On the stability with probability one for a class of stochastic semigroups
The formulated theorem specifies the result of A.V. Skorokhod concerning the stability with probability one for the solution of a linear stochastic system built on a continuous homogeneous semimartingale with independent increments.
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| Цитувати: | On the stability with probability one for a class of stochastic semigroups / A.C. Chani // Theory of Stochastic Processes. — 2005. — Т. 11 (27), № 3-4. — С. 58–62. — Бібліогр.: 1 назв.— англ. |
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irk-123456789-44252009-11-10T12:00:30Z On the stability with probability one for a class of stochastic semigroups Chani, A.C. The formulated theorem specifies the result of A.V. Skorokhod concerning the stability with probability one for the solution of a linear stochastic system built on a continuous homogeneous semimartingale with independent increments. 2005 Article On the stability with probability one for a class of stochastic semigroups / A.C. Chani // Theory of Stochastic Processes. — 2005. — Т. 11 (27), № 3-4. — С. 58–62. — Бібліогр.: 1 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4425 519.21 en Інститут математики НАН України |
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The formulated theorem specifies the result of A.V. Skorokhod concerning the stability
with probability one for the solution of a linear stochastic system built on a
continuous homogeneous semimartingale with independent increments. |
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Article |
| author |
Chani, A.C. |
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Chani, A.C. On the stability with probability one for a class of stochastic semigroups |
| author_facet |
Chani, A.C. |
| author_sort |
Chani, A.C. |
| title |
On the stability with probability one for a class of stochastic semigroups |
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On the stability with probability one for a class of stochastic semigroups |
| title_full |
On the stability with probability one for a class of stochastic semigroups |
| title_fullStr |
On the stability with probability one for a class of stochastic semigroups |
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On the stability with probability one for a class of stochastic semigroups |
| title_sort |
on the stability with probability one for a class of stochastic semigroups |
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Інститут математики НАН України |
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2005 |
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http://dspace.nbuv.gov.ua/handle/123456789/4425 |
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On the stability with probability one for a class of stochastic semigroups / A.C. Chani // Theory of Stochastic Processes. — 2005. — Т. 11 (27), № 3-4. — С. 58–62. — Бібліогр.: 1 назв.— англ. |
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2025-07-02T07:40:27Z |
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2025-07-02T07:40:27Z |
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1836520075674255360 |
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Theory of Stochastic Processes
Vol. 11 (27), no. 3–4, 2005, pp. 58–62
UDC 519.21
A. C. CHANI
ON THE STABILITY WITH PROBABILITY ONE
FOR A CLASS OF STOCHASTIC SEMIGROUPS
The formulated theorem specifies the result of A.V. Skorokhod concerning the sta-
bility with probability one for the solution of a linear stochastic system built on a
continuous homogeneous semimartingale with independent increments.
We specify the result by A.V. Skorokhod published in monograph [1] concerning the
conditions of the asymptotic stability with probability one of a homogeneous stochastic
semigroup U0
t with values in the space L(Rd). The semigroup U0
t is given by the solution
of the linear stochastic differential equation
dU0
t = AU0
t dt +
n∑
k=1
BkU0
t dwk(t), 0 ≤ t < ∞, (1)
where A, B1, B2, ..., Bn ∈ L(Rd), w1(t), w2(t), ..., wn(t) are independent Wiener processes
with values in R1, and the stability is understood as follows: for each x ∈ Rd,
P
(
lim
t→∞U0
t x = 0
)
= 1. (2)
The mentioned result ([1], p. 244, Theorem 18) means that it is sufficient for the
stability of the semigroup U0
t for all x ∈ Rd lying on a unit sphere with the center at
zero that the following inequality be fulfilled:
D(x) = (Ax, x) +
1
2
n∑
k=1
|Bkx|2 −
n∑
k=1
(Bkx, x)2 < 0, (3)
where | · | is the norm in Rd generated by a scalar product (·, ·). In the particular case
with B1 = B2 = ... = Bn = 0, hypothesis (3) appears as (Ax, x) < 0.
