Existence and uniqueness of solution of mixed stochastic differential equation driven by fractional Brownian motion and wiener process
The existence and uniqueness of solution of stochastic differential equation driven by standard Brownian motion and fractional Brownian motion with Hurst parameter H belongs (3/4, 1) is established.
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Інститут математики НАН України
2007
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| Cite this: | Existence and uniqueness of solution of mixed stochastic differential equation driven by fractional Brownian motion and wiener process / Y. Mishura, S. Posashkov // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 152-165. — Бібліогр.: 8 назв.— англ. |
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| author | Mishura, Y. Posashkov, S. |
| author_facet | Mishura, Y. Posashkov, S. |
| citation_txt | Existence and uniqueness of solution of mixed stochastic differential equation driven by fractional Brownian motion and wiener process / Y. Mishura, S. Posashkov // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 152-165. — Бібліогр.: 8 назв.— англ. |
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| description | The existence and uniqueness of solution of stochastic differential equation driven by standard Brownian motion and fractional Brownian motion with Hurst parameter H belongs (3/4, 1) is established.
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Theory of Stochastic Processes
Vol.13 (29), no.1-2, 2007, pp.152-165
YULIA MISHURA AND SERGIY POSASHKOV
EXISTENCE AND UNIQUENESS OF SOLUTION
OF MIXED STOCHASTIC DIFFERENTIAL
EQUATION DRIVEN BY FRACTIONAL
BROWNIAN MOTION AND WIENER PROCESS
The existence and uniqueness of solution of stochastic differential
equation driven by standard Brownian motion and fractional Brow-
nian motion with Hurst parameter H ∈ (3/4, 1) is established.
1. Introduction
We will consider the following stochastic differential equation with non-
homogeneous coefficients, defined on the stochastic basis (Ω,F , (Ft, t ∈
[0, T ]),P) :
Xt = X0 +
∫ t
0
a(s, Xs)ds +
∫ t
0
b(s, Xs)dWs +
∫ t
0
c(s, Xs)dBH
s , t ∈ [0, T ],
(1)
where X0 is F0-measurable random variable, EX2
0 < ∞, W = (Wt, t ∈
[0, T ]) is standard Brownian motion (sBm), BH = (BH
t , t ∈ [0, T ]) is frac-
tional Brownian motion (fBm) with Hurst parameter H ∈ (3/4, 1). Coef-
ficients a, b, c : [0, T ] × R −→ R are deterministic functions, which satisfy
well-known conditions of existence of integrals in the right-hand of (1),
where the integral
∫ t
0
c(s, Xs)dBH
s is considered in pathwise sense.
If b = 0 we obtain the equation, which contains only fBm. The condi-
tions of existence and uniqueness of solution of such equations were shown
in [1]. The case b(t, x) = bx and c(t, x) = cx was considered in [7].
Using the similarity to sBm of process MH,ε
t := Vt + 1/εBH
t , ε > 0, for
H ∈ (3/4, 1), which was proven in [2], and the representation of processes
similar to sBm from [8], the existence and uniqueness of solution of auxiliary
stochastic differential equation
XN
t = X0 +
∫ t
0
a(s, XN
s )dt +
∫ t
0
b(s, XN
s )dWt
+
∫ t
0
c(s, XN
s )dBH
s +
1
N
∫ t
0
c(s, XN
s )dVs, t ∈ [0, T ], (2)
2000 Mathematics Subject Classifications. 60G15, 60H05, 60H10
Key words and phrases. Stochastic differential equation, fractional Brownian motion
152
SOLUTION OF STOCHASTIC DIFFERENTIAL EQUATION 153
where V = (Vt, t ∈ [0, T ]) is another standard Brownian motion, indepen-
dent on W and BH , for any N ∈ N, was proved in the paper [3].
The main aim of present work is to find the solution of equation (1) as
the limit in some complete space of solutions of (2) when N tends to ∞.
2. Main part
Let the coefficients of equation (1) satisfy, in addition, the following
assumptions
(A) There exists A > 0 such that |a(t, x)| ≤ A, |b(t, x)| ≤ A, |c(t, x)| ≤ A
∀t ∈ [0, T ], ∀x ∈ R.
(B) There exists L > 0 such that
(a(t, x)−a(t, y))2 +(b(t, x)−b(t, y))2 +(c(t, x)−c(t, y))2 ≤ L2(x−y)2,
for ∀t ∈ [0, T ], ∀x, y ∈ R.
