On the φ-asymptotic behaviour of solutions of stochastic differential equations
In this paper we study the a.s. asymptotic behaviour of the solution of the stochastic dfferential equation dX(t) = g(X(t))dt +σ(X(t))dW(t), X(0) = b > 0, where g and σ are positive continuous functions and W is a Wiener process. Making use of the theory of pseudo-regularly varying (PRV) function...
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| Цитувати: | On the φ-asymptotic behaviour of solutions of stochastic differential equations / V.V. Buldygin, O.I. Klesov, J.G. Steinebach, O.A. Tymoshenko // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 1. — С. 11–29. — Бібліогр.: 28 назв.— англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860186491258404864 |
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| author | Buldygin, V.V. Klesov, O.I. Steinebach, J.G. Tymoshenko, O.A. |
| author_facet | Buldygin, V.V. Klesov, O.I. Steinebach, J.G. Tymoshenko, O.A. |
| citation_txt | On the φ-asymptotic behaviour of solutions of stochastic differential equations / V.V. Buldygin, O.I. Klesov, J.G. Steinebach, O.A. Tymoshenko // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 1. — С. 11–29. — Бібліогр.: 28 назв.— англ. |
| collection | DSpace DC |
| description | In this paper we study the a.s. asymptotic behaviour of the solution of the stochastic dfferential equation dX(t) = g(X(t))dt +σ(X(t))dW(t), X(0) = b > 0, where g and σ are positive continuous functions and W is a Wiener process. Making use of the theory of pseudo-regularly varying (PRV) functions, we find conditions on g, σ and φ, under which φ(X(•)) can be approximated a.s. by φ(μ(•), where μ is the solution of the ordinary differential equation dμ(t) = g(μ(t))dt, μ(0) = b. As an application of these results we discuss the problem of φ-asymptotic equivalence for solutions of stochastic differential equations.
|
| first_indexed | 2025-12-07T18:04:17Z |
| format | Article |
| fulltext |
Theory of Stochastic Processes
Vol. 14 (30), no. 1, 2008, pp. 11–29
UDC 519.21
V. V. BULDYGIN, O. I. KLESOV, J. G. STEINEBACH, AND O. A. TYMOSHENKO
ON THE ϕ-ASYMPTOTIC BEHAVIOUR OF SOLUTIONS
OF STOCHASTIC DIFFERENTIAL EQUATIONS
In this paper we study the a.s. asymptotic behaviour of the solution of the stochastic
differential equation dX(t) = g(X(t))dt + σ(X(t))dW (t), X(0) = b > 0, where g and
σ are positive continuous functions and W is a Wiener process. Making use of the
theory of pseudo-regularly varying (PRV) functions, we find conditions on g, σ and
ϕ, under which ϕ(X(�)) can be approximated a.s. by ϕ(μ(�)), where μ is the solution
of the ordinary differential equation dμ(t) = g(μ(t))dt, μ(0) = b. As an application
of these results we discuss the problem of ϕ-asymptotic equivalence for solutions of
stochastic differential equations.
1. Introduction
Gihman and Skorohod [16], Keller et al. [19], and later Buldygin et al. [7–11] consid-
ered the asymptotic behaviour, as t → ∞, of solutions of certain stochastic differential
equations (SDE’s) and gave conditions, under which the asymptotics of these solutions
are determined by nonrandom functions. In this paper, we reconsider this problem and
study conditions, under which solutions of two SDE’s are asymptotically equivalent.
Consider, for k = 1, 2, the stochastic differential equations
dXk(t) = gk(Xk(t)) dt + σk(Xk(t)) dWk(t), t ≥ 0, Xk(0) = bk > 0. (1.1)
Here {Wk, k = 1, 2} are standard Wiener processes defined on a common probability
space; {bk, k = 1, 2} are nonrandom positive constants; {gk, σk, k = 1, 2} are continuous
functions defined on the set R = (−∞,∞) and such that, for each k = 1, 2, the functions
σk and (gk(u), u > 0) are positive, and (1.1) has almost surely (a.s.) a unique and
continuous Itô-solution Xk = (Xk(t), t ≥ 0) with
lim
t→∞Xk(t) =∞ a.s. (1.2)
For k = 1, 2 denote by μk = (μk(t), t ≥ 0) the solution of the Cauchy problem for the
ordinary differential equations (ODE’s) corresponding to (1.1) with σk ≡ 0, i.e.
dμk(t) = gk(μk(t)) dt, t ≥ 0, μk(0) = bk > 0 (k = 1, 2). (1.3)
We assume that, for each k = 1, 2, the function gk is such that the solution μk exists, is
unique and satisfies
lim
t→∞μk(t) =∞. (1.4)
The following four main problems will be considered in this paper for given functions
ϕ1 and ϕ2.
2000 AMS Mathematics Subject Classification. Primary 60H10; Secondary 34D05, 60F15, 60G17.
This work has partially been supported by Deutsche Forschungsgemeinschaft under DFG grants 436
UKR 113/41/0-2 and 436 UKR 113/68/0-1
11
12 V. V. BULDYGIN ET AL.
The first problem (Problem I) is to investigate, under which conditions it follows that
solutions of the SDE’s (1.1) and their corresponding ODE’s (1.3) are ϕ-asymptotically
equivalent, that is
lim
t→∞
ϕk(Xk(t))
ϕk(μk(t))
= 1 a.s., k = 1, 2. (1.5)
The second problem (Problem II) is to study, under which conditions it holds that
solutions of the ODE’s (1.3) are ϕ-asymptotically equivalent, that is
lim
t→∞
ϕ1(μ1(t))
ϕ2(μ2(t))
= 1. (1.6)
The next problem (Problem III) is a modification of Problem I. The question is, under
which conditions it follows that the solution of the first SDE in (1.1) is ϕ-asymptotically
equivalent to the solution of the second ODE in (1.3), that is
lim
t→∞
ϕ1(X1(t))
ϕ2(μ2(t))
= 1 a.s. (1.7)
And finally, Problem IV is to verify, under which conditions it holds that solutions of
the SDE’s (1.1) are ϕ-asymptotically equivalent, that is
lim
t→∞
ϕ1(X1(t))
ϕ2(X2(t))
= 1 a.s. (1.8)
All these problems are closely connected, of course, and it is clear that the solutions
of Problems III and IV follow from those of Problems I and II.
Gihman and Skorohod [16], §17, and Keller et al. [19] considered some versions of
Problem I for a single equation, while Buldygin et al. [7–12] considered some versions of
all problems above. Here, we further study these problems in more detail.
In order to solve Problems I and II, we follow the general approach developed in
Buldygin et al. [5–7]. This approach allows for solving the following general problem:
Find conditions on a given function, under which its inverse or quasi-inverse function
preserves the equivalence of functions.
The paper is organized as follows. In Section 2, we formulate and discuss the results
concerning Problem I. Subsequently, Problems II, III and IV are considered in Sections 3,
4 and 5, respectively. The main problems of this paper are closely connected with the
relations between limits of ratios of functions and their (quasi-) inverse functions from
various classes of regularly varying (RV) functions and their extensions. These relations
are discussed in Section 6. In Section 7, some of our main results are proved.
2. The ϕ-Asymptotic Equivalence of Solutions of SDE’s and ODE’s.
In this section we consider the asymptotic behaviour, as t → ∞, of the Itô-solution
X = (X(t), t ≥ 0) of the SDE
dX(t) = g(X(t))dt+ σ(X(t))dW (t), t ≥ 0, X(0) = b > 0. (2.1)
Here W is a standard Wiener process. We assume that σ = (σ(x),−∞ < x < ∞) is
a positive function and g = (g(x),−∞ < x < ∞) is positive on (0,∞) (or ultimately
positive), and we shall only be interested in situations, in which limt→∞X(t) = ∞ a.s.
and such that infinity will not be reached in finite time.
Denote by μ = (μ(t), t ≥ 0) the solution of the ODE corresponding to (2.1) for σ ≡ 0,
i.e.
dμ(t) = g(μ(t))dt, t ≥ 0, μ(0) = b. (2.2)
We assume that the function g is such that the solution μ exists, is unique, tends to ∞
as t→∞, and that infinity will not be reached in finite time.
