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On the Classification of the Asymptotic Behaviour of Solutions of Globally Stable Scalar Differential Equations with Respect to State--Independent Stochastic Perturbations

In this paper we characterise the global stability, global boundedness and recurrence of solutions of a scalar nonlinear stochastic differential equation. The differential equation is a perturbed version of a globally stable autonomous equation with unique equilibrium where the diffusion coefficient is independent of the state. We give conditions which depend on the rate of decay of the noise intensity under which solutions either (a) tend to the equilibrium almost surely, (b) are bounded almost surely but tend to zero with probability zero, (c) or are recurrent on the real line almost surely. We also show that no other types of asymptotic behaviour are possible. Connections between the conditions which characterise the various classes of long--run behaviour and simple sufficient conditions are explored, as well as the relationship between the size of fluctuations and the strength of the mean reversion and diffusion coefficient, in the case when solutions are a.s. bounded.


Introduction
In this paper, we characterise the global asymptotic stability of the unique equilibrium of a scalar deterministic ordinary differential equation when it is subjected to a stochastic perturbation independent of the state.
(1.3) These conditions ensure the existence of a continuous adapted process which obeys (1.1) on [0, ∞), and we will refer to any such process as a solution. We do not rule out the existence of more than one process, but part of our analysis will show that all solutions share the same asymptotic properties. Hypotheses such as local Lipschitz continuity or monotonicity can be imposed in order to guarantee that there is a unique such solution.
In this paper, we improve upon the results in [5] and [9,8] in a number of directions. First, we show that neither the Lipschitz continuity of f nor the condition (1.9) is needed in order to guarantee that any solution X of (1.1) obeys (1.6). Moreover, we give necessary and sufficient conditions for the convergence of solutions which do not require the monotonicity of σ 2 . One of our main results shows that if f obeys (1.2) and σ is also continuous, then any solution X of (1.1) obeys (1.6) if and only if S ′ (ǫ) := ∞ n=0 n+1 n σ 2 (s) ds exp − ǫ 2 2 1 n+1 n already been mentioned above. An important ingredient in establishing the asymptotic stability in the case when σ ∈ L 2 (0, ∞) is the fact that process M defined by M (t) = t 0 σ(s) dB(s) can be considered as a one-dimensional Brownian motion on [0, ∞), and therefore returns to the origin infinitely often. For the one-dimensional SDE, this causes the solution to return to an arbitrarily small neighbourhood of the equilibrium infinitely often, and once other stochastic terms fade, ensures convergence to the equilibrium.
Once we turn to consider higher dimensional equations, the corresponding stochastic process M (t) = t 0 σ(s) dB(s) in higher dimensions may start to inherit some properties of finite-dimensional Brownian motion, for dimension greater than or equal to three. However, this means that M can be transient, obeying lim t→∞ |M (t)| = +∞, a.s.
In this situation, therefore, one can no longer use the scalar method of proof to show that the solution of the finite dimensional SDE returns to an arbitrarily small neighbourhood of the equilibrium infinitely often. At this moment, it is unclear to the authors whether this is merely a technical problem, or if it presages different asymptotic behaviour, with solutions losing stability more readily than in the scalar case.
Other interesting questions which can be attacked by means of the methods in this paper include an analysis of local stability, where there are a finite number of equilibria of the underlying deterministic dynamical system (1.4). Some work in this direction has been conducted in a discrete-time setting in [1]. Numerical methods under monotonicity methods have been studied in [3]. We expect that the sharper information on the asymptotic behaviour of the linear SDE will lead to improved results for the corresponding nonlinear equations. Section 2 deals with preliminary results, including the proof that solutions of (1.1) exist. Results for an auxiliary affine SDE, proven in [3], are recapitulated in Section 3, along with some new results for the stability of affine equations. Section 4 considers general results, including the classification of the almost sure behaviour of solutions under the additional assumption (1.9) on f . Section 5 considers the characterisation of asymptotic stability using only the assumption (1.2). Proofs of many results are deferred to the end of the paper, and these proofs are presented in Sections 6,7,8,9 and 10. 2. Preliminaries 2.1. Notation. In advance of stating and discussing our main results, we introduce some standard notation. We denote the maximum of the real numbers x and y by x ∨ y and the minimum of x and y by x ∧ y. Let C(I; J) denote the space of continuous functions f : I → J where I and J are intervals contained in R. We denote by 2.2. Remarks on existence and uniqueness of solutions of (1.1). There is an extensive theory regarding the existence and uniqueness of solutions of stochastic differential equations under a variety of regularity conditions on the drift and diffusion coefficients. Perhaps the most commonly quoted conditions which ensure the existence of a strong local solution are the Lipschitz continuity of the drift and diffusion coefficients. However, in this paper, we would like to establish our asymptotic results under weaker hypotheses on f . We do not concern ourselves greatly with relaxing conditions on σ, because σ being continuous proves sufficient to ensure the existence of solutions in many cases.
