PRV property and the asymptotic behaviour of solutions of stochastic differential equations
We consider the a.s. asymptotic behaviour of a solution of the stochastic differential
 equation (SDE) dX(t) = g(X(t))dt + σ(X(t))dW(t), with X(0) ≡ b > 0, where g(.)
 and σ(.) are positive continuous functions and W(.) is the standard Wiener process.
 By applying the theo...
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| Cite this: | PRV property and the asymptotic behaviour of solutions of stochastic differential equations / V.V. Buldygin, O.I. Klesov, J.G. Steinebach // Theory of Stochastic Processes. — 2005. — Т. 11 (27), № 3-4. — С. 42–57. — Бібліогр.: 17 назв.— англ. |
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| author | Buldygin, V.V. Klesov, O.I. Steinebach, J.G. |
| author_facet | Buldygin, V.V. Klesov, O.I. Steinebach, J.G. |
| citation_txt | PRV property and the asymptotic behaviour of solutions of stochastic differential equations / V.V. Buldygin, O.I. Klesov, J.G. Steinebach // Theory of Stochastic Processes. — 2005. — Т. 11 (27), № 3-4. — С. 42–57. — Бібліогр.: 17 назв.— англ. |
| collection | DSpace DC |
| description | We consider the a.s. asymptotic behaviour of a solution of the stochastic differential
equation (SDE) dX(t) = g(X(t))dt + σ(X(t))dW(t), with X(0) ≡ b > 0, where g(.)
and σ(.) are positive continuous functions and W(.) is the standard Wiener process.
By applying the theory of PRV and PMPV functions, we find the conditions on g(.)
and σ(.), under which X(.) resp. f(X(.)) may be approximated a.s. on {X(t)→∞}
by μ(.) resp. f(μ(.)), where μ( ) is a solution of the deterministic differential equation
dμ(t) = g(μ(t))dt with μ(0) = b, and f(.) is a strictly increasing function. Moreover,
we consider the asymptotic behaviour of generalized renewal processes connected
with this SDE.
|
| first_indexed | 2025-12-07T18:31:32Z |
| format | Article |
| fulltext |
Theory of Stochastic Processes
Vol. 11 (27), no. 3–4, 2005, pp. 42–57
UDC 519.21
V. V. BULDYGIN, O. I. KLESOV, AND J. G. STEINEBACH
PRV PROPERTY AND THE ASYMPTOTIC BEHAVIOUR OF
SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS
We consider the a.s. asymptotic behaviour of a solution of the stochastic differential
equation (SDE) dX(t) = g(X(t))dt + σ(X(t))dW (t), with X(0) ≡ b > 0, where g( )
and σ( ) are positive continuous functions and W ( ) is the standard Wiener process.
By applying the theory of PRV and PMPV functions, we find the conditions on g( )
and σ( ), under which X( ) resp. ϕ(X( )) may be approximated a.s. on {X(t) → ∞}
by μ( ) resp. ϕ(μ( )), where μ( ) is a solution of the deterministic differential equation
dμ(t) = g(μ(t))dt with μ(0) = b, and ϕ( ) is a strictly increasing function. Moreover,
we consider the asymptotic behaviour of generalized renewal processes connected
with this SDE.
1. Introduction
Gikhman and Skorohod [8], §17, and later Keller et al. [11] considered the asymptotic
behaviour, as t → ∞, of a solution X(·) = (X(t), t ≥ 0) of the stochastic differential
equation (SDE)
(1.1) dX(t) = g(X(t))dt + σ(X(t))dW (t), t ≥ 0, X(0) ≡ b > 0.
Here, W (·) is the standard Wiener process and X(·) denotes the Itô-solution of the
SDE (1.1). One of the basic assumptions in the above works was that both σ(·) =
(σ(x),−∞ < x < ∞) and g(·) = (g(x),−∞ < x < ∞) are positive functions, and the
authors were only interested in situations, in which the event {limt→∞ X(t) = ∞} occurs
with positive probability and such that infinity will not be reached in finite time.
Gikhman and Skorohod [8], §17, and Keller et al. [11] gave the conditions, under
which the asymptotics of X(·) is determined by a nonrandom function. In this paper,
we reconsider this problem under the same basic conditions.
Denote, by μ(·) = (μ(t), t ≥ 0), a solution of the deterministic differential equation
corresponding to (1.1) for σ(·) ≡ 0, i.e.
(1.2) dμ(t) = g(μ(t))dt, t ≥ 0, μ(0) = b.
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. An interesting question
then is under which conditions it holds that
(1.3) lim
t→∞
X(t)
μ(t)
= 1 a.s. on
{
lim
t→∞ X(t) = ∞
}
or, more generally, that
(1.4) lim
t→∞
ϕ(X(t))
ϕ(μ(t))
= 1 a.s. on
{
lim
t→∞X(t) = ∞
}
,
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 113/41/0-3 and 436 UKR 113/68/0-1.
42
ASYMPTOTIC BEHAVIOUR OF SOLUTIONS OF SDE’S 43
for a given function (ϕ(x),−∞ < x < ∞). Here, “a.s.” stands for “almost surely”.
The methods used in Gikhman and Skorohod [8], §17, Theorem 4, and in Keller et
al. [11] are similar and consist of two main steps. First, they study the process
(1.5) Y (t) = G(X(t)), t ≥ 0,
where
(1.6) G(t) =
∫ t
1
ds
g(s)
, t ≥ 0,
and prove that, under some conditions (see Section 2 below),
(1.7) lim
t→∞
G(X(t))
t
= 1 a.s. on
{
lim
t→∞X(t) = ∞
}
.
