CUBO A Mathematical Journal

Vol.10, N
o
¯ 03, (145–159). October 2008

Asymptotic Constancy and Stability in Nonautonomous

Stochastic Differential Equations

John A.D. Appleby∗

School of Mathematical Sciences, Dublin City University,

Dublin 9, Ireland

email: john.appleby@dcu.ie

URL: http://webpages.dcu.ie/~applebyj

James P. Gleeson

Department of Applied Mathematics, University College Cork,

Cork, Ireland

email: j.gleeson@ucc.ie

and

Alexandra Rodkina

Department of Mathematics and Computer Science,

The University of the West Indies, Mona, Kingston 7, Jamaica

email: alexandra.rodkina@uwimona.edu.jm

ABSTRACT

This paper considers the asymptotic behaviour of a scalar non-autonomous stochastic

differential equation which has zero drift, and whose diffusion term is a product of

a function of time and space dependent function, and which has zero as a unique

∗The first author is partially supported by an Albert College Fellowship awarded by Dublin City University’s
Research Advisory Panel. The third author is supported by the Mona Research Fellowship Programm awarded by
the University of the West Indies, Jamaica and by London Mathematical Society



146 John A.D. Appleby et al. CUBO
10, 3 (2008)

equilibrium solution. We classify the pathwise limiting behaviour of solutions; solution

either tends to a non-trivial, non-equilibrium and random limit, or the solution hits

zero in finite time. In the first case, the exact rate of decay can always be computed.

These results can be inferred from the square integrability of the time dependent factor,

and the asymptotic behaviour of the corresponding autonomous stochastic equation,

where the time dependent multiplier is unity.

RESUMEN

Este art́ıculo considera el comportamiento asintótico de una ecuación diferencial esto-

castica escalar no-autónoma la cual tiene cero desviación y cujo término de difusión

es un producto de una función de equilibrio. Nosotros clasificamos el comportamiento

limite por caminos de las soluciones; la solución atiende a un no equilibrio y ĺımite ran-

don no trivial, o la solución encuentra cero en tiempo finito. En el primer caso, las tasas

de decaimiento siempre pueden ser calculadas. Estos resultados pueden ser inferidos de

la integrabilidad al cuadrado del factor dependiente del tiempo, y el comportamiento

asintótico de la correspondiente ecuación estocatica autónoma, donde el multiplicador

dependiente del tiempo es la unidad.

Key words and phrases: Brownian motion, almost sure asymptotic stability, asymptotic con-

stancy, stochastic differential equation, nonautonomous, Feller’s test, explosions.

Math. Subj. Class.: 60H10, 93E15.

1 Introduction

This note considers the asymptotic behaviour of solutions of the “separable” stochastic differential

equation

dX(t) = σ(t)g(X(t)) dB(t). (1.1)

A solution of this equation with initial condition ξ is denoted by X(·, ξ). It is presumed that zero

is a point equilibrium, so X(t, 0) = 0 is a solution of (1.1). A standard deterministic change of

time scale reduces this equation to an autonomous equation

dX̃(t) = g(X̃(t)) dB̃(t), (1.2)

from which it can be shown that the condition that σ ∈ L2([0, ∞); R) largely determines whether

the solution tends to the equilibrium or to a non–trivial and non–equilibrium limit. Another feature

which is examined is the relationship between the process X̃ hitting zero in a finite amount of time,

or tending to zero as t → ∞ (in the case when X̃ remains strictly positive) and the corresponding

properties of X. As will be seen, a complete picture of the dynamics of (1.1) can be deduced in

terms of conditions on g and σ.



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Asymptotic constancy and stability in SDEs 147

Even though the time–change technique employed is well–known, some novel features appear in

the analysis. First, we are unaware of an extensive literature concerning the pathwise convergence

of solutions of stochastic differential equations to non–equilibrium limits. Second, we determine

here sharp upper and lower estimates in terms of the rate at which the noise intensity fades on

the almost sure rate of convergence of the solution to this non–equilibrium limit, in the case when

σ ∈ L2([0, ∞); R). This requires a delicate use of the law of the iterated logarithm, partly correcting

an error on the asymptotic behaviour of a tail martingale established in [1] and used in [2]. Finally,

in the case where σ 6∈ L2([0, ∞); R), the results here, taken in conjunction with work in [3, 4] would

enable exact almost sure rates of convergence to zero of solutions of (1.1) to be established.