The existence of a unique solution of can be asserted in the case when |σ(t, x)| ≥ c > 0 for some c > 0 for all (t, x) and f being bounded, so no continuity assumption is required on f . However, assuming such a lower bound on σ would not natural in the context of this paper: for asymptotic stability results, we would typically require that lim inf t→∞ σ 2 (t) = 0. Moreover, f being bounded excludes the important category of strongly meanreverting functions f that have been investigated for this stability problem in [9] and [5].
One of the aims of this paper and of [5] is to relax monotonicity assumptions on σ which are required in [9]. Therefore, although we are often interested in functions σ which tend to zero in some sense, we do not want to exclude the cases when σ(t) = 0 for all t in a given interval (or indeed union of intervals). Our analysis will show that in these cases, the behaviour of σ on the intervals where it is nontrivial can give rise to solutions of (1.1) obeying (1.6) or (1.11). However, on those time intervals I for which σ is zero, the process X obeys the differential equation where X(inf I) is a random variable. On such an interval, it is conceivable that a lack of regularity in f could give rise to multiple solutions of the ordinary equation (and hence the SDE (1.1)), so our most general existence results which make assertions about the existence of solutions (but say nothing about unicity of solutions), and which use the weakest hypotheses on f that we impose in this work, do not appear to be especially conservative.
For these reasons, we prove that there is a continuous and adapted process which obeys (1.1) by using very elementary methods, rather than by appealing to a result from the substantial body of sophisticated theory concerning the existence of solutions of (2.1). Of course, these methods are of very limited utility in establishing existence and uniqueness for more general equations of the form (2.1): our method of proof works because the diffusion coefficient is independent of the state. In fact, our method of proof gives a weaker conclusion for the existence of solutions of (2.1) in the case when f is bounded and σ(t, x) = σ(t) for all (t, x) and |σ(t)| ≥ c > 0. Our result states that there is a solution of (1.1) (or (2.1)) while existing results guarantee the existence of a unique solution under the assumption that f be continuous. Despite these general limitations, however, our proof does ensure existence of solutions for all the problems that are of concern in this paper, while existing results cannot always be applied without the imposition of additional hypotheses.
When f obeys (1.2) and σ obeys (1.3), we now demonstrate that there exists a continuous and adapted process X which satisfies (1.1). The existence of a local solution is ensured by the continuity of f and σ, while the fact that any such solution is well-defined for all time follows from the mean-reverting condition xf (x) > 0 for x = 0 which is part of (1.2). In the paper, the spirit of our approach is to show that any solution of (1.1) has the stated asymptotic properties, even though multiple solutions exist, without paying particular concern as to whether solutions are unique. The proof is postponed to Section 6. In order to ensure that solutions of (1.1) are unique, it is often necessary to impose additional regularity properties on f . One common and mild assumption which ensures uniqueness is that f is locally Lipschitz continuous.
(2.2) See e.g., [12]. Another assumption which guarantees the uniqueness of the solution is that the drift coefficient −f obeys a one-sided Lipschitz condition. More precisely, imposing such an assumption on f implies 3) It is to be noted that if f is non-decreasing, it obeys (2.3), because the righthand side is non-negative, and we can choose K = 0. Since non-decreasing functions do not have to be Lipschitz continuous, we see that in general (2.3) does not imply (2.2), so these additional assumptions can be used to cover different situations. Again the proof is deferred to Section 6. In the proof of Proposition 1, and elsewhere throughout the paper, it is helpful to introduce the following processes and notation. Consider the affine stochastic differential equation Since σ is continuous, there is a unique continuous adapted process which obeys (2.4), and we identify the a.s. event Ω Y on which this solution is defined: It is also helpful throughout the paper to identify the event Ω X on which the continuous adapted process X obeys (1.1), so we therefore define Ω X = ω ∈ Ω : the continuous adapted process X is such that the realisation X(·, ω) obeys (1.1) . (2.6) By virtue of Proposition 1, Ω X is an almost sure event.
