Solutions of Stochastic Differential Equations obeying the Law of the Iterated Logarithm, with Applications to Financial Markets ∗

By using a change of scale and space, we study a class of stochastic differential equations (SDEs) whose solutions are drift–perturbed and exhibit asymptotic behaviour similar to standard Brownian motion. In particular sufﬁcient conditions ensuring that these processes obey the Law of the Iterated Logarithm (LIL) are given. Ergodic–type theorems on the average growth of these non-stationary processes, which also depend on the asymptotic behaviour of the drift coefﬁcient, are investigated. We apply these results to inefﬁcient ﬁnancial market models. The techniques extend to certain classes of ﬁnite–dimensional equation.


Introduction
The following Law of the Iterated Logarithm is one of the most important results on the asymptotic behaviour of finite-dimensional standard Brownian motion: Classical work on iterated logarithm-type results, as well as associated lower bounds on the growth of transient processes, date back to Dvoretzky and Erdős [5]. There is an interesting literature on iterated logarithm results and the growth of lower envelopes for self-similar Markov processes (cf. e.g., Rivero [16], Chaumont and Pardo [4]) which exploit a Lamperti representation [13], processes conditioned to remain positive (cf. Hambly et al. [10]), and diffusion processes with special structure (cf. e.g. Bass and Kumagi [3]).
In contrast to these papers, the analysis here is inspired by work of Motoo [15] on iterated logarithm results for Brownian motions in finite dimensions, in which the asymptotic behaviour is determined by means of time change arguments which reduce the process under study to a stationary one. Our paper concentrates mainly on iterated logarithm upper bounds of solutions of stochastic differential equations, as well as obtaining lower envelopes for the growth rate. Our goal is to establish these results under the minimum continuity and asymptotic conditions on the drift and diffusion coefficients. An advantage of this approach is that it enables us to analyse a class of equations of the form d X (t) = f (X (t)) d t + g(X (t)) d B (t) for which x f (x)/g 2 (x) tends to a finite limit as x → ∞ in the case when f and g are regularly varying at infinity. Ergodic-type theorems are also presented. We also show how results can be extended to certain classes of non-autonomous and finite-dimensional equations. We employ extensively comparison arguments of various kinds throughout.
In [1], Appleby et al. studied general conditions which ensure a scalar stochastic differential equation with Markov switching obeys the Law of the Iterated Logarithm. In our work here, we are concerned with similar problems for SDEs without switching. In particular, for a parameterised family of SDEs, we observe that solutions can change from being recurrent to transient when a critical value of the bifurcation parameter is exceeded. Despite this, the solutions still obey the Law of the Iterated Logarithm in the sense of (1.1). Between this paper and [1], we examine the extent to which the drift can be perturbed so that in the long-run the size of the large deviations remains the same as those of standard Brownian motion.
In [14], Mao shows that if X is the solution of the d-dimensional equation The main steps of the Mao's proof are as follows: first, make a suitable Itô transformation; then estimate the size of the Itô integral term by a Riemann integral using the exponential martingale inequality (EMI); and finally apply Gronwall's inequality (GI) to determine the asymptotic rate of growth.
In contrast, the results in this paper are established through a combination of comparison principles and Motoo's theorem. Motoo's theorem (cf. [15]) determines the exact asymptotic growth rate of the partial maxima of a stationary or asymptotically stationary process governed by an autonomous SDE. Motoo [15] also gives a proof of the Law of the Iterated Logarithm for a finite-dimensional Brownian motion. This proof is crucially reliant on applying a change in both space and scale. He considers an autonomous non-stationary δ-dimensional Bessel process R δ , which is governed by the scalar equation with R δ (0) = r 0 ≥ 0. The Bessel process R δ is transformed into an autonomous process with finite speed measure (i.e., a process that possesses a limiting distribution) to which the Motoo's theorem can be applied. More precisely, if we let S δ (t) = e −t R 2 δ (e t − 1), (1.4) then dS δ (t) = (δ − S δ (t)) d t + 2 S δ (t) dB(t). (1.5) It is reasonable to ask whether a combination of space and scale transformations of this classic type could reduce a general non-stationary autonomous SDE to one with finite speed measure to which Motoo's theorem could then be applied. If we consider general transformations of the form Y (t) = λ(t)P(X (γ(t))) where γ : + → + is increasing, P ∈ C 2 ( ; ) and λ ∈ C 1 ( + ; + ) (and is related to γ), the resulting SDE for Y will be non-autonomous, and in particular, will have non-autonomous diffusion coefficient. Adapting the proof of Motoo's theorem to cope with SDEs with non-autonomous diffusion coefficients introduces formidable difficulties, because the independence of excursions, on which the proof relies, can no longer be assured.
However, in this paper, with the well-known stochastic comparison principle (which assumes an order on the drift coefficients), we are able to investigate a much wider class of SDEs which are related to (1.3) through (1.4) and which give rise to equations of the type (1.5). In addition, with ordinary Itô transformations, we could map an even wider class of nonlinear equations onto a family of SDEs whose asymptotic behaviour is understood. This is shown in [2]. A detailed discussion on the relative advantages and disadvantages of this comparison-Motoo technique with the existing EMI-Gronwall approach can also be found in [2].
Also in [1], Appleby et al. applied processes obeying the Law of the Iterated Logarithm to financial market models which are inefficient in the sense of Fama. In this paper, we further investigate some ergodic-like properties of these processes. Under some reasonable assumptions of regarding the market, we establish two main results. First, we show that the largest fluctuations from the trend growth rate of the price are of the same size as in a related efficient market model. Second, we show that these fluctuations are "on average" greater than those in the efficient model, in a sense later made precise.
This paper considers a number of closely related equations, and proves a number of diverse asymptotic results. In order to understand the relationships between these results and to ease the readers' path through the paper, we give a synopsis and discussion about the main results, as well as their applications in Section 3. Full statements of the theorems and detailed proofs are found in succeeding sections.