(C) The function c(t, x) is differentiable by c, and there exist constants
B > 0 and β ∈ (1 − H, 1) such that ∀s, t ∈ [0, T ], ∀x ∈ R
|c(s, x) − c(t, x)| + |∂xc(s, x) − ∂xc(t, x)| ≤ Bs − t|β.
(D) Holder continuity of ∂xc(t, x) in x :
|∂xc(t, x) − ∂xc(t, y)| ≤ D|x − y|ρ,
for ∀t ∈ [0, T ], ∀x, y ∈ R with some parameter ρ ∈ (3/2 − H, 1).
Note that this conditions imply the conditions from [3] of existence and
uniqueness of solution of equation (2).
Let consider for any 1−H < α < min(1/2, β, ρ−1/2) the space of Besov
type
Wα([0, T ]) := {Y = Yt(ω) : (t, ω) ∈ [0, T ] × Ω, ||Y ||α < ∞} (3)
with the norm
||Y ||α := sup
t∈[0,T ]
(
E(Yt)
2 + E
(∫ t
0
|Yt − Ys|
(t − s)1+α
ds
)2
)
, (4)
and prove that the solution of SDE (2) belongs to this space for any N ∈ N.
We shall denote different constants as C if it is unimportant for stated
results.
From [5] we have for pathwise integral
∫ t
0
f(s)dBH
s the representation∫ t
0
f(s)dBH
s =
∫ t
0
Dα
0+f(s)D1−α
t− BH
t−(s)ds,
154 YULIA MISHURA AND SERGIY POSASHKOV
therefore this integral can be estimated as∣∣∣∣∫ t
0
f(s)dBH
s
∣∣∣∣ ≤ Ct(ω)
(∫ t
0
|f(s)|
sα
ds +
∫ t
0
∫ r
0
|f(r) − f(u)|
(r − u)1+α
dudr
)
, (5)
where
Ct(ω) = sup
0≤u≤s≤t
|D1−α
s− BH
s−(u)|. (6)
The existence of all moments E|Ct(ω)|p for any p ≥ 1 has also been proved
in [1].
Lemma 1. The process |Ct(ω)| is dominated by some continuous process.
Proof. To prove this statement we estimate |D1−α
t− BH
t−(s)|:
|D1−α
t− BH
t−(s)| ≤ 1
Γ(α)
( |BH
t − BH
s |
(t − s)1−α
+ (1 − α)
∫ t
s
|BH
s − BH
u |
(s − u)2−α
du
)
.
Using Garsia - Rodemich - Rumsey inequality from [1] that has a form
|f(t) − f(s)|p ≤ Cλ,p|t − s|λp−1
∫ t
0
∫ t
0
|f(x) − f(y)|p
|x − y|λp+1
dxdy (7)
with f ∈ C[0, T ], t, s ∈ [0, T ], p ≥ 1 and λ > p−1, we obtain
|BH
t − BH
s | ≤ Cλ,p|t − s|λ−1/p
(∫ t
0
∫ t
0
|BH
u − BH
v |p
|u − v|λp+1
dudv
)1/p
.
Let λ = H − ε/2, p = 2/ε for 0 < ε < H + α − 1, then
|BH
t − BH
s | ≤ Cε|t − s|H−ε
(∫ t
0
∫ t
0
|BH
u − BH
v |2/ε
|u − v|2H/ε
dudv
)ε/2
,
note that 1 − α < H − ε. Denote
ψt,ε =
(∫ t
0
∫ t
0
|BH
u − BH
v |2/ε
|u − v|2H/ε
dudv
)ε/2
, (8)
The process ψt,ε is continuous and nondecreasing in t, and Eψq
t,ε < ∞ for
any q > 0 and any t ∈ [0, T ] [1].
So,
|BH
t − BH
s |
(t − s)1−α
≤ CH,ε(t − s)H−ε−1+αψt,ε ≤ Cψt,ε,
and∫ t
s
|BH
u − BH
s |
(u − s)2−α
du ≤ CH,ε
∫ t
s
(u − s)H−ε−2+αψt,εdu
≤ CH,ε(t − s)H+α−1−εψt,ε ≤ Cψt,ε,
SOLUTION OF STOCHASTIC DIFFERENTIAL EQUATION 155
where H + α − 1 − ε > 0.