2.3. Preliminary asymptotic results. We first consider hypotheses on the data i.e., on σ under which any solution X of (1.1) obeys (1.6). We note that when σ ∈ L 2 (0, ∞), we have X obeys (1.6). However, we cannot apply directly the semimartingale convergence theorem of Lipster-Shiryaev (see e.g., [11,Theorem 7,p.139]) to the non-negative semimartingale X 2 , because it is not guaranteed that E[X 2 (t)] < +∞ for all t ≥ 0. The proof of the following theorem, which is deferred to the next section, uses the ideas of [11, Theorem 7, p.139] heavily, however.
The proof is relegated to Section 7.1. Our next result shows that if, on the contrary, σ ∈ L 2 (0, ∞), we can only guarantee that X visits a neighbourhood of the equilibrium infinitely often.
Again the proof is postponed to Section 7.1.
By virtue of this observation, it can be seen that the following result characterises the asymptotic behaviour of Y . Roughly speaking, if S ′ (ǫ) is always finite, Y tends to zero with probability one; if S ′ (ǫ) can be finite or infinite, it fluctuates asymptotically within interval of the form Theorem 3. Suppose that σ obeys (1.3) and that Y is the unique continuous adapted process which obeys (2.4). Let θ be defined by (3.3) and S ′ (·) by (3.5).
(A) If θ is such that then the event Ω 1 defined by is almost sure and there exist deterministic 0 < Y ≤ Y < +∞ defined by Remark 1. The existence of deterministic bounds on |Y | in (3.12) and (3.13) in part (B) was established as part of Theorem 1 in [2]. Moreover, it was established as part of the proof that explicit bounds on Y and Y can be given in terms of the critical value of ǫ = ǫ ′ in (3.10). The estimates given by the analysis in [2] are It remains an open question as to whether in general the explicit bounds y and y on Y and Y can be improved. In part of Theorem 4 in which case (B) holds, it can be shown by an independent argument that y = y = ǫ ′ and therefore that Remark 2. If σ obeys (1.3) and σ ∈ L 2 (0, ∞), and Y is the solution of (2.4), then Y obeys lim t→∞ Y (t) = 0 a.s. by Theorem 1. Moreover, if σ ∈ L 2 (0, ∞), then σ obeys (3.8). If σ obeys either (3.10) or (3.14), then σ ∈ L 2 (0, ∞).
The condition that S ′ (ǫ) is finite or infinite can be difficult to check. However, in the case when each of the conditions (3.8), (3.10) and (3.14) is possible according as to whether the limit L is zero, non-zero and finite, or infinite. In this case therefore, the asymptotic behaviour of any solution of (1.1) can be classified completely.
Scrutiny of the proof reveals that we can replace the condition (3.17) with the weaker condition and still obtain the same trichotomy in Proposition 3. The proof of Proposition 3 is postponed to Section 8. The conditions of Theorem 3 can be quite difficult to check in practice. In [2], easily-checked sufficient conditions on σ for which Y is bounded, stable or unstable, are developed. These results are extended slightly here, and will also be used to analyse the nonlinear equation (1.1). For this reason, they are stated afresh here.
In the case when σ ∈ L 2 (0, ∞) we have that Y tends to zero. Therefore, we confine attention to the case where σ ∈ L 2 (0, ∞). In this case, we can define a number T > 0 such that t 0 e 2s σ 2 (s) ds > e e for t > T and so one can define a function Σ : Our main result in this direction can now be stated. Apart from part (C) it appears in [2, Theorem 3.2].
Theorem 4. Suppose that σ obeys (1.3) and that Y is the unique continuous adapted process which obeys (2.4). Let Σ be given by (3.19).
The proof of part (C) uses the methods of [2, Theorem 3.2], so is not given. It is now clear that part (D) is merely a corollary of parts (B) and (C). Parts (A) and (E) may also be thought of as limiting cases of part (D) as L → 0 and L → ∞, respectively. We note that when σ obeys (3.17), then Σ 2 (t) → L as t → ∞, so that in part (D), we have from the proof of part (B) of Proposition 3 that S ′ obeys (3.10) with ǫ ′ = √ 2L and by (3.23 .12) and (3.13). This strengthens the general estimates given on Y and Y in (3.16).
