Asymptotic equivalence of the solutions of the linear stochastic ito equations in the Hilbert space
We obtain the suffcient conditions of asymptotic equivalence in mean square and with probability one of linear ordinary and stochastic Ito equations in the Hilbert space.
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Інститут математики НАН України
2007
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| Cite this: | Asymptotic equivalence of the solutions of the linear stochastic ito equations in the Hilbert space / A. Krenevych // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 103-109. — Бібліогр.: 4 назв.— англ. |
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| author | Krenevych, A. |
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| citation_txt | Asymptotic equivalence of the solutions of the linear stochastic ito equations in the Hilbert space / A. Krenevych // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 103-109. — Бібліогр.: 4 назв.— англ. |
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| description | We obtain the suffcient conditions of asymptotic equivalence in mean square and with probability one of linear ordinary and stochastic Ito equations in the Hilbert space.
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Theory of Stochastic Processes
Vol.13 (29), no.1-2, 2007, pp.103-109
ANDRIY KRENEVYCH
ASYMPTOTIC EQUIVALENCE OF THE
SOLUTIONS OF THE LINEAR STOCHASTIC ITO
EQUATIONS IN THE HILBERT SPACE
We obtain the sufficient conditions of asymptotic equivalence in mean
square and with probability one of linear ordinary and stochastic Ito
equations in the Hilbert space.
1. Introduction
Qualitative theory of linear stochastic differential equations takes significant
place in the general questions of stochastic equations. Stability investigation
is one of the most important parts of this theory. A lot of papers are devoted
to this subject.
A new approach to the investigation of asymptotical behavior of the
solutions of linear stochastic equations is used. More precisely, finding an
ordinary differential equation, whose solutions have asymptotical behavior
similar to the one of the solutions of the stochastic equation, is proposed.
Hence, the question of the stochastic equation stability is reduced to the
one of the ordinary differential equation.
Stochastic systems, whose solutions have similar asymptotical behavior,
analogically to the ordinary differential equations, are called asymptotically
equivalent.
The paper [1] is devoted to this approach for the systems of stochastic
equations in the space Rn. The present work is a generalization of the former
result to the Hilbert phase space. The sufficient conditions of the stochastic
differential Ito equations asymptotical equivalence in mean square and with
probability 1 are obtained.
2000 Mathematics Subject Classifications.60H10.
Key words and phrases. Ito equations, asymptotic equivalence.
103
104 ANDRIY KRENEVICH
2. Definitions.
Let (H, (·, ·), ‖ · ‖) be a real separable Hilbert space (with scalar product
(·, ·) and norm ‖ · ‖). Let also {en|n ≥ 1} be some orthonormal basis in H .
We consider a stochastic differential equation in Hilbert space H [3,p.247]
dy = (A + B(t))ydt + D(t)ydWt (1)
on the probability space (Ω,F, P ), where t ≥ 0, x ∈ H , and A : H → H is a
linear bounded operator; linear operators B(t), D(t) : H → H are bounded
for each t ≥ 0; Wt is a scalar Wiener process, defined for t ≥ 0 on probability
space (Ω,F, P ); {Ft, t ≥ 0} is a filtration corresponding to Wt.
Under conditions of the theorem below, the stochastic equation (1)
has the (strong) unique solution y(t) ≡ y(t, ω) ∈ H with initial condition
y(0) = y0.
Consider a linear differential equation in H
dx
dt
= Ax. (2)
We compare solutions of the stochastic differential equation (1) and solu-
tions of the ordinary differential equation (2) with some properly chosen
random initial conditions.
Definition. If for each solution y(t) of the equation (1) there is solution
x(t) of the equation (2), such that
lim
t→∞E|x(t) − y(t)|2 = 0,
then the equation (1) is called asymptotically equivalent to the equation (2)
in mean square.
Definition. If for each solution y(t) of the equation (1) there is solution
x(t) of the equation (2), such that
P{ lim
t→∞ |x(t) − y(t)| = 0} = 1,
then the equation (1) is called asymptotically equivalent to the equation (2)
with probability 1.
3. Decomposition of Hilbert space.
Consider a complex covering of space H (see [2,p.26]). Let H̃ be a
complex covering of space H , i.e. H̃ = {x1 + ix2|x1, x2 ∈ H}.
The spectrum of A we denote by σ(A). Since A is a real operator, we
obtain that σ(A) is symmetrical set relative to real axis.
