Journal of Probability and Statistics

Volume 2016, Article ID 5191583, 8 pages

http://dx.doi.org/10.1155/2016/5191583

## Parameter Estimation in Mean Reversion Processes with Deterministic Long-Term Trend

Department of Mathematical Sciences, Universidad Eafit, Carrera 49 No. 7 Sur 50, Medellin, Colombia

Received 12 April 2016; Accepted 23 July 2016

Academic Editor: Chin-Shang Li

Copyright © 2016 Freddy H. Marín Sánchez and Verónica M. Gallego. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper describes a procedure based on maximum likelihood technique in two phases for estimating the parameters in mean reversion processes when the long-term trend is defined by a continued deterministic function. Closed formulas for the estimators that depend on observations of discrete paths and an estimation of the expected value of the process are obtained in the first phase. In the second phase, a reestimation scheme is proposed when* a priori* knowledge exists of the long-term trend. Some experimental results using simulated data sets are graphically illustrated.

#### 1. Introduction

In the literature it is common to find studies and applications of mean reversion models, at both practical and theoretical levels, especially in fields such as energy markets, commodities, and interest rates. In these models there is a long-term trend which acts as an attractor making the process to oscillate around it and there is a random component that adds volatility to the movement. The specific characteristics of the models will depend on the structure of the trend and the volatility of the process.

Some common models of mean reversion are those in which all the parameters in the process are constant, such as Ornstein-Uhlenbeck model, the CIR model proposed by Cox et al. [1], and in general the models obtained from the CKLS structure proposed by Chan et al. [2]. In addition to these models there are other methods not so common in the literature in which the reversion is determined not by a parameter but a deterministic function or another stochastic process [3, 4]. In general, the fit of the models to specific situations is done by assigning values to the parameters and functions, either from* a priori* knowledge of the problem or from parameter estimation techniques based on historical data series that help to detect unobservable information from the process.

One of the first difficulties in the application of parameter estimation techniques is that it is necessary to have information with the same time resolution that is specified in the model. Since models of stochastic differential equations are formulated in continuous time and that the observed paths of the process can be obtained only in discrete time, it is necessary to discretize the EDE model. An approach frequently used for this purpose is given by Euler-Maruyama. With this methodology a new discrete-time process is obtained, with which it is possible to make inferences that are valid for continuous time model, as in the case of estimating the parameters.

There are different procedures that can be implemented such as methods based on distributional moments [5], Kernel methods [6, 7], least squares, and maximum likelihood [8–10]. Parameter estimation becomes fundamental too for modeling control systems. In the literature it is possible to find different methodologies for those issues such as those developed for Meng and Ding [11], where the parameter estimation in an equation-error autoregressive (EEAR) system is done through a transformation of the model to an equivalent one removing the autoregressive term and to obtain the parameters of the original model the principle of equivalence is used. Another methodology is proposed for Cao and Liu [12] where the parameter estimation in a power system is carried out from a recursive procedure to estimate simultaneously all the parameters from techniques based on the hierarchical identification principle, using gradient algorithms and least squared algorithms. Complementarily, Ding et al. [13] used methods based on the gradient algorithms and least squares iterative algorithms for system identification in output error (OE) and output error moving average (OEMA) systems, where the parameters that depend on unknown variables are computed using estimates of these unknown variables from previously estimated parameters.

In the area of multirate system identification, Ding et al. [14, 15] used the polynomial transformation technique to deal with the identification problem for dual-rate stochastic systems, while Sahebsara et al. [16] discussed the parameter estimation of multirate multi-input multioutput systems. Also, Ding and Chen [17] presented the combined parameter and state estimation algorithms of the lifted state-space models for general dual-rate systems based on the hierarchical identification method. Shi et al. [18] gave a crosstalk identification algorithm for multirate xDSL FIR systems.

In general, the auxiliary model identification idea has been used to solve the identification problem of dual-rate single-input single-output systems as shown in [19].

For more detailed information about multi-innovation stochastic gradient algorithm for multi-input multioutput systems and for multirate multi-input systems as well as auxiliary models based multi-innovation extended stochastic identification theory (see [20] and references therein). We will concentrate on one-factor stochastic model in which the parameters are estimated using maximum likelihood based on the discretized model when the long-term trend is given by a deterministic function. In this case we must estimate both the trend function and the parameters. To estimate the long-term trend function some convolution techniques and numerical differentiation are used, whereas the normality properties of residuals resulting from discretization are used for estimating parameters by maximum likelihood and in this way it is possible to obtain closed formulas for estimators based on the observations of a path of the process and the estimation of the long-term trend.

In the estimation process there may be some bias in the estimates with respect to the actual values despite the fact that they are obtained by the method of maximum likelihood. When this situation occurs it is necessary to develop alternative methodologies to have a greater adjustment to the estimates based on the initial estimates [21].

This paper is organized as follows. In Section 2 a description of mean reversion processes is presented for cases when the long-term trend is given by a continued deterministic function. The first phase of the estimation technique is shown in Section 3. Section 4 presents some examples and preliminary results for the estimation technique and in Section 5 (second phase), a procedure for reestimating the parameters from additional* a priori* knowledge is described.

#### 2. Mean Reversion Processes with Deterministic Long-Term Trend

Mean reversion processes of one factor with constant parameters and mean given by a deterministic function can be written aswith initial condition , where , , and are constants, is a deterministic function, and is a Unidimensional Standard Brownian Motion defined on a complete probability space .

parameter is called reversion rate, is the mean reversion level or trend of long-run equilibrium, is the parameter associated with the volatility, and determines the sensitivity of the variance to the level of .

Model (1) is a generalization of the models CKLS, Chan et al. [2], where the mean reversion level is a deterministic function that captures the trend of the