Theorem 3 gives necessary and sufficient conditions in terms of the sequence θ for Y to exhibit certain types of asymptotic behaviour, while Theorem 4 gives sufficient conditions in terms of the function Σ. In the next result, we explore the relationship between Σ and θ, and the conditions in Theorems 3 and 4. One consequence of this analysis is to give simpler sufficient conditions equivalent to those in part (A) of Theorem 4 under which Y tends to zero.

Proposition 4.
Suppose that Σ is given by (3.19), that θ is given by (3.3), and that Θ is given by The following statements are equivalent: Moreover, all imply that S ′ (ǫ) < +∞ for all ǫ > 0. (ii) The following statements are equivalent: The following statements are equivalent: Moreover, all imply that lim inf n→∞ θ 2 (n) log n ∈ [0, ∞). (iv) The following statements are equivalent: Moreover, all imply that lim sup n→∞ θ 2 (n) log n = ∞.
Once again, the proof is relegated to Section 8.

Nonlinear Equation
In this section we explore the asymptotic behaviour of the nonlinear differential equation (1.1). In the first part of this section, we establish a connection between the solution of (2.4) and solutions of (1.1). This enables us to state the main results of the paper, which appear, together with interpretation and examples, in the second part of this section.

Connection between the linear and nonlinear equation.
In our first result, we show that knowledge of the pathwise asymptotic behaviour of Y (t) as t → ∞ enables us to infer a great deal about the asymptotic behaviour of X(t) as t → ∞. Indeed, we show in broad terms that X inherits the asymptotic behaviour exhibited by Y , when f obeys (1.2).
Proposition 5. Suppose that f satisfies (1.2) and that σ obeys (1.3). Let X be any solution of (1.1), and Y the solution of (2.4), and suppose that the a.s. events Ω X and Ω Y are defined as in (2.6) and (2.5) respectively.
(A) Suppose that there is an a.s. event defined by Then lim t→∞ X(t) = 0 a.s. (B) Suppose that the event Ω 1 defined by (3.11) is almost sure. Then the event is almost sure, and there exists a positive and deterministic X given by Then lim sup t→∞ |X(t)| = +∞ a.s.
In the proof of part (B), we can even determine an explicit lower bound for X. If the event Ω 1 is defined by (3.11), we may define as in (3.12) and (3.13) the deterministic numbers 0 < Y ≤ Y < +∞. For any f obeying (1.2) it can be shown that there is function y → x(y) = x(f, y) which, for y ≥ 0, obeys This leads to the estimate where Y is given by (3.12). Moreover, as it transpires that x(f, ·) is an increasing function, by (3.16), we can estimate X explicitly according to where y is given explicitly by (3.16). An interesting implication of part (C) is that an arbitrarily strong mean-reverting force (as measured by f ) cannot keep solutions of (1.1) within bounded limits if the noise perturbation is so intense that a linear mean-reverting force cannot keep solutions bounded. Therefore, the system will run "out of control" (in the sense of becoming unbounded) however strongly the function f pushes it back towards the equilibrium state.

4.2.
Main results. Due to Theorem 3, we can readily use Proposition 5 to characterise the asymptotic behaviour of solutions of (1.1).

(B)
If θ is such that (3.10) holds, then there exists an almost sure event Ω 2 = Ω 1 ∩ Ω X , and a deterministic X > 0 defined by (4.2) such that where x(f, ·) is the unique solution of (4.3), and Y is defined by (3.12). Proof. If θ is such that (3.8) holds, then from Theorem 3, we have lim t→∞ Y (t) = 0, a.s. Taking this together with Proposition 5, part (A) holds. If θ is such that (3.10) holds, or if θ is such that (3.14) holds, then taken together with Theorem 3 and Proposition 5 we have that the first part (B) and of (C) is true. For the second part of (B) and (C), we recall that if (3.10) or (3.14) hold, Remark 2 implies that σ / ∈ L 2 (0, ∞). In this case, we already know that lim inf t→∞ |X(t)| = 0, a.s. by Theorem 2.
The formula (4.3), which is established in the proof of part (B) of Proposition 5, relates the lower bound on the large fluctuations x to the size of the diffusion coefficient σ and the nonlinearity in f . Thus, we may view x = x(f, Y ) = x(f, σ), because Y depends on σ but not on f . It is clear that the larger the diffusion coefficient, the larger the value of Y . We now show for fixed f that x is increasing and that x(f, y) → ∞ as y → ∞. Moreover, we show for fixed y that (4.5) These ordering results seem to make intuitive sense, as we would expect weaker mean reversion and a larger diffusion coefficient to lead to larger fluctuations in X.