Preliminaries
Throughout the paper, the set of non-negative real numbers is denoted by + . The space of d × m matrices with real entries is denoted by d×m ; in the case when m = 1, we write d×1 = d . Let |h(x)|d x < ∞. If x and y are two real numbers, then the maximum and minimum of x and y are denoted by x ∨ y and x ∧ y respectively. Let |x| be the Euclidean norm of a column or a row vector x ∈ d ; ||A|| and ||A|| op denote the Frobenius norm and operator norm respectively for any A ∈ d×m . Note that ||A|| op ≤ ||A|| and ||A|| ≤ m||A|| op .
Moreover, we use the Landau symbol for functions f : → and g : → : We use (Ω, , { (t)} t≥0 , ) to denote a complete filtered probability space. The abbreviation a.s. stands for almost surely. We always assume that both the drift and the diffusion coefficients of SDEs being studied satisfy the local Lipschitz condition even if this is not explicitly stated. If an autonomous scalar SDE has drift coefficient f (·) and non-degenerate diffusion coefficient g(·), then a scale function and speed measure of the solution of this SDE are defined by , c, x ∈ I := (l, r) (2.1) respectively, where I is the state space of the process. These functions help us to determine the recurrence and stationarity of a process on I (cf. e.g. [12]). Moreover, Feller's test for explosions (cf. e.g. [12]) allows us to examine whether a process will never escape from its state space in finite time. This in turn relies on whether As mentioned in the introduction, Motoo's Theorem is an important tool in determining the pathwise largest deviations for stationary or asymptotically stationary processes. We state Motoo's theorem in this section for future use.
Theorem 2.1. (Motoo) Let X be the unique continuous real-valued process satisfying the following equation with X (0) = x 0 . Let s and m be the scale function and speed measure of X as defined in (2.1), and let h : (0, ∞) → (0, ∞) be an increasing function with h(t) → ∞ as t → ∞. If X is recurrent on (l, ∞) (or [l, ∞) in the case when l is an instantaneous reflecting point) and m(l, ∞) < ∞, then The following lemma may be proven by applying Motoo's theorem directly to it. We will use it frequently.

Lemma 2.2. Let U be the unique continuous adapted solution of the following equation
, a.s.
Throughout the paper, we repeatedly use Doob's theorem for the representation of a continuous martingale in terms of standard one-dimensional Brownian motion. We state the theorem in this section for notational convenience and future reference.

Theorem 2.3. (Doob) Suppose M is a continuous local martingale defined on a probability space
(Ω, , ), and the square variation 〈M 〉 is an absolutely continuous function of t for -almost every ω. Then there is an extended space (Ω,˜ ,˜ ) of (Ω, , ) on which is defined a one-dimensional Brownian motion W = {W (t),˜ (t); 0 ≤ t < ∞} and a˜ (t)-adapted process X with˜ -a.s.
such that we have the representations˜ -a.s.
In the proof of the above theorem, the new Brownian motion W is constructed by a continuous local martingale with respect to the original probability space (Ω, , ) and another Brownian motion, say B, which is defined on the extended part of (Ω, , ) in (Ω,˜ ,˜ ). Moreover, B is independent of M . Therefore in this paper, any conclusion made with respect to the extended measure˜ about the underlying semimartingale (with martingale component M ) defined on (Ω, , ) coincides with that for the measure . Therefore we do not make explicit reference to the probability spaces when stating results.

Synopsis and Discussion of Main Results
In this section, we give a brief discussion of the results proven in this paper. First, we state the Law of the Iterated Logarithm and other results on asymptotic growth bounds for transient solutions of autonomous SDEs. Second, we discuss general non-autonomous equations for which the LIL holds, under some uniform estimates on the drift. Third, we give comprehensive results for a parameterised family of autonomous SDEs with constant diffusion coefficient which do not require uniform estimates on the drift. Finally, we discuss some extensions of these results to multi-dimensional SDEs, as well as applications of the results to weakly inefficient financial markets.

Transient processes
Our first main result, Theorem 4.3, concerns transient solutions of the scalar autonomous stochastic differential equation These results are established by comparing X with a general Bessel process which has similar behaviour to X . The asymptotic behaviour of the Bessel process is given in Lemma 4.1. The modulus of a finite-dimensional Brownian motion (i.e., a Bessel process) with dimension greater than two is known to be transient, and when the dimension is less than or equal to two, the process is recurrent on the positive real line. However, for general Bessel processes, the critical dimension altering its behaviour does not have to be an integer. This fact is eventually captured in Theorem 4.3 by condition (3.2) (or (3.4)). More precisely, if exactly one of the parameters L ∞ and L −∞ is greater than the critical value σ 2 /2, then the process tends to infinity or minus infinity almost surely while still obeying the Law of the Iterated Logarithm. If on the other hand L ∞ and L −∞ are both greater than σ 2 /2, and we denote the event {ω : lim t→∞ X (t, ω) = −∞} byÃ, we have that Furthermore both probabilities are positive and can be computed explicitly in terms of the scale function and the deterministic initial value of the process (cf. [12,Proposition 5.5.22]). Motoo's theorem also helps us to find an exact pathwise lower bound on the growth rate of the process.
This result could also be very useful in determining the pathwise decay rates of asymptotically stable SDEs. In Theorem 4.5, the constant diffusion coefficient σ is replaced by a state-dependent coefficient g(·) tending to σ as x tends to infinity, and similar results are obtained by means of a random time-change argument. Theorem 4.3 lays the foundation for further results concerning transient solutions of more general equations with unbounded diffusion coefficients. For example, suppose X obeys (3.1), where g is strictly positive and regularly varying at infinity with index β (0 < β < 1), and f and g are related via lim x→∞ x f (x) Then by Itô's rule, if A is as previously defined, it is easy to show that lim sup t→∞ X (t) where G is defined as We leave the details of this result to the interested reader. Another application of these results is given in the next section: we make use of the upper envelope of the growth rate (3.3) to determine upper bounds for a more general type of equation whose solutions obey the Law of the Iterated Logarithm.

