . . . , B(ϕn)), where
f ∈ C∞
b (Rn), ϕi ∈ H the stochastic derivative, or Malliavin derivative, is
defined as the H-valued random variable
DF :=
n∑
i=1
∂xi
f(B(ϕ1), . . . , B(ϕn))ϕi.
This operator is closable from Lp(Ω) to Lp(Ω;H). It is also convenient to
write DF = {DsF, s ≥ 0} in many cases when DF has usual, not gener-
alized, meaning. The space D
k,p is defined as the closure of the space of
ON DIFFERENTIABILITY OF SOLUTION TO SDE WITH FBM 245
smooth random variables with respect to the norm
‖F‖k,p :=
(
E [ |F |p ] +
k∑
j=1
E
[ ∥∥DjF
∥∥p
H⊗j
] ) 1
p
.
We denote by D
k,p
loc the corresponding local domain, i.e. the set of random
variables F such that there exists a sequence {(Ωn, Fn), n ≥ 1} ⊂ F × D
k,p
satisfying Ωn ↑ Ω, n → ∞ and F = Fn on Ωn. In [8], the following fact is
proved.
Theorem 1. Let H > 1/2 and assume that the coefficients b and σ are
infinitely differentiable functions which are bounded together with all their
derivatives, then the solution of the SDE (1) belongs to D
k,p
loc(H), for any
p > 0, k ≥ 1.
Remark 2. It is easily seen from the argument of paper [8] that for a given
k ≥ 1 one needs no infinite differentiability of b and σ to prove the fact
Xt ∈ D
k,p
loc , it is only enough that b, σ ∈ Ck+1(Rd).
We will consider two cases when the global differentiability can be proved
for the solution of (1).
2. Differentiability of the solution to quasilinear SDE
Consider equation (1), in which H > 3/4 and the coefficient σ is linear,
that is σj(x) = σjx is some linear operator:
Xt = X0 +
∫ t
0
b(Xs)ds +
m∑
j=1
∫ t
0
σjXsdBj,H
s , i = 1, ..., d. (2)
We will assume that the coefficient b ∈ C1
b (Rd) and that X0 is F0-measurable
bounded random variable. Together with the linearity of σ, these conditions
is enough to assure that equation (2) has unique solution, which belongs to
all Lp(Ω), see [7].
Theorem 3. Under the above assumptions the solution of (2) belongs to
D
1,p for any p > 0.
Proof. We will assume for simplicity that d = 1, all argumentation transfers
easily to arbitrary dimension. Throughout the proof we will denote by
C all constants which may depend on the coefficients b and σ, but are
independent of everything else. We remind that the unique solution of (2)
can be constructed as the limit of successive approximations
{
X
(n)
t , n ≥ 1
}
,
where X
(0)
t ≡ X0. Now we are going to prove by induction that for every
p > 0 X
(n)
t ∈ D
1,p and DsX
(n)
t is Hölder continuous of order 1 − α for
some α ∈ (1 − H, 1/2). This is, of course, obvious for n = 0. Assume this
246 YU. S. MISHURA AND G. M. SHEVCHENKO
is true for n. Then the integrals
∫ t
s
σDsX
(n)
r dBH
r and
∫ t
s
b(X
(n)
r )dBH
r are
well-defined and due to the closedness of the stochastic derivative we can
write
DsX
(n+1)
r = σX(n)
s +
∫ t
s
b′(X(n)
r )DsX
(n)
r dr +
∫ t
s
σDsX
(n)
r dBH
r .
Now we can write, using the Hölder continuity assumption and well-known
estimates for an integral with respect to the fBm [7],
∣∣∣DsX
(n+1)
t
∣∣∣ ≤ C1(ω) + C2(ω)
∫ t
s
∣∣∣DsX
(n)
r
∣∣∣
(r − s)α
dr
+ C2(ω)
∫ t
s
∫ r
s
∣∣∣DsX
(n)
r − DsX
(n)
u
∣∣∣
(r − u)1+α
du dr,
where C1(ω) = C exp{CGα}, C2(ω) = CGα, κ = 1/(1 − 2α), Gα is certain
random variable s.t. E
[
exp
{
pGδ
α
} ]
< ∞ for all p > 0, δ ∈ (0, 2). Similarly,
∣∣DsX
(n+1)
r − DsX
(n+1)
u
∣∣
≤ C2(ω)
∫ r
u
∣∣∣DsX
(n)
z
∣∣∣
(z − u)α
dz + C2(ω)
∫ r
u
∫ z
u
∣∣∣DsX
(n)
z − DsX
(n)
v
∣∣∣
(z − v)1+α
dv dz.