Proposition 6. Suppose that f obeys (1.2). Let x be the unique solution of (4.3).
Since h f1 is an increasing function, we have x(f 1 , y) ≥ x(f 2 , y) as required.
Just as the conditions of Theorem 3 can be quite difficult to check in practice for Y , the same is true for the conditions of Theorem 5 on θ for X. As in Theorem 4, and because of Proposition 5, we can supply easily checked sufficient conditions on σ for which X is bounded, stable or unstable.
In the case when σ ∈ L 2 (0, ∞) we have that X tends to zero. Therefore, we confine attention to the case where σ ∈ L 2 (0, ∞). In this case, we can define a number T > 0 such that t 0 e 2s σ 2 (s) ds > e e for t > T and so one can define, as before, the function Σ : [T, ∞) → [0, ∞) by (3.19). that f obeys (1.2) and that σ obeys (1.3). Let X be any solution of (1.1). Let Σ be given by (3.19). We finish this Section by giving a sufficient condition on f for which solutions of (1.1) do not tend to zero but are nonetheless bounded. In the case when σ is such that either parts (A) or (C) apply, we have unambiguous information about the asymptotic behaviour of solutions: either almost all sample paths tend to zero, or almost all sample paths exhibit unbounded fluctuations. However, scrutiny of the statement of Proposition 5 shows that part (B) does not rule out the possibility that lim sup t→∞ |X(t)| = +∞ with positive probability (or even almost surely). We make a further hypothesis on f , under which this is impossible, and X is forced to be bounded. The hypothesis is (4.8) An estimate on the lower bound X in case (B) is given in (4.3), which is found as part of the proof of Proposition 5. X is given in terms of f and σ. Similarly, an estimate can be determined for the upper bound. Towards this end, we introduce functions which are a type of generalised inverse of f by defining the functions f − and f + by These functions are well-defined if f obeys (1.2) and (4.8). We notice also that if f is increasing, then f ± are exactly the inverse of f . We may therefore define for any f the function y → x(f, y) by x(f, y) = 2y + max(f + (y), −f − (−y)), y ≥ 0. The main conclusion of the following theorem is that an explicit upper bound can be found for lim sup t→∞ |X(t)|. In fact, it can be shown that if Y obeys (3.13), then lim sup t→∞ |X(t, ω)| ≤ x(f, Y ), for each ω ∈ Ω 2 , (4.12) where Ω 2 is given by (4.1).
We are finally in a position to state the main result of this section. ∩ Ω X where Ω 1 defined in (3.11), and deterministic 0 < X ≤ X < +∞ such that

13)
Moreover, , where x(f, ·) is the unique solution of (4.3), and Y is defined by (3.12), and X , where x(f, ·) is defined by (4.11) and Y is defined by (3.13). Furthermore, where y and y are given by (3.16). It is interesting to determine the effect of weaker mean reversion and an increasing diffusion coefficient on the upper bound of the large deviations of X, given by x(f, Y ), just as we did for the lower bound on the size of the largest fluctuations in Proposition 6, given by x(f, Y ). As before, it can be shown that weaker mean reversion and increasing diffusion coefficients increase the bound x. Also, if the effect of the diffusion coefficient alone is negligible (so that Y → 0), or unboundedly large (so that Y → ∞), we see that cases (A) and (C) in Theorem 7 can be viewed as limiting cases of the asymptotic behaviour described in case (B). These properties of the bounds are established in the following result. The proof is relegated to the final section. We finish the section with an example which shows how estimates of X and X can be obtained in practice.

Example 8.
We see how these estimates on the fluctuations behave for a specific class of examples. Suppose that f (x) = x n where n is an odd integer and that σ 2 (t) log t → L ∈ (0, ∞) as t → ∞. Then by Theorem 4 it follows that From this, we readily see that Notice that lim L→0 + x(L) = 0 and lim L→∞ x(L) = ∞. It is clear that these asymptotic bounds are widely spaced, because It would be an interesting question to determine whether either of these bounds is satisfactory, but we do not pursue this here. We suspect that the upper bound x(L) as L → ∞ is very conservative, however, as it does not take into account the strong mean reversion of f .