THE ϕ-ASYMPTOTIC BEHAVIOUR OF SOLUTIONS OF SDE’S 13
Put
G(x) =
∫ x
b
ds
g(s)
, x ∈ [b,∞). (2.3)
Note that G = (G(x), x ≥ b) is the inverse function of μ, i.e., G = μ−1, if g is positive
and continuous for x ≥ b, and limt→∞ μ(t) =∞ if and only if limt→∞G(t) =∞.
The main problem in this section is to study conditions, under which
lim
t→∞
ϕ(X(t))
ϕ(μ(t))
= 1 a.s. (2.4)
for a given function (ϕ(x),−∞ < x <∞).
As a first step for solving this problem we use the Skorohod method. In Gihman and
Skorohod [16], §17, Theorem 4 (see also Keller et al. [19], Theorem 5) the process
Y (t) = G(X(t)), t ≥ 0,
is studied and it is proved that
lim
t→∞
G(X(t))
t
= 1 a.s. (2.5)
under certain conditions (see Remark 2.3 below).
The General Statement. Let g, σ and ϕ be functions satisfying the following condi-
tions:
(A1) g is continuous and positive on (0,∞) and σ is continuous and positive on
(−∞,∞) and such that (2.1) has a.s. a unique and continuous solution as well
as (2.2) has a unique and continuously differentiable solution;
(A2) ϕ = (ϕ(x), x > 0) is a positive and continuously differentiable function, strictly
increasing to infinity as x→∞.
Put
G(ϕ)(·) = G(ϕ−1(·)), g(ϕ)(·) = g(ϕ−1(·))ϕ′(ϕ−1(·)),
where G is as in (2.3), the function ϕ−1 is inverse to ϕ, and ϕ′ is the first derivative of
ϕ.
Observe that (G(ϕ)(t), t ≥ ϕ(b)) is the inverse function of ϕ(μ(·)).
For example, if ϕ(·) = log(·), then G(log)(·) = G(e(·)) and g(log)(·) = e−(·)g(e(·)).
If ϕ(x) ≡ x, then G(ϕ) = G and g(ϕ) = g.
Now our goal is to find conditions on g, σ and ϕ, under which relation (2.4) holds. To
do so, we first consider the following general statement, which describes extra conditions
for relation (2.5) to imply or being equivalent to (2.4). Note that the result below holds
for nonrandom functions.
Theorem 2.1. Assume conditions (A1), (A2) and∫ ∞
b
du
g(u)
=∞. (2.6)
Let g and ϕ be such that
lim inf
t→∞
∫ ct
t
du
g(ϕ)(u)G(ϕ)(u)
= lim inf
t→∞
∫ ϕ−1(ct)
ϕ−1(t)
du
g(u)G(u)
> 0 for all c > 1. (2.7)
Then,
1) if (2.5) holds, then (2.4) follows;
14 V. V. BULDYGIN ET AL.
2) if
lim
c↓1
lim sup
t→∞
∫ ct
t
du
g(ϕ)(u)G(ϕ)(u)
= lim
c↓1
lim sup
t→∞
∫ ϕ−1(ct)
ϕ−1(t)
du
g(u)G(u)
= 0, (2.8)
then (2.5) and (2.4) are equivalent.
Recall that, by condition (2.6), μ(t) → ∞ as t → ∞. Moreover, (2.6) excludes the
possibility of explosions, that is, the solution does not reach infinity in finite time.
Theorem 2.1 with ϕ(x) ≡ x describes extra conditions for relation (2.5) to imply or
being equivalent to
lim
t→∞
X(t)
μ(t)
= 1 a.s. (2.9)
(see Buldygin et al. [8]).
Corollary 2.1. Assume (A1), (A2) and (2.6). Let g and G be such that
lim inf
t→∞
∫ ct
t
du
g(u)G(u)
> 0 for all c > 1.
Then,
1) if (2.5) holds, then (2.9) follows;
2) if
lim
c↓1
lim sup
t→∞
∫ ct
t
du
g(u)G(u)
= 0,
then (2.5) and (2.9) are equivalent.
Next, we consider some sufficient conditions, for both (2.7) (Proposition 2.1) and (2.8)
(Proposition 2.2), which may be more suitable for practical use.
Proposition 2.1. Let g be a positive and continuous function on (0,∞) such that (2.6)
holds, and let ϕ satisfy (A2). Assume that at least one of the following conditions holds:
(i) lim supt→∞ g(ϕ)(t)G(ϕ)(t)/t = lim supt→∞ g(t)G(t)ϕ′(t)/ϕ(t) <∞;
(ii) g(·)ϕ′(·) is eventually nonincreasing;
(iii) there exists α < 1 such that 0 < inft≥1 g
(ϕ)(t)t−α, supt≥1 g
(ϕ)(t)t−α <∞;
(iv) (g(ϕ))∗(c) < c for all c > 1, with (g(ϕ))∗(c) = lim supt→∞ g(ϕ)(ct)/g(ϕ)(t);
(v) g(ϕ) is an RV function with index α < 1 (see Section 6 below).
Then, condition (2.7) is satisfied.
Remark 2.1. Under (2.6), condition (i) of Proposition 2.1 is equivalent to (2.7), if the
function g is eventually nondecreasing.
Remark 2.2. Condition (i) of Proposition 2.1 does not hold for g(ϕ)(t) ≡ t, since
lim sup
t→∞
g(ϕ)(t)G(ϕ)(t)/t = lim
t→∞
∫ t
1
ds
s
=∞.
Moreover, this condition does not hold for any regularly varying function g(ϕ) of index
1, that is, for a function g(ϕ) such that g(ϕ)(t) ≡ t�(t), where � is slowly varying. This is
due to a result of Parameswaran [24], which proves that
lim
t→∞ �(t)
∫ t
1
ds
s�(s)
=∞.
THE ϕ-ASYMPTOTIC BEHAVIOUR OF SOLUTIONS OF SDE’S 15
Proposition 2.2. Let g be a positive and continuous function on (0,∞) such that (2.6)
holds, and let ϕ satisfy (A2). Assume that at least one of the following conditions holds:
(i) lim inft→∞ g(ϕ)(t)G(ϕ)(t)/t = lim inft→∞ g(t)G(t)ϕ′(t)/ϕ(t) > 0;
(ii) g(·)ϕ′(·) is eventually nondecreasing;
(iii)
∫ 1
0+ dc/
(
g(ϕ)
)∗
(c) > 0, with
(
g(ϕ)
)∗
(c) = lim supt→∞ g(ϕ)(ct)/g(ϕ)(t);
(iv) the set {c ∈ (0, 1] :
(
g(ϕ)
)∗
(c) <∞} has positive Lebesgue measure;
(v) at least one of the conditions (iii), (iv), or (v) of Proposition 2.1 holds.
Then, condition (2.8) is satisfied.
Remark 2.3. Under (2.6), condition (i) of Proposition 2.2 is equivalent to (2.8), if the
function g is eventually nonincreasing.
Example 2.1. Let g(x) = ϕ(x) = x, x > 0. Clearly condition (2.6) holds, but
condition (2.7) does not, since, for all c > 1,
lim inf
t→∞
∫ ct
t
du
g(ϕ)(u)G(ϕ)(u)
= lim inf
t→∞
∫ ct
t
du
u log u
≤ lim inf
t→∞
c− 1
log t
= 0.
Next, if g(x) = x, x > 0, and ϕ(x) = log x, x > 0, then
lim
t→∞
g(ϕ)(t)G(ϕ)(t)
t
= 1, t > 0.
Thus, by Propositions 2.1 and 2.2, conditions (2.6), (2.7) and (2.8) hold.
The Gihman–Skorohod Condition. Theorem 2.2 below provides some conditions,
under which relation (2.4) holds true.
First, consider the following condition of Gihman and Skorohod [16], §17:
(GS) g is continuous and positive on (0,∞), σ is continuous and positive on (−∞,∞),
and g and σ are such that (2.1) has a.s. a unique and continuous solution with ar-
bitrary initial condition and with limt→∞X(t) =∞, as well as (2.2) has a unique
and continuously differentiable solution with arbitrary positive initial condition.
Let σ/g be bounded and let g′(x) exist for all x > 0 with g′(x)→ 0 as x→∞.
Remark 2.4. Recall that, under (GS), relation (2.5) holds true a.s., that is
lim
t→∞
G(X(t))
t
= 1 a.s.
(see Gihman and Skorohod [16], §17, Theorem 4 and Remark 1).