Define
ϕ1
n(t, s) =
∣∣∣DsX
(n)
t
∣∣∣ , ϕ2
n(t, s) =
∫ t
s
∣∣∣DsX
(n)
t − DsX
(n)
u
∣∣∣
(t − u)1+α
du,
ϕn(t, s) = ϕ1
n(t, s) + ϕ2
n(t, s),
so that we can write
ϕ1
n+1(t, s) ≤ C1(ω) + C2(ω)
∫ t
s
ϕ1
n(r, s)(r − s)−αdr + C2(ω)
∫ t
s
ϕ2
n(r, s)dr,
ϕ2
n+1(t, s) ≤ C2(ω)
∫ t
s
ϕ1
n(v, s)(t − v)−2αdv + C2(ω)
∫ t
s
ϕ2
n(v, s)(t − v)−αdv,
whence
ϕn+1(t, s) ≤ C1(ω) + C2(ω)
∫ t
s
(
(u − s)−2α + (t − u)−2α
)
ϕn(u, s)du
and one gets easily by induction that
ϕn(t, s) ≤ C1(ω) exp
{
Cα
(
C2(ω)
)
(t − s)
}
,
ON DIFFERENTIABILITY OF SOLUTION TO SDE WITH FBM 247
where C(α) =
(
4Γ(1 − 2α)
)
. Since H > 3/4, we can choose α > 1 − H
so that κ < 2. Then we have E [ |ϕn|p ] ≤ Cp due to the properties of the
random variable Gα. Therefore, we have for any p > 0
∥∥∥X
(n)
t
∥∥∥
1,p
≤ Cp with
constant independent of n.
Further,
∣∣∣X(n+1)
t − Xt
∣∣∣ ≤ C2(ω)
∫ t
0
∣∣∣X(n)
s − Xs
∣∣∣
sα
ds
+ C2(ω)
∫ t
0
∫ r
0
∣∣∣Xr − X
(n)
r − Xu + X
(n)
u
∣∣∣
(r − u)1+α
du dr,
∫ t
0
∣∣∣X(n+1)
t − Xt − X
(n+1)
u + Xu
∣∣∣
(t − u)1+α
du
≤
∫ t
0
du
(t − u)1+α
(
C2(ω)
∫ t
u
∣∣X(n)
s − Xs
∣∣ s−αds
+ C2(ω)
∫ t
u
∫ r
u
∣∣∣Xr − Xv − X
(n)
r + X
(n)
v
∣∣∣
(r − v)1+α
dv dr
)
≤ C2(ω)
∫ t
0
∣∣X(n)
s − Xs
∣∣ s−α(t − s)−αds
+ C2(ω)
∫ t
0
(t − s)−α
∫ s
0
∣∣∣Xs − Xv − X
(n)
s + X
(n)
v
∣∣∣
(s − v)1+α
ds,
or
ξ1
n+1(t) ≤ C2(ω)
∫ t
0
ξ1
n(s)s
−αds + C2(ω)
∫ t
0
ξ2
n(s)ds,
ξ2
n+1(t) ≤ C2(ω)
∫ t
0
ξ1
n(s)s
−α(t − s)−αds + C2(ω)
∫ t
0
ξ2
n(s)(t − s)−αds,
where
ξ1
n(t) =
∣∣∣X(n)
t − Xt
∣∣∣ , ξ2
n(t) =
∫ t
0
∣∣∣X(n)
t + X
(n)
u − Xt + Xu
∣∣∣
(t − u)1+α
du.
Define ξn = ξ1
n + ξ2
n and write
ξn+1(t) ≤ C2(ω)
∫ t
0
s−α(t − s)−αξn(s)ds
≤ C2(ω)t2α
∫ t
0
s−2α(t − s)−2αξn(s)ds.
248 YU. S. MISHURA AND G. M. SHEVCHENKO
It is easily proved by induction that
sup
0≤t≤T
|ξn(t)| ≤ Cn+1Gn+1
α C3(ω)
Γ(n(1 − 2α))
,
where
C3(ω) = sup
0≤t≤T
|ξ0(t)| = sup
0≤t≤T
(
|X0| + |Xt| +
∫ t
0
|Xt − Xr|
(t − r)1+α
drdu
)
.
It is well-known that for all p > 0 E [ (C3(ω))p ] < ∞. Further, we can
write for some δ ∈ (κ, 2) the inequality gne−gδ ≤ (n/(eδ))
n
δ (the right-hand
side is the maximal value for the left-hand side for g ≥ 0) and apply that
Γ(α) ≥ Cααα for α large enough to obtain
sup
0≤t≤T
|ξn(t)| ≤ (
Cα,δ)
nnn(1/δ−1/ ) exp
{
Gδ
}
. (3)
Since for all p > 0 E
[
exp
{
pGδ
} ]
< ∞, inequality (3) yields
E
[
sup
0≤t≤T
|ξn(t)|p
]
→ 0, n → ∞.