General conditions and ergodicity
In Section 5, we state and prove three theorems which give sufficient conditions ensuring Law of the Iterated Logarithm-type asymptotic behaviour, and which enable us to prove further results later in the paper. We will study the one-dimensional non-autonomous equation with X (0) = x 0 . From the results in Section 4, in Theorem 5.1 it can be shown that Furthermore, in Theorem 5.3, we prove that 2t log log t ≥ |σ|, a.s. (3.9) Hence if both (3.6) and (3.8) are satisfied, we can determine the exact growth rate of the partial maxima. Moreover, we establish an ergodic-type theorem on a suitably scaled the average value of the process, as described by the following inequalities: It appears that a condition of the form (3.6) is necessary to ensure that the solution obeys the LIL. Suppose for instance in equation Then X (t) → ∞ on some event Ω ′ with positive probability and lim t→∞ X (t) It is worth noticing that ρ does not appear in the estimate in (3.7). This fact is used in Theorem 7.3 which deals with multi-dimensional systems. However ρ does affect the average value of X in the long-run, as seen in (3.10). As mentioned in the introduction, by the Motoo-comparison approach, the estimate on the constant on the righthand side of (3.7) has been reduced by a factor of e compared to that obtained by the EMI-Gronwall method. In addition, this approach enables us to find the lower estimate (3.9), which is the same size as the upper estimate. This has not been achieved to date by the exponential martingale inequality approach. Condition (3.8) is sufficient but not necessary for securing an LIL-type of lower bound, as will be seen in Theorem 5.6.
We noted already that the parameters ρ and µ in the drift do not affect the growth of the partial maxima as given by (3.7) and (3.9). However, (3.10) and (3.11) show that these parameters are important in determining the "average" size of the process, with larger contributions from the drift leading to larger average values. To cast further light on this we consider the related deterministic differential equation Then it is easy to verify that x 2 (t)/2t → C as t → ∞, which implies This simple example is interesting for a number of reasons. Firstly, it can be seen as motivating the stochastic results (3.10) and (3.11), or as an easily and independently verified corollary of (3.10) and (3.11). Secondly, it gives insight into the "average" long-run value of X : the fact that x(t)/ t → 2C as t → ∞ obeys (3.13) suggests, in the sense of (3.10) and (3.11), that |X (t)| 2 is "on average" C 2 1 t as t → ∞ for some constant C 1 . (3.10) and (3.11) may also be seen as a generalisation of a known result for Brownian motions without drift. Indeed, using (3.10) and (3.11) with ρ = µ = 0, the Brownian motion X (t) := σB(t) must also obey lim t→∞ t 0 X 2 (s) (1+s) 2 ds log t = σ 2 , a.s. (3.14) (3.14) indicates that the large excursions of Brownian motion excursions in the solution of (3.5) contributes the σ 2 term in (3.10) and (3.11). In this case, the statement that |X (t)| 2 is "on average" σ 2 t is justified not only in the sense of (3.14), but because [X 2 (t)] = σ 2 t. These two extreme cases (where there is no diffusion in the first, and no drift in the second) indicate that the contributions of drift and diffusion are of similar magnitude, and this is reflected in (3.10) and (3.11). Finally, it is an easy consequence of Theorem 5.1 and 5.3 that |X (t)| 2 grows "on average" like t as t → ∞, because Theorem 5.6 deals with processes with integrable drift coefficients. For an autonomous equation with drift coefficient f ∈ L 1 ( ; ) and constant diffusion coefficient, there exist positive constants 2t log log t ≤ −C 4 , a.s.
Formulae for the constants in these estimates can be found in Section 5. These processes are recurrent and can be transformed to other processes which are drift-free, have bounded diffusion coefficient, and which preserve the largest fluctuation size. This result is consistent with those in [9,Chapter 4], which roughly say that if the drift coefficient is zero on average along the real line and the diffusion coefficient g(x) has a positive limit σ as |x| → ∞, the process has a limiting normal distribution with mean zero and variance σ 2 t. This is precisely the distribution of the Brownian motion σB(t) at time t.