Asymptotic Stability
It should be remarked that one consequence of Theorem 5 is that sample paths of X tend to zero with non-zero probability if and only if θ obeys (3.8), in which case almost all sample paths tend to zero. Therefore, we have the following immediate corollary of Theorem 5.
It should also be noted that no monotonicity conditions are required on σ in order for this result to hold, and that a.s. global stability is independent of the form of f . The conditions and form of Theorem 5 and 9 are inspired by those of [9, Theorem 1] and by [7,Theorem 6,Corollary 7].
An interesting fact of Theorem 9 is that it is unnecessary for σ(t) → 0 as t → ∞ in order for X to obey (1.6). In fact, we can even have lim sup t→∞ |σ(t)| 2 = ∞ and still have X(t) → 0 as t → ∞ a.s. Some examples are supplied in [5].
(i) If σ obeys lim t→∞ σ 2 (t) log t = 0, then X obeys (1.6). Theorem 11. Suppose that f satisfies (1.2). Suppose that σ obeys (1.3) and t → σ 2 (t) is decreasing. Let X be any solution of (1.1). Then the following are equivalent: The remark preceding this result points to the importance of the condition θ(n) 2 log n → 0 as n → ∞, as indeed does Proposition 4 part (a). We now supply an example in which θ(n) 2 log n → 0 as n → ∞, but t → σ 2 (t) has "spikes" which prevents it from satisfying the condition lim t→∞ σ 2 (t) log t = 0.
By the continuity of ϕ in both arguments, by the Peano existence theorem, there exists a continuous local solution t → z(t, ω) for each ω ∈ Ω Y and 0 ≤ t < τ e (ω). Presently, it will be shown that τ e (ω) = +∞ a.s. on Ω Y .
Moreover, as Y is adapted to (F B (t)) t≥0 , z is also adapted to (F B (t)) t≥0 . Now consider the process X defined on Ω Y by X(t) = z(t) + Y (t) for t ∈ [0, τ e ). By construction it is continuous and adapted. Furthermore, we have for t ∈ [0, τ e ) Hence X(·, ω) obeys (1.1) for each ω ∈ Ω Y on the interval [0, ∞). The proof that τ e is infinite a.s. was given in the Appendix of [5].
6.2. Proof of Proposition 2. The proof is inspired by an observation in e.g., [10]. Note first by Proposition 1 that the continuity of f together with (1.2) guarantees the existence of a continuous adapted process which obeys (1.1). Suppose therefore that X 1 and X 2 are any two solutions of (1.1). Then and by Itô's rule we have that Since f obeys (2.3), we have (X 1 (t) − X 2 (t)) 2 ≤ 2K t 0 (X 1 (s) − X 2 (s)) 2 ds, t ≥ 0.
If K ≤ 0, we can conclude automatically that X 1 (t) = X 2 (t) for all t ≥ 0 a.s., and that therefore the solution is unique. If K > 0, by applying Gronwall's inequality to the non-negative continuous function t → (X 1 (t) − X 2 (t)) 2 , we conclude that X 1 (t) = X 2 (t) for all t ≥ 0 a.s., and once again we have uniqueness.
By Proposition 6, h f is an increasing function, so we have X * (ω) ≥ h −1 f (Y * (ω)). Now by the definition of X * , X and the fact that h −1 f is increasing, we have Since Ω 2 ⊆ Ω 1 , inf ω∈Ω2 Y * (ω) ≥ inf ω∈Ω1 Y * (ω) = Y , by the definition of Y . Thus as h −1 f is increasing, using (4.7) at the last step. Notice lastly that part (i) of Proposition 6 implies that x(f, Y ) > 0 because Y > 0, by hypothesis. We prefer the estimate in (4.12) in part because the estimate on the right hand side of (10.1) is difficult to analyse in general, due to the complexity of x + and x − . Moreover, there is no loss of sharpness in the estimate in (4.12) relative to (10.1) in the case when f is increasing. To see this, first note that when f is increasing on R, it can readily be seen that x + (y) = y + f −1 (y) and x − (y) = y − f −1 (−y). Therefore, if we grant that (10.1) holds, it follows that Therefore, if we define x * (f, y) = 2y + max(f −1 (y), −f −1 (−y)), (10.4) it can be seen that lim sup t→∞ |X(t, ω)| ≤ x * (f, Y ), for each ω ∈ Ω 2 .