Remark 2.5. Problem (2.1) has a.s. a unique and continuous solution X with arbitrary
initial condition and with limt→∞X(t) = ∞ a.s., as well as problem (2.2) has a unique
and continuous solution with arbitrary positive initial condition, if, for example, the
functions g and σ satisfy the following assumptions:
a) for some K and for all x ∈ (−∞,∞),
|g(x)|+ |σ(x)| ≤ K(1 + |x|);
b) for each C > 0 there exists an LC such that, for |x| ≤ C and |y| ≤ C,
|g(x)− g(y)|+ |σ(x)− σ(y)| ≤ LC |x− y|;
c) for all x ∈ (−∞,∞),∫ x
−∞
exp
{
−
∫ z
0
2g(u)
σ2(u)
du
}
dz =∞ and
∫ ∞
x
exp
{
−
∫ z
0
2g(u)
σ2(u)
du
}
dz <∞
(see Gihman and Skorohod [16], §15, and §16, Theorem 1).
16 V. V. BULDYGIN ET AL.
Theorem 2.2. Assume conditions (GS), (A2) and (2.6), and let condition (2.7) or at
least one of the conditions (i)–(v) of Proposition 2.1 hold. Then relation (2.4) follows.
The Keller–Kersting–Rösler Conditions. Theorem 2.3 below provides further con-
ditions, under which relation (2.4) holds true.
Here we discuss the conditions of Keller et al. [19]. For t > 0, put
h(t) =
g′(t)σ2(t)
2g2(t)
, ψ(t) =
∫ t
1
σ2(u)
g3(u)
du .
First consider the following general condition:
(K0) g is continuous and positive on (0,∞), σ is continuous and positive on (−∞,∞),
and g and σ are such that (2.1) has a.s. a unique and continuous solution with
arbitrary initial condition and limt→∞X(t) = ∞ with positive probability, as
well as (2.2) has a unique and continuously differentiable solution with arbitrary
positive initial condition.
The following five conditions have been used in Keller et al. [19]:
(K1) g : (0,∞)→ (0,∞) is strictly positive and twice continuously differentiable such
that
∫∞
1
(g(u))−1du =∞.
(K2) h(t)→ 0 as t→∞.
(K3) σ : (0,∞)→ (0,∞) is strictly positive and continuously differentiable such that∫∞
0
(tg(μ(t)))−2σ2(μ(t))dt <∞.
(K4) The functions g(·), g′(·), σ2(μ(·))/g2(μ(·)) and h(μ(·)) are eventually concave or
convex. If ψ(∞) =∞, we require the same behaviour for the function h(ψ−1(·)).
(K5) There is a constant C > 0 such that logμ(2t) ≤ C logμ(t) for large t. Further-
more, the function e−(·)g(e(·)) together with its derivative is eventually concave
or convex.
Remark 2.6. Under the above conditions, the following two statements hold (see
Theorem 1 and Theorem 5 in Keller et al. [19]).
I) Under (K0)–(K4), relation (2.5) holds true.
II) Under (K0)–(K5), relation (2.4) holds true with ϕ(t) = log t, t > 0.
Theorem 2.3. Assume conditions (K0)–(K4), and (A2), and let condition (2.7) or at
least one of the conditions (i)–(v) of Proposition 2.1 holds. Then relation (2.4) follows.
3. The ϕ-Asymptotic Equivalence of the Solutions of ODE’s.
In this section we consider the ODE’s (1.3) and discuss conditions under which it holds
that solutions μ1 and μ2 of these ODE’s are ϕ-asymptotically equivalent, that is (1.6)
holds true.
Consider functions gk and ϕk, k = 1, 2, satisfying the following conditions: for each
k = 1, 2,
(B1) gk is continuous and positive on (0,∞) and such that (1.3) has a unique and
continuously differentiable solution;
(B2) ϕk = (ϕk(x), x > 0) is a positive and continuously differentiable function, strictly
increasing to infinity as x→∞.
Put
G
(ϕk)
k (·) = Gk(ϕ−1
k (·)), g
(ϕk)
k = gk(ϕ−1
k (·))ϕ′
k(ϕ−1
k (·)), k = 1, 2,
where, for each k = 1, 2,
Gk(x) =
∫ x
bk
ds
gk(s)
, x ∈ [bk,∞),
THE ϕ-ASYMPTOTIC BEHAVIOUR OF SOLUTIONS OF SDE’S 17
the function ϕ−1
k is inverse to ϕk, and ϕ′
k is the first derivative of ϕk.
Note that, for k = 1, 2, the function Gk = (Gk(x), x ≥ bk) is inverse to μk, i.e.,
Gk = μ−1
k , and (G(ϕk)
k (x), x ≥ ϕk(bk)) is the inverse function of ϕk(μk(·)), that is
G
(ϕk)
k (x) =
∫ x
ϕk(bk)
(
ds
g
(ϕk)
k (s)
)
, x ∈ [ϕk(bk),∞).
In the sequel we make use of the condition∫ ∞
bk
du
gk(u)
=∞, k = 1, 2. (3.1)
It follows from (B1) that (3.1) is equivalent to∫ ∞
bk
du
g
(ϕk)
k (u)
=∞, k = 1, 2.
The latter condition means that limx→∞G
(ϕk)
k (x) = ∞, k = 1, 2. Thus, under (B1),
condition (3.1) holds if and only if (1.4) holds.
Our goal in this section is to find conditions on gk and ϕk, k = 1, 2, under which
relation (1.6) holds. Theorem 3.1 below gives conditions, under which the following
three relations hold:
lim
t→∞
G
(ϕ1)
1 (t)
G
(ϕ2)
2 (t)
= 1 =⇒ lim
t→∞
ϕ1(μ1(t))
ϕ2(μ2(t))
= 1, (3.2)
lim
t→∞
G
(ϕ1)
1 (t)
G
(ϕ2)
2 (t)
= 1 ⇐= lim
t→∞
ϕ1(μ1(t))
ϕ2(μ2(t))
= 1, (3.3)
lim
t→∞
G
(ϕ1)
1 (t)
G
(ϕ2)
2 (t)
= 1 ⇐⇒ lim
t→∞
ϕ1(μ1(t))
ϕ2(μ2(t))
= 1. (3.4)
Consider the next two conditions: for k = 1, 2,
lim inf
t→∞
∫ ct
t
du
g
(ϕk)
k (u)G(ϕk)
k (u)
= lim inf
t→∞
∫ ϕ−1
k (ct)
ϕ−1
k (t)
du
gk(u)Gk(u)
> 0 for all c > 1; (3.5)
lim
c↓1
lim sup
t→∞
∫ ct
t
du
g
(ϕk)
k (u)G(ϕk)
k (u)
= lim
c↓1
lim sup
t→∞
∫ ϕ−1
k (ct)
ϕ−1
k (t)
du
gk(u)Gk(u)
= 0. (3.6)
Remark 3.1. Note that, for k = 1, 2, conditions (3.5) and (3.6), respectively, coincide
with conditions (2.7) and (2.8), where g = g
(ϕk)
k and G = G
(ϕk)
k . Hence, if at least one
of the conditions (i) – (v) of Proposition 2.1 [Proposition 2.2] holds with g = g
(ϕk)
k and
G = G
(ϕk)
k , then condition (3.5) [(3.6)] follows.
Example 3.1. For k = 1, 2, let the functions gk and ϕk be positive and continuous on
(0,∞) and such that condition (B2) holds, and let g(ϕk)
k be an RV function with index
α < 1 (see Section 6 below). Then condition (v) of both Propositions 2.1 and 2.2 holds
together with condition (3.1) and, by Remark 3.1, conditions (3.5) and (3.6) are satisfied.
Theorem 3.1. Let gk and ϕk, k = 1, 2, be such that conditions (B1), (B2) and (3.1)
hold. Then,
1) if, at least for one k = 1, 2, condition (3.5) holds, then (3.2) follows;
2) if, at least for one k = 1, 2, condition (3.6) holds, then (3.3) follows;
3) if, at least for one k = 1, 2, condition (3.5) holds and also, at least for one
k = 1, 2, condition (3.6) holds, then (3.4) follows.
18 V. V. BULDYGIN ET AL.
By Theorem 3.1 and Example 3.1, we have the following result.
Corollary 3.1. Let gk and ϕk, k = 1, 2, be such that conditions (B1) and (B2) hold.