2 Existence of solutions

In this paper we deal with highly nonlinear stochastic differential equations, SDE, whose solutions

can hit zero at finite time, due to the non-Lipshitz behavior of the diffusion coefficients. Moreover,

for non-autonomous equations, it is convenient for the completeness of our exposition, to state

carefully and to prove an existence result. This result is a corollary of well-known existence result

and martingale time changing theorem.

Let (Ω, F, (F(t))t≥0, P) be a complete probability space with filtration (F(t))t≥0 satisfying

the usual conditions (i.e. it is increasing and right continuous while F(0) contains all P-null

sets). Let (B(t))t≥0 be a scalar standard Brownian motion, defined on the probability space

(Ω, F, (F(t))t≥0, P). Since we will consider equations with deterministic initial conditions, it is

enough to work with the natural filtration of B: that is F(t) ≡ FB(t) where FB(t) = σ(B̃(s) :

0 ≤ s ≤ t).

Suppose that the function σ obeys

σ ∈ C([0, ∞); R). (2.1)

We define the local martingale M = {M (t), 0 ≤ t < ∞, FB(t)} by

M (t) =

∫ t

0

σ(s) dB(s), t ≥ 0, (2.2)

with square variation 〈M〉 given by

〈M〉(t) =

∫ t

0

σ2(s) ds.

Let T ∗ be given by

T ∗ =

∫

∞

0

σ2(s) ds, (2.3)

where we define T ∗ = ∞ if σ 6∈ L2([0, ∞); R).



148 John A.D. Appleby et al. CUBO
10, 3 (2008)

Define also, for each 0 ≤ s ≤ T ∗, the FB—stopping time

T (s) = inf{t ≥ 0 : 〈M〉(t) ≥ s}. (2.4)

By the martingale time change theorem, there exists a standard Brownian motion {B̃(t); 0 ≤ t ≤

T ∗; G(t)}, such that

B̃(s) = M (T (s)), G(s) = FB(T (s)), 0 ≤ s ≤ T ∗,

and moreover

M (t) = B̃(〈M〉(t)), FB(t) = G(〈M〉(t)), t ≥ 0.

In what follows we presume that g : R → R obeys

g(0) = 0. (2.5)

Since we do not want the equation to have any other equilibria in (0, ∞) we ask that the non–

degeneracy condition

g(x) > 0, for all x 6= 0, (2.6)

also be satisfied.

We are now in a position to state an existence result for the solution of the autonomous

stochastic differential equation.

Proposition 2.1. Suppose that g obeys (2.5) and (2.6) and that

there exists a strictly increasing function (2.7a)

q : [0, ∞) → [0, ∞) with q(0) = 0 such that
∫ ǫ

0

1

q2(u)
du = ∞, for all ǫ > 0,

and that g and q both obey

|g(x) − g(y)| ≤ q(|x − y|), x, y ∈ R. (2.7b)

Then there exists a unique strong non-exploding solution X̃ of

dX̃(t) = g(X̃(t)) dB̃(t), t ≥ 0, X(0) = ξ > 0, (2.8)

on the complete probability space (Ω, F, (G(t))t≥0, P).

This result is a corollary of the result of Yamada and Watanabe (see e.g. [7], Proposition 2.13,

page 291, and [7], Theorem 5.4., page 332).

The main concern of this paper is the asymptotic behaviour of solutions of the nonautonomous

equations

dX(t) = σ(t)g(X(t)) dB(t), t ≥ 0, X(0) = ξ > 0. (2.9)

Before conducting this asymptotic analysis however, we must verify that this equation has a well-

defined solution. This is accomplished by the following result.



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Asymptotic constancy and stability in SDEs 149

Proposition 2.2. Suppose that g and q obey (2.5), (2.6), (2.7a) and (2.7b) and σ obeys (2.1).

Then there exists a unique strong non-exploding solution X of (2.9) on the complete probability

space (Ω, F, (FB(t))t≥0, P).