ASYMPTOTIC EQUIVALENCE OF THE ITO EQUATIONS 105
Assume that σ(A) is not connected as a set. The closed part of the
spectrum, whose complement is closed in σ(A) is said to be a spectral set
(see [2,p.32]).
Let
σ(A) = σ1(A)
⋃
σ2(A),
where σ1(A), σ2(A) are disjoint spectral sets.
It follows from [2,p.33] that the space H̃ can be decomposed in a direct
sum of subspaces invariant with respect to operator A
H̃ = H̃1 ⊕ H̃2,
such that the spectrum σ(A|H̃i) of the restriction of A on the subspace H̃i
coincides with σi(A), i = 1, 2.
Let Hi = ReH̃i, i = 1, 2. If σ1,2(A) are symmetric sets w.r.t. real axis
(further, we will use this construction for symmetric spectral sets only),
then Hi are subspaces invariant with respect to operator A. Thus,
H = H1 ⊕ H2.
Let Pi be a projector on the subspace Hi, i = 1, 2. It is easy to see that
I = P1 + P2, where I is identity operator on H . It is proved in [2,p.25] that
one can modify the scalar product (·, ·) in H to the topologically equivalent
one (·, ·)′, where projectors P1 and P2 become orthoprojectors and the scalar
product in H is given by the formula
(x, y)′ = (P1x, P1y) + (P2x, P2y).
Obviously, (Pix, Piy) is a scalar product in Hi, i = 1, 2. Without loss of
generality we can assume that (·, ·)′ = (·, ·).
It is clear that H1, H2 are Hilbert subspaces in H . From here we will
call the projector Pi and the subspace Hi = PiH the spectral projector and
invariant subspace, corresponding each other and spectral set σi(A), i = 1, 2.
4. Asymptotical equivalence of the stochastic equations in
Hilbert space
The following theorem is a generalization of Levinson theorem (see, for
instance, [4,p.159]) on the asymptotical equivalence of the ordinary differ-
ential equation to the case of stochastic differential equations in a Hilbert
space.
Theorem. Assume the solutions of equation (2) are bounded on [0,∞) and
the spectrum of operator A consists of two spectral sets
σ(A) = σ−(A)
⋃
σ0(A),
106 ANDRIY KRENEVICH
such that σ−(A) ⊂ {z ∈ C|Rez < 0} and σ0(A) ⊂ {z ∈ C|Rez = 0}.
Let A0 be a restriction of operator A on invariant subspace H0, correspond-
ing to the spectral set σ0(A). If A0 is similar to some skew-hermitian oper-
ator (i.e. A0 = S−1(iQ)S, Q = Q∗) and∫ ∞
0
‖B(t)‖dt ≤ K1 < ∞,
∫ ∞
0
‖D(t)‖2dt ≤ K1 < ∞, (3)
for some k > 0, then equation (1) is asymptotically equivalent to equa-
tion (2) in mean square and with probability 1.
Proof. It follows from the conditions of the theorem and [3,p.272] that there
exists the unique solution of equation (1) with initial condition y(0) = y0,
which can be represented in the form
y(t) = X(t)y(0) +
t∫
0
X(t − τ)B(τ)y(τ)dτ +
t∫
0
X(t − τ)D(τ)y(τ)dWτ , (4)
where t ≥ 0 and X(t) = eAt is an operator exponent (see [2,p.41]). Taking
into account the boundedness of the solutions on the positive semiaxis, we
get that there is a constant K2 > 0 such that for all t ≥ 0 the following
estimate holds
‖eAt‖ ≤ K2.
Let’s prove that all the solutions of equation (1) are bounded in mean
square under condition (3).
For this we use the presentation of the solution of equation (1) in the
form (4). Let’s estimate expectation of |y(t)|2.