If at least one of the functions g(ϕ1)
1 and g(ϕ2)
2 is an RV function with index less than 1,
then (3.4) holds true.
On the Mutual Relation Between the ϕ-Asymptotic Equivalence of the Func-
tions g1, g2 and the Solutions of ODE’s.
Here we discuss a new Problem II*. The question is to find conditions, under which
it holds that the following three relations are satisfied:
lim
t→∞
g
(ϕ1)
1 (t)
g
(ϕ2)
2 (t)
= 1 =⇒ lim
t→∞
ϕ1(μ1(t))
ϕ2(μ2(t))
= 1, (3.7)
lim
t→∞
g
(ϕ1)
1 (t)
g
(ϕ2)
2 (t)
= 1 ⇐= lim
t→∞
ϕ1(μ1(t))
ϕ2(μ2(t))
= 1, (3.8)
lim
t→∞
g
(ϕ1)
1 (t)
g
(ϕ2)
2 (t)
= 1 ⇐⇒ lim
t→∞
ϕ1(μ1(t))
ϕ2(μ2(t))
= 1. (3.9)
It is clear that, in general, these relations do not hold. For instance, consider the
following counterexample to relation (3.7).
Example 3.2. Let ϕ1(x) = ϕ2(x) = x,
g1(x) = x, g2(x) = x+
√
x, x > 0,
and μ1(0) = μ2(0) = 1. Then
μ1(t) = et, μ2(t) =
(
2et/2 − 1
)2
, t ≥ 0.
Thus
lim
t→∞
g1(t)
g2(t)
= 1, but lim
t→∞
μ2(t)
μ1(t)
= 4.
Observe that g1 and g2 are both RV functions with index 1.
On an Application of Karamata’s Theorem. Theorem 3.1 shows that Problem II*
is directly connected with the next one.
Consider two functions (f1(t), t > 0) and (f2(t), t > 0), which are nonnegative and
Lebesgue-integrable on finite intervals, and, for given positive numbers a1 and a2, put
Fk(t) =
∫ t
ak
fk(u) du, t ≥ ak, k = 1, 2.
Assume that limt→∞ Fk(t) = ∞, k = 1, 2. The question is, under which conditions the
following three relations hold:
lim
t→∞
f1(t)
f2(t)
= 1 =⇒ lim
t→∞
F1(t)
F2(t)
= 1, (3.10)
lim
t→∞
f1(t)
f2(t)
= 1 ⇐= lim
t→∞
F1(t)
F2(t)
= 1, (3.11)
lim
t→∞
f1(t)
f2(t)
= 1 ⇐⇒ lim
t→∞
F1(t)
F2(t)
= 1, (3.12)
It is clear that (3.10) always holds. But the inverse relation (3.11) does not hold in
general. A simple counterexample is the following one: f1(t) = 2t, f2(t) = 2t(1 + cos t2),
THE ϕ-ASYMPTOTIC BEHAVIOUR OF SOLUTIONS OF SDE’S 19
t ≥ 0, and F1(t) = t2, F2(t) = t2 + sin t2, t ≥ 0. Note that the function f1 is an RV
function with index 1.
So, for the relation (3.11) to hold one needs additional conditions. On applying Kara-
mata’s theorem (see Bingham et al. [4], p. 26) we get the next result.
Lemma 3.1. If f1 and f2 are RV functions with indices α1 and α2 greater than −1,
then (3.11) and (3.12) hold true.
Now we return to Problem II*. By relation (3.10) and Theorem 3.1, the following
result holds.
Theorem 3.2. Let gk and ϕk, k = 1, 2, be such that conditions (B1), (B2) and (3.1)
hold. If, at least for one k = 1, 2, condition (3.5) holds, then (3.7) follows.
From Theorem 3.2, with ϕ1 = ϕ2 = ϕ, we conclude the following result.
Corollary 3.2. Let g1, g2 and ϕ be such that conditions (B1), (3.1) and (A2) hold.
If, at least for one k = 1, 2, condition (3.5) holds, with ϕk = ϕ, then
lim
t→∞
g1(t)
g2(t)
= 1 =⇒ lim
t→∞
ϕ(μ1(t))
ϕ(μ2(t))
= 1.
Lemma 3.1 in combination with Corollary 3.1 gives the next theorem.
Theorem 3.3. Let gk and ϕk, k = 1, 2, be such that conditions (B1) and (B2) hold.
If g(ϕ1)
1 and g(ϕ2)
2 are RV functions with indices less than 1, then (3.9) follows.
From Theorem 3.3, with ϕ1 = ϕ2 = ϕ, we conclude the following results.
Corollary 3.3. Let g1, g2 and ϕ be such that conditions (B1) and (A2) hold. If g(ϕ)
1
and g(ϕ)
2 are RV functions with indices less than 1, then
lim
t→∞
g1(t)
g2(t)
= 1 ⇐⇒ lim
t→∞
ϕ(μ1(t))
ϕ(μ2(t))
= 1.
Corollary 3.4. Let g1 and g2 be RV functions with indices less than 1, and let (B1)
hold. Then,
lim
t→∞
g1(t)
g2(t)
= 1 ⇐⇒ lim
t→∞
μ1(t)
μ2(t)
= 1.
Remark 3.2. Example 3.2 shows that the RV functions g1 and g2 in Corollary 3.4 (and
in other statements above) cannot have indices equal to 1.
4. More about Asymptotic Equivalence of the
Solutions of SDE’s and Their Corresponding ODE’s.
On applying the results above, we can now discuss, under which conditions it holds
that the solution X1 of the first SDE in (1.1) and the solution μ2 of the second ODE
in (1.3) are ϕ-asymptotically equivalent, i.e. that (1.7) holds true.
This problem is more general than Problem I (see Section 1), but its solution follows
from the results of Sections 2 and 3, since
lim
t→∞
ϕ1(X1(t))
ϕ2(μ2(t))
= lim
t→∞
ϕ1(X1(t))
ϕ1(μ1(t))
· lim
t→∞
ϕ1(μ1(t))
ϕ2(μ2(t))
. (4.1)
The following results demonstrate that, under certain conditions, the statements of
Theorems 2.1, 2.2 and 2.3 are stable with respect to a change of the initial condition and
a change of the function g(ϕ) to an asymptotically equivalent version.
20 V. V. BULDYGIN ET AL.
Theorem 4.1. Assume (B1), (B2) and (3.1), and let g = g1 and σ = σ1 be such
that (A1) and (2.5) hold. Then,
1) if, for at least one k = 1, 2, condition (3.5) holds, then
lim
t→∞
G
(ϕ1)
1 (t)
G
(ϕ2)
2 (t)
= 1 =⇒ lim
t→∞
ϕ1(X1(t))
ϕ2(μ2(t))
= 1 a.s., (4.2)
and moreover,
lim
t→∞
g
(ϕ1)
1 (t)
g
(ϕ2)
2 (t)
= 1 =⇒ lim
t→∞
ϕ1(X1(t))
ϕ2(μ2(t))
= 1 a.s.; (4.3)
2) if, for each k = 1, 2, conditions (3.5) and (3.6) hold, then
lim
t→∞
G
(ϕ1)
1 (t)
G
(ϕ2)
2 (t)
= 1 ⇐⇒ lim
t→∞
ϕ1(X1(t))
ϕ2(μ2(t))
= 1 a.s. (4.4)
Remark 4.1. Theorem 4.1 remains valid if (2.5) is replaced by (GS) or (K0)–(K4).
We consider some corollaries of Theorem 4.1 under the Gihman–Skorohod condition
(GS).
Theorem 4.2. Assume (B1), (B2) and (3.1), and let g = g1 and σ = σ1 be such
that (GS) holds.
1) If, for at least one k = 1, 2, condition (3.5) holds, then (4.2) and (4.3) follow;
2) If, for each k = 1, 2, conditions (3.5) and (3.6) hold, then (4.4) follows.
Theorem 4.3. Assume (B1) and (B2), and let g = g1 and σ = σ1 be such that (GS)
holds.
1) If at least one of g(ϕ1)
1 and g
(ϕ2)
2 is an RV function with index less than 1 (see
Section 6 below), then (4.2) and (4.3) follow;
2) If both g(ϕ1)
1 and g(ϕ2)
2 are RV functions with indices less than 1, then (4.4) follows
and, moreover,
lim
t→∞
g
(ϕ1)
1 (t)
g
(ϕ2)
2 (t)
= 1 ⇐⇒ lim
t→∞
ϕ1(X1(t))
ϕ2(μ2(t))
= 1 a.s. (4.5)
Observe that Theorems 2.2, 4.2 and 4.3 generalize and complement Theorem 4 in
Gihman and Skorohod [16], §17.