Proof. By Proposition 2.1, X̃ is the unique strong solution of (2.8). Consider X̃(t) for t ∈ [0, T ∗),

where T ∗ is defined as in (2.3), and define

X(t) = X̃

(
∫ t

0

σ2(s)ds

)

, t ≥ 0. (2.10)

Then, as X̃(s) is G(s)-measurable, X̃(t) is G
(

∫ t

0
σ2(s)ds

)

-measurable. But G (〈M〉(t)) = FB(t),

so X(t) is FB(t)-measurable. Now, by [7], Proposition 3.4.8, we get for t ≥ 0

∫

〈M〉(t)

0

g(X̃(u))dB̃(u) =

∫ t

0

g(X(s))dM (s) =

∫ t

0

σ(s)g(X(s))dB(s),

and, therefore, as 〈M〉(t) =
∫ t

0
σ2(s)ds, we get

X(t) = X̃

(
∫ t

0

σ
2(s)ds

)

= X̃ (〈M〉(t))

= X̃(0) +

∫

〈M〉(t)

0

g(X̃(u))dB̃(u) = ξ +

∫ t

0

σ(s)g(X(s))dB(s).

Therefore X defined by (2.10) is a strong solution of (2.9) on (Ω, F, (FB(t))t≥0, P).

To show uniqueness, suppose that there is another solution of (2.9), Y . Then Ỹ defined by

Ỹ (s) = Y (T (s)) obeys (2.8). Hence, as (2.8) has a unique solution, we have Ỹ = X̃, and therefore

it follows that Y = X.

This completes the proof.

In this paper, we choose to write explicitly the dependence of solutions on their initial condi-

tions, which are always assumed to be deterministic. Thus, the value at time t ≥ 0 of the process

Y with initial condition Y (0) = ξ is denoted by Y (t, ξ).

3 Main result

In this section, we state and discuss the main results of the paper concerning the asymptotic

behaviour of non–autonomous equation. At the end of the section we present an example of a non–

autonomous linear equation which can be analysed without the use of the theorems established

here, but whose behaviour illustrates the results proven.

As seen in the proof of Proposition 2.2 the non–autonomous equation (2.9) is equivalent to

(2.8), under a deterministic time change. However the subject of Proposition 2.2 is the existence



150 John A.D. Appleby et al. CUBO
10, 3 (2008)

for non–autonomous equation. The following Proposition by contrust, focusses on the relation

between the solutions of the two equations.

Proposition 3.1. Let ξ > 0 be deterministic. Suppose that g obeys (2.5), (2.6). Let σ obey (2.1).

Suppose that X(·, ξ) is the unique strong solution of (2.9) with X(0, ξ) = ξ. Let T ∗ be given by

(2.3), and T be defined by (2.4).

(i) If σ ∈ L2([0, ∞); R), then there exists a standard Brownian motion B̃ = {B̃(t); 0 ≤ t ≤

T ∗; G(t)} where G(t) = FB(T (t)) and B̃(t) = B(T (t)) such that the process X̃ = {X̃(t); 0 ≤

t ≤ T ∗; G(t)} defined by X̃(t) = X(T (t)) obeys (2.8).

(ii) If σ 6∈ L2([0, ∞); R), then there exists a standard Brownian motion B̃ = {B̃(t); 0 ≤ t <

∞; G(t)} where G(t) = FB (T (t)) and B̃(t) = B(T (t)) such that the process X̃ = {X̃(t); 0 ≤

t < ∞; G(t)} defined by X̃(t) = X(T (t)) obeys (2.8).

Before stating the first result on asymptotic behaviour we present some notation and an

important auxiliary result.

Suppose that X̃(·, ξ) is the solution of (2.8), where ξ > 0. Define

S̃0(ξ) = inf{t ≥ 0 : X̃(t, ξ) = 0}. (3.1)

Let us suppose that for δ > 0 we may define the function v : (0, ∞) → (0, ∞) by

v(x) = 2

∫ δ

x

∫ δ

y

dz

g2(z)
dy, x > 0. (3.2)

The following result is due to Feller (see e.g. [7], Theorem 5.5.29, page 348).

Proposition 3.2. Let ξ > 0 be deterministic, and X̃(·, ξ) be a strong solution of (2.8). If S̃0(ξ)

is as defined in (3.1), then

lim
t→S̃0(ξ)

X̃(t, ξ) = 0, sup
0≤t<S̃0(ξ)

X̃(t, ξ) < ∞, a.s.

Let v be defined by (3.2). Then

(i) limx→0+ v(x) < ∞ implies S̃0(ξ) < ∞, a.s.;

(ii) limx→0+ v(x) = ∞ implies S̃0(ξ) = ∞, a.s.

We define also

S0(ξ) = inf{t ≥ 0 : X(t, ξ) = 0}. (3.3)

We can determine whether S0(ξ) is finite or infinite with the help of Proposition 2.2.

