Parameter Estimation for Fractional Diffusion Process with Discrete Observations

This paper deals with the problem of estimating the parameters for fractional diffusion process from discrete observations when the Hurst parameterH is unknown.With combination of several methods, such as the Donsker type approximate formula of fractional Brownian motion, quadratic variation method, and the maximum likelihood approach, we give the parameter estimations of the Hurst index, diffusion coefficients, and volatility and then prove their strong consistency. Finally, an extension for generalized fractional diffusion process and further work are briefly discussed.


Introduction
In recent years, many scholars have found that some financial time series data tend to be shown as biased random walk, long memory, and self-similarity, etc., which made the stochastic differential equation model driven by Brownian motion no longer applicable to describe financial data.Perhaps the most popular approach for modeling long memory is the use of fractional Brownian motion (hereafter fBm) that has been verified as a good model to describe the long memory property of some time series.
Compared with the traditional efficient market hypothesis theory, fractional market theory can accurately depict the actual law of financial market, such as the Ornstein-Uhlenbeck process driven by fractional Brownian motion, which is more consistent with the characteristics of long-term memory, in place of Vasicek model that is suitable to simulate the short-term interest rate model.
Although the study of fractional Brownian motion has been going on for decades, statistical inference problems related are just in its infancy.Such questions have been recently treated in several papers [1][2][3]: in general, the techniques used to construct maximum likelihood estimators (MLE) for the drift parameter are based on Girsanov transforms for fBm and depend on the properties of the deterministic fractional operators related to the fBm.Generally speaking, these papers focused on the problems of estimating the unknown parameters in the continuous-time case.Prakasa Rao [4] gave an extensive review on most of the recent developments related to the parametric and other inference procedures for stochastic models driven by fBm.The latest study can be found in Xiao and Yu [5,6], who developed the asymptotic theory for least square estimators for two parameters in the drift function in the fractional Vasicek model with a continuous record of observations.Another possibility is to use Euler-type approximations for the solution of the above equation and to construct an MLE estimator based on the density of the observations given "the past", for the case of stochastic equations driven by Brownian motion."Real-world" data is, however, typically discretely sampled (e.g., stock prices collected once a day or, at best, at every tick).Therefore, statistical inference for discretely observed diffusions is of great interest for practical purposes and at the same time it poses a challenging problem.Some papers are devoted to the parameter estimation for the models with fBm and discrete observations; see, e.g., Hu and Nualart [1], Hu and Song [7], Mishura and Ralchenko [8], Zhang, Xiao, Zhang and Niu [9], and Sun and Shi [10].
In this paper, we shall consider the parameter estimation problem for fractional linear diffusion process (FLDP).Assume that we have the model which can describe the intrinsic characteristics of interest rate more accurately in practical problem.The drift parameter ,  can characterize, respectively, the long-term equilibrium interest rate level and the rate of the short-term interest rates deviate from long-term interest rates.In general, the parameters of long-term equilibrium level of short-term interest rate are unknown.We assume  > 0 throughout the paper so that the process is ergodic (when  < 0 the solution to (1) will diverge),  describes the volatility of interest rates, and (   ) ≥0 is a fBm with Hurst parameter  ∈ (0, 1).In this paper, we suppose the Hurst index , the diffusion coefficients , , and the volatility  are unknown parameters to be estimated.We will furthermore show the strong consistence of these estimators.
In the case of diffusion process driven by Brownian motion, the most important methods are either maximum likelihood estimation or least square estimation.Since fBm is not a Markov process, the Kalman filter method cannot be applied to estimate the parameters of stochastic process driven by fBm.Consequently, it is a convenient way to handle the estimation problem by replacing fBm with its associated disturbed random walk.In this paper, we follow Zhang et.al. [9] to use discrete expressions of fractional Bronwnian motion with Donsker type approximate formula, which can, to some extent, simplify calculation and simulation.Although we do not have martingales in the model, this construction involving random walks allows using martingales arguments to obtain the asymptotic behaviour of the estimators.
Our paper is organized as follows.In Section 2, we propose MLE estimators for FLDP from discrete observations.The almost sure convergence of the estimators is provided in the latter part of this section.In Section 4, an extension for generalized fractional diffusion process is briefly discussed.Finally, Section 5 includes conclusions and directions of further work.

