Estimation of the Hurst index of the solutions of fractional SDE with locally Lipschitz drift

Abstract. Strongly consistent and asymptotically normal estimate of the Hurst index H are obtained for stochastic differential equations (SDEs) that have a unique positive solution. A strongly convergent approximation of the considered SDE solution is constructed using the backward Euler scheme. Moreover, it is proved that the Hurst estimator preserves its properties, if we replace the solution with its approximation.


Introduction
The models defined by SDE Currently, much attention is paid to models with fractional Brownian motion (fBm) B H since it introduces a memory element into the model under consideration. Consider SDE with H ∈ (1/2, 1). The stochastic integral in equation (1) is a pathwise Riemann-Stieltjes integral. SDE (1) cannot be treated directly since the function h(x) = x β , 1/2 β < 1, does not satisfy the usual Lipschitz conditions that are commonly imposed. For fractional CIR and CKLS models, the existence of a unique positive solution of equation (1) was obtained in [3,8,9,11,12,16]. The proof is based on several approaches. One approach is based on the consideration of the conditions under which the equation admits a unique positive solution, where f (t, x) is a locally Lipschitz function with respect to the space variable x on x ∈ (0, ∞). This approach was used in [3,8,16], where the inverse Lamperti transform was used to obtain conditions under which equation (1) admits a unique positive solution for fractional CIR and CKLS models. Unfortunately, we cannot apply the proof of the positivity of the solution of equation (1) given in [16]. The proof must be revised because it is not applicable, for example, for the Ait-Sahalia model with 1/2 β < 1.
Marie [9] used rough-path approach to find the existence of the unique positive solution of the fractional CKLS model. One more approach for fractional CIR process was suggested in [11,12], where the integral with respect to fractional Brownian motion is considered as the pathwise Stratonovich integral. In [10], it was proved that equation has a unique solution for H ∈ (1/(1 + β), 1), β ∈ [1/2, 1), and X t = 0 a.s. for all t τ , where τ = inf{t > 0: X t = 0}. The problem of the statistical estimation of the long-memory parameter H is of a great importance. This parameter determines the mathematical properties of the model and consequently describes the behavior of the underlying physical system.
Our goal is to construct strongly consistent and asymptotically normal estimator of the Hurst index H for SDE (1), which has a unique positive solution. For such processes, we can do this in the same way as done for the diffusion coefficient satisfying the usual Lipschitz conditions (see [6,7]). More results on parameter estimations for stochastic differential equations can be found in the book [5]. Since the existence of a unique positive solution for general form SDE (1) is unknown, we will pay attention to this problem.
To model the estimator of the Hurst index, we need an approximation of the SDE (1) solution. The approximation of the solution X is based on the use of the backward Euler scheme, which is positivity preserving. Moreover, the Hurst index H estimator preserves its properties if we replace the solution X with its approximation.
The paper is organized in the following way. In Section 2, we present the main results of the paper. In Section 3, we prove the main auxiliary result about the existence and uniqueness of positive solution for SDE (1). Section 4 contains proofs of main theorems. In Section 5, fractional CKL and Ait-Sahalia models are considered as examples. Section 6 gives some examples of simulating the fractional CKLS model to illustrate the results. Finally, in Appendix, we recall same results for fBm and the Love-Young inequality.