We may now state the first main result on the asymptotic behaviour in this paper. It is a

direct consequence of Proposition 3.1 and Proposition 3.2.



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Asymptotic constancy and stability in SDEs 151

Theorem 3.3. Let ξ > 0 be deterministic. Suppose that g obeys (2.5), (2.6). Let σ obey (2.1).

Suppose that X(·, ξ) is the unique strong solution of (2.9) with X(0, ξ) = ξ.

Let v be defined as in (3.2), and suppose that

lim
x→0+

v(x) = ∞.

Then we have the following case distinction:

(a) If σ ∈ L2((0, ∞); R), then there exists an almost surely positive and FB(∞)–measurable

random variable L = L(ξ, ω) such that

lim
t→∞

X(t, ξ) = L(ξ) > 0, a.s. (3.4)

(b) If σ 6∈ L2((0, ∞); R), then

lim
t→∞

X(t, ξ) = 0, a.s.

and S0(ξ) defined by (3.3) obeys S0(ξ) = ∞, a.s.

The proof of this result and proofs of subsequent results in the this section, are postponed to

the final section of the paper.

When σ ∈ L2([0, ∞); R) the rate at which convergence to L(ξ) occurs can be determined

exactly.

Theorem 3.4. Let ξ > 0 be deterministic. Suppose that g obeys (2.5), (2.6), and let g ∈

C1((0, ∞); (0, ∞)). Let σ obey (2.1) and σ ∈ L2([0, ∞); R). Suppose that X(·, ξ) is the unique

strong solution of (2.9) with X(0, ξ) = ξ.

Let v be defined as in (3.2), and suppose that

lim
x→0+

v(x) = ∞.

Let L(ξ) be the almost surely positive and FB(∞)–measurable random variable defined by (3.4).

Then:

(i) If σ obeys
∫

∞

t

σ
2(s) ds > 0, for all t ≥ 0, (3.5)

then

lim sup
t→∞

X(t, ξ) − L(ξ)
√

2
∫

∞

t
σ2(s) ds log log

(∫

∞

t
σ2(s) ds

)−1

= g(L(ξ)), a.s., (3.6a)

lim inf
t→∞

X(t, ξ) − L(ξ)
√

2
∫

∞

t
σ2(s) ds log log

(∫

∞

t
σ2(s) ds

)−1

= −g(L(ξ)), a.s. (3.6b)



152 John A.D. Appleby et al. CUBO
10, 3 (2008)

(ii) If σ does not obey (3.5) i.e., if there exists τ ≥ 0 such that
∫

∞

t
σ2(s) ds = 0 for all t ≥ τ ,

then

X(t, ξ) = X(τ, ξ) = L(ξ), for all t ≥ τ , a.s.

Of course, in case (b) in Theorem 3.3, the rate of convergence cannot be so easily computed.

However pathwise rates of decay to zero for nonlinear autonomous stochastic differential equations

have been found in [3, 4], and could readily be applied here.

Finally, the distribution of the random limit L is in principle well–understood, by using the

forward Kolmogorov equation for the process X̃.

Theorem 3.5. Let ξ > 0 be deterministic. Suppose that g obeys (2.5), (2.6). Let σ obey (2.1)

and σ ∈ L2([0, ∞); R). Let T ∗ be given by (2.3). Suppose that X(·, ξ) is the unique strong solution

of (2.9) with X(0, ξ) = ξ.

Let v be defined as in (3.2), and suppose that

lim
x→0+

v(x) = ∞,

and let L(ξ) be the almost surely positive and FB(∞)–measurable random variable defined by (3.4).

Then

P[L(ξ) ≤ x] =

∫ x

0

Γ(T ∗; y) dy, x ≥ 0,

where
∂Γ

∂t
(t; y) =

1

2

∂2

∂y2

(

g2(y)Γ(t; y)
)

, (t, y) ∈ [0, T ∗] × (0, ∞),

and Γ(0; y) = δξ(y), y ∈ R, where δξ is the δ-function.

The result holds because L(ξ) = limt→T ∗+ X̃(t, ξ) = X̃(T
∗, ξ). Moreover, as X̃ is a diffusion

process with known infinitesimal generator and deterministic initial condition ξ, we can deduce

its distribution function from the forward Kolmogorov equation, and therefore, the distribution of

L(ξ) is also known.

It remains merely to classify the behaviour in the case when limx→0 v(x) < ∞.