By Theorem 4.3, with ϕ1 = ϕ2 = ϕ, we have the following results.
Corollary 4.1. Assume (GS) with g = g1 and σ = σ1, and let g2 and ϕ be such that
conditions (B1) and (A2) hold. If g(ϕ)
1 and g(ϕ)
2 are RV functions with indices less than
1, then
lim
t→∞
g1(t)
g2(t)
= 1 ⇐⇒ lim
t→∞
ϕ(X1(t))
ϕ(μ2(t))
= 1 a.s.
Corollary 4.2. Assume (B1) and (GS), with g = g1 and σ = σ1. If g1 and g2 are RV
functions with indices less than 1, then
lim
t→∞
g1(t)
g2(t)
= 1 ⇐⇒ lim
t→∞
X1(t)
μ2(t)
= 1 a.s.
THE ϕ-ASYMPTOTIC BEHAVIOUR OF SOLUTIONS OF SDE’S 21
Example 4.1. (See Gihman and Skorohod [16], §17, Corollary 1). Assume (GS) with
g(x) = g1(x) ∼ Cxβ as x → ∞, where 0 ≤ β < 1 and C > 0. Then, by Corollary 4.2,
with g2(x) = Cxβ for x > 0, we have
lim
t→∞
X1(t)
(C(1− β)t)1/(1−β)
= 1 a.s.,
since μ2(t) ∼ (C(1 − β)t)1/(1−β) as t→∞.
Observe that, in view of Remark 2.2, we cannot use Theorem 4.2 with ϕ1(x) ≡ ϕ2(x) ≡
x and g1(x) ∼ Cx as x→∞.
Example 4.2. Assume (GS) with g(x) = g1(x) ∼ Cx/(log x)γ as x→∞, where γ > 0
and C > 0. Put ϕ1(x) = ϕ2(x) = (log(x+ 1))1+γ for x > 0. Then g(ϕ1)
1 (t) ∼ C(1 + γ) as
t→∞. Thus, by Corollary 4.1, with g2(x) = C(x + 1)/(log(x+ 1))γ for x > 0, we have
lim
t→∞
(logX1(t))
1+γ
C(1 + γ)t
= 1 a.s.,
since ϕ2(μ2(t)) ∼ (C(1 + γ)t) as t→∞.
Example 4.3. Assume (GS) with g(x) = g1(x) ∼ Cx exp (−(log x)r) as x→∞, where
0 < r < 1 and C > 0. Note that exp ((log x)r) , x > 1, is a slowly varying function, and
exp ((logx)r) /(log x)γ →∞ as x→∞ for all γ > 0. Put ϕ1(x) = ϕ2(x) = exp ((logx)r)
for x > 0. Then g
(ϕ1)
1 (t) ∼ r(log t)(r−1)/r as t → ∞. Thus, by Corollary 4.1, with
g2(x) = Cx exp (−(log x)r) for x > 0, we have
lim
t→∞
exp ((logX1(t))r)
exp ((logμ2(t))r)
= 1 a.s.
Remark 4.2. Theorems 4.2 and 4.3, and Corollaries 4.1 and 4.2 remain valid if (2.5) is
replaced by (K0)–(K4).
Example 4.4. (See Gihman and Skorohod [16], §17, Corollary 2). Assume (K0)–
(K4) with g(x) = g1(x) ∼ Cx as x → ∞, where C > 0. Put ϕ1(x) = ϕ2(x) = log x
and g2(x) = Cx for x > 0. Then g
(ϕ1)
1 (t) ∼ C as t → ∞. Thus, by Theorem 4.3 and
Remark 4.2, we have
lim
t→∞
logX1(t)
Ct
= 1 a.s.,
since ϕ2(μ2(t)) ∼ Ct as t→∞.
5. The ϕ-Asymptotic Equivalence of the Solutions of SDE’s.
In this section we consider the SDE’s (1.1) and discuss, under which conditions it
holds that the solutions X1 and X2 of these SDE’s are ϕ-asymptotically equivalent, i.e.
that (1.8) holds true. This problem is the last one in the list of our main problems (see
Section 1), and its solution follows from the results of Sections 2 and 3, since
lim
t→∞
ϕ1(X1(t))
ϕ2(X2(t))
= lim
t→∞
ϕ1(X1(t))
ϕ1(μ1(t))
· lim
t→∞
ϕ1(μ1(t))
ϕ2(μ2(t))
· lim
t→∞
ϕ2(μ2(t))
ϕ2(X2(t))
. (5.1)
In this section, we study some new statements under the Gihman-Skorokhod condition
(GS) only (see Section 2). Consider the following version of (GS):
(GS*) Condition (GS) holds for the functions g = gk and σ = σk, k = 1, 2.
The following statements provide conditions, under which solutions of the SDE’s (1.1)
are ϕ-asymptotically equivalent.
22 V. V. BULDYGIN ET AL.
Theorem 5.1. Assume (GS*), (B2) and (3.1).
1) If, for at least one k = 1, 2, condition (3.5) holds, then
lim
t→∞
G
(ϕ1)
1 (t)
G
(ϕ2)
2 (t)
= 1 =⇒ lim
t→∞
ϕ1(X1(t))
ϕ2(X2(t))
= 1 a.s., (5.2)
and moreover,
lim
t→∞
g
(ϕ1)
1 (t)
g
(ϕ2)
2 (t)
= 1 =⇒ lim
t→∞
ϕ1(X1(t))
ϕ2(X2(t))
= 1 a.s. ; (5.3)
2) if, for each k = 1, 2, conditions (3.5) and (3.6) hold, then
lim
t→∞
G
(ϕ1)
1 (t)
G
(ϕ2)
2 (t)
= 1 ⇐⇒ lim
t→∞
ϕ1(X1(t))
ϕ2(X2(t))
= 1 a.s. (5.4)
Theorem 5.2. Assume (GS*) and (B2).
1) If at least one of g(ϕ1)
1 and g
(ϕ2)
2 is an RV function with index less than 1 (see
Section 6 below), then (5.2) and (5.3) follow;
2) If both g(ϕ1)
1 and g(ϕ2)
2 are RV functions with indices less than 1, then (5.4) follows
and, moreover,
lim
t→∞
g
(ϕ1)
1 (t)
g
(ϕ2)
2 (t)
= 1 ⇐⇒ lim
t→∞
ϕ1(X1(t))
ϕ2(X2(t))
= 1 a.s. (5.5)
Theorem 5.2, with ϕ1 = ϕ2 = ϕ, implies the following result.
Corollary 5.1. Assume (GS*) and (A2). If g(ϕ)
1 and g
(ϕ)
2 are RV functions with
indices less than 1, then
lim
t→∞
g1(t)
g2(t)
= 1 ⇐⇒ lim
t→∞
ϕ(X1(t))
ϕ(X2(t))
= 1 a.s.
For ϕ(x) ≡ x one has the following statement.
Corollary 5.2. Assume (GS*). If g1 and g2 are RV functions with indices less than
1, then
lim
t→∞
g1(t)
g2(t)
= 1 ⇐⇒ lim
t→∞
X1(t)
X2(t)
= 1 a.s.
6. Properties of Some Classes of Functions
In this section we recall the definitions and some properties of various classes of reg-
ularly varying functions and their extensions. Also asymptotic quasi-inverse functions
and relations between limits of the ratio of functions and their quasi-inverse functions
are discussed.
Let R+ be the set of nonnegative reals. Also let F = F(R+) be the space of real-valued
functions f = (f(t), t ≥ 0), and F+ =
⋃
A>0{f ∈ F : f(t) > 0, t ∈ [A,∞)}. Thus f ∈ F+
if and only if f is eventually positive.
Let F(∞) be the space of functions f ∈ F+ such that
(i) sup0≤t≤T f(t) <∞ ∀ T > 0;
(ii) lim supt→∞ f(t) =∞.
THE ϕ-ASYMPTOTIC BEHAVIOUR OF SOLUTIONS OF SDE’S 23
Further let F∞ be the space of functions f ∈ F(∞) such that limt→∞ f(t) = ∞. We
also make use of the subspaces C
(∞) and C
∞ of continuous functions in F
(∞) and F
∞,
respectively.