Note that G(·) = (G(t), t ≥ 1) is the inverse function of μ(·) (G(·) = μ−1(·)) if g(·) is
positive and continuous. In the second step, relation (1.7) is used to prove (1.3).
For the second step, Gikhman and Skorohod [8], §17, Theorem 4, assume that, for
some C > 0,
(1.8) lim
ε→0
sup
z>C
sup
| z
u−1|≤ε
∣∣∣∣μ(z)
μ(u)
− 1
∣∣∣∣ = 0.
We will explain in Section 5, Corollary 5.1, that, under condition (1.8), the function
μ(·) preserves the equivalence of functions (see Definition 5.4), so that (1.7) implies in
this case that
lim
t→∞
X(t)
μ(t)
= lim
t→∞
μ(G(X(t)))
μ(t)
= 1 a.s. on
{
lim
t→∞X(t) = ∞
}
,
that is, relation (1.3) holds. Note that Gikhman and Skorohod [8] use another reasoning;
the general idea above, however, simplifies the proof considerably.
Condition (1.8) is formulated in terms of the function μ(·), i.e. in terms of the solution
of Eq. (1.2). It is more natural, however, to give conditions in terms of the functions g(·)
and G(·).
Keller et al. [11], Theorem 5, use another set of conditions and prove the implica-
tion (1.7)=⇒ (1.4) for ϕ(x) = log x, x > 0. Their conditions are expressed in terms of
both μ(·) and g(·). Again the disadvantage therein is the part involving the function
μ(·).
Our goal in this paper is to consider the conditions for implications (1.7)=⇒ (1.3)
and (1.7)=⇒ (1.4) expressed in terms of the functions g(·) and G(·). For doing so,
we must find the conditions, under which the functions μ(·) and ϕ(μ(·)) preserve the
equivalence of functions. To achieve this goal, we follow the general approach developed
in Buldygin et al. [3]–[5]. This approach allows for solving the following general problem:
Find the conditions on a given function, under which its inverse or quasi-inverse function
preserves the equivalence of functions. Note that the approach in [3]–[5] works not only
for continuous and increasing functions (like the ones considered in this paper), but
also for discontinuous and/or non-monotone functions, where the so-called asymptotic
quasi-inverse functions substitute the inverse functions.
Further in this paper, we study the asymptotic behaviour of generalized renewal pro-
cesses related to the solution of SDE (1.1).
PRV and PMPV functions play an important role in this paper. For example, one of
the key results is that PRV functions, and only they, preserve the equivalence of functions
and sequences (see Buldygin et al. [3]). For the further studies of PRV functions and their
44 V. V. BULDYGIN, O. I. KLESOV, AND J. G. STEINEBACH
applications, confer, e.g., Korenblyum [13], Stadtmüller and Trautner [16], Yakymiv [17],
Klesov et al. [12], and Buldygin et al. [3]–[5].
Korenblyum [13] as well as Stadtmüller and Trautner [16] considered nondecreasing
PRV functions and studied their properties in order to obtain analogs of the Tauberian
theorems for Laplace transforms. In particular, Stadtmüller and Trautner [16] proved
that the Tauberian theorem for the Laplace transform of a nondecreasing positive func-
tion f is valid if and only if f is a PRV function. A substantial progress has been achieved
by Yakymiv [17], who investigated the multivariate PRV-property, but his results are of
particular interest in the one-dimensional case, too. Note that PRV functions are called
weakly oscillating in Yakymiv [17].
In Klesov et al. [12], the relationship between the strong law of large numbers for
sequences of random variables and its counterpart for renewal processes constructed
from these sequences has been studied. Some results of Klesov et al. [12] have been
extended in Buldygin et al. [3]–[5], where the PRV property was studied in more details.
PMPV functions were introduced and studied in Buldygin et al. [3]–[5]. One of the
key results is that inverse or quasi-inverse functions of PMPV functions preserve the
equivalence of functions.
This paper makes use of results from Buldygin et al. [6]–[7] and is organized as follows.
In Sections 2–4, we formulate and discuss the main results of the paper. In Section 5, we
introduce the necessary definitions and give some auxiliary results on PRV and PMPV
functions . The main problems of this paper are closely connected with the problem of
finding out when differentiable functions satisfy PRV or PMPV conditions. In Section 6,
we discuss these questions in some details. The proofs of some of our main results are
sketched in Section 7.
2. Asymptotic Behaviour of the Solution
of a Stochastic Differential Equation
General statements. Consider functions g(·) and σ(·) satisfying the following condi-
tions:
(C1) g(·) is continuous and positive on (0,∞) and σ(·) is continuous and positive on
(−∞,∞) and such that (1.1) has a.s. a unique and continuous solution X(·) with
arbitrary initial condition, as well as (1.2) has a unique and continuous solution
with arbitrary positive initial condition.
Remark 2.1. Problem (1.1) has a.s. a unique and continuous solution X(·) with
arbitrary initial condition, as well as problem (1.2) has a unique and continuous solution
with arbitrary positive initial condition, if, for example, the functions g(·) and σ(·) satisfy
the following conditions:
a) for some K and for all x ∈ (−∞,∞),
|g(x)| + |σ(x)| ≤ K(1 + |x|);
b) for each C > 0, there exists LC such that, for |x| ≤ C and |y| ≤ C,
|g(x) − g(y)| + |σ(x) − σ(y)| ≤ LC |x − y|
(see Gikhman and Skorohod [8], §15).