Theorem 3.6. Let ξ > 0 be deterministic. Suppose that g obeys (2.5), (2.6). Let σ obey (2.1).

Suppose that X(·, ξ) is the unique strong solution of (2.9) with X(0, ξ) = ξ.

Let v be defined as in (3.2), and suppose that

lim
x→0+

v(x) < ∞.

Then we have the following case distinction:

(a) If σ ∈ L2((0, ∞); R), then there exists an almost surely positive and FB(∞)–measurable

random variable L = L(ξ, ω) such that

lim
t→∞

X(t, ξ) = L(ξ) > 0, a.s. on {S̃0(ξ) ≥ T
∗},



CUBO
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Asymptotic constancy and stability in SDEs 153

and

lim
t→S0(ξ)

−

X(t, ξ) = 0, a.s. on {S̃0(ξ) < T
∗},

where S̃0(ξ) defined by (3.1) and S0(ξ) is defined by (3.3).

(b) If σ 6∈ L2((0, ∞); R), then

lim
t→S0(ξ)

−

X(t, ξ) = 0, a.s.,

where S0(ξ) defined by (3.3) obeys S0(ξ) < +∞, a.s.

In case (a) the rate of convergence to the non–trivial random limit is the same as given in

Theorem 3.4, but only a.s. on the event {S̃0(ξ) > T
∗}.

The probability of the event {S̃0(ξ) < T
∗} can be computed for the process X̃ obeying (2.8),

by considering the limit

P[S̃0(ξ) < T
∗] = lim

a→0+
P[S̃a(ξ) < T

∗]

where for ξ > a > 0 we define S̃a(ξ) = inf{t ≥ 0 : X̃(t, ξ) = a}. It is possible to compute the

moment generating function of S̃a(ξ), λ 7→ E[e
−λS̃a(ξ)] for λ ≥ 0, by solving an appropriate Sturm–

Liouville problem, from which the probability P[S̃a(ξ) < T
∗] can in principle be determined by

inverse transform methods. The interested reader can refer to [6, Chapter 4.11] for further details

on computation of the moment generating function.

3.1 An example

A simple example of a process which can be analysed completely without appealing to these results

(but which is consistent with them) is the unique strong solution of

X(t) = ξ +

∫ t

0

σ(s)X(s) dB(s), t ≥ 0,

where ξ > 0 and σ ∈ C([0, ∞); R). This equation has explicit solution

X(t, ξ) = ξ exp

(
∫ t

0

σ(s) dB(s) −
1

2

∫ t

0

σ2(s) ds

)

, t ≥ 0.

Here we identify g(x) = x, x ≥ 0, and have limx→0+ v(x) = ∞. Hence Theorems 3.3, 3.4 and 3.5

can be applied to this stochastic differential equation.

In the case when σ ∈ L2([0, ∞); R), the martingale convergence theorem (cf., e.g., [8, Propo-

sition IV.1.26]) ensures that limt→∞
∫ t

0
σ(s) dB(s) exists and is almost surely finite. Therefore,

there is an almost surely positive and almost surely finite FB(∞)—measurable random variable

L(ξ) given by

L(ξ) = ξ exp

(
∫

∞

0

σ(s) dB(s) −
1

2

∫

∞

0

σ2(s) ds

)



154 John A.D. Appleby et al. CUBO
10, 3 (2008)

such that

lim
t→∞

X(t, ξ) = L(ξ) > 0, a.s.

This chimes with part (a) of Theorem 3.3.

In the case when σ 6∈ L2([0, ∞); R), we have that

lim
t→∞

∫ t

0

σ2(s) ds = +∞, lim
t→∞

∫ t

0
σ(s) dB(s)

∫ t

0
σ2(s) ds

= 0, a.s.

The latter fact resulting from the Strong Law of Large Numbers for martingales (cf., e.g., [8,

Exercise V.I.16]). Therefore

lim
t→∞

1
∫ t

0
σ2(s) ds

log X(t) = −
1

2
, a.s.

Hence limt→∞ X(t, ξ) = 0, a.s., which agrees with part (b) of Theorem 3.3.

Moreover, in the case when σ ∈ L2([0, ∞); R), L(ξ) is lognormally distributed; this is obvious

by observation of the formula for L(ξ), but can also be confirmed by solving the partial differential

equation for the transition density Γ in Theorem 3.5.