Finally, the space C∞
inc contains all functions f ∈ C∞, which are strictly increasing for
large t.
For a given f ∈ F+, we consider the upper and lower limit functions
f∗(c) = lim sup
t→∞
f(ct)
f(t)
and f∗(c) = lim inf
t→∞
f(ct)
f(t)
, c > 0,
which take values in [0,∞].
RV and ORV Functions. Recall that a measurable function f ∈ F+ is called regularly
varying (RV) if f∗(c) = f∗(c) = κ(c) ∈ R+ for all c > 0 (see Karamata [17]). In
particular, if κ(c) = 1 for all c > 0, then the function f is called slowly varying (SV). For
any RV function f , κ(c) = cα, c > 0, for some number α ∈ R, which is called the index
of the function f . Moreover, f(t) = tα�(t), t > 0, where � is a slowly varying function.
A measurable function f ∈ F+ is called O-regularly varying (ORV) if f∗(c) < ∞ for
all c > 0 (see Avakumović [1] and Karamata [18]). It is obvious that any RV function is
an ORV function. The theory of RV functions and later extensions and generalizations
turned out to be fruitful in various fields of mathematics (cf. Seneta [25] and Bingham et
al. [4] for excellent surveys on this topic and for the history of its theory and applications).
PRV Functions. For any RV function f , we have f∗(c) → 1 as c → 1. In order to
generalize this property to a wider class of functions, we introduce the following notion
(see Buldygin et al. [5]).
Definition 6.1. A measurable function f ∈ F+ is called pseudo-regularly varying
(PRV) if
lim sup
c→1
f∗(c) = 1. (6.1)
Any PRV function is ORV, but not vice versa. Moreover, any RV function is PRV,
but not vice versa. Corresponding examples have been given in Buldygin et al. [5].
PRV functions and their various applications have been studied by Korenblyum [21],
Matuszewska [22], Matuszewska and Orlicz [23], Stadtmüller and Trautner [26], Berman
[2, 3], Yakymiv [28], Cline [13], Djurčić [14], Djurčić and Torgašev [15], Klesov et al. [20],
and Buldygin et al. [5–10]. Note that PRV functions are called regularly oscillating in
Berman [2], weakly oscillating in Yakymiv [28], intermediate regularly varying in Cline [13]
and CRV in Djurčić [14]. We stick to the notion PRV introduced in Buldygin et al. [5].
One of the well-known properties of PRV functions is that they and only they preserve
the equivalence of functions (see, for example, Buldygin et al. [5]).
Two functions u and v are called (asymptotically) equivalent if u(t) ∼ v(t) as t→∞,
that is, limt→∞ u(t)/v(t) = 1. The equivalence of functions is denoted by u ∼ v. Recall
that a function f preserves the equivalence of functions if f(u(t))/f(v(t))→ 1 as t→∞
for all nonnegative functions u and v such that u ∼ v and limt→∞ u(t) = limt→∞ v(t) =
∞.
Lemma 6.1. A measurable function f ∈ F+ preserves the equivalence of functions if
and only if it is PRV.
SQI Functions. Next we define a further class of functions playing an important role
in the context of this paper (see also Buldygin et al. [5, 6]).
24 V. V. BULDYGIN ET AL.
Definition 6.2. A measurable function f ∈ F+ is called sufficiently quickly increasing
(SQI) if
f∗(c) > 1 for all c > 1. (6.2)
These functions have also been used by Yakymiv [27], Djurčić and Torgašev [15], and
Buldygin et al. [5–10].
Note that any slowly varying function f cannot be an SQI function. On the other
hand, any RV function of positive index as well as any quickly increasing monotone
function, for example f(t) = et, t ≥ 0, is SQI.
Quasi-Inverse Functions. First, we recall the definition of a quasi-inverse function,
which will be useful for our considerations below (cf. Buldygin et al. [5, 6]).
Definition 6.3. Let f ∈ F(∞). A function f (−1) ∈ F∞ is called a quasi-inverse
function of f if f(f (−1)(s))) = s for all large s.
For any f ∈ C
(∞), a quasi-inverse function exists, but may not be unique (see Buldygin
et al. [5, 6]). If f ∈ C∞
inc, then its inverse function f−1 exists, that is, f(f−1(s)) = s and
f−1(f(t)) = t for all sufficiently large s and t.
Quasi-Inverse Functions Preserving the Equivalence of Functions. Next we dis-
cuss conditions under which quasi-inverse functions preserve the equivalence of functions
(see Buldygin et al. [5, 6]).
Theorem 6.1. Assume f ∈ C∞
inc. Then, its inverse function f−1 preserves the
equivalence of functions if and only if condition (6.2) holds.
Finally we consider relations between limits of the ratio of functions and their quasi-
inverse functions (see Buldygin et al. [5, 6]).
Theorem 6.2. Assume f ∈ C∞
inc and let f satisfy condition (6.2). If, for some
function x ∈ F
∞,
lim
t→∞
x(t)
f(t)
= a with some a ∈ (0,∞),
then, for any quasi-inverse function x(−1) of x, we have
lim
s→∞
x(−1)(s)
f−1(s/a)
= 1,
where f−1 is the inverse function of f.
7. Auxiliary Results
The proofs of some statements in this paper are closely connected with the questions
of when differentiable functions satisfy PRV or SQI conditions (see Section 6). These
questions were studied in Buldygin et al. [8, 9, 11]. In this section, some results from
these papers are collected.
Conditions for Differentiable Functions to be PRV or SQI. Consider the follow-
ing five conditions on a function f and its derivative f ′:
(D) f ∈ F∞ and f is positive and continuously differentiable for all t ≥ t0 > 0;
(DM) Condition (D) holds and f ′(t) ≥ 0 for all t ≥ t0 > 0;
(DM+) Condition (D) holds and f ′(t) > 0 for all t ≥ t0 > 0;
(DM1) Condition (DM+) holds and f ′ is nonincreasing for all t ≥ t0 > 0;
(DM2) Condition (DM+) holds and f ′ is nondecreasing for all t ≥ t0 > 0.
THE ϕ-ASYMPTOTIC BEHAVIOUR OF SOLUTIONS OF SDE’S 25
For a function f satisfying condition (D), the following integral representation holds:
f(t) = f(t0) exp
{∫ t
t0
f ′(u)
f(u)
du
}
(7.1)
for any t > t0.
The next result provides some simple inequalities between the limit functions of f and
f ′.
Proposition 7.1. If condition (DM+) holds, then
c(f ′)∗(c) ≤ f∗(c) ≤ f∗(c) ≤ c(f ′)∗(c)
for all c ≥ 0.
In particular, Proposition 7.1 demonstrates that under condition (DM+), if f ′(·) is an
ORV (PRV, SQI, RV) function, then f possesses the same property. Below, these results
will be strengthened for PRV and SQI functions.
Lemma 7.1. Assume condition (D). Then f is a PRV function if and only if
lim
c→1
lim sup
t→∞
∫ ct
t
f ′(u)
f(u)
du = 0.
Lemma 7.2. Assume condition (DM). Then f is a PRV function if and only if
lim
c↓1
lim sup
t→∞
∫ ct
t
f ′(u)
f(u)
du = 0.
Let us consider some corollaries of the above lemmas.
Corollary 7.1. Assume condition (D).
1) If
lim sup
t→∞
t|f ′(t)|
f(t)
<∞,
then f is a PRV function.
2) If f is a PRV function, then
lim inf
t→∞
tf ′(t)
f(t)
<∞.
3) If condition (DM) holds and
lim sup
t→∞
tf ′(t)
f(t)
<∞, (7.2)
then f is a PRV function.
4) If condition (DM1) holds, then f is a PRV function.
Remark 7.1. If condition (D) holds and lim supt→∞ t|f ′(t)| < ∞, then f∗(c) = 1 for
all c > 0. This means that f is an SV function, and hence it is a PRV function. Thus we
can confine ourselves to the case, when lim supt→∞ t|f ′(t)| =∞.
Corollary 7.2. Assume condition (DM2). Then f is a PRV function if and only
if (7.2) holds true.
The integral in the next statement means the Lebesgue integral.
26 V. V. BULDYGIN ET AL.
Corollary 7.3. Assume condition (DM+). If∫ 1
0+
(f ′)∗(c)dc > 0, (7.3)
then f is a PRV function.