Consequently, X
(n)
t → Xt in Lp(Ω). Moreover, we can show similarly that
for any p > 0
E
[
sup
s,t
∣∣∣DsX
(n)
t − DsX
(m)
t
∣∣∣p
]
→ 0, m, n → ∞.
Thus, the derivatives DX
(n)
t converge in Lp(Ω;H), but the closedness of the
stochastic derivative then gives that the limit is DXt and that ‖Xt‖1,p < ∞,
which concludes the proof. �
Remark 4. Theorem 1 requires boundedness of σ and provides only local
differentiability of solution. Theorem 3 above gives global differentiability
for an unbounded σ, but the price we pay is the assumptions that H > 3/4
and that σ is linear.
2. Differentiability of the solution in one-dimensional case
Consider one-dimensional equation
Xt = X0 +
∫ t
0
b(Xs)ds +
∫ t
0
σ(Xs)dBH
s . (4)
We assume that b, σ ∈ C∞
b (R). We remind that these assumptions guaran-
tee existence and uniqueness of solution, and moreover by Theorem 1 the
ON DIFFERENTIABILITY OF SOLUTION TO SDE WITH FBM 249
solution will be in D
k,p
loc for any p > 0, k ≥ 1. We are going to prove that Xt
is globally differentiable and all derivatives are bounded under additional
assumption of non-degeneracy of σ.
Theorem 5. Assume that the coefficients of equation (4) satisfy b, σ ∈
C∞
b (R), inf |σ| > 0. Then for all k ≥ 1 the solution Xt of this equation
belongs to D
k,∞, i.e., Xt is infinitely stochastically differentiable and all its
derivatives are essentially bounded.
Proof. We assume without loss of generality that σ(x) > 0. It is proved in
[8] that the stochastic derivative of Xt satisfies the linear equation
DrXt = σ(Xr) +
∫ t
r
b′(Xs)DrXsds +
∫ t
r
σ′(Xs)DrXsdBH
s , t ≥ r (5)
which is easily solved, and the solution is
DrXt = σ(Xr) exp
{∫ t
r
b′(Xs)ds +
∫ t
r
σ′(Xs)dBH
s
}
, t ≥ r.
According to Itô formula
log σ(Xt) = log σ(Xr) +
∫ t
r
σ′(Xs)σ
−1(Xs)
[
b(Xs)ds + σ(Xs)dBH
s
]
,
whence
σ(Xr) exp
{∫ t
r
σ′(Xs)dBH
s
}
= σ(Xt) exp
{∫ t
r
σ′(Xs)σ
−1(Xs)b(Xs)ds
}
,
and thus the derivative DrXt is uniformly bounded, which already means
that Xt is differentiable in usual sense rather then local. Moreover, we can
write
DrXt = σ(Xt) exp
{∫ t
r
b′(Xs)ds +
∫ t
r
σ′(Xs)σ
−1(Xs)b(Xs)ds
}
= σ(Xt)E
and differentiate this equation, getting for u ∨ r ≤ t
DuDrXt = E ×
(
σ′(Xt)DuXt
+
∫ t
r∨u
b′′(Xs)DuXs ds +
∫ t
r∨u
[
σ′′(Xs)σ
−1(Xs)b(Xs)
− (σ′(Xs))
2σ−2(Xs)b(Xs) + σ′(Xs)σ
−1(Xs)b
′(Xs)
]
DuXs ds
)
.
Then DuDrXt exists and is uniformly bounded. Going on, we can easily
prove by induction that
Ds1 . . .Dsk
Xt = E
(
Pk +
∫ t
k
i=1 si
Qk ds
)
,
250 YU. S. MISHURA AND G. M. SHEVCHENKO
where Pk and Qk are polynomials of Dsi1
. . .Dsil
Xt, l < k, σ(j)(Xt), j ≤ k
and Dsi1
. . .Dsil
Xs, l < k, b(j)(Xs), σ(j)(Xs), j ≤ k, σ−1(Xs) respectively.
Then existence and boundedness of all derivatives of Xt can be proved by
induction. �
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Department of Probability Theory and Mathematical Statistics,
Kyiv National Taras Shevchenko University, Kyiv, Ukraine
E-mail address: myus@univ.kiev.ua
Department of Probability Theory and Mathematical Statistics,
Kyiv National Taras Shevchenko University, Kyiv, Ukraine
E-mail address: zhora@univ.kiev.ua
|