Main results
To state our main results, we use the following requirements on function f : (C2) There exist constants a > 0 and α 0 such that f (x) a/x 1+α for all sufficiently small x. (C3) There exists a constant K ∈ R such that the derivative is bounded above by K, i.e., f (x) K.
To estimate the Hurst index for SDE (1), we need conditions when this equation admits a unique positive solution. The following theorem solves this problem. C γ ([0, T ]) denotes the space of Hölder continuous functions of order γ > 0 on [0, T ].
Our goal is to construct strongly consistent and asymptotically normal estimator of the Hurst parameter H for the solution X of equation (2) from discrete observations of a single sample path.
Let π = {t n k = (k/n)T, 1 k n} be a sequence of uniform partitions of the interval [0, T ] and h = t n k − t n k−1 , 1 k n. For a real-valued process X = {X t , t ∈ [0, T ]}, we define the first and second-order increments along uniform partitions as ∆ n,k X = X t n k − X t n k−1 , 1 k n, ∆ (2) n,k X = X t n k+1 − 2X t n k + X t n k−1 , 1 k n − 1.
To avoid cumbersome expressions, we introduce symbol O ω . Let (Z n ) be a sequence of r.v., ς is an a.s. nonnegative r.v. and (a n ) ⊂ (0, ∞) vanishes. Z n = O ω (a n ) means that |Z n | ς · a n . In particular, Z n = O ω (1) corresponds to the sequence (Z n ), which is a.s. bounded.
Theorem 2. Assume that X is a unique positive solution of SDE (2) with H ∈ (1/2, 1). Then In practice, it is very interesting to compare different estimators. Therefore, we consider approximations of discrete time that can be used in modeling. To construct an approximation scheme for the SDE (2) solution, we use the solution of the SDE The solutions of SDEs (2) and (3) satisfy the relation (see the proof of Theorem 1).
The backward Euler approximation scheme for Y is defined as follows: For the well-definedness of the backward Euler approximation scheme, we need the following assumption: Assume that the function f (x) satisfies conditions (C1), (C3) and there exists h 0 > 0 such that lim x→+∞ F (x) = +∞, Remark 1. Note that under condition (C3) the function F (x) is strictly monotone on (0, ∞) for small h. Thus, from the conditions (C3) and (C4) it follows that for each b ∈ R, the equation F (x) = b has a unique positive solution for 0 < h < h 0 . Consequently, the backward Euler approximation scheme preserves positivity.

Theorem 3.
Let the function f (x) satisfies conditions (C1)-(C4), and f is continuous on (0, ∞). If the sequence of uniform partitions π of the interval [0, T ] is such that h < h 0 , then for any T > 0, where where X is the solution of equation (2).
The following result states that if we replace the solution X with its approximation, then the estimator of the Hurst index H will remain strongly consistent and asymptotically normal.

Auxiliary result
We are interested in conditions under which the SDE has a unique positive solution, where k 1 and k 2 are positive constants. As mentioned in the introduction, this type of equation was considered in [3,8,16]. We provide conditions of a different kind than in the above papers under which equation (6) has a unique positive solution and which are easily applicable to the fractional CKLS and Ait-Sahalia models.
Consider the following deterministic differential equation driven by a continuous function ϕ: where y 0 is a constant.

From (9) and inequality |ϕ
Now we show that there exists ε * ∈ (0, ε) such that, for all ε < ε * and for all x 0, Then we get contradiction, which proves that the solution of equation (7) is positive.
http://www.journals.vu.lt/nonlinear-analysis It is easy to check that F ε (0) = ε > 0 and F ε is convex on (0, +∞) (its second derivative is strictly positive on this set), so it is enough to examine the sign of the function in its critical points. So After some calculations, we get Choosing the corresponding ε * for and choosing an arbitrary ε < ε * , we obtain that F ε (x) > 0 for all x > 0. The contradiction obtained proves that τ = +∞. It remains to prove that y ∈ C λ ([0, T ]). Indeed, Uniqueness. Let y and y be two positive solutions of equation (7). Then and from condition (C3) From Gronwall 's inequality it follows that y t = y t for all t T , T > 0. As an immediate consequence of Proposition 1, we have the following result.