So sup
0≤u≤s≤t
|D1−α
s− BH
s−(u)| ≤ Cψt,ε, and the statement of lemma follows
from continuity and strictly increasing property of ψt,ε.
Introduce the random variable
C(ω) := sup
0≤t≤T
Ct(ω). (9)
Note that C(ω) ≤ ψT,ε and E|C(ω)|q < ∞ for any q ≥ 1.
First of all we prove the Hölder continuity of the solution of equation
(2).
Theorem 2. For any δ ∈ (0, 1/2) the solution of equation (2) is Hölder
continuous with parameter 1/2 − δ.
Proof. At first, establish Hölder properties of the integral {∫ t
0
bsdWs, t ∈
[0, T ]}, where bs is a predictable bounded process. For any 0 < δ < 1/4 put
p = 2
δ
, θ = 1/2 − δ/2 in Garsia - Rodemich - Rumsey inequality (7). Then∣∣∣∣∫ t
s
budWu
∣∣∣∣ ≤ Cδ|t − s|1/2−δξb
t,δ,
where
ξb
t,δ :=
(∫ t
0
∫ t
0
| ∫ y
x
budWu|p
|x − y|θp+1
dxdy
)1/p
=
(∫ t
0
∫ t
0
| ∫ y
x
budWu|2/δ
|x − y|1/δ
dxdy
)δ/2
(10)
and for any q > p from Hölder and Burkholder inequalities
E(ξb
t,δ)
q = E
(∫ t
0
∫ t
0
| ∫ y
x
budWu|2/δ
|x − y|1/δ
dxdy
)qδ/2
≤
(∫ t
0
∫ t
0
E| ∫ y
x
budWu|qdxdy
|x − y|q/2
)
tqδ−2
≤ Cq
(∫ t
0
∫ t
0
| ∫ y
x
b2
udu|q/2dxdy
|x − y|q/2
)
tqδ−2 ≤ Θt,q.
So, the process ξb
t,δ from (10) is continuous, strictly increasing and has the
moments of any order.
156 YULIA MISHURA AND SERGIY POSASHKOV
Now consider |XN
r − XN
z | for 0 < z < r < T :
|XN
r −XN
z | ≤ |
∫ r
z
a(u, Xu)du|+ |
∫ r
z
b(u, Xu)dWu|+ 1
N
|
∫ r
z
c(u, Xu)dVu|
+ |
∫ r
z
c(u, Xu)dBH
u | ≤ A(r − z) + Cξb
r,δ|r − z|1/2−δ +
C
N
ξc
r,δ|r − z|1/2−δ
+ Cr(ω)
∫ r
z
|c(u, XN
u )|du
uα
+ Cr(ω)
∫ r
z
∫ u
z
|c(u, XN
u ) − c(v, XN
v )|
(u − v)1+α
dvdu
≤ C ′
r(ω)(r − z)1/2−δ + C ′
r(ω)
∫ r
z
∫ u
z
|XN
u − XN
v |
(u − v)1+α
dvdu,
where
C ′
t(ω) := C(ψt,ε ∨ ξb
t,δ ∨ ξc
t,δ ∨ 1), (11)
ψt,ε is defined by (8), C ′
t(ω) ≤ C ′
T (ω) and C ′
T (ω) has the moments of any
order.
Therefore, for δ < 1/2 − α
φr,s :=
∫ r
s
|XN
r − XN
z |
(r − z)1+α
dz ≤ C ′
r(ω)
∫ r
s
(r − z)−1/2−δ−αdz
+ C ′
r(ω)
∫ r
s
1
(r − z)1+α
∫ r
z
∫ u
z
|XN
u − XN
v |
(u − v)1+α
dvdudz
= C ′
r(ω)(r − s)1/2−α−δ + C ′
r(ω)
∫ r
s
(r − u)−α
∫ u
s
|XN
u − XN
v |
(u − v)1+α
dvdu
≤ C ′
r(ω)(r − u)1/2−α−δ + C ′
r(ω)
∫ r
s
(r − u)−αφu,sdu.
From modified Gronwall inequality (Lemma 7.6 [1])
φr,s ≤ C ′
r(ω)(r − s)1/2−α−δ exp[C ′
r(ω)
1
1−α ].