E|y(t)|2 ≤ 3‖X(t)‖2E|y(0)|2 + 3E
∣∣∣∣∣∣
t∫
0
X(t − τ)B(τ)y(τ)dτ
∣∣∣∣∣∣
2
+
+3E
∣∣∣∣∣∣
t∫
0
X(t − τ)D(τ)y(τ)dWτ
∣∣∣∣∣∣
2
≤ 3K2
2E|y(0)|2+
+3E
⎛⎝ t∫
0
√
‖X(t − τ)‖
√
‖X(t − τ)‖
√
‖B(τ)‖
√
‖B(τ)‖|y(τ)|dτ
⎞⎠2
+
+3
t∫
0
E |X(t − τ)D(τ)y(τ)|2dτ ≤ 3K2
2E|y(0)|2+
+3
t∫
0
‖X(t− τ)‖‖B(τ)‖E|y(τ)|2dτ
t∫
0
‖X(t − τ)‖‖B(τ)‖dτ+
ASYMPTOTIC EQUIVALENCE OF THE ITO EQUATIONS 107
+3
t∫
0
‖X(t − τ)‖2‖D(τ)‖2E|y(τ)|2dτ ≤
≤ 3K2
2
⎛⎝E|y(0)|2 +
∞∫
0
‖B(τ)‖dτ
t∫
0
‖B(τ)‖E|y(τ)|2dτ+
+
t∫
0
‖D(τ)‖2E|y(τ)|2dτ
⎞⎠ .
The Gronuoll-Bellman inequality implies
E|y(t)|2 ≤ 3K2
2E|y(0)|2e
3K2
2
t∫
0
(K1‖B(τ)‖+‖D(τ)‖2)dτ
≤
≤ 3K2
2E|y(t0)|2e
3K2
2
∞∫
0
(K1‖B(τ)‖+‖D(τ)‖2)dτ
≤ K̃E|y(0)|2, (5)
where K̃ = 3K2
2e
3K2
2
∞∫
0
(K1‖B(τ)‖+‖D(τ)‖2)dτ
.
Furthermore, the conditions of the theorem and the computations above
imply that space H is decomposable in the direct sum of Hilbert subspaces
H = H− ⊕ H0 = P−H ⊕ P0H,
where H− is an invariant subspace corresponding to the spectral set σ−(A),
P− is an orthoprojector on H−, H0 is an invariant subspace, corresponding
to the spectral set σ0(A), P0 is an orthoprojector on H0.
It can be obtained from [2,p.122] that equation (2) is equivalent to the
system of two independent equations
dx−
dt
= A−x−,
dx0
dt
= A0x0, (6)
where x− = P−x, x0 = P0x, A− = P−A, A0 = P0A.
Then
eAt =
(
eA−t 0
0 eA0t
)
= X−(t) + X0(t),
where
X−(t) =
(
eA−t 0
0 0
)
, X0(t) =
(
0 0
0 eA0t
)
.
The evolutionary property of matrix exponent allows to transform the equa-
tion (4) in the following form
y(t) = X(t)
⎡⎣y(t0) +
∞∫
0
X0(0 − τ)B(τ)y(τ)dτ+
108 ANDRIY KRENEVICH
+
∞∫
0
X0(0 − τ)D(τ)y(τ)dWτ
⎤⎦ +
+
t∫
t
X−(t − τ)B(τ)y(τ)dτ +
t∫
0
X−(t − τ)D(τ)y(τ)dWτ− (7)
−
∞∫
t
X0(t − τ)B(τ)y(τ)dτ −
∞∫
t
X0(t − τ)D(τ)y(τ)dWτ .
For each solution y(t) ≡ y(t, ω) of the equation (1) with the initial condition
y(0) = y0 we correspond the solution x(t) of the equation (2) with the initial
condition
x(0) = y(0) +
∞∫
0
X0(−τ)B(τ)y(τ)dτ +
∞∫
0
X0(−τ)D(τ)y(τ)dWτ . (8)
Since A0 is similar to skew-hermitian operator, it follows from [2,p.113]
that there is a constant K3 > 0, t ∈ (−∞,∞) such that ‖X0‖ ≤ K3.
From the stated above and inequality (5) we get that all improper inte-
grals involved in (7) are convergent in mean square.
Since the solution x(t) of linear equation (2) and the solution y(t) ≡
y(t, ω) of stochastic equation (1) are defined by initial conditions, equa-
tion (8) defines correspondence modulo stochastic equivalence between the
solution set {y(t) ≡ y(t, ω)} of equation (1) and solution set {x(t)} of equa-
tion (2).
The rest of the proof is analogical to the one in [1].
5. Example.
Consider equation (2) in the space H = (L2[0, 1])3, such that x =
x(t, u) = (x1, x2, x3), xi ∈ L2[0, 1], t ≥ 0, u ∈ [0, 1] and operator A is de-
fined as follows
A =
⎛⎜⎝ 0 I 0
T 2 0 0
0 0 V
⎞⎟⎠
where I is the identity operator in L2[0, 1], operators T and V are defined
as
(Tz)(u) = z(u) −
1∫
0
ϕ(u)ϕ(s)z(s)ds, (9)
(V z)(u) = (u − 2)z(u),
here ϕ(u) is a continuous not identically zero function on [0, 1] such that
c0 =
1∫
0
ϕ2(s)ds �= 1
ASYMPTOTIC EQUIVALENCE OF THE ITO EQUATIONS 109
It is easy to see, that σ(V ) = [−2,−1].