Now our goal is to find the conditions on g(·), G(·), and σ(·), under which relation (1.3)
holds. To do so, we first consider the following general statement which describes the
extra conditions for relation (1.7) to imply or being equivalent to (1.3).
ASYMPTOTIC BEHAVIOUR OF SOLUTIONS OF SDE’S 45
Theorem 2.1. Assume condition (C1) and
(2.1) lim
t→∞G(t) =
∫ ∞
1
du
g(u)
= ∞.
Let g(·) and G(·) (see (1.6)) be such that
(2.2) lim inf
t→∞
∫ ct
t
du
g(u)G(u)
> 0 for all c > 1.
Then,
1) if (1.7) holds, then also (1.3) holds true;
2) if
(2.3) lim
c↓1
lim sup
t→∞
∫ ct
t
du
g(u)G(u)
= 0,
then (1.7) and (1.3) are equivalent.
Recall that G(·) = (G(t), t ≥ 1) is the inverse function of μ(·) (G(·) = μ−1(·)).
By condition (2.1), μ(t) → ∞ as t → ∞. Moreover, (2.1) excludes the possibility of
explosions (that is, the solution does not reach infinity in finite time). Note that the
function g(u) = u, u > 0, satisfies (2.1), but does not satisfy condition (2.2).
Remark 2.2. In view of Definition 5.2 below, condition (2.2) means that the function
G(·) is a PMPV function. Observe that, by Theorem 5.2, the function μ(·) preserves,
under condition (2.1), the equivalence of functions (see Definition 5.4) if and only if (2.2)
holds.
Condition (2.3) means that the function G(·) is a PRV function (see Definition 5.1)
and, by Theorem 5.1, this condition is equivalent to the condition that G(·) preserves
the equivalence of functions. The set of conditions (2.1), (2.2) and (2.3) means that both
G(·) and μ(·) preserve the equivalence of functions.
Next, we consider some sufficient conditions for (2.2) (Proposition 2.1) and (2.3)
(Proposition 2.2), which can be expressed in terms of the function g(·), and thus are
more suitable for practical use. For more details see Section 6.
Proposition 2.1. Let g(·) be a positive and continuous function such that (2.1) holds.
Assume that at least one of the following conditions holds:
(i) lim supt→∞ g(t)G(t)/t < ∞;
(ii) g(·) is eventually nonincreasing;
(iii) there exists α < 1 such that 0 < infs≥1 g(s)s−α, sups≥1 g(s)s−α < ∞;
(iv) g∗(c) < c for all c > 1, with g∗(c) = lim supt→∞ g(ct)/g(t);
(v) g(·) is an RV function with index α < 1 (see Section 5).
Then, g(·) satisfies condition (2.2).
Remark 2.3. Under (2.1), condition (i) of Proposition 2.1 is equivalent to (2.2), if the
function g(·) is eventually nondecreasing.
Remark 2.4. Substituting t → G(t), we get from (1.3):
lim
t→∞
X(G(t))
t
= 1 a.s. on
{
lim
t→∞X(t) = ∞
}
.
This means that, under the conditions of Theorem 2.1, if (1.7) holds, then
lim
t→∞
G(X(t))
t
= lim
t→∞
X(G(t))
t
= 1 a.s. on
{
lim
t→∞ X(t) = ∞
}
,
46 V. V. BULDYGIN, O. I. KLESOV, AND J. G. STEINEBACH
that is, G(·) is an asymptotic inverse function for the process X(·) a.s. on the set
{limt→∞ X(t) = ∞}.
Remark 2.5. Condition (i) of Proposition 2.1 does not hold for any regularly varying
function g(·) of index 1, that is, for functions g(t) = t�(t), where �(·) is slowly varying.
This is due to a result of Parameswaran [14] which proves that
lim
t→∞ �(t)
∫ t
1
ds
s�(s)
= ∞.
Proposition 2.2. Let g(·) be a positive and continuous function such that (2.1) holds.
Assume that at least one of the following conditions holds:
(i) lim inft→∞ g(t)G(t)/t > 0;
(ii) g(·) is eventually nondecreasing;
(iii)
∫ 1
0+
dc/g∗(c) > 0, with g∗(c) = lim supt→∞ g(ct)/g(t);
(iv) the set {c ∈ (0, 1] : g∗(c) < ∞} has positive Lebesgue measure;
(v) at least one of conditions (iii), (iv), or (v) of Proposition 2.1 holds.
Then g(·) satisfies condition (2.3).
Remark 2.6. Under (2.1), condition (i) of Proposition 2.2 is equivalent to (2.3), if the
function g(·) is eventually nonincreasing.
Asymptotic Behaviour of the Solution of a Stochastic Differential Equation
under the Gikhman–Skorohod conditions. Our next result (Theorem 2.2) contains
the sufficient conditions for relation (1.3) to hold. The following one is a condition from
Gikhman and Skorohod [8], §17.
(GS) Let g(·) and σ(·) be positive and continuous functions such that (1.1) has a.s.
a unique and continuous solution X(·) with arbitrary initial condition and with
limt→∞ X(t) = ∞ a.s., as well as (1.2) has a unique and continuous 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 2.7. Under (GS), relation (1.7) holds true a.s., that is,
lim
t→∞
G(X(t))
t
= 1 a.s.