Return to |XN
r − XN
z |:
|XN
r − XN
z | ≤ C ′
r(ω)(r − z)1/2−δ
+ C ′
r(ω) exp[C ′
r(ω)
1
1−α ]
∫ r
z
(v − z)1/2−α−δdv ≤ C̃r(ω)(r − z)1/2−δ, (12)
where C̃r(ω) = C ′
r(ω) exp[C ′
r(ω)
1
1−α ], and the theorem is proved for 0 < δ <
1/2 − α consequently for 0 < δ < 1/2.
Introduce the random variable
C̃(ω) := sup
0≤t≤T
C̃t(ω), (13)
SOLUTION OF STOCHASTIC DIFFERENTIAL EQUATION 157
it also has moments of any order.
Now prove that the solution of (2) belongs to space (3) with norm (4)
for all N ∈ N.
Theorem 3. Under assumptions (A) - (D) the solution of equation (2)
belongs to space of Besov type (3) with norm (4) for all N ∈ N.
Proof. To prove the statement of this theorem we estimate
E(XN
t )2 + E
(∫ t
0
|XN
t − XN
s |
(t − s)1+α
ds
)2
=: A1(t) + A2(t). (14)
At first, for E(XN
t )2 we have
E(XN
t )2 ≤ 5E(X0)
2 + 5E(
∫ t
0
a(s, XN
s )ds)2 + 5E(
∫ t
0
b(s, XN
s )dWs)
2
+ 5E(
∫ t
0
c(s, XN
s )dBH
s )2 + 5E(
1
N
∫ t
0
c(s, XN
s )dVs)
2.
Evidently
E
(∫ t
0
a(s, XN
s )ds
)2
≤ A2T 2,
E
(∫ t
0
b(s, XN
s )dWs
)2
≤ A2T,
E
(
1
N
∫ t
0
c(s, XN
s )dVs
)2
≤ A2T
N2
≤ A2T .
Further, using the estimate (5), (12) and with the help of random variables
defined by (9), (13), E
(∫ t
0
c(s, XN
s )dBH
s
)2
can be estimated as
E
(∫ t
0
c(s, XN
s )dBH
s
)2
≤ E
(
C
2
(ω)
(∫ t
0
c(s, XN
s )
sα
ds +
∫ t
0
∫ s
0
|c(s, XN
s ) − c(u, XN
u )|
(s − u)1+α
duds
)2
)
≤ CE
(
C
2
(ω)
(
t
∫ t
0
A2
s2α
ds
+
(∫ t
0
∫ s
0
B(s − u)β + LC̃(ω)(s − u)1/2−δ
(s − u)1+α
duds
)2
⎞⎠⎞⎠
≤ C(A2t2−2αEC
2
(ω)+B2EC
2
(ω)t2(1−α+β) +L2E(C̃2(ω)C
2
(ω))T 3−2α−2δ)).
158 YULIA MISHURA AND SERGIY POSASHKOV
So, we have
A1(t) ≤ C(A2T 2 + 2A2T + A2T 2−2αEC
2
(ω)
+ B2EC
2
(ω)T 2(1−α+β) + L2E(C̃2(ω)C
2
(ω))T 3−2α−2δ) < ∞. (15)
Consider now A2(t). We have that
A2(t) ≤ 4E
(∫ t
0
| ∫ t
s
a(u, XN
u )du|
(t − s)1+α
ds
)2
+ 4E
(∫ t
0
| ∫ t
s
b(u, XN
u )dWu|
(t − s)1+α
ds
)2
+ 4N−2E
(∫ t
0
| ∫ t
s
c(u, XN
u )dVu|
(t − s)1+α
ds
)2
+ 4E
(∫ t
0
| ∫ t
s
c(u, XN
u )dBH
u |
(t − s)1+α
ds
)2
.
Evidently,
E
(∫ t
0
| ∫ t
s
a(u, Xu)du|
(t − s)1+α
ds
)2
≤ CA2t2−2α.
Now, let γ ∈ (α, 1/2), then
E
(∫ t
0
| ∫ t
s
b(u, Xu)dWu|
(t − s)1+α
ds
)2
≤ Ct1−2γ
∫ t
0
E| ∫ t
s
b(u, Xu)dWu|2
(t − s)2+2α−2γ
ds
≤ Ct1−2γ
∫ t
0
∫ t
s
b2(u, Xu)du
(t − s)2+2α−2γ
ds ≤ Ct1−2γ
∫ t
0
A2
(t − s)1+2α−2γ
ds
≤ CA2t1−2α,
and similarly
E
(∫ t
0
| ∫ t
s
c(u, Xu)dVu|
(t − s)1+α
ds
)2
≤ CA2t1−2α.