Recurrent processes
In Section 6, we investigate the scalar autonomous equation where the drift coefficient satisfies These hypotheses are complementary to those discussed in Section 3.1. Using Feller's test (cf. [12]), it can be shown that under condition (3.16), X is no longer transient but in fact recurrent on the real P P P P P P P P P asymptotically stationary   A signifies that X obeys the Law of the Iterated Logarithm exactly; B that |X (t)| is bounded above and below by 2t log 2 t as t → ∞; C that X has a polynomial upper Liapunov exponent equal to 1/2; and D that the asymptotic behaviour is consistent with the Law of the Iterated Logarithm.
line. However results in Section 4 together with Theorem 5.6 (which deals with integrable drift) suggest that solutions should still have asymptotic behaviour similar to the LIL. This idea motivates us to prove similar results in the recurrent case to those already obtained for transient processes. The upper bound (3.3) given by Theorem 5.1 automatically applies, while difficulties arise in finding the lower bound on the limsup without condition (3.8), particularly when L ∞ and L −∞ are of the same sign. The subdivision of the main result into various theorems is necessitated by slight distinctions in the proofs, which in turn depends on the value of both L ∞ and L −∞ . The results are summarised in the case σ = 1 in Figure 1. Theorem 6.1 is a direct result of Motoo's theorem: it shows that −σ 2 /2 is another critical value at which the behaviour of the process changes from being stationary (or asymptotically stationary) to non-stationary. The LIL is no longer valid when L ±∞ < −σ 2 /2. By constructing another asymptotically stationary process as a lower bound for X 2 and X in Theorem 6.3 and 6.4 respectively, we obtain the following exact estimate on the polynomial Liapunov exponent of |X |: is of course a less precise result than the LIL. It shows that the partial maxima of the solution grows at least as fast as K ǫ t (1−ǫ)/2 for ǫ ∈ (0, 1) and some positive K ǫ . However, (3.17) is still consistent with the LIL. Using the same construction (see Lemma 6.6) and comparison techniques, together with Theorem 5.6, we obtain Theorem 6.7, which gives upper and lower estimates on the growth rate of the partial maxima.
Note that we have excluded the cases L ±∞ = 0 from Figure 1 for the purpose of stating consistent results on pairs of intervals for L ∞ and L −∞ . Nonetheless Theorem 6.7 covers the case when at least one of L ±∞ = 0 and the drift coefficient f changes sign an even number of times. In particular, if f remains non-negative or non-positive on the real line, X can be pathwise compared with the Brownian motion {σB(t)} t≥0 directly, so an exact estimate can be obtained (Corollary 6.8). Otherwise, Theorem 6.3 and 6.4 are sufficient to cover the rest of the cases (Remark 6.5).

Multi-dimensional processes
In Section 7, we generalise results from Section 5 to the following d-dimensional equation driven by an m-dimensional Brownian motion In a similar manner, Theorem 7.2 extends Theorem 5.3 in d . The generalisation of these results to unbounded diffusion coefficients can be found in [2]. Moreover under some additional assumptions, the largest fluctuations of the norm is given by the co-ordinate process with the largest fluctuations. This result is an extension of the LIL for a ddimensional Brownian motion (1.1). Mao (cf. [14]) pointed out the fact that the independent individual components of the multi-dimensional Brownian motion are not simultaneously of the order 2t log log t, for otherwise we would have d rather than unity on the right-hand side of (1.1). We establish these facts for drift-perturbed finite-dimensional Brownian motions. To simplify the analysis, we look at the following equation in d : In the more general case that Γ is any invertible matrix, with the same conditions as above, the proof of this result can be easily adapted to show that with respect to the norm |x| Γ := |Γ −1 x|, the solution of (3.19) satisfies