Consider operator
A =
(
0 I
T 2 0
)
which can be represented in the form A = S(iQ)S−1, where
Q =
(
T 0
0 −T
)
, S =
(
I I
−iT iT
)
, S−1 =
1
2
(
I iT−1
I −iT−1
)
.
Under stated above conditions the inverse to T operator exists and is defined
by relation
(T−1y)(u) = y(u) +
1
1 − c0
1∫
0
ϕ(u)ϕ(s)y(s)ds.
Thus, operator A is similar to skew-hermitian operator, which implies that
its spectrum belongs to the imaginary axis.
Hence, the space H can be decomposed into a direct sum of subspaces
H = H1 ⊕ H2,
where H1 = {(x1, x2, 0)|x1, x2 ∈ L2[0, 1]}, H2 = {(0, 0, x3)|x3 ∈ L2[0, 1]}.
The restriction of operator A on the subspace H1 is the operator A, and
the restriction on H2 is the operator V . Therefore, σ(A) = σ−(A)
⋃
σ0(A),
where σ−(A) = [−2,−1] and σ0(A) is some closed subset of imaginery axis.
Hence, equation (2) satisfies the conditions of the theorem.
Thus, for arbitrary operators B(t), D(t) : H → H , which satisfy esti-
mates (3) stochastic equation (1) will be asymptotically equivalent to equa-
tion (2).
Bibliography
1. Krenevich A.P., Asymptotic Equivalence Of the solutions of The quasilinear
Stochastic Ito systems. Bulletin of the University of Kiev.Series: Physics
& Mathematics. (2006), N1, 69–76. (Ukrainian)
2. Daletsky Yu.L., Kreyn M.G., Stability of the solution of the differential
equations in the Banach space, Moscow. Nauka, (1970), 534p. (Russian)
3. Dorogovtsev A.Ya., Periodical and stationary states of infinite-dimensional
determinate and stochastic dynamic systems, Kiev.
Vyscha Shkola, (1992), 319p. (Russian)
4. Demidovich B.P., Mathematical stability theory lectures, Moscow.
Nauka, (1967), 472p. (Russian)
Department of Mathematical Physics,
Kyiv National Taras Shevchenko University, Kyiv, Ukraine
E-mail address: krenevich@univ.kiev.ua
|
| id | nasplib_isofts_kiev_ua-123456789-4481 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0321-3900 |
| language | English |
| last_indexed | 2025-12-07T16:04:07Z |
| publishDate | 2007 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Krenevych, A. 2009-11-19T10:14:29Z 2009-11-19T10:14:29Z 2007 Asymptotic equivalence of the solutions of the linear stochastic ito equations in the Hilbert space / A. Krenevych // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 103-109. — Бібліогр.: 4 назв.— англ. 0321-3900 https://nasplib.isofts.kiev.ua/handle/123456789/4481 We obtain the suffcient conditions of asymptotic equivalence in mean square and with probability one of linear ordinary and stochastic Ito equations in the Hilbert space. en Інститут математики НАН України Asymptotic equivalence of the solutions of the linear stochastic ito equations in the Hilbert space Article published earlier |
| spellingShingle | Asymptotic equivalence of the solutions of the linear stochastic ito equations in the Hilbert space Krenevych, A. |
| title | Asymptotic equivalence of the solutions of the linear stochastic ito equations in the Hilbert space |
| title_full | Asymptotic equivalence of the solutions of the linear stochastic ito equations in the Hilbert space |
| title_fullStr | Asymptotic equivalence of the solutions of the linear stochastic ito equations in the Hilbert space |
| title_full_unstemmed | Asymptotic equivalence of the solutions of the linear stochastic ito equations in the Hilbert space |
| title_short | Asymptotic equivalence of the solutions of the linear stochastic ito equations in the Hilbert space |
| title_sort | asymptotic equivalence of the solutions of the linear stochastic ito equations in the hilbert space |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/4481 |
| work_keys_str_mv | AT krenevycha asymptoticequivalenceofthesolutionsofthelinearstochasticitoequationsinthehilbertspace |