SOLUTION OF STOCHASTIC DIFFERENTIAL EQUATION 159
Now we estimate E
(∫ t
0
| t
s c(u,Xu)dBH
u |
(t−s)1+α ds
)2
.
E
(∫ t
0
| ∫ t
s
c(u, Xu)dBH
u |
(t − s)1+α
ds
)2
≤ E
⎛⎝C(ω)
∫ t
0
∫ t
s
|c(u,Xu)|
(u−s)α du +
∫ t
s
∫ u
s
|c(u,xN
u )−c(r,XN
r )|
(u−r)1+α drdu
(t − s)1+α
ds
⎞⎠2
≤ E
⎛⎝C(ω)
∫ t
0
∫ t
s
|c(u,Xu)|
(u−s)α du +
∫ t
s
∫ u
s
B(u−r)β+LC(ω)(u−r)1/2−δ
(u−r)1+α drdu
(t − s)1+α
ds
⎞⎠2
≤ E
(
C(ω)
∫ t
0
A(t − s)1−α + B(t − s)1+β−α + LC̃(ω)(t − s)3/2−δ−α
(t − s)1+α
ds
)2
≤ C(A2t2−4αEC
2
(ω) + B2t2+2β−4αEC
2
(ω) + L2t3−2δ−4αEC
2
(ω)C̃2(ω)).
Therefore A2(t) satisfies the inequality
A2(t) ≤ C(A2T 2−2α + A2T 1−2α + A2T 1−2α
+(A2T 2−4αEC
2
(ω)+B2T 2+2β−4αEC
2
(ω)+L2T 3−2δ−4αEC
2
(ω)C̃2(ω)) < ∞.
(16)
At last, the statement of our theorem follows from inequalities (15) and
(16).
Introduce for any R > 1 the stopping time τR by
τR := inf{t : C ′
t(ω) ≥ R} ∧ T, (17)
where C ′
t(ω) is defined by (11). For any ω ∈ Ω τR = T , for any R > R(ω).
Define the processes {XN
t∧τR
, N ∈ N, t ∈ [0, T ]} as the solutions of equa-
tion (2) stopped an the moment τR, and prove that they are fundamental
in the norm (4) of the space (3).
Theorem 4. Under assumptions (A) - (D) the sequence {XN
t∧τR
, N ≥ 1, t ∈
[0, T ]} of solutions of equations (2) is fundamental in the norm (4).
Proof. Consider
E(XN
t∧τR
− XM
t∧τR
)2 + E
(∫ t
0
|XN
t∧τR
− XM
t∧τR
− XN
s∧τR
+ XM
s∧τR
|
(t − s)1+α
ds
)2
= E(XN
t∧τR
− XM
t∧τR
)2 + E
(∫ t∧τR
0
|XN
t∧τR
− XM
t∧τR
− XN
s + XM
s |
(t − s)1+α
ds
)2
=: AN,M
1 (t) + AN,M
2 (t). (18)
160 YULIA MISHURA AND SERGIY POSASHKOV
First, for AN,M
1 (t) we have
AN,M
1 (t) ≤ 4E
(∫ t∧τR
0
(a(s, XN
s ) − a(s, XM
s ))ds
)2
+ 4E
(∫ t∧τR
0
(b(s, XN
s ) − b(s, XM
s ))dWs
)2
+ 4E
(∫ t∧τR
0
(c(s, XN
s ) − c(s, XM
s ))dBH
s
)2
+ 4E
(∫ t∧τR
0
(
c(s, XN
s )
N
− c(s, XM
s )
M
)
dVs
)2
=: 4(I1 + I2 + I3 + I4).
Then
I1 ≤ CTL2
∫ t
0
E(XN
s∧τR
− XM
s∧τR
)2ds,
I2 ≤ CL2
∫ t
0
E(XN
s∧τR
− XM
s∧τR
)2ds,
I4 ≤ CA2T (N−2 + M−2).