(see Gikhman and Skorohod [8], §17, Theorem 4 and Remark 1).
Remark 2.8. Problem (1.1) has a.s. a unique and continuous solution X(·) with
arbitrary initial condition and with limt→∞ X(t) = ∞ a.s., as well as problem (1.2) has a
unique and continuous solution with arbitrary positive initial condition, if, for example,
conditions a), b) of Remark 2.1 hold and the functions g(·) and σ(·) satisfy the following
condition:
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 Gikhman and Skorohod [8], §16, Theorem 1).
Together with condition (GS), we will consider
(C2) condition (2.1) holds and condition (2.2) or at least one of conditions (i)–(v) of
Proposition 2.1 holds.
ASYMPTOTIC BEHAVIOUR OF SOLUTIONS OF SDE’S 47
Theorem 2.2. Assume conditions (C2) and (GS). Then relation (1.3) follows a.s.,
that is,
lim
t→∞
X(t)
μ(t)
= 1 a.s..
Observe that Theorem 2.2 complements Theorem 4 in Gikhman and Skorohod [8],
§17.
Example 2.1. (see Gikhman and Skorohod [8], §17, Corollary 1). Assume (GS) with
g(x) = Cxβ for x > 0, where 0 ≤ β < 1 and C > 0. Then (2.1) and (v) of Proposition 2.1
hold. Thus, by Theorem 2.2, we have
lim
t→∞
X(t)
(C(1 − β)t)1/(1−β)
= 1 a.s.
for all b > 0, since μ(t) ∼ (C(1 − β)t)1/(1−β) as t → ∞, with any b > 0.
Observe that, by Remark 2.4, we cannot use Theorem 2.2 with ϕ(x) ≡ x and g(x) ∼
Cx as x → ∞.
Asymptotic Behaviour of the Solution of a Stochastic Differential Equation
under the Keller–Kersting–Rösler conditions. Here, we discuss the conditions of
Keller et al. [11]. 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:
(A0) g(·) is continuous and positive on (0,∞), σ(·) is continuous and positive on
(−∞,∞), and g(·) and σ(·) are such that (1.1) has a.s. a unique and continuous
solution with arbitrary initial condition and limt→∞ X(t) = ∞ with positive
probability, as well as (1.2) has a unique and continuously differentiable solution
with arbitrary positive initial condition.
The following five conditions have been used in Keller et al. [11]:
(A1) g : (0,∞) → (0,∞) is strictly positive and twice continuously differentiable such
that
∫ ∞
1 (g(u))−1du = ∞.
(A2) h(t) → 0 as t → ∞.
(A3) σ : (0,∞) → (0,∞) is strictly positive and continuously differentiable such that∫ ∞
0
(tg(μ(t)))−2σ2(μ(t))dt < ∞.
(A4) 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(·)).
(A5) There is a constant C > 0 such that log μ(2t) ≤ C log μ(t) for large t. Further-
more, the function e−(·)g(e(·)) together with its derivative is eventually concave
or convex.
Remark 2.9. Under the above conditions, the following two statements follow (see
Theorem 1 and Theorem 5 in Keller et al. [11]).
I) Under (A0)–(A4), relation (1.7) holds true.
II) Under (A0)–(A5), relation (1.4) holds true with ϕ(t) = log t, t > 0.
Theorem 2.3. Assume conditions (C2) and (A0)–(A4). Then relation (1.3) follows.
48 V. V. BULDYGIN, O. I. KLESOV, AND J. G. STEINEBACH
3. ϕ-Asymptotic Behaviour of the Solution
of a Stochastic Differential Equation
General statements. Consider a function ϕ(·) satisfying the following condition:
(C3) ϕ(·) = (ϕ(x), x > 0) is an eventually 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 (1.6), the function ϕ−1(·) is inverse to ϕ(·), and ϕ′(·) is the first
derivative of ϕ(·).
Observe that (G(ϕ)(t), t ≥ tb) is the inverse function of ϕ(μ(·)), where μ(·) is the
solution of problem (1.2), with tb = ϕ(b).
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 main goal is to find the conditions, under which relation (1.4) holds. To do
so, we first consider the following general statement which describes the extra conditions
for relation (1.7) to imply or being equivalent to (1.4).
Theorem 3.1. Assume (C1), (C3), and (2.1). Let g(·) and ϕ(·) be such that
(3.1) 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.
Then
1) if (1.7) holds, (1.4) holds also true;
2) if
(3.2) 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,
then (1.7) and (1.4) are equivalent.
Remark 3.1. In view of Definition 5.2 below, condition (3.2) means that the function
G(ϕ)(·) is a PMPV function. Observe that, by Theorem 5.2, the function ϕ(μ(·)) pre-
serves, under condition (2.1), the equivalence of functions (see Definition 5.3) if and only
if (3.2) holds.
Condition (3.1) means that the function G(ϕ)(·) is a PRV function (see Definition 5.1)
and, by Theorem 5.1, the function G(ϕ)(·) preserves, under condition (2.1) the equivalence
of functions if and only if (3.1) holds.
Next, we consider some sufficient conditions for both (3.1) (Proposition 3.1) and (3.2)
(Proposition 3.2) which can be expressed in terms of the functions g(·), G(·), and ϕ(·)
and thus are more suitable for practical use. For more details, see Section 6.
Proposition 3.1. Let g(·) be a positive and continuous function on (0,∞) such
that (2.1) holds, and let ϕ(·) satisfy (C2). 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 5).