Applications to inefficient financial markets
According to Fama [6;7], when efficiency refers only to historical information which is contained in every private trading agent's information set, the market is said to be weakly efficient (cf. [8,Definition 10.17]. Weak efficiency implies that successive price changes (or returns) are independently distributed. Formally, let the market model be described by a probability triple (Ω, , ). Suppose that trading takes place in continuous time, and that there is one risky security. Let h > 0, t ≥ 0 and let r h (t + h) denote the return of the security from t to t + h, and let S(t) be the price of the risky security at time t. Also let (t) be the collection of historical information available to every market participant at time t. Then the market is weakly efficient if Here the information (t) which is publicly available at time t is nothing other than the generated σ-algebra of the price S (t) = σ{S(u) : 0 ≤ u ≤ t}. An equivalent definition of weak efficiency in this setting is that Geometric Brownian Motion is the classical stochastic process that is used to describe stock price dynamics in a weakly efficient market. More concretely, it obeys the linear SDE is the price of the risky security at time t, µ is the appreciation rate of the price, and σ is the volatility. It is well-known that the logarithm of S grows linearly in the long-run. The increments of log S are stationary and Gaussian, which is a consequence of the driving Brownian motion. That is, for a fixed time lag h, , it follows that the market is weakly efficient. To see this, note that S being a strong solution of (3.21) implies that S (t) ⊆ B (t). On the other hand, since we can rearrange for B in terms of S to get that B (t) ⊆ S (t), and hence B (t) = S (t). Due to this reason, equation (3.21) has been used to model stock price evolution under the classic Efficient Market Hypothesis.
In order to reflect the phenomenon of occasional weak inefficiency resulting from feedback strategies widely applied by investors, in [1] SDEs whose solutions obey the Law of the Iterated Logarithm are applied to inefficient financial market models. More precisely, a semi-martingale X , which is slightly drift-perturbed and obeys the Law of the Iterated Logarithm, is introduced into equation (3.21) as the driving semimartingale instead of Brownian motion. It is shown that if a process S * satisfies then S * preserves some of the main characteristics of the standard Geometric Brownian Motion S. More precisely, the size of the long-run large deviations from the linear trend of the cumulative returns is preserved, along with the exponential growth of S. This is despite the fact that the increments of log S * are now correlated and non-Gaussian.
In this paper, we further investigate the effect of this drift perturbation on the cumulative returns in (3.22) with the process X satisfying (3.5) or (3.15), say. We do not wish to provide a complicated and empirically precise model, but rather a simple and tractable model, and to interpret the mathematical results.
With a modest bias in the trend (e.g. captured by condition (3.6) and (3.8)), the excursions in prices from the linear trend are no longer independent. The largest possible sizes of these excursions coincide with those under no bias (as seen in (3.7) and (3.9)). However, by ergodic-type results (e.g. (3.10) and (3.11)), the stronger the positive bias that the investors have, the larger the average values of price excursions, and consequently the smaller the volatility that arises around the average values. This causes the price to persist on average further from the long-run growth trend that the GBM model would allow. This is made precisely in (3.24) below. This persistence could make investors believe that the cumulative returns are close to their true values and are unbiased, which might cause a more dramatic fall in cumulative returns later on. Moreover, if the market is even more pessimistic after a relatively large drop in returns, the bias tends to have a longer negative impact on the market.
In the model presented below, we presume that the returns evolve according to the strength of the various agents trading in the market. At a given time, each agent determines a threshold which signals whether the market is overbought or oversold. The agents become more risk averse in their trading strategies when these overbought or oversold thresholds are breached. If we make the simplifying assumption that one agent is representative of all, then the threshold level is simply the weighted average of the threshold for all the individuals.
Using these ideas, we are led to study the equation Here f is assumed continuous and odd on so that the positive and negative returns are treated symmetrically. Moreover, in order that the bias be modest, we require lim |x|→∞ x f (x) = L ∈ (0, σ 2 /2]. In (3.23), I is the indicator function, and α ∈ (0, 1] measures the extent of short-selling or "going long" in the market. Here an increased α is associated with an increased tendency to sell short or go long. We presume that investors believe that the de-trended security returns are given by Brownian motion without drift, and the returns obey the Law of the Iterated Logarithm. Moreover, we assume that the investors can estimate the value of σ by tracking the size of the largest deviations. We briefly indicate how the threshold level is arrived. The standard Brownian motion (which the investors believe models the security return) is scaled by σ, and therefore, at time t, has standard deviation σ t. If each agent i chooses a multiple k i of this standard deviation as his/her threshold level, and assuming that all agents are representative, there exists a weighted coefficient k, such that kσ t measures the overall market threshold level. In practice, the value of k might be different for price increases and falls. We treat two situations with one fixed k here for simplicity.
Given these assumptions, we prove the following. First, X is recurrent on and obeys the Law of the Iterated Logarithm by the results in Section 5 and 6. Second, we determine the long-run average value of the de-trended cumulative returns by proving the following ergodic-type theorem: Here, Λ L,σ,α,k measures the market bias from the unbiased value of σ 2 . It can be computed and is given in Section 8. Our assumptions on parameters ensure that Λ L,σ,α,k > σ 2 . This means that the presence of bias increases the "average size" of the departures of the returns from the trend growth rate.
To establish (3.24), we first transform the solution X of (3.23) into a process Y by a change in both time and scale; second, we construct two equations with continuous and time-homogenous drift coefficients and with finite speed measures, such that Y is trapped between the solutions of these equations; third, by adjusting certain auxiliary parameters, we obtain an ergodic-type theorem for Y , which in turn implies (3.24). From a mathematical point of view, we have proved an ergodic-type theorem for a non-autonomous equation using the stochastic comparison principle.
Finally, we confirm that equation (3.22) with X satisfying (3.23) does represent an inefficient market in the weak sense, i.e., we want to show that where r * is the return. It is easy to verify that In the proof of the main result of this section, we establish the strong existence and uniqueness of the solution of equation (3.23) (this requires a little care because of the discontinuity of the drift coefficient). Since X (0) = 0 is deterministic, and X is a strong solution, we have t+h t F (s, X (s))ds must also be B (t)-independent. However, by the Markov property of X , t+h t F (s, X (s))ds is a functional of X (t) and the increments of B.
Hence, t+h t F (s, X (s))ds is X (t)-dependent, and since X (t) = B (t), this gives a contradiction. Therefore (3.25) is proved.