Now we are in position to estimate I3 :
I3 = E
(∫ t∧τR
0
(c(s, XN
s ) − c(s, XM
s ))dBH
s
)2
≤ R2E
(∫ t∧τR
0
|c(s, XN
s ) − c(s, XM
s )|
sα
ds
+
∫ t∧τR
0
∫ s
0
|c(s, XN
s ) − c(s, XM
s ) − c(u, XN
u ) + c(u, XM
u )|
(s − u)1+α
duds
)2
≤ 2R2
(
E
(∫ t∧τR
0
|c(s, XN
s ) − c(s, XM
s )|
sα
ds
)2
+ E
(∫ t∧τR
0
∫ s
0
|c(s, XN
s ) − c(s, XM
s ) − c(u, XN
u ) + c(u, XM
u )|
(s − u)1+α
duds
)2
)
= 2R2(I4 + I5).
Further,
I4 ≤ CL2T 1−2αE
∫ t∧τR
0
(XN
s − XM
s )2ds = CL2T 1−2α
∫ t
0
(AN,M
1 (s))2ds.
SOLUTION OF STOCHASTIC DIFFERENTIAL EQUATION 161
Using Lemma 7.1 [1] we estimate I5 as
I5 ≤ E
(∫ t∧τR
0
∫ s
0
A|XN
s − XM
s − XN
u + XM
u |
(s − u)1+α
duds
+
∫ t∧τR
0
∫ s
0
AB|XN
s − XM
s |(s − u)β
(s − u)1+α
duds
+
∫ t∧τR
0
∫ s
0
D|XN
s − XM
s |(|XN
s − XN
u |ρ + |XM
s − XM
u |ρ)
(s − u)1+α
duds
)2
≤ 3E
(∫ t∧τR
0
∫ s
0
A|XN
s − XM
s − XN
u + XM
u |
(s − u)1+α
duds
)2
+ 3E
(∫ t∧τR
0
∫ s
0
AB|XN
s − XM
s |(s − u)β
(s − u)1+α
duds
)2
+ 3E
(∫ t∧τR
0
∫ s
0
D|XN
s − XM
s |(|XN
s − XN
u |ρ + |XM
s − XM
u |ρ)
(s − u)1+α
duds
)2
= 3(I6 + I7 + I8),
where
I6 ≤ CTA2
∫ t
0
E
(∫ s∧τR
0
|XN
s − XM
s − XN
u + XM
u |
(s − u)1+α
du
)2
ds,
I7 ≤ CTA2
∫ t
0
s2(β−α)E
(|XN
s∧τR
− XM
s∧τR
|)2
ds,
I8 ≤ E
(∫ t∧τR
0
∫ s
0
B|XN
s − XM
s |(2R(s − u)ρ(1/2−δ))
(s − u)1+α
duds
)2
≤ CTD2R2
∫ t
0
sρ−2ρδ−2αE
(|XN
s∧τR
− XM
s∧τR
|)2
ds,
where we choose δ in such a way that ρ− 2ρδ − 2α > 0. It is possible since
α < ρ − 1/2 so ρ − 2α > 1/2 − α > 0. At last,
I5 ≤ C
∫ t
0
E
(∫ s∧τR
0
|XN
s − XM
s − XN
u + XM
u |
(s − u)1+α
du
)2
ds
+ C
∫ t
0
s2(β−α)E
(|XN
s∧τR
− XM
s∧τR
|)2
ds
+ CR2
∫ t
0
sρ−2ρδ−2αE
(|XN
s∧τR
− XM
s∧τR
|)2
ds,
162 YULIA MISHURA AND SERGIY POSASHKOV
and
AN,M
1 (t) ≤ CR2
∫ t
0
AN,M
1 (s)ds + CR2
∫ t
0
AN,M
2 (s)ds
+ C(N−2 + M−2). (19)
Return to AN,M
2 (t). It admits the following estimate
AN,M
2 (t) ≤ C
⎛⎝E
(∫ t∧τR
0
∫ t∧τR
s
(a(u, XN
u ) − a(u, XM
u ))du
(t − s)1+α
ds
)2
+ E
(∫ t∧τR
0
∫ t∧τR
s
(b(u, XN
u ) − b(u, XM
u ))dWu
(t − s)1+α
ds
)2
+ E
(∫ t∧τR
0
∫ t∧τR
s
(c(u, XN
u ) − c(u, XM
u ))dBH
u
(t − s)1+α
ds
)2
+ E
(∫ t∧τR
0
∫ t∧τR
s
( c(u,XN
u )
N
− c(u,XM
u )
M
)dVu
(t − s)1+α
ds
)2
⎞⎠
≤ C(I9 + I10 + I11 + I12).