On applying Corollary 7.3, we get the following result.
Corollary 7.4. Assume condition (DM+). If the set {c ∈ (0, 1] : (f ′)∗(c) > 0} has
positive Lebesgue measure, then f is a PRV function. In particular, this condition holds
if f ′ is an ORV function.
Now we discuss conditions for differentiable functions to be SQI.
Lemma 7.3. Assume condition (D). Then f is an SQI function if and only if
lim inf
t→∞
∫ ct
t
f ′(u)
f(u)
du > 0 for all c > 1.
Next, we consider some corollaries of Lemma 7.3.
Corollary 7.5. Assume condition (DM).
1) If
lim inf
t→∞
tf ′(t)
f(t)
> 0, (7.4)
then f is an SQI function.
2) If f is an SQI function, then
lim sup
t→∞
tf ′(t)
f(t)
> 0.
3) If f is an SQI function, then
lim sup
t→∞
tf ′(t) =∞.
4) If condition (DM2) holds, then f is an SQI function.
Corollary 7.6. Assume condition (DM1). Then f is an SQI function if and only
if (7.4) holds true.
The next result gives a condition in terms of the function (f ′)∗(·).
Corollary 7.7. Assume condition (DM+). If
c(f ′)∗(c) > 1 for all c > 1,
then f is an SQI function.
8. Proofs of the Main Results
Proof of Theorem 2.1. By conditions (2.6), (2.7) and Lemma 7.3, with f = G(ϕ) and
f ′ = 1/g(ϕ), we have that G(ϕ) is an SQI function, that is, it satisfies (6.2). Moreover,
G(ϕ) ∈ C∞
inc. Hence, by Theorem 6.1, the function ϕ(μ(·)) = (G(ϕ))−1(·) preserves the
equivalence of functions (see Section 6). Therefore, in view of (2.5),
lim
t→∞
ϕ(X(t))
ϕ(μ(t))
= lim
t→∞
ϕ(μ(G(X(t))))
ϕ(μ(t))
= lim
t→∞
G(X(t))
t
= 1 a.s.,
since μ = G−1. Thus, relation (2.4) holds and statement 1) is proved.
THE ϕ-ASYMPTOTIC BEHAVIOUR OF SOLUTIONS OF SDE’S 27
By conditions (2.6), (2.8) and Lemma 7.2, with f = G(ϕ) and f ′ = 1/g(ϕ), we have
that G(ϕ) is a PRV function (see Definition 6.1). Hence, by Lemma 6.1, the function
G(ϕ) = G(ϕ−1) preserves the equivalence of functions. Therefore, in view of (2.4),
lim
t→∞
G(X(t))
t
= lim
t→∞
G(ϕ)(ϕ(X(t)))
G(ϕ)(ϕ(μ(t)))
= lim
t→∞
ϕ(X(t))
ϕ(μ(t))
= 1 a.s.,
that is, relation (2.5) holds. Thus, statement 2) follows from the last implication in
combination with 1). �
Proof of Proposition 2.1. Condition (2.7) follows from
a) (2.6) and (i), in view of Corollary 7.5, with f = G(ϕ) and f ′ = 1/g(ϕ), since
lim inf
t→∞
tf ′(t)
f(t)
= lim inf
t→∞
t
G(ϕ)(t)g(ϕ)(t)
= lim inf
t→∞
ϕ(t)
G(ϕ)(ϕ(t))g(ϕ)(ϕ(t))
= lim inf
t→∞
ϕ(t)
G(t)g(t)ϕ′(t)
=
(
lim sup
t→∞
G(t)g(t)ϕ′(t)
ϕ(t)
)−1
> 0;
b) (2.6) and (ii), since, by (ii), g(ϕ) is eventually nonincreasing and thus (i) holds;
c) (iii), since (i) (and also (2.6)) follows from (iii);
d) (2.6) and (iv), in view of Corollary 7.7, with f = G(ϕ) and f ′ = 1/g(ϕ), since(
1/g(ϕ)
)
∗ = 1/(g(ϕ))∗;
e) (2.6) and (v), since (iv) follows from (v).
�
Proof of Proposition 2.2. Condition (2.8) follows from
a) (2.6) and (i), in view of Corollary 4.1, with f = G(ϕ) and f ′ = 1/g(ϕ), since
lim sup
t→∞
tf ′(t)
f(t)
= lim sup
t→∞
t
g(ϕ)(t)G(ϕ)(t)
= lim sup
t→∞
ϕ(t)
G(ϕ)(ϕ(t))g(ϕ)(ϕ(t))
= lim sup
t→∞
ϕ(t)
G(t)g(t)ϕ′(t)
<∞ =
(
lim inf
t→∞
G(t)g(t)ϕ′(t)
ϕ(t)
)−1
<∞;
b) (2.6) and (ii), since, by (ii), g(ϕ) is eventually nondecreasing and thus (i) holds;
c) (2.6) and (iii), in view of Corollary 7.3, with f = G(ϕ) and f ′ = 1/g(ϕ), since
(1/g(ϕ))∗(c) = 1/(g(ϕ))∗(c) for all c > 0;
d) (2.6) and (iv), since (iii) follows from (iv);
e) (2.6) and (v), since (iv) follows from (v).
�
Proof of Theorem 2.2. Theorem 2.2 follows from Theorem 2.1 in combination with Re-
mark 2.4 and Proposition 2.1. �
Proof of Theorem 2.3. Theorem 2.3 follows from Theorem 2.1 in combination with Re-
mark 2.6 and Proposition 2.1. �
Proof of Theorem 3.1. Assume that (3.5) holds for at least one k = 1, 2. Then, by
conditions (3.1), (3.5) and Lemma 7.3, with f = G
(ϕk)
k and f ′ = 1/g(ϕk)
k , we have that
G
(ϕk)
k is an SQI function, i.e. it satisfies (6.2). Moreover, G(ϕj)
j ∈ C∞
inc and (G(ϕj)
j )−1 =
ϕj ◦ μj , j = 1, 2. Hence, by Theorem 6.2, relation (3.2) follows and statement 1) is
proved.
In order to prove statement 2) we assume that (3.6) holds for at least one k = 1, 2.
Then, by conditions (3.1), (3.6) and Lemma 7.2, with f = G
(ϕk)
k and f ′ = 1/g(ϕk)
k ,
we have that G(ϕk)
k is a PRV function (see Definition 6.1). Hence, by Lemma 6.1, the
function G
(ϕk)
k preserves the equivalence of functions. Therefore, by Theorem 6.1, the
28 V. V. BULDYGIN ET AL.
function ϕk ◦μk is an SQI function, since G(ϕj)
j ∈ C∞
inc and (G(ϕj)
j )−1 = ϕj ◦μj , j = 1, 2.
Hence, by Theorem 6.2, relation (3.3) follows and statement 2) is proved.
Statement 3) follows from statements 1) and 2). �
Proof of Lemma 3.1. For each k = 1, 2, the function Fk is an RV function with index
αk + 1 > 0, since fk is an RV function with index αk > −1. Assume that
lim
t→∞
F1(t)
F2(t)
= 1.
Then α1 + 1 = α2 + 1 = β > 0 and, by Karamata’s theorem (see Bingham et al. [4], p.
26), we have that
lim
t→∞
tfk(t)
βFk(t)
= 1, k = 1, 2.
Hence
lim
t→∞
f1(t)
f2(t)
= lim
t→∞
tf1(t)
βF1(t)
· lim
t→∞
F1(t)
F2(t)
· lim
t→∞
βF2(t)
tf2(t)
= 1.
Thus relation (3.11) is proved.
Relation (3.12) follows from relations (3.10) and (3.11). �
Proof of Theorem 4.1. In view of relation (4.1), Theorem 4.1 follows from Theorems 2.1,
3.1 and 3.2. �
Proof of Theorem 4.2. Theorem 4.2 follows from Theorem 4.1 and Remark 2.4. �
Proof of Theorem 4.3. Theorem 4.3 follows from Theorem 4.2 in combination with Ex-
ample 3.1 and Lemma 3.1, with fk = 1/g(ϕk)
k , k = 1, 2. �
Proof of Theorem 5.1. In view of relation (5.1), Theorem 5.1 follows from Theorems 2.2,
3.1 and 3.2. �
Proof of Theorem 5.2. Theorem 5.2 follows from Theorem 5.1 in combination with Ex-
ample 3.1 and Lemma 3.1, with fk = 1/g(ϕk)
k , k = 1, 2. �
Acknowledgement. The authors are grateful to Prof. A. Yu. Pilipenko for useful
comments concerning the results of the paper.