Asymptotic Behaviour of Transient Processes
In this section, we study processes which obey (3.1) and are transient, obeying |X (t)| → ∞ as t → ∞. To do this, we introduce an auxiliary process: let δ > 2 and consider where y 0 is deterministic. The solution of the above equation is a generalised Bessel process of dimension higher than 2. δ > 2 does not have to be an integer. If δ > 2 is an integer, then Y (t) = σ|W (t)| where W is a δ-dimensional Brownian motion. Therefore, in the general case, we expect Y to grow to infinity like e.g. a three-dimensional Bessel process. This can be confirmed by [12, Chapter 3.3 Section C]. In fact, as proven in the following lemma, Y should also obey the Law of the Iterated Logarithm. The proof is the same in spirit as that in Motoo [15], but is briefly given here in the language of stochastic differential equations in order to be consistent with the techniques of this paper. We moreover employ Motoo's techniques to establish a lower bound on the growth rate. and where by Doob's martingale representation theorem, we have replaced the original Brownian motion B by B in an extended probability space. Therefore where W is again another Brownian motion. If Z(t) = Z(e t − 1), then It is easy to show that the scale function of H * satisfies s H * (x) = K 1 Let h 1 (t) = t 2/(δ−2) . Then for some t 1 > 0, On the other hand, for ε ∈ (0, δ − 2), a.s. on an a.s. event Ω ε,θ := Ω ε ∩ Ω θ , where Ω ε and Ω θ are both a.s. events. From this by letting ε ↓ 0 and θ ↓ 0 through rational numbers, it can be deduced that Using the relation between H * and Y , we get the desired result (4.3). Proof. Letting Y * (t) = −Y (t) and applying the same analysis as Lemma 4.1 to Y * , the results can be easily shown. The details are omitted.
We are now in a position to determine the asymptotic behaviour of (3.1) when the diffusion coefficient is constant. Proof. First note that given L ∞ > σ 2 /2, the existence of such a non-null event A in the sample space is guaranteed by Feller's test [12,Proposition 5.5.22]. From now on, we assume that we are working in A, and will frequently suppress ω-dependence and A a.s. qualifications accordingly. We compare X with Y +ε , where Y +ε is given by . We now consider three cases: Suppose to the contrary there exists a minimal t which gives a contradiction.
Combining the above results, for almost all ω in A, there exists a random variable C + such that 2t log log t . (4.14) A lower estimate on X can now be deduced by a similar argument. For the same ε, define Y −ε by is guaranteed to be positive. Then, by arguing as above, we obtain an analogous result to (4.13), namely that for some T 4 (ε, ω) > 0 and random variable C − . This implies We are now in a position to prove (4.9). Using (4.14), and letting Ω * ε be the a. With Ω * * = ∩ ε∈ ∩(0,1) Ω * −ε , it follows that To prove (4.10), notice that Y +ε obeys (4.1) with δ = δ ε = 1 + 2(L ∞ + ε)/σ 2 . Then, by (4.3) we have where Ω + ε is an almost sure event. Therefore by (4.13), a.s. on A ∩ Ω + ε we have Proceeding similarly with Y −ε and using (4.15) we can prove that That is for almost all ω ∈ A, Combining (4.23) with these limits, the desired assertion can be obtained.
A similar result can be developed in the case when X (t) → −∞ as t → ∞ under the assumptions that x f (x) → L −∞ > σ 2 /2 and g(x) → σ as x → −∞. The proof is essentially the same as that of Theorem 4.5, and hence omitted.

General Conditions Ensuring the Law of the Iterated Logarithm and Ergodicity
Theorem 5.1. Let X be the unique continuous adapted process satisfying (3.5). If there exists a positive real number ρ such that Let Z(t) = X 2 (t), t ≥ 0. Define γ(x) = x/|x| for x = 0 and γ(0) = 1. Then is a standard Brownian motion with respect to B , and we have Now consider the process X u defined by with X u (0) > X 2 (0). Arguing as in the forthcoming Theorem 5.3, it can be shown that X u (t) ≥ 0 for all t ≥ 0 a.s. This means that the absolute values in the diffusion coefficient in (5.4) can be omitted. Hence by the Ikeda and Watanabe comparison theorem (cf. [11]), X u (t) ≥ X 2 (t) for all t ≥ 0 a.s. From the proof of Lemma 4.1, we know that [lim t→∞ X u (t) = ∞] = 1. Moreover, X u obeys lim sup t→∞ X u (t) 2t log log t ≤ σ 2 a.s.
Hence the assertion (5.2) is obtained.
The second part of the theorem can be easily deduced from the fact X u (t) ≥ X 2 (t) for all t ≥ 0 a.s., and (5.4) by Exercise XI.1.32 in [17], which is stated below as Lemma 5.2.

Lemma 5.2. Suppose that Q is the unique continuous adapted process satisfying
We now establish lower bounds corresponding to the upper bounds given in the previous theorem.
Theorem 5.3. Let X be the unique continuous adapted process satisfying (3.5). If there exists a real number µ such that Proof. We begin with a change in both time and scale on X to transform it to a process which can be compared with a stationary process. Let Y (t) = e −t X ( 1 2 (e 2t − 1)). By Itô's rule, it can be shown that for t ≥ 0 with Y 2 (0) = x 2 0 where W is the B -adapted standard Brownian motion introduced in the proof of Theorem 5.1. Consider the processes governed by the following two equations, This implies that there exists x * > 0 such that for all x > x * , s Y 1 (x) < e x/σ 2 . Since h is an increasing function, there exists t 0 > 0 such that for all t > t 0 , log t ≥ σ 2 , a.s.
Using the relation between X and Y , we get the desired result (5.6).
For the second part of the conclusion, consider the following equation Again, by applying Lemma 5.2 to Z, (5.7) is proved.
The following corollary combines Theorem 4.5 with Theorem 5.1, and shows that the condition that the diffusion coefficient be constant can be relaxed.

Corollary 5.4. Let X be the unique continuous adapted process satisfying the equation
where h : + → + is increasing and 0 + h −2 (u)du = ∞. If there exists a positive constant ρ such that f satisfies (5.1), then X obeys (5.2).
Besides being of independent interest, the following result will be used extensively in Section 6 to prove comparison results. It is a special case of a result in [1]. The result in [1] covers equations with Markovian switching.