I9 ≤ CT 1−2γE
∫ t∧τR
0
(t − s)
∫ t∧τR
0
L2|XN
u − XM
u |2du
(t − s)2+2α−2γ
ds
≤ CT 1−2α
∫ t
0
E
(
XN
s∧τR
− XM
s∧τR
)2
ds ≤ CT 1−2α
∫ t
0
AN,M
1 (s)ds,
I10 ≤ CT 1−2γ
∫ t
0
∫ t
s
E|XN
u∧τR
− XM
u∧τR
|2du
(t − s)2+2α−2γ
ds
≤ CT 1−2γ
∫ t
0
AN,M
1 (s)
(t − s)1+2α−2γ
ds,
where we choose γ in such a way that α < γ < 1/2.
For I12 we have
I12 ≤ CT 1−2α(N−2 + M−2).
SOLUTION OF STOCHASTIC DIFFERENTIAL EQUATION 163
Now consider I11 :
I11 ≤ CR2T 1−2γ
⎛⎜⎝E
∫ t∧τR
0
(∫ t∧τR
s
c(u,XN
u )−c(u,XM
u )
(u−s)α du
)2
(t − s)2+2α−2γ
ds
+ E
∫ t∧τR
0
(∫ t∧τR
s
∫ u
s
|c(u,XN
u )−c(u,XM
u )−c(v,XN
v )+c(v,XM
v )|
(u−v)1+α dvdu
)2
(t − s)2+2α−2γ
ds
⎞⎟⎠
=: CR2T 1−2γ(I12 + I13).
I12 ≤ C
∫ t
0
(t − s)
∫ t
s
E(XN
u∧τr
−XM
u∧τr)
2
(u−s)2α du
(t − s)2+2α−2γ
≤ C
∫ t
0
AN,M
1 (s)
(t − s)1+2α−2γ
ds.
I13 ≤ C
⎛⎜⎝E
∫ t∧τR
0
(∫ t∧τR
s
∫ u
s
L|XN
u −XM
u −XN
v +XM
v |
(u−v)1+α dvdu
)2
(t − s)2+2α−2γ
ds
+ E
∫ t∧τR
0
(∫ t∧τR
s
∫ u
s
D|XN
u −XM
u |(u−v)β
(u−v)1+α dvdu
)2
(t − s)2+2α−2γ
ds
+ E
∫ t∧τR
0
(∫ t∧τR
s
∫ u
s
B|XN
u −XM
u |(|XN
u −XN
v |ρ+|XM
u −XM
v |ρ)
(u−v)1+α dvdu
)2
(t − s)2+2α−2γ
ds
⎞⎟⎠
=: C(I14 + I15 + I16).
I14 ≤ CT 2γ−2α
∫ t
0
E(
∫ s∧τR
0
|XN
s − XM
s − XN
u + XM
u |
(s − u)1+α
du)2ds
= CA2T 2γ−2α
∫ t
0
AN,M
2 (s)ds,
I15 ≤ C
∫ t
0
E
(∫ t∧τR
s
|XN
u − XM
u |(u − s)β−α
)2
(t − s)2+2α−2γ
ds
≤ C
∫ t
0
(t − s)1+2β−2α
∫ t
s
E
(
XN
u∧τR
− XM
u∧τR
)2
du
(t − s)2+2α−2γ
ds
≤ CT 2β+2γ−4α
∫ t
0
AN,M
1 (s)ds,
164 YULIA MISHURA AND SERGIY POSASHKOV
note that α < β.
I16 ≤ CR2E
∫ t∧τR
0
(∫ t∧τR
s
∫ u
s
|XN
u −XM
u |(u−v)ρ(1/2−δ)
(u−v)1+α dvdu
)2
(t − s)2+2α−2γ
ds,
where we chose 0 < δ < 1/2−α/ρ, note that α < ρ− 1/2. Similarly to I15,
I16 ≤ CT 2γ−2α
∫ t
0
AN,M
1 (s)ds.
Therefore we have
I13 ≤ CR2(
∫ t
0
AN,M
1 (s)ds +
∫ t
0
AN,M
2 (s)ds).
Hence
I11 ≤ CR4(
∫ t
0
AN,M
1 (s)
(t − s)1+2α−2γ
ds +
∫ t
0
AN,M
2 (s)ds).