In other words, for the stability of the nonrandom semigroup U0
t which is a solution
of the equation
dU0
t = AU0
t dt, (4)
it is required that the matrix A∗ + A be negatively defined
((A∗ + A)x, x) < 0, |x| = 1, (5)
where A∗ is the matrix symmetric to A.
Condition (5) for systems of the form (4) is overstated because the classical result of
Lyapunov only requires in this case that the matrix A be stable. This means that we
can found a symmetric positive definite matrix H such that
((A∗H + HA)x, x) < 0, |x| = 1.
2000 AMS Mathematics Subject Classification. Primary 60F15, 60H25.
Key words and phrases. Stability, stochastic system.
58
ON THE STABILITY FOR A CLASS OF STOCHASTIC SEMIGROUPS 59
Thus, it is natural to deduce the conditions for system (1) to be stable, from which
the result of Lyapunov will follow in the absence of stochastic additives. The present
work is devoted to the solution of the above-mentioned problem.
We now pass to exact statements.
All processes are defined on a space (Ω,F ,Ft, P ), t ≥ 0, which satisfies the usual
conditions. We consider the stochastic process
Yt = At +
n∑
k=1
Bkwk(t)
receiving the values in the space (L(Rd),M), d ≥ 1, where L(Rd) is the set of real d× d-
matrices alloted with a Borel σ-algebra M with respect to the Euclidean norm generated
by a scalar product (A, B) = Sp(B∗A). We define Ut as a solution of the equation
Ut = I +
∫ t
0
dYsUs, (6)
where I is the identity matrix from L(Rd). By L+
s (Rd), we denote the set of symmetric
positive definite matrices from L(Rd) and, with the help of V ∈ L+
s (Rd), apply it to the
similarity transformation
Ut(V ) = V UtV
−1.
Consider the matrix V as a parameter and define the family of processes
ϑ = {Ut(V ); V ∈ L+
s (Rd)}.
Let V be an arbitrary matrix from L+
s (Rd), and let
λ1 ≥ λ2 ≥ ... ≥ λd > 0
be its eigenvalues. Then the well-known inequalities
λd|Uty| ≤ |Ut(V )x| ≤ λ1|Uty|,
y = V −1x, x ∈ Rd,
are valid. The mappings x → y = V −1x and y → x = V y are one-to-one mutually
inverse. Therefore, the stability in the sense of definition (2) of the process Ut yields the
stability of Ut(V ), and conversely. Thus, one can judge the stability of the ϑ family of
processes by the stability of any of its representatives.
On the other hand, each process Ut(V ) from ϑ is the solution of an equation of the
form (6):
Ut(V ) = I +
∫ t
0
dYs(V )Us(V ),
where
Yt(V ) = A(V )t +
n∑
k=1
Bk(V )wk(t),
A(V ) = V AV −1, Bk(V ) = V BkV −1, k = 1, 2, ..., n.
Therefore, the result of A.V. Skorokhod is applicable to the process Ut(V ), i.e. Ut(V )
will be stable with probability one, if the following inequality is fulfilled:
DV (x) = (V AV −1x, x) +
1
2
n∑
k=1
|V BkV −1x|2 −
n∑
k=1
(V BkV −1x, x)2 < 0, |x| = 1. (7)
The mentioned arguments allow us to formulate the following theorem.
60 A. C. CHANI
Theorem. The process Ut defined as a solution of Eq. (6) will be stable with probability
one, if there exists a symmetric positive definite matrix V , for which inequality (7) is
fulfilled.