Estimation Procedure
It is worth emphasizing that the solution of (1) is given by where the unknown parameters included , ,  and .We now proceed to estimate these parameters based on quadratic variation method and maximum likelihood approach.
In applications, the estimation of  ∈ (0, 1) (called the Hurst index) is a fundamental problem.Its solution depends on the theoretical structure of a model under consideration.Therefore particular models usually deserve separate analysis.
According to the notation of Kubilius Skorniakov [11], suppose there are two hypotheses: for all  ∈ (0,  − 1/2), where =   (  ) means for a sequence of r.v.  , and   ⊂ (0, ∞), and there exists a.s.non-negative r.v., such that |  | ≤ ⋅  .These two conditions are used to prove the strongly consistent and asymptotically normality of the estimator  from discrete observatios. Denote where 1 ≤  ≤  and    = (/  ), Then, the estimator of Hurst parameter  can be written as Next, we turn to the estimation problem of the diffusion parameter  2 .When  is known, Xiao et al. [12] obtained the estimators based on approximating integrals via Riemann sums with Hurst index  ∈ (1/2, 3/4).In contrast, we suppose in this paper the Hurst index is unknown.Therefore in the next estimation, the estimator of  will be embedded in the equation.For simplicity, denote ,  = 1, . . .,   , ℎ = /  .Thus, the full sequence of   observations can be written as { ℎ ,  2ℎ , . . .,    ℎ }.
For the diffusion parameter, we easily obtain an estimator for the diffusion parameter by using quadratic variations, such σ2 = ∑ which converges (in  2 and almost surely) to  2 .Finally, we are in a position to estimate the drift parameter.Note that    −   −1 is not independent and the process    is not a semimartingale; therefore the martingale type techniques cannot be used to study this estimator.This problem will be avoided by the use of the random walks that approximate    .Based on the results on Sottinen [13], the fractional Brownian motion can be approximated by a "disturbed" random walk, which was called Donsker type approximation for fBm.
Lemma 1.The fBm with Hurst parameter  > 1/2 can be represented by its associated disturbed random walk: with converges weakly in the skorohod topology to the fractional Brownian motion.With the estimators Ĥ, σ plug-in, the replacing model still kept the main properties of the original process, such as long range dependence and asymptotic self-similar.Therefore, the martingales can be used to treat this replacing model.
In general, numerical approximation of model ( 1) can be presented by Euler scheme: to denote the contribution of the  − 1 first jumps of the random walk and to denote the contribution of the last jump.
With the approximation of fBm (Lemma 1), we can write with which (8) can be written as Hence we have We assume that random variables   follow a standard normal law (0, 1).Then, the random variable  (+1)ℎ is conditionally Gaussian and the conditional density of  (+1)ℎ given  ℎ ,  2ℎ , . . .,  ℎ can be written as The likelihood function can be expressed as This leads to the MLE of  and  α = ∑ where Remark 2. Note that the parameter estimators of drift coefficients are related to the volatility , while, in fact,  2 can be (at least theoretically) computed on any finite time interval.Furthermore, fBm is self-similar to stationary increments and it satisfies |   −   | = |−| 2 for every ,  ∈ [0, ].For this reason, we may assume that the diffusion coefficient is equal to 1.

The Asymptotic Properties
In this section, we turn to study the strong consistency of these estimators by ( 5), ( 6), (16), and (17).

Theorem 3. Assume that solution of (1) satisfies hypotheses (C1) and (C2), then estimator Ĥ converges to 𝐻 almost surely as 𝑚 𝑛 goes to infinity.
Detailed proof can be found in Kubilus and Skorniakov [11].Proof.Clearly, the consistency of α can be inferred combined with (16) and consistency of β.We just prove that β is strong consistent.
A simple calculation shows that where Using the assumption of  ∈ (1/2,1) and fractional integral, we have the explicit solution of (1) that can be expressed as where the integral can be understood in the Skorohod sense.As a consequence, for any , we have Hence, for any , we obtain that [ 2 ℎ ] is bounded.Moreover, by using Cauchy-Schwartz inequality, we show that (see also in [14], with a slight modification below)
Remark 6.The asymptotic normality of estimators is not involved in the results of this paper.In fact, Kubilius and Skorniakov [11] proposed the asymptotic normality of the estimators Ĥ; in view of Remark 2, the asymptotic of σ is trivial.For the parameter estimation of fractional diffusion process (1), there are usually two key challenges: the likelihood is intractable and the data is not Markovian.With the Donsker type approximation formula, the statistical inference of fractional diffusion process (FDP) can be simplified to a certain extent.It has proved that the estimator of drift parameter is   ( ≥ 1)consistent and the asymptotic normality may be obtained with more complex operations by the future studies of this area.

Extension
Fractional stochastic differential equations have been widely used in the fields of finance, hydrology, information, and stochastic networks.Although model ( 1) is concerned of simpler linear function, our method can be expected to be applicable for general fractional diffusion processes, such as where (.; ) is drift functions representing the conditional mean of the infinitesimal change of   at time ;    is the random perturbation.Here we suppose the diffusion function is constant for simplicity.As far as we know, for a general smooth and elliptic coefficient (.),only the uniqueness of the invariant measure is shown in Haier and Ohashi [15], with an interesting extension to the hypoelliptic case in Haier and Pillai [16].Nothing is known about the convergence of estimate equation, not to mention rates.Suppose   is observed at a discrete set of instants    = (/  ),  = 1, 2, . . .,   .With the Donsker type approximate formula and the above estimation procedure, we use the following global estimation equations to estimate parameter : where ġ (,  ℎ ) is the derivative of (,  ℎ ) on .The asymptotic property of the estimators is expected to be studied in the future and how to obtain the asymptotic theory is still an open question.
On the other hand, our method can extend to another self-similar process still with long memory (but not Gaussian), which is called Rosenblatt process    .In contrast to the fBm model, the density of Rosenblatt process is not explicitly known any more.However, it can be written as a double integral of a two-variable deterministic with respect to the Wiener process.The method based on random walks approximation offers a solution to the problem of estimating the parameters in fractional diffusion process driven by Rosenblatt process.

Concluding Remarks
In this paper, we proposed the estimators of FLDP, such as the Hurst index, drift coefficients, and volatility, and provided the strong consistency for these estimators.With the Donsker representation of fractional Brownian motion, the statistical inference of FLDP may be simplified.However, it is important to note that this approximation is satisfied in the sense of weak convergence.This means only when with large number of samples can the simulation be much better.On the other hand, the approximate representation of FLDP is based on the Euler scheme, which is the main source of the error in the computations.There is always a trade-off between the number of Euler steps and the number of simulations, but what is usually computationally costly is the number of Euler steps.The rate of convergence depended on  and the closer the value of  to 1/2.This study also suggests several important directions for further research.How to estimate parameters in FDP from discrete time observations and how to obtain the asymptotic theory are open questions.