Recurrent Processes with Asymptotic Behaviour Close to the Law of the Iterated Logarithm
In this section, we again study solutions of (3.15), where the drift coefficient satisfies As mentioned previously, the solutions are no longer transient but are now recurrent on the real line. The results obtained vary according to the values of L ∞ and L −∞ . We classify these results into four main cases. The first result is a direct and easy application of Motoo's theorem. However, we state it as a theorem here for two reasons: first, it shows that −σ 2 /2 is another critical value for the process; second, it provides a way to construct a process with known behaviour to which we can compare processes in the other three cases.
Finally combining the above limit with (6.5), we get The previous theorem is not the main focus of this paper, as it applies to stationary or asymptotic stationary processes. It shows that such processes do not behave asymptotically in a manner close to the LIL. However, taking the results of Theorem 6.1, Theorem 4.3 and Theorem 4.5 together, we can exclude the necessity to study these regions of (L ∞ , L −∞ , σ 2 ) parameter space further.
The rest of our analysis focusses on the regions of (L ∞ , L −∞ , σ 2 ) parameter space not covered by these results. Before moving on to the next theorem, we give a lemma which allows us to construct appropriate comparison processes.
Proof. For every ε > 0 there exists x ε > 1 such that Since f is locally Lipschitz continuous, there is a constant K > 0 such that Then and By the second inequality in (6.10), and the fact that L ∞ ∧ L −∞ ≥ −σ 2 /2, we may define δ ε : Now we define the candidate function q ε . It is given for x ≥ 0 by x > x ε , and extended for x ≤ 0 according to q ε (x) = −q ε (−x). Clearly q ε is odd by definition, and is obviously Lipschitz continuous on (−x ε , x ε ). Since we have that q ε is locally Lipschitz continuous on . Noting that Since q ε is odd, the same limit pertains as x → −∞. .
where we have used the first inequality of (6.10) to deduce the third inequality in each case, and the definition of q ε and the fact that it is an odd function at the last steps.
By (6.11), for x ∈ [1, x ε ] we have q ε (x) = −C ε < f (x), and as q ε is odd, for x ∈ [−x ε , −1] using (6.12) we get q ε (x) = C ε > f (x). It remains to establish inequalities on (x ε , ∞) and (−∞, −x ε ). We noted earlier that δ ε (x) > 0 for x > x ε . Hence, by the definition of q ε , this fact and (6.7) yield for x > x ε , as required. We now consider the case when x < −x ε . Since q ε is odd, we get the last step coming from the fact that δ ε (−x) > 0 for −x > x ε . By the definition of f ε , we have Thus, as x < 0, we get We conclude by dealing with the continuity of is Lipschitz continuous. Since M > 1 is arbitrary, recalling that G ε is Lipschitz continuous on [0, 1) and continuous at x = 1, we have that G ε is locally Lipschitz continuous on [0, ∞). Moreover, as · and q ε (·) are actually globally Lipschitz continuous on [1, ∞), and G ε is Lipschitz continuous on [0, 1], it follows that G ε is globally Lipschitz continuous on [0, ∞). Finally since G ε is an even function, it is also globally Lipschitz continuous on .
Since q ε is odd, we can rewrite the above equation governing X 2 (t) =: where Y (0) = (x 0 − x * ) 2 and W is another Brownian motion in an extended space (Ω,˜ ,˜ ). Consider now the processes governed by the following two equations Since the drift coefficient of Y 0 is globally Lipschitz continuous by the previous lemma, we can use Yamada and Watanabe's uniqueness theorem, as well as the Ikeda-Watanabe comparison theorem to show that for every ε ∈ (0, 1), there exists an a.s. event Ω ε , such that Y (t) ≥ Y ε (t) ≥ Y 0 (t) = 0 for all t ≥ 0 a.s. on Ω ε . Therefore all the absolute values can be removed. Now by the definition and properties of q ε , it is easy to check that the scale function and the speed measure of Y ε satisfy s(∞) = ∞, s(0) > −∞, and m(0, ∞) < ∞ respectively. A similar argument to that used in Theorem 5.3 shows that zero is a reflecting barrier for Y ε . Therefore Y ε is a recurrent process on + with finite speed measure to which we can apply Motoo's theorem in order to determine the growth rate of its largest deviations. Now since lim x→∞ xq( x) = −σ 2 /2 − ε, for the same ε, there exists x ε such that for all x > x ε , Let s be the scale function of Y ε , then for some real positive constants K 1,ε , If we choose h(t) = t 1 1+ε(1+ε)/σ 2 , then Hence on the a.s. event Considering the a.s. event Ω * = ∩ ε∈ Ω * * ε , we have and hence the result.
Instead of comparing pathwise with X 2 , we construct a solution with drift coefficient q ε and directly compare it with X . The proof is left to the reader. Even though zeros are not included on the intervals for L ±∞ in Figure 1 in Section 3, the construction of q ε in either Theorem 6.3 or Theorem 6.4 covers the case when at least one of L ∞ and L −∞ is zero. Therefore (6.13) always holds provided the drift coefficient f reaches zero at least once. However, if f changes its sign an even number of times, more precise estimates on the growth rate can be obtained, even when at least one of L ∞ and L −∞ is zero. Lemma 6.6 and Theorem 6.7 deal with this case. In particular, if f remains non-negative (or non-positive) on the real line, we can compare X with the Brownian motion {σB(t)} t≥0 directly. This fact is stated in Corollary 6.8 without proof.
In order to apply a comparison argument to the next category of parameter values, we need to construct an appropriate drift coefficient, just as was done in Lemma 6.2 and Theorem 6.3. Lemma 6.6. Suppose f : → is locally Lipschitz continuous and satisfies (6.1).
Moreover, q x * in either case is globally Lipschitz continuous.
Proof. Under the conditions in Part (i), define C := min x∈[−x * ,x * ] f (x) ∧ 0 and construct q x * according to: It is obvious that q x * is even, globally Lipschitz continuous, and f (x) ≥ q x * (x) for all x ∈ . By a similar argument, we get the second part of the assertion.
Theorem 6.7. Let X be the unique continuous adapted process satisfying (3.15), and suppose f satisfies (6.1). Proof. We show assertion (i) first. Consider another process Y governed by the equation where q x * is the function defined in Lemma 6.6. Note that q x * ∈ L 1 ( ; ), so by Theorem 5.6, we have By Lemma 6.6 part (i), f (x) ≥ q x * (x) for all x ∈ , so a comparison argument gives 2t log log t , a.s.
Combining this with the result of Theorem 5.1, we get the first part of the theorem. For part (ii), let Similarly lim y→−∞ yf ( y) = L ∞ < 0. Therefore by the first part of the proof we get 2t log log t ≤ −ς, a.s.
Combining this limit with the result of Theorem 5.1, the second assertion is proved. The lower estimate on the asymptotic growth rate of partial maxima of |X | in this section can also be obtained when the limit in condition (6.1) is replaced by a limit superior or limit inferior in the appropriate way. For example, in Theorem 6.3, we can amend the condition (6.1) to lim inf x→−∞ x f (x) = L −∞ and lim sup x→∞ x f (x) = L ∞ . Hence we are able to estimate the growth rate of the partial maxima (or minima) of solutions in this section in terms of either the Law of the Iterated Logarithm or the polynomial Liapunov exponent for all real values of L ∞ and L −∞ .