Bibliography
1. V. G. Avakumović, Über einen O–Inversionssatz, Bull. Int. Acad. Youg. Sci. 29–30 (1936),
107–117.
2. S. M. Berman, Sojourns and extremes of a diffusion process on a fixed interval, Adv. Appl.
Prob. 14 (1982), 811–832.
3. S. M. Berman, The tail of the convolution of densities and its application to a model of HIV-
latency time, Ann. Appl. Prob. 2 (1992), 481–502.
4. N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, Cambridge University
Press, Cambridge, 1987.
5. V. V. Buldygin, O. I. Klesov, and J. G. Steinebach, Properties of a subclass of Avakumović
functions and their generalized inverses, Ukrain. Math. J. 54 (2002), 179–206.
6. V. V. Buldygin, O. I. Klesov, and J. G. Steinebach, On some properties of asymptotically quasi-
inverse functions and their applications I, Teor. Imov. Mat. Stat. 70 (2004), 9–25 (Ukrainian);
English transl. in Theory Probab. Math. Statist. 70 (2005), 11–28.
7. V. V. Buldygin, O. I. Klesov, and J. G. Steinebach, On some properties of asymptotically
quasi-inverse functions and their applications II, Teor. Imov. Mat. Stat. 71 (2004), 34–48
(Ukrainian); English transl. in Theory Probab. Math. Statist. 71 (2005), 37–52.
8. V. V. Buldygin, O. I. Klesov, and J. G. Steinebach, The PRV property of functions and the
asymptotic behavior of solutions of stochastic differential equations, Teor. Imov. Mat. Stat. 72
(2005), 10–23 (Ukrainian); English transl. in Theory Probab. Math. Statist. 72 (2006), 11–25.
9. V. V. Buldygin, O. I. Klesov, and J. G. Steinebach, PRV property and the asymptotic behaviour
of solutions of stochastic differential equations, Theory Stoch. Process. 11(27) (2005), 42–57.
10. V. V. Buldygin, O. I. Klesov, and J. G. Steinebach, On some extensions of Karamata’s theory
and their applications, Publ. Inst. Math. (Beograd) (N. S.) 80(94) (2006), 59–96.
THE ϕ-ASYMPTOTIC BEHAVIOUR OF SOLUTIONS OF SDE’S 29
11. V. V. Buldygin, O. I. Klesov, and J. G. Steinebach, PRV property and the � –Asymptotic
behaviour of solutions of stochastic differential equations, Lithuanian Math. J. (2007), no. 4,
1–21.
12. V. V. Buldygin and O. A. Tymoshenko, On the asymptotic stability of stochastic differential
equations, Naukovi Visti NTUU ”KPI” (2007), no. 1, 126-129.
13. D. B. H. Cline, Intermediate regular and Π-variation, Proc. London Math. Soc. 68 (1994),
594–616.
14. D. Djurčić, O-regularly varying functions and strong asymptotic equivalence, J. Math. Anal.
Appl. 220 (1998), 451–461.
15. D. Djurčić and A. Torgašev, Strong asymptotic equivalence and inversion of functions in the
class Kc, J. Math. Anal. Appl. 255 (2001), 383–390.
16. I. I. Gihman, A. V. Skorohod, Stochastic Differential Equations, Springer-Verlag, Berlin-Heidel-
berg-New York, 1972.
17. J. Karamata, Sur un mode de croissance régulière des fonctions, Mathematica (Cluj) 4 (1930),
38–53.
18. J. Karamata, Bemerkung über die vorstehende Arbeit des Herrn Avakumović, mit näherer
Betrachtung einer Klasse von Funktionen, welche bei den Inversionssätzen vorkommen, Bull.
Int. Acad. Youg. Sci. 29–30 (1936), 117–123.
19. G. Keller, G. Kersting, and U. Rösler, On the asymptotic behaviour of solutions of stochastic
differential equations, Z. Wahrsch. Verw. Geb. 68 (1984), 163–184.
20. O. Klesov, Z. Rychlik, and J. Steinebach, Strong limit theorems for general renewal processes,
Theory Probab. Math. Statist. 21 (2001), 329–349.
21. B. H. Korenblyum, On the asymptotic behaviour of Laplace integrals near the boundary of a
region of convergence, Dokl. Akad. Nauk. USSR (N.S.) 109 (1956), 173–176.
22. W. Matuszewska, On a generalization of regularly increasing functions, Studia Math. 24 (1964),
271–279.
23. W. Matuszewska and W. Orlicz, On some classes of functions with regard to their orders of
growth, Studia Math. 26 (1965), 11–24.
24. S. Parameswaran, Partition functions whose logarithms are slowly oscillating, Trans. Amer.
Math. Soc. 100 (1961), 217–240.
25. E. Seneta, Regularly Varying Functions, Springer-Verlag, Berlin, 1976.
26. U. Stadtmüller and R. Trautner, Tauberian theorems for Laplace transforms, J. Reine Angew.
Math. 311/312 (1979), 283–290.
27. A. L. Yakymiv, Asymptotics properties of the state change points in a random record process,
Theory Probab. Appl. 31 (1987), 508–512.
28. A. L. Yakymiv, Asymptotics of the probability of nonextinction of critical Bellman–Harris
branching processes, Proc. Steklov Inst. Math. 4 (1988), 189–217.
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|
| id | nasplib_isofts_kiev_ua-123456789-4532 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0321-3900 |
| language | English |
| last_indexed | 2025-12-07T18:04:17Z |
| publishDate | 2008 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Buldygin, V.V. Klesov, O.I. Steinebach, J.G. Tymoshenko, O.A. 2009-11-25T11:00:57Z 2009-11-25T11:00:57Z 2008 On the φ-asymptotic behaviour of solutions of stochastic differential equations / V.V. Buldygin, O.I. Klesov, J.G. Steinebach, O.A. Tymoshenko // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 1. — С. 11–29. — Бібліогр.: 28 назв.— англ. 0321-3900 https://nasplib.isofts.kiev.ua/handle/123456789/4532 519.21 In this paper we study the a.s. asymptotic behaviour of the solution of the stochastic dfferential equation dX(t) = g(X(t))dt +σ(X(t))dW(t), X(0) = b > 0, where g and σ are positive continuous functions and W is a Wiener process. Making use of the theory of pseudo-regularly varying (PRV) functions, we find conditions on g, σ and φ, under which φ(X(•)) can be approximated a.s. by φ(μ(•), where μ is the solution of the ordinary differential equation dμ(t) = g(μ(t))dt, μ(0) = b. As an application of these results we discuss the problem of φ-asymptotic equivalence for solutions of stochastic differential equations. This work has partially been supported by Deutsche Forschungsgemeinschaft under DFG grants 436 UKR 113/41/0-2 and 436 UKR 113/68/0-1 en Інститут математики НАН України On the φ-asymptotic behaviour of solutions of stochastic differential equations Article published earlier |
| spellingShingle | On the φ-asymptotic behaviour of solutions of stochastic differential equations Buldygin, V.V. Klesov, O.I. Steinebach, J.G. Tymoshenko, O.A. |
| title | On the φ-asymptotic behaviour of solutions of stochastic differential equations |
| title_full | On the φ-asymptotic behaviour of solutions of stochastic differential equations |
| title_fullStr | On the φ-asymptotic behaviour of solutions of stochastic differential equations |
| title_full_unstemmed | On the φ-asymptotic behaviour of solutions of stochastic differential equations |
| title_short | On the φ-asymptotic behaviour of solutions of stochastic differential equations |
| title_sort | on the φ-asymptotic behaviour of solutions of stochastic differential equations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/4532 |
| work_keys_str_mv | AT buldyginvv ontheφasymptoticbehaviourofsolutionsofstochasticdifferentialequations AT klesovoi ontheφasymptoticbehaviourofsolutionsofstochasticdifferentialequations AT steinebachjg ontheφasymptoticbehaviourofsolutionsofstochasticdifferentialequations AT tymoshenkooa ontheφasymptoticbehaviourofsolutionsofstochasticdifferentialequations |