The formulated theorem makes sense, if there exist the stable processes Ut with proba-
bility one, for which condition (3) is not fulfilled. We provide two simple examples which
show that such processes exist. In addition, it is useful to simultaneously clarify the issue
about the stability of the corresponding processes in the square mean sense. The point
is that neglecting the third term on the left-hand side of inequality (7) turns it into the
necessary and sufficient condition of stability in the quadratic mean sense:
(V AV −1x, x) +
1
2
n∑
k=1
|V BkV x−1|2 < 0, |x| = 1. (8)
In fact, if we introduce the designation V 2 = C, then inequality (8) is equivalent to
((A∗C + CA)y, y) +
n∑
k=1
(B∗
kCBky, y) < 0, |y| = 1, (9)
where y = |V −1x|−1V −1x. The fact of the existence of a symmetric positive definite
matrix C, for which inequality (9) is fulfilled, is the necessary and sufficient condition of
stability in the quadratic mean sense for the solution of Eq. (1) ([1], p. 227, Theorem
12 and p. 221, formula (47)).
Example 1. Let Ut be a solution of Eq. (6) in the phase space L(R2). We choose the
stable matrices
A =
( −λ a
0 −λ
)
, λ > 0,
B1 =
(
μ b
0 μ
)
, B2 = B3 = ... = Bn = 0.
Concerning the parameters a, μ, b, we assume that they are connected by the equation
a + μb = 0. (10)
Let us deal with the stability of the process Ut in the square mean sense. To this end,
we rewrite condition (8) in the matrix form:
D = V AV −1 + V −1A∗V + V −1B∗
1V 2B1V
−1 < 0.
The left-hand side of this inequality is a homogeneous function of the zero order in V .
Therefore, it is sufficient to search for a matrix V in the form
V =
(
1 β
β α
)
, Δ = α − β2 > 0.
We have
V AV −1 =
1
Δ
(
1 β
β α
)( −λ a
0 −λ
)(
α −β
−β 1
)
=
=
1
Δ
( −λΔ − aβ a
−aβ2 −λΔ + aβ
)
,
V −1A∗V = (V AV −1)∗ =
1
Δ
( −λΔ − aβ −aβ2
a −λΔ + aβ
)
,
V AV −1 + V −1A∗V =
1
Δ
( −2(λΔ + aβ) a(1 − β2)
a(1 − β2) −2(λΔ − aβ)
)
.
ON THE STABILITY FOR A CLASS OF STOCHASTIC SEMIGROUPS 61
Further,
V B1V
−1 =
1
Δ
(
1 β
β α
)(
μ b
0 μ
)(
α −β
−β 1
)
=
1
Δ
(
μΔ − bβ b
−bβ2 μΔ + bβ
)
,
V −1B∗
1V = (V B1V
−1)∗ =
1
Δ
(
μΔ − bβ −bβ2
b μΔ + bβ
)
,
V −1B∗
1V 2B1V
−1 =
1
Δ2
(
(μΔ − bβ)2 + b2β4 μb(1 − β2)Δ − b2β(1 + β2)
μb(1 − β2)Δ − b2β(1 + β2) (μΔ + bβ)2 + b2
)
.
After simple transformations with regard for equality (10), we get finally
D =
1
Δ2
(
(μ2 − 2λ)Δ2 + b2β(1 + β2) −b2β(1 + β2)
−b2β(1 + β2) (μ2 − 2λ)Δ2 + b2(1 + β2)
)
.
For the matrix D to be negative definite, it is necessary and sufficient that the system
of inequalities {
(μ2 − 2λ)Δ2 + b2β2(1 + β2) < 0,
det D = (μ2 − 2λ)
[
(μ2 − 2λ)Δ2 + b2(1 + β2)2
]
> 0
be satisfied. Treating Δ ∈ (0,∞) and β ∈ (−∞,∞) as independent variables, we draw
conclusion that this system of inequalities has a solution if and only if
μ2 < 2λ.
For arbitrary β, it is sufficient to take
Δ >
b2(1 + β2)2
2λ − μ2
.
Thus, the relevant solution Ut will be stable in the square mean in the region of param-
eters
Q2 = {(λ, a, μ, b) : λ > 0, a + μb = 0, μ2 < 2λ}.
We now examine the question of the stability of Ut with probability one.