Generalisation to multi-dimensional Systems
In this section, we generalize some of the main results in the scalar case to finite-dimensional processes. We show that analogous results can be obtained by using the same technique under adjusted conditions.
The filtration˜ (t) in the extended space is such thatX is˜ (t)-adapted. Similarly, we obtain Therefore Next define a : (7.7) By (7.1b), a(x, t) > 0 for all t ≥ 0 and x = 0. Define for j = 1, . . . , m the functions Φ j : and Φ j (x, t) = 1/ m for x = 0 and t ≥ 0. Then Now, applying Itô's rule to Z(t) = |X (t)| 2 , we get so by (7.8) and Finally define By (7.9) and e.g. [12,Theorem 3.3.16], W is a standard Brownian motion adapted to (˜ (t)) t≥0 such that Now it is easy to see that the drift coefficient of (7.10) is bounded above by K u := (2ρ + mC 2 a )/C 2 b due to (7.1). Consider the process governed by the equation A similar argument as given in the proof of Theorem 5.3 shows that Z u is nonnegative. Applying the comparison theorem again, we have, for almost all ω ∈ Ω, 0 ≤ Z(t) ≤ Z u (t) for all t ≥ 0. Let V u (t) := e −t Z u (e t − 1). By Itô's rule, it can be shown that where W is another one-dimensional Brownian motion. Applying Lemma 2.2, we obtain lim sup t→∞ V u (t) 2 log t = 1, a.s.
Using the relation between V u and Z u , and then comparing Z u with Z, we get 2t log log t ≤ 1, a.s.
We now establish the corresponding lower bound.
Proof. Proceeding in the same way as in the previous theorem, we arrive at the process Z governed by where Φ is as defined in (7.3). By condition (7.11), it is obvious that the drift coefficient is bounded below by K l := µ/(mC 2 a ). Let Z l be the non-negative process with Then Z(t) ≥ Z l (t), for all t ≥ 0 a.s. Applying the same change in time and scale to Z l as in the previous proof, and defining V l (t) := e −t Z l (e t − 1), we get Applying Lemma 2.2 again yields lim sup t→∞ V l (t) 2 log t = 1, a.s.
Following a similar argument as in Theorem 7.1, we get the desired result (7.12).
Our last theorem covers the special case where the diffusion coefficient is constant, diagonal and invertible. In this result, we use the notation 〈x, y〉 to denote the standard inner product of x and y in d , and e i as the i-th standard basis vector.   where W is another one-dimensional Brownian motion. If we can show that ∀ ( y, t) ∈ d × + , y Tf ( y, t) ≤ K, (7.18) for some positive K, then the non-negative process governed by d Z u (t) = (2K + d) d t + 2 Z u (t) dW (t), t ≥ 0, with Z u (0) ≥ x 2 0 satisfies Z u (t) ≥ Z(t) for all t ≥ 0 almost surely. As in the proof of the previous theorem, we have

Theorem 7.3. Let B be a d-dimensional Brownian motion and X be the unique continuous adapted process satisfying the d-dimensional equation
The proof of part (i) is complete. For part (ii), note for each 1 ≤ i ≤ d and all t ≥ 0, that |X (t)| ≥ |X i (t)|. Consider a particular X i which is governed by Here by (7.16)

Application to a Financial Market Model
In this section, for the purposes mentioned in Section 3.5, we present an ergodic-type theorem for the solution of the equation In the case when f (x) = 0, then L = 0, and we can independently prove (3.14), which is consistent with (8.3) (Λ L,σ,α,k = σ 2 ). On the other hand, letting L → 0 in (8.4) yields lim L→0 + Λ L,σ,α,k = σ 2 .
Proof of Theorem 8.1. We first discuss the existence of a strong solution of (8.1), which is not immediately obvious because the drift coefficient of (8.1) is discontinuous. However, by condition (8.2) and the continuity of f , the drift coefficient of X is uniformly bounded on [0, ∞) × . Therefore, we may apply Proposition 5.3.6 and Remark 5.3.7 in [12] (which are based on Girsanov's theorem) to obtain a weak solution. Moreover, by Corollary 5.3.11 in [12], the weak solution of (8.1) is unique in the sense of probability law. On the other hand, Theorem V.41.1 in [18] by Nakao and Le Gall gives us the pathwise uniqueness of the solution. This, together with the weak existence implies the existence of a strong solution by Corollary 5.3.23 in [12]. For a given initial value x 0 , and a fixed Brownian motion B, this strong solution is unique. Now we let the parameters tend to zero through rational numbers in the order ǫ, θ 1 , θ 2 and η. We consider each term in the square brackets in (8.8) in turn. As ǫ ↓ 0, the first integral on the interval (0, η) becomes