ASYMPTOTIC BEHAVIOUR OF SOLUTIONS OF SDE’S 49
Then, condition (3.1) holds true.
Remark 3.2. Under (2.1), condition (i) of Proposition 3.1 is equivalent to (3.1), if the
function g(·) is eventually nondecreasing.
Remark 3.3. Condition (i) of Proposition 3.1 does not hold for any regularly varying
function g(ϕ)(·) of index 1, that is, for functions g(ϕ)(·) such that g(ϕ)(t) = t�(t), where
�(·) is slowly varying (see Remark 2.5).
Proposition 3.2. Let g(·) be a positive and continuous function on (0,∞) such
that (2.1) holds, and let ϕ(·) satisfy (C2). 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 conditions (iii), (iv), or (v) of Proposition 3.1 holds.
Then, g(ϕ)(·) satisfies condition (3.2).
Remark 3.4. Under (2.1), condition (i) of Proposition 3.2 is equivalent to (3.2), if the
function g(·) is eventually nonincreasing.
Example 3.1. Let g(x) = ϕ(x) = x, x > 0. Clearly condition (2.1) holds, but
condition (3.2) 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
g(ϕ)(t)G(ϕ)(t)
t
= 1, t > 0.
Thus, by Propositions 3.1 and 3.2, conditions (2.1), (3.1), and (3.2) hold.
ϕ-Asymptotic Behaviour of the Solution of a Stochastic Differential Equation
under the Gikhman–Skorohod conditions. The following Theorem 3.2 provides
some conditions, under which relation (1.4) holds true, and thus generalizes Theorem 4
in Gikhman and Skorohod [8], §17.
Theorem 3.2. Assume conditions (C2), (C3), and (GS). Then relation (1.4) follows
a.s., that is,
(3.3) lim
t→∞
ϕ(X(t))
ϕ(μ(t))
= 1 a.s.
Remark 3.5. If (1.4) (or (3.3)) holds for a given function ϕ(·), then, by Theorem 5.1
below, this relation also holds true for the function f(ϕ(·)), where f(·) is a PRV function
(cf. Definition 5.1).
Example 3.2. Assume (GS) with g(x) = Cx/(log(x+1)γ for x > 0, where γ > 0 and
C > 0. Put ϕ(x) = (log(x + 1))1+γ for x > 0. Then g(ϕ)(t) ∼ C(1 + γ) as t → ∞, that
is, (2.1) and (v) of Proposition 3.1 hold. Thus, by Theorem 3.2, we have
lim
t→∞
(log X(t))1+γ
C(1 + γ)t
= 1 a.s.
for all b > 0, since ϕ(μ(t)) ∼ (C(1 + γ)t) as t → ∞.
50 V. V. BULDYGIN, O. I. KLESOV, AND J. G. STEINEBACH
ϕ-Asymptotic Behaviour of the Solution of a Stochastic Differential Equation
under the Keller–Kersting–Rösler conditions. The following result provides the
conditions, under which relation (1.4) holds true.
Theorem 3.3. Assume conditions (C2), (C3), and (A0)–(A4). Then relation (1.4)
follows.
Observe that Theorem 3.3 generalizes statement i) of Theorem 5 in Keller et al. [11].
Example 3.3. (see Gikhman and Skorohod [8], §17, Corollary 2). Assume (A0)–(A4)
with g(x) = Cx for x > 0, where C > 0. Put ϕ(x) = log x for x > 0. Then g(ϕ)(t) ∼ C
as t → ∞, that is, (2.1) and (v) of Proposition 3.1 hold. Thus, by Theorem 3.3, we have
lim
t→∞
log X(t)
Ct
= 1 a.s.
for all b > 0, since ϕ(μ(t)) ∼ Ct as t → ∞.
4. Asymptotic Behaviour of Generalized Renewal Processes
As above, we assume that both the functions g(·) and σ(·) are positive and continuous
and such that problem (1.1) has a.s. a unique continuous solution X(·), as well as
problem (1.2) has a unique continuous solution μ(·).
Let us consider the following three generalized renewal processes for the process X(·):
F (s) = inf{t ≥ 0 : X(t) = s}, s > 1,
i.e. the first time when the stochastic process X(·) crosses the level s,
L(s) = sup{t ≥ 0 : X(t) = s}, s > 1,
the last time when the process X(·) crosses the level s, and
T (s) = meas({t ≥ 0: X(t) ≤ s}) =
∫ ∞
0
I(X(t) ≤ s)dt, s > 1,
the total time spent by the process X(·) in (−∞, s], where “meas” denotes the Lebesgue
measure.
The next results describe the asymptotic behaviour of these generalized renewal pro-
cesses.
Theorem 4.1. Assume conditions (C1) and (1.7). Then
(4.1) lim
t→∞
F (t)
G(t)
= lim
t→∞
T (t)
G(t)
= lim
t→∞
L(t)
G(t)
= 1 a.s. on
{
lim
t→∞X(t) = ∞
}
,
and
(4.2) lim
t→∞
F (μ(t))
t
= lim
t→∞
T (μ(t))
t
= lim
t→∞
L(μ(t))
t
= 1 a.s. on
{
lim
t→∞X(t) = ∞
}
;
Moreover, if conditions (C1), (C2), and (1.7) are satisfied, then
(4.3) lim
t→∞
μ(F (t))
t
= lim
t→∞
μ(T (t))
t
= lim
t→∞
μ(L(t))
t
= 1 a.s. on { lim
t→∞X(t) = ∞}.