In the case when σ ∈ L2([0, ∞); R), the rate of convergence in Theorem 3.4 can be obtained,

if it is shown that

lim sup
t→∞

∫

∞

t
σ(s) dB(s)

√

2
∫

∞

t
σ2(s) ds log log

(∫

∞

t
σ2(s) ds

)−1

= 1, a.s., (3.7a)

lim inf
t→∞

∫

∞

t
σ(s) dB(s)

√

2
∫

∞

t
σ2(s) ds log log

(∫

∞

t
σ2(s) ds

)−1

= −1, a.s. (3.7b)

This can be established as follows: define T ∗ =
∫

∞

0
σ2(s) ds, and let M be the local martingale

defined in (2.2), and with square variation 〈M〉. Then, by the martingale time change theorem

there exists a Brownian motion B̃ such that M (t) = B̃(〈M〉(t)), 0 ≤ t < ∞. Thus

lim sup
t→∞

∫

∞

t
σ(s) dB(s)

√

2
∫

∞

t
σ2(s) ds log log

(∫

∞

t
σ2(s) ds

)−1

= lim sup
t→∞

M (∞) − M (t)
√

2(〈M〉(∞) − 〈M〉(t)) log log (〈M〉(∞) − 〈M〉(t))
−1

= lim sup
t→∞

B̃(〈M〉(∞)) − B̃(〈M〉(t))
√

2(〈M〉(∞) − 〈M〉(t)) log log (〈M〉(∞) − 〈M〉(t))
−1

= lim sup
s→T ∗−

B̃(T ∗) − B̃(s)
√

2(T ∗ − s) log log(T ∗ − s)−1

= lim sup
u→0+

B̃(T ∗) − B̃(T ∗ − u)
√

2u log log(u)−1
= 1, a.s.,



CUBO
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Asymptotic constancy and stability in SDEs 155

since B̄ defined by B̄(t) = B̃(T ∗) − B̃(T ∗ − t), 0 ≤ t ≤ T ∗, is also a standard Brownian motion,

and therefore subject to the Law of the Iterated Logarithm. (3.7) was stated in [2], but a proof

was not supplied there. A variant of this result is proven in [1] but this proof contains an error as

it is incorrectly stated there that

∫

∞

t

X(s) dB(s) =

∫

1/t

0

1

s
X(1/s) dW (s)

for a process X ∈ L2([0, ∞); R) a.s., where W is the standard Brownian motion given by W (t) =

tB(1/t) for t > 0 and W (0) = 0.

4 Proofs

4.1 Proof of Theorem 3.3

By Proposition 3.2, it follows that limx→0+ v(x) = ∞ implies S̃0(ξ) = +∞, a.s.

In case (a), when σ ∈ L2([0, ∞); R), T (s) → ∞ as s → T ∗−. Hence, as X̃(s) = X(T (s)) for

s ∈ [0, T ∗],

lim
t→∞

X(t) = lim
s→T ∗−

X(T (s)) = lim
s→T ∗−

X̃(s) = X̃(T ∗) > 0, a.s.,

because T ∗ < +∞ = S̃0(ξ), a.s.

In case (b), when σ 6∈ L2([0, ∞); R), T (s) → ∞ as s → ∞. Hence, as X̃(s) = X(T (s)) for

s ∈ [0, ∞), we have S0(ξ) = T (S̃0(ξ)) = +∞, a.s., and

lim
t→∞

X(t) = lim
s→∞

X(T (s)) = lim
s→∞

X̃(s) = lim
s→S̃0(ξ)

X̃(s) = 0, a.s.,

because S̃0(ξ) = ∞, a.s.

4.2 Proof of Theorem 3.4

The proof of part (ii) is straightforward, because for t ≥ τ we have

X(t) = X(τ ) +

∫ t

τ

σ(s)g(X(s)) dB(s) = X(τ ),

as
∫

∞

τ
σ2(s) ds = 0 and the continuity of σ imply that σ(t) = 0 for all t ∈ [τ, ∞).

To prove part (i) we proceed as follows. Because σ ∈ L2([0, ∞); R), and

limx→0+ v(x) = ∞, by Proposition 3.1 and Proposition 3.2, the process X̃ = {X(t); 0 ≤ t ≤

T ∗; G(t)} defined by X̃(t) = X(T (t)) is strictly positive a.s. and obeys

X̃(t) = ξ +

∫ t

0

g(X̃(s)) dB̃(s), 0 ≤ t ≤ T ∗,



156 John A.D. Appleby et al. CUBO
10, 3 (2008)

where T ∗ and T are defined by (2.3) and (2.4).