Proofs of main theorems
Proof of Theorem 1. Set , where Y is a solution of equation (6) with k 1 = k 2 = 1 − β. Since the process Y is positive Hölder continuous process up to the order γ ∈ (1/2, H) and β/(1 − β) 1, then X β s is Hölder continuous process up to the order γ ∈ ( 1 2 , H). This follows from the inequality for 1/2 β < 1, where we applied the mean value theorem. Thus, the integral t 0 X β s dB H s is well defined as a pathwise Riemann-Stieltjes integral for γ ∈ (1/2, H) and equation (2) is well defined. Now we verify that X is a solution of equation (2). By chain rule we obtain . Thus, equation (2) has a continuous positive solution. Since The proof of Theorem 2 is based on the following lemma.
n,i X = σX β (t n i )∆ where γ ∈ (1/2, H). http://www.journals.vu.lt/nonlinear-analysis Proof. Second-order increments of the process X we write as follows: Applying inequality (10), condition (C1) and the fact that Thus, Moreover, by Love-Young inequality, (10) and Hölder continuity of B H we get s. Thus, we get the statement of the lemma.
Proof of Theorem 2. Since X is a positive solution, it follows from Lemma 1 and (A.1) that if γ is slightly different from H. Thus, by Maclaurin's expansion Repeating the proof of Theorem 3.17 in [5] and using (11), we get Applying the limit results of Section A.2, we obtain H n → H a.s. and Now, to finish the proof, it is enough to apply the Slutsky's theorem and the results obtained above. Note that the limit variance σ 2 H of H n equals that to H n .
Proof of Theorem 3. First, we prove (4). By Remark 1 the values of the approximation ( Y k ) are strictly positive for y 0 > 0 and 0 < h < h 0 .
http://www.journals.vu.lt/nonlinear-analysis By definition of Y n , for any t ∈ (t n k , t n k+1 ], The asymptotic behavior of the first two terms is O ω (n −γ ). Thus, it remains for us to obtain the asymptotic of the last two terms. Note that where Continuity of the function f and the positivity of the process Y ∈ C γ ([0, T ]), γ ∈ (1/2, H), gives an estimate From (12) it follows that This finishes the proof of (4). It remains to prove (5). By applying inequality . From (4) and finiteness of sup 0 t T |Y t | we have Thus, This finishes the proof of (5).
To prove Theorem 4, we need the following statement. Proof. We now recall a well-known equality of algebra From this equality we get Thus, Since from Theorem 3 we have To prove the assertion of the lemma, it remains to prove that The last equality follows from (13). Indeed, applying formula (12), we get where ∆ n,i+1 Y = Y n,i+1 − Y n,i . By Remark 1 the values of the approximation ( Y k ) are strictly positive for small h. Since Y is continuous and Y t > 0 for all t ∈ [0, T ], then from (14) it follows that Since for small h, it follows that ζ i (1 − β)h Kh < 1, then From (13), (16) and (17) it follows that So, we obtain (15).
Proof of Theorem 4. The proof is a similar to that of Theorem 2.

Examples
Example 1. The fractional CKLS model has a unique positive solution, and Theorem 3 holds.
Thus, conditions (C1) and (C2) are satisfied. Moreover, since g(x) = x β f (x 1−β ) = a 1 − a 2 x, then from Theorem 1 we get that the CKLS model has a unique positive solution.
Since the function f (x) is continuous on (0, ∞) and then condition (C3) is fulfilled and Theorem 3 holds.
Example 2. The Ait-Sahalia model has a unique positive solution, and Theorem 3 holds.

Simulation results
The purpose of this section is to provide some simulations in order to illustrate various aspects of the suggested estimator. We consider CKLS model. The simulation of the obtained estimate presented below was performed using Wolfram Mathematica. The values of the constants involved in these simulations were x 0 = 4, a = 1, b = 2, σ = 1. We considered these sample paths on the unit interval, hence, T = 1. The number of batches were 200 in all of the considered cases.
The CKLS model after Lamperti transform has the form Applying Theorem 4, we calculate the estimator H E n with β = (m − 1)/m, m ∈ N. The asymptotic behavior of the variance of the difference H − H E n for different m, n, and H is shown in Figs. 1, 2. Figure 1 shows that the variance of the difference H − H E n decreases as the sample size increases. Figure 2 shows how the variance of the difference H − H E n for different H depends on the sample size. We see that with increasing sample size, the variance decreases for all H values.

A.2 Several results on fBm
Recall that fBm B H = {B H t , t 0} with the Hurst index H ∈ (0, 1) is a real-valued continuous centered Gaussian process with covariance given by For consideration of strong consistency and asymptotic normality of the given estimators, we need several facts regarding B H . Limit results. Let Then (see [4], [5, pp. 46 for all s, t ∈ [0, T ], where γ ∈ (0, H) (see [5, p. 4]).