At last,
AN,M
2 (t) ≤ CR4
(∫ t
0
AN,M
1 (s)
(t − s)1+2α−2γds
+
∫ t
0
AN,M
2 (s)ds
)
+ C(N−2 + M−2). (20)
From (19) and (20) we obtain that the sum AN,M
1 (t) + AN,M
2 (t) admits the
same estimate as AN,M
2 (t), i.e.
AN,M
1 (t) + AN,M
2 (t) ≤ CR4
(∫ t
0
AN,M
1 (s)
(t − s)1+2α−2γds
+
∫ t
0
AN,M
2 (s)ds
)
+ C(N−2 + M−2), (21)
and from modified Gronwall lemma [1]
AN,M
1 (t) + AN,M
2 (t)
≤ CR4(N−2 + M−2) exp{t(CR4)1/(2γ−2α)}, (22)
taking, for example, γ := (1/2 + α)/2. As choosing N, M → 0 see that
right-hand side of (22) tends to zero whence the proof follows.
Theorem 5. The SDE (1) has the solution on interval [0, T ], and this
solution is unique.
SOLUTION OF STOCHASTIC DIFFERENTIAL EQUATION 165
Proof. Since the space (3) is complete, and from Theorem 4 we can define
Xt∧τR
:= lim
N→∞
XN
t∧τR
, (23)
where the limit is taken in space Wα[0, T ] (in particular, we have that the
limit exists in L2(Ω× [0, T ])).Using the similar estimates an Theorem 4, we
can prove that Xt∧τR
is the unique solution of the original equation (1) on
interval [0, τR].
From definition (17) of τR we have τR1 ≤ τR2 for R1 ≤ R2. So XτR1
and XτR2
coincide a.s. on interval [0, τR1 ]. Where R → ∞ we obtain the
existence and uniqueness of solution of SDE (1) on interval [0, T ].
3. Conclusion
So, we proved the existence and uniqueness of solution of stochastic
differential equation driven by standard and fractional Brownian motions
(1).
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2. Cheridito, P. Regularizing fractional Brownian motion with a view towards
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3. Mishura, Yu., Posashkov, S., Existence and uniqueness of solution of sto-
chastic differential equation driven by fractional Brownian motion with sta-
bilizing term Theory Prob. Math. Stat, 76, (2007).
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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: corlagon@univ.kiev.ua
|
| id | nasplib_isofts_kiev_ua-123456789-4486 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0321-3900 |
| language | English |
| last_indexed | 2025-11-27T19:28:04Z |
| publishDate | 2007 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Mishura, Y. Posashkov, S. 2009-11-19T10:19:03Z 2009-11-19T10:19:03Z 2007 Existence and uniqueness of solution of mixed stochastic differential equation driven by fractional Brownian motion and wiener process / Y. Mishura, S. Posashkov // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 152-165. — Бібліогр.: 8 назв.— англ. 0321-3900 https://nasplib.isofts.kiev.ua/handle/123456789/4486 The existence and uniqueness of solution of stochastic differential equation driven by standard Brownian motion and fractional Brownian motion with Hurst parameter H belongs (3/4, 1) is established. en Інститут математики НАН України Existence and uniqueness of solution of mixed stochastic differential equation driven by fractional Brownian motion and wiener process Article published earlier |
| spellingShingle | Existence and uniqueness of solution of mixed stochastic differential equation driven by fractional Brownian motion and wiener process Mishura, Y. Posashkov, S. |
| title | Existence and uniqueness of solution of mixed stochastic differential equation driven by fractional Brownian motion and wiener process |
| title_full | Existence and uniqueness of solution of mixed stochastic differential equation driven by fractional Brownian motion and wiener process |
| title_fullStr | Existence and uniqueness of solution of mixed stochastic differential equation driven by fractional Brownian motion and wiener process |
| title_full_unstemmed | Existence and uniqueness of solution of mixed stochastic differential equation driven by fractional Brownian motion and wiener process |
| title_short | Existence and uniqueness of solution of mixed stochastic differential equation driven by fractional Brownian motion and wiener process |
| title_sort | existence and uniqueness of solution of mixed stochastic differential equation driven by fractional brownian motion and wiener process |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/4486 |
| work_keys_str_mv | AT mishuray existenceanduniquenessofsolutionofmixedstochasticdifferentialequationdrivenbyfractionalbrownianmotionandwienerprocess AT posashkovs existenceanduniquenessofsolutionofmixedstochasticdifferentialequationdrivenbyfractionalbrownianmotionandwienerprocess |