We have
2DV (x) =
(
(V AV −1 + V −1A∗V + V −1B∗
1V 2B1V
−1)x, x
) − 2(V B1V
−1x, x)2 =
=
1
Δ2
{ [
(μ2 − 2λ)Δ2 + b2β2(1 + β2)
]
x2
1 − 2b2β(1 + β2)x1x2 +
[
(μ2 − 2λ)Δ2+
+b2(1 + β2)
]
x2
2 − 2
[
(μΔ − bβ)x2
1 + b(1 − β2)x1x2 + (μΔ + bβ)x2
2
]2} =
=
1
Δ2
{ [
(μ2 − 2λ)Δ2 + b2β2(1 + β2)
]
x2
1 − 2b2β(1 + β2)x1x2 +
[
(μ2 − 2λ)Δ2+
+b2(1 + β2)
]
(1 − x2
1) − 2
[
μΔ + bβ − 2bβx2
1 + b(1 − β2)x1x2
]2} =
=
1
Δ2
{
(μ2 − 2λ)Δ2 + b2(1 + β2) − 2(μΔ + bβ)2 + b
[
b(β4 − 1) + 8β(μΔ + bβ)
]
x2
1−
−2b
[
bβ(1 + β2) + 2(1 − β2)(μΔ + bβ)
]
x1x2 − 2b2
[
(1 − β2)x1x2 − 2βx2
1
]2
.
For β = 0,
2DV (x) = −(μ2 + 2λ) +
1
Δ2
(b2x2
2 − 4bμΔx1x2 − 2b2x2
1x
2
2), x2
1 + x2
2 = 1.
Having selected a rather big value Δ = α, we define the symmetric positive definite
matrix
V =
(
1 0
0 α
)
, Δ = α > 0,
62 A. C. CHANI
for which condition (7) of Theorem will be fulfilled:
DV (x) < 0 with |x| = 1.
This condition quarantees the stability of the process Ut on the set
Q0 = {(λ, a, μ, b) : λ > 0, a + μb = 0}.
Concerning condition (3) with Δ = 1, we get
2D(x) = 2DI(x) = −(μ2 + 2λ) + b2x2
2 − 4bμx1x2 − 2b2x2
1x
2
2, x2
1 + x2
2 = 1.
Calculating the functional 2D(x) at the points
x1 = (0; 1), x2 = (1; 0), x3 =
(
1√
2
;
1√
2
)
, x4 =
(
1√
2
;− 1√
2
,
)
we obtain
−(μ2 + 2λ) + b2, −(μ2 + 2λ), −(μ2 + 2λ) − 4bμ, −(μ2 + 2λ) + 4bμ,
respectively. Therefore, for the fulfillment of condition (3), it is necessary that b2∨4|bμ| <
μ2 + 2λ. This means that the set of stability Q1 which is defined by inequality (3) lies
in the region
Q = {(λ, a, μ, b) : λ > 0, a + μb = 0, b2 ∨ 4|μb| < μ2 + 2λ}.
In this case, Q1 ⊂ Q ⊂ Q0 and Q
= Q0. The exact description of the set Q1 is not the
purpose of the present work and demands a separate study. As a result, we obtain the
following picture:
Q2 ⊂ Q0 and Q2
= Q0,
Q1 ⊂ Q0 and Q1
= Q0.
Example 2. In Example 1, we change only the matrix A and choose it to be unstable:
A =
(
λ a
0 λ
)
, λ > 0,
B1 =
(
μ b
0 μ
)
, B2 = B3 = ... = Bn = 0, a + μb = 0.
Then
Q2 = ∅, Q0 = {(λ, a, μ, b) : λ > 0, a + μb = 0, 2λ < μ2},
Q1 ⊂ Q = {(λ, a, μ, b) : λ > 0, a + μb = 0, 2λ + (b2 ∨ 4|bμ|) < μ2}.
Bibliography
1. A. V. Skorokhod, Asymptotic methods of theory of stochastic differential equations, Naukova
Dumka, Kiev, 1987.
E-mail : chani@iamm.ac.donetsk.ua
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