Since g in C1((0, ∞); (0, ∞)) we may define the function h ∈ C2((0, ∞); R) by

h(x) =

∫ x

1

1

g(u)
du, x ∈ R.

Then the process Ỹ = {Ỹ (t); 0 ≤ t ≤ T ∗; G(t)} defined by Ỹ (t) = h(X̃(t)) is well–defined. Since

h ∈ C2((0, ∞); R) and X̃(t) > 0 for all t ∈ [0, T ∗] a.s. Ỹ is an Itô–process, which by Itô’s rule, has

decomposition for 0 ≤ t ≤ T ∗ given by

Ỹ (t) = h(X̃(t)) = h(ξ) + B̃(t) −
1

2

∫ t

0

g′(X̃(s)) ds.

Since X̃ is almost surely positive and continuous, g ∈ C1((0, ∞); (0, ∞)), it follows that

lim
t→T ∗−

1

T ∗ − t

∫ T
∗

t

g′(X̃(s)) ds = g′(X̃(T ∗)), a.s.

Therefore

lim
t→T ∗−

1
√

2(T ∗ − t) log log(1/(T ∗ − t))

∫ T
∗

t

g′(X̃(s)) ds = 0, a.s.

Then for t ∈ [0, T ∗]

h(X̃(T ∗)) − h(X̃(t)) = B̃(T ∗) − B̃(t) −
1

2

∫ T
∗

t

g′(X̃(s)) ds,

and

lim sup
t→T ∗−

h(X̃(T ∗)) − h(X̃(t))
√

2(T ∗ − t) log log(1/(T ∗ − t))

= lim sup
t→T ∗−

B̃(T ∗) − B̃(t)
√

2(T ∗ − t) log log(1/(T ∗ − t))

= lim sup
s→0+

B̃(T ∗) − B̃(T ∗ − s)
√

2s log log(1/s)
.

Now, because B̄ = {B̄(t) : 0 ≤ t ≤ T ∗; FB̄(t)} defined by B̄(t) = B̃(T ∗) − B̃(T ∗ − t) is a standard

Brownian motion, by the Law of the Iterated Logarithm for Brownian motion we have

lim sup
s→0∗+

B̃(T ∗) − B̃(T ∗ − s)
√

2s log log(1/s)
= lim sup

s→0∗+

B̄(s)
√

2s log log(1/s)
= 1, a.s.

Hence

lim sup
t→T ∗−

h(X̃(T ∗)) − h(X̃(t))
√

2(T ∗ − t) log log(1/(T ∗ − t))
= 1, a.s.

Since h is in C1((0, ∞); R) and X̃ has continuous sample paths, we have

lim
t→T ∗−

h(X̃(T ∗)) − h(X̃(t))

X̃(T ∗) − X̃(t)
= h′(X̃(T ∗)) =

1

g(X̃(T ∗))
, a.s.



CUBO
10, 3 (2008)

Asymptotic constancy and stability in SDEs 157

Hence

lim sup
t→T ∗−

X̃(T ∗) − X̃(t)
√

2(T ∗ − t) log log(1/(T ∗ − t))

= lim sup
t→T ∗−

X̃(T ∗) − X̃(t)

h(X̃(T ∗)) − h(X̃(t))

h(X̃(T ∗)) − h(X̃(t))
√

2(T ∗ − t) log log(1/(T ∗ − t))

= g(X̃(T ∗)), a.s.

Therefore, as T (s) → ∞ as s ↑ T ∗, and 〈M〉(T (s)) = s for s ∈ [0, T ∗], we have

lim sup
t→∞

L(ξ) − X(t)
√

2
∫

∞

t
σ2(s) ds log log(1/

∫

∞

t
σ2(s) ds)

= lim sup
t→∞

X(∞) − X(t)
√

2(〈M〉(∞) − 〈M〉(t)) log log(1/〈M〉(∞) − 〈M〉(t))

= lim sup
s↑T ∗

X(T (T ∗)) − X(T (s))
√

2(〈M〉(T (T ∗)) − 〈M〉(T (s))) log log(1/〈M〉(T (T ∗)) − 〈M〉(T (s)))

= lim sup
s↑T ∗

X̃(T ∗) − X̃(s)
√

2(T ∗ − s) log log(1/(T ∗ − s))

= g(X̃(T ∗))

= g(L(ξ)), a.s.