ASYMPTOTIC BEHAVIOUR OF SOLUTIONS OF SDE’S 51
Theorem 4.2. Under condition (GS), we have
lim
t→∞
F (t)
G(t)
= lim
t→∞
T (t)
G(t)
= lim
t→∞
L(t)
G(t)
= 1 a.s.,
and
lim
t→∞
F (μ(t))
t
= lim
t→∞
T (μ(t))
t
= lim
t→∞
L(μ(t))
t
= 1 a.s.
Moreover, if conditions (GS) and (C2) are satisfied, then
lim
t→∞
μ(F (t))
t
= lim
t→∞
μ(T (t))
t
= lim
t→∞
μ(L(t))
t
= 1 a.s.
Theorem 4.3. Under conditions (A0)–(A4), we have that relations (4.1) and (4.2)
hold true. Moreover, if (A0)–(A4) and (C2) are satisfied, then relation (4.3) follows.
5. Properties of PRV and PMPV Functions
Let R be the set of real numbers, and let R+ be the set of nonnegative real numbers.
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) = ∞.
Further, let F∞ be the space of functions f(·) ∈ F(∞) such that limt→∞ f(t) = ∞. We
also use 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 make use of 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 reg-
ularly varying (RV) if f∗(c) = f∗(c) = κ(c) ∈ R+ for all c > 0 (see Karamata [9]). 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 [10]). 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 [15]
and Bingham et al. [2] for excellent surveys on this topic and for the history of the 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. [3]).
52 V. V. BULDYGIN, O. I. KLESOV, AND J. G. STEINEBACH
Definition 5.1. A measurable function f(·) ∈ F+ is called pseudo-regularly varying
(PRV) if
(5.1) lim sup
c→1
f∗(c) = 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. [3].
PRV functions and their applications have been studied by Korenblyum [13], Stadt-
müller and Trautner [16], Yakymiv [17], Klesov et al. [12], and Buldygin et al. [3]–[5].
Lemma 5.1. (Buldygin et al. [3]) Assume f(·) ∈ F+. Then,
1) condition (3.1) is equivalent to any of the following four conditions:
(1) (i) lim infc→1 f∗(c) = 1,
(2) (ii) limc→1 lim supt→∞
∣∣∣ f(ct)
f(t) − 1
∣∣∣ = 0,
(3) (iii) limc↓1 f∗(c) = limc↓1 f∗(c) = 1,
(4) (iv) limc↑1 f∗(c) = limc↑1 f∗(c) = 1;
2) condition (3.1) holds if and only if the upper limit function f∗(·) (or the lower
limit function f∗(·)) is continuous at the point c = 1, that is, either
lim
c→1
f∗(c) = 1
or limc→1 f∗(c) = 1;
3) if f(·) is a function with a nondecreasing upper limit function f∗(·), then con-
dition (3.1) holds if and only if limc↓1 f∗(c) = 1 or limc↑1 f∗(c) = 1; moreover,
under these conditions, f∗(·) is continuous at every point c ∈ (0,∞).
PMPV Functions. Next we define further classes of functions playing an important
role in the context of this paper (see also Buldygin et al. [3], [4]).
Definition 5.2. A measurable function f(·) ∈ F+ is called pseudo-monotone of
positive variation (PMPV) if
(5.2) f∗(c) > 1 for all c > 1.
Note that any slowly varying function f(·) cannot be a PMPV 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 PMPV.
Functions Preserving Asymptotic Equivalence. In this subsection, functions u(·)
and v(·) are nonnegative and eventually positive.
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(·).
Definition 5.3. 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) = ∞.
One of the most important properties of PRV functions is that they and only they
preserve the equivalence of functions.
ASYMPTOTIC BEHAVIOUR OF SOLUTIONS OF SDE’S 53
Theorem 5.1. (Buldygin et al. [3]) Assume f(·) ∈ F+. Then the function f(·) preserves
the equivalence of functions if and only if condition (5.1) holds.
In view of Theorem 5.1 and Lemma 5.1, the following statement holds true.
Corollary 5.1. Under condition (1.7), the function μ(·) (see Section 1) preserves the
equivalence of functions.
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. [3] and [4]).
Definition 5.4. Let f(·) ∈ F
(∞). A function f (−1)(·) ∈ F∞ is called a quasi-inverse
function for 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. 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.
Example 5.1. Let x(·) ∈ C(∞). Put
x
(−1)
1 (s) = inf{t ≥ 0 : x(t) = s},
for s ≥ s0 = x(0), and x
(−1)
1 (s) = 0, for 0 ≤ s < s0, if s0 > 0. The function x
(−1)
1 (·) is a
quasi-inverse function for x(·). If x(·) ∈ C∞
inc, then x
(−1)
1 (·) = x−1(·). �
Example 5.2. Let x(·) ∈ C∞. Put
x
(−1)
2 (s) = sup{t ≥ 0 : x(t) = s},
for s ≥ s0 = x(0), and x
(−1)
2 (s) = 0, for 0 ≤ s < s0, if s0 > 0. The function x
(−1)
2 (·) is a
quasi-inverse function for x(·). Observe that x
(−1)
1 (s) ≤ x
(−1)
2 (s), s > 0, and, in general,
x
(−1)
1 (·) �= x
(−1)
2 (·). If x(·) ∈ C∞
inc, then x
(−1)
2 (·) = x
(−1)
1 (·) = x−1(·). �
Quasi-inverse Functions Preserving the Equivalence of Functions. Next we dis-
cuss the conditions, under which quasi-inverse functions preserve the equivalence of func-
tions.
Theorem 5.2. (Buldygin et al. [3]) Assume f(·) ∈ C
∞
inc. Then, its inverse function
f−1(·) preserves the equivalence of functions if and only if condition (5.2) holds.
Theorem 5.3. (Buldygin et al. [3]) Assume f(·) ∈ C∞
inc, and let f(·) satisfy condi-
tion (3.2). If, for some function x(·) ∈ F∞,
(5.3) lim
t→∞
x(t)
f(t)
= a with some a ∈ (0,∞),
then, for any quasi-inverse function x(−1)(·) of x(·), we have
(5.4) lim
s→∞
x(−1)(s)
f−1(s/a)
= 1,
where f−1(·) is the inverse function of f(·).
54 V. V. BULDYGIN, O. I. KLESOV, AND J. G. STEINEBACH
6. Conditions for Differentiable Functions to be PRV or PMPV
The solutions of the main problems in this paper are closely connected with the ques-
tion of when the differentiable functions satisfy PRV or PMPV conditions. In this sec-
tion, we will discuss this question. For the proofs of all statements of this section, confer
Buldygin et al. [6].
In the sequel, the following five conditions on a function f(·) and its derivative f ′(·)
will play a certain role:
(D) f(·) ∈ F∞, and there exists t0 = t0(f(·)) > 0 such that f(·) is positive and
continuously differentiable for all t ≥ t0;
(DM) condition (D) holds and f ′(t) ≥ 0 for all t ≥ t0;
(DM+) condition (D) holds and f ′(t) > 0 for all t ≥ t0;
(DM1) condition (DM+) holds and f ′(·) is nonincreasing for all t ≥ t0;
(DM2) condition (DM+) holds and f ′(·) is nondecreasing for all t ≥ t0.
For a function f(·) satisfying condition (D), the integral representation
(6.1) f(t) = f(t0) exp
{∫ t
t0
f ′(u)
f(u)
du
}
holds for any t > t0.
Conditions for Differentiable Functions to be PRV. The following statement is
immediate from Definition 5.1 in combination with (6.1).
Lemma 6.1. 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.
On applying Lemma 6.1 in combination with Lemma 5.1, we get the following result.
Lemma 6.2. 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 6.1. 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
(6.2) lim sup
t→∞
tf ′(t)
f(t)
< ∞,
then f(·) is a PRV function.
4) If condition (DM1) holds, then f(·) is a PRV function.
ASYMPTOTIC BEHAVIOUR OF SOLUTIONS OF SDE’S 55
Remark 6.1. 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 where lim supt→∞ t|f ′(t)| = ∞.
Corollary 6.2. Assume condition (DM2). Then f(·) is a PRV function if and only
if (6.2) holds true.
Proof of Corollary 6.2. Its sufficiency follows from Corollary 6.1.
Now, let f(·) be a PRV function. By condition (DM2), we have, for any c > 1,
1
c − 1
lim sup
t→∞
∫ ct
t
f ′(u)
f(u)
du ≥ lim sup
t→∞
tf ′(t)
f(ct)
≥ lim sup
t→∞
tf ′(t)
f(t)
lim inf
t→∞
f(t)
f(ct)
=
1
f∗(c)
lim sup
t→∞
tf ′(t)
f(t)
.
Hence,
lim sup
t→∞
tf ′(t)
f(t)
≤ f∗(c)
(c − 1)
lim sup
t→∞
∫ ct
t
f ′(u)
f(u)
du,
since any PRV function is an ORV function (see Buldygin et al. [3], Remark 2.1), that
is, f∗(c) < ∞ for all c > 0. So, (6.2) follows from Lemma 6.2. �
The integral in the following statement means the Lebesgue integral.
Corollary 6.3. Assume condition (DM+). If
(6.3)
∫ 1
0+
(f ′)∗(c)dc > 0,
then f(·) is a PRV function.
By applying Corollary 6.3, we get the following result.
Corollary 6.4. 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.
Conditions for Differentiable Functions to be PMPV. The following statement
is also immediate from Definition 5.2 in combination with (6.1).
Lemma 6.3. Assume condition (D). Then f(·) is a PMPV 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 6.3.
Corollary 6.5. Assume condition (DM).
1) If
(6.4) lim inf
t→∞
tf ′(t)
f(t)
> 0,
then f(·) is a PMPV function.
2) If f(·) is a PMPV function, then
lim sup
t→∞
tf ′(t)
f(t)
> 0.
3) If f(·) is a PMPV function, then
lim sup
t→∞
tf ′(t) = ∞.
56 V. V. BULDYGIN, O. I. KLESOV, AND J. G. STEINEBACH
4) If condition (DM2) holds, then f(·) is a PMPV function.
Corollary 6.6. Assume condition (DM1). Then f(·) is a PMPV function if and only
if (6.4) holds true.
The following result gives a condition in terms of the function (f ′)∗(·).
Lemma 6.4. Assume condition (DM+). If
(6.5) c(f ′)∗(c) > 1 for all c > 1,
then f(·) is a PMPV function.
7. Proofs of Some Main Results
In this section, we consider only the proofs of some results from Section 2. For further
details, we refer to Buldygin et al. [6], [7].
Proof of Theorem 2.1. By conditions (2.1) and (2.2) and by Lemma 6.3, with f(·) = G(·)
and f ′(·) = 1/g(·), we have that G(·) is a PMPV function, that is, it satisfies (5.2).
Moreover, G(·) ∈ C∞
inc. Hence, by Theorem 5.2, the function μ(·) = G−1(·) preserves the
equivalence of functions (see Definition 5.4). Therefore, in view of (1.7),
lim
t→∞
X(t)
μ(t)
= lim
t→∞
μ(G(X(t)))
μ(t)
= 1 a.s. on
{
lim
t→∞X(t) = ∞
}
,
that is, relation (1.3) holds. Thus, statement 1) is proved.
By conditions (2.1) and (2.3) and by Lemma 6.2, with f(·) = G(·) and f ′(·) = 1/g(·),
we have that G(·) is a PRV function (see Definition 5.1). Hence, by Theorem 5.1, the
function G(·) = μ−1(·) preserves the equivalence of functions. Therefore, in view of (1.3),
lim
t→∞
G(X(t))
t
= lim
t→∞
G(X(t))
G(μ(t))
= 1 a.s. on
{
lim
t→∞ X(t) = ∞
}
.
that is, relation (1.7) holds. Thus, statement 2) follows from the last implication in
combination with 1). �
Proof of Proposition 2.1. Condition (2.2) follows from
a) (2.1) and (i) (or (ii)), in view of Corollary 6.5, with f(·) = G(·) and f ′(·) = 1/g(·).
b) (iii), since (i) (and even (2.1)) follows from (iii).
c) (2.1) and (iv), in view of Lemma 6.4, with f(·) = G(·) and f ′(·) = 1/g(·), since
(1/g(·))∗ = 1/g∗(·).
d) (2.1) and (v), since (iv) follows from (v).
�
Proof of Remark 2.3. It follows from Corollary 6.6 with f(·) = G(·) and f ′(·) = 1/g(·).
�
Proof of Proposition 2.2. Condition (2.3) follows from
a) (2.1) and (i) (or (ii)), in view of Corollary 6.1, with f(·) = G(·) and f ′(·) = 1/g(·).
b) (2.1) and (iii), in view of Corollary 6.3, with f(·) = G(·) and f ′(·) = 1/g(·), since
(1/g)∗(c) = 1/g∗(c) for all c > 0.
c) (2.1) and (iv), since (iii) follows from (iv).
d) (2.1) and (v), since (iv) follows from (v).
�
Proof of Remark 2.6. It follows from Corollary 6.2, with f(·) = G(·) and f ′(·) = 1/g(·).
�
Proofs of Theorems 2.2 and 2.3. They follow from Remarks 2.7 and 2.9, respectively, in
combination with Theorem 2.1 and Proposition 2.1. �
ASYMPTOTIC BEHAVIOUR OF SOLUTIONS OF SDE’S 57
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E-mail : valbuld@comsys.ntu-kpi.kiev.ua
E-mail : oleg@tbimc.freenet.kiev.ua
E-mail : jost@math.uni-koeln.de
|
| id | nasplib_isofts_kiev_ua-123456789-4424 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0321-3900 |
| language | English |
| last_indexed | 2025-12-07T18:31:32Z |
| publishDate | 2005 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Buldygin, V.V. Klesov, O.I. Steinebach, J.G. 2009-11-09T15:30:54Z 2009-11-09T15:30:54Z 2005 PRV property and the asymptotic behaviour of solutions of stochastic differential equations / V.V. Buldygin, O.I. Klesov, J.G. Steinebach // Theory of Stochastic Processes. — 2005. — Т. 11 (27), № 3-4. — С. 42–57. — Бібліогр.: 17 назв.— англ. 0321-3900 https://nasplib.isofts.kiev.ua/handle/123456789/4424 519.21 We consider the a.s. asymptotic behaviour of a solution of the stochastic differential
 equation (SDE) dX(t) = g(X(t))dt + σ(X(t))dW(t), with X(0) ≡ b > 0, where g(.)
 and σ(.) are positive continuous functions and W(.) is the standard Wiener process.
 By applying the theory of PRV and PMPV functions, we find the conditions on g(.)
 and σ(.), under which X(.) resp. f(X(.)) may be approximated a.s. on {X(t)→∞}
 by μ(.) resp. f(μ(.)), where μ( ) is a solution of the deterministic differential equation
 dμ(t) = g(μ(t))dt with μ(0) = b, and f(.) is a strictly increasing function. Moreover,
 we consider the asymptotic behaviour of generalized renewal processes connected
 with this SDE. This work has partially been supported by Deutsche Forschungsgemeinschaft under DFG grants 436 UKR 113/41/0-3 and 436 UKR 113/68/0-1. en Інститут математики НАН України PRV property and the asymptotic behaviour of solutions of stochastic differential equations Article published earlier |
| spellingShingle | PRV property and the asymptotic behaviour of solutions of stochastic differential equations Buldygin, V.V. Klesov, O.I. Steinebach, J.G. |
| title | PRV property and the asymptotic behaviour of solutions of stochastic differential equations |
| title_full | PRV property and the asymptotic behaviour of solutions of stochastic differential equations |
| title_fullStr | PRV property and the asymptotic behaviour of solutions of stochastic differential equations |
| title_full_unstemmed | PRV property and the asymptotic behaviour of solutions of stochastic differential equations |
| title_short | PRV property and the asymptotic behaviour of solutions of stochastic differential equations |
| title_sort | prv property and the asymptotic behaviour of solutions of stochastic differential equations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/4424 |
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