An analogous argument gives

lim inf
t→T ∗−

X̃(T ∗) − X̃(t)
√

2(T ∗ − t) log log(1/(T ∗ − t))
= −g(X̃(T ∗)), a.s.,

from which we can infer that

lim inf
t→∞

L(ξ) − X(t)
√

2
∫

∞

t
σ2(s) ds log log(1/

∫

∞

t
σ2(s) ds)

= −g(L(ξ)), a.s.,

as required.

4.3 Proof of Theorem 3.6

By Proposition 3.2, it follows that limx→0+ v(x) < +∞ implies S̃0(ξ) < +∞, a.s.

In case (b) when σ 6∈ L2([0, ∞); R), T (s) → ∞ as s → ∞. Hence, as X̃(s) = X(T (s)) for

s ∈ [0, ∞), we have S0(ξ) = T (S̃0(ξ)) < +∞, a.s., and

lim
s→S0(ξ)

−

X(s) = lim
t→S̃0(ξ)

−

X(T (t)) = lim
t→S̃0(ξ)

−

X̃(t) = 0, a.s.,

because S̃0(ξ) < +∞, a.s.



158 John A.D. Appleby et al. CUBO
10, 3 (2008)

In case (a) when σ ∈ L2([0, ∞); R), T (s) → ∞ as s → T ∗−. Define the event

A = {ω : S̃0(ξ) ≥ T
∗}.

Then because X̃(s) = X(T (s)) for s ∈ [0, T ∗], we have S0(ξ) = T (S̃0(ξ)), and so

A = {ω : S̃0(ξ) ≥ T
∗} = {ω : S0(ξ) = +∞}.

Clearly, as S̃0(ξ) ≥ T
∗ on A, we have

lim
t→T ∗−

X̃(t) = X̃(T ∗) > 0, a.s. on A.

Thus, as X̃(s) = X(T (s)) for s ∈ [0, T ∗],

lim
t→∞

X(t) = lim
s→T ∗−

X(T (s)) = lim
s→T ∗−

X̃(s) = X̃(T ∗) > 0, a.s. on A.

Hence
{

lim
t→∞

X(t, ξ) = L(ξ) > 0, S0(ξ) = +∞
}

= {S̃0(ξ) ≥ T
∗}, a.s.

On the other hand, consider the event Ā, where

Ā = {ω : S̃0(ξ) < T
∗} = {ω : S0(ξ) < +∞},

by virtue of the fact that S0(ξ) = T (S̃0(ξ)) < T (T
∗) = +∞. Then

lim
t→S̃0(ξ)

−

X̃(t) = 0, a.s. on Ā.

Thus, as X̃(s) = X(T (s)) for s ∈ [0, T ∗],

lim
t→S0(ξ)

−

X(t) = lim
s→S̃0(ξ)

−

X(T (s)) = lim
s→S̃0(ξ)

−

X̃(s) = 0, a.s. on Ā.

Hence
{

lim
t→S0(ξ)

−

X(t, ξ) = 0, S0(ξ) < +∞

}

= {S̃0(ξ) < T
∗}, a.s.

Received: July 2008. Revised: August 2008.

References

[1] J.A.D. Appleby Almost sure subexponential decay rates of scalar Itô-Volterra equations,
Electron. J. Qual. Theory Differ. Equ., Proc. 7th Coll. QTDE, Paper no.1, 1–32, 2004.

[2] J.A.D. Appleby and D. Mackey, Almost sure polynomial asymptotic stability of scalar
stochastic differential equations with damped stochastic perturbations, Electron. J. Qual.
Theory Differ. Equ., Proc. 7th Coll. QTDE, Paper no.2, 1–33, 2004.



CUBO
10, 3 (2008)

Asymptotic constancy and stability in SDEs 159

[3] J.A.D. Appleby, X. Mao and A. Rodkina, Pathwise superexponential decay rates of
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[4] J.A.D. Appleby, A. Rodkina and H. Schurz, Pathwise non-exponential decay rates of
solutions of scalar nonlinear stochastic differential equation, Disc. Con. Dynam. Sys. Ser. B.,
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[5] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes,
North Holland, New York, 2nd edition, 1981.

[6] K. Itô and H.P. McKean, Diffusion processes and their sample paths, Springer, Berlin,
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[7] I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus, volume 113 of
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[8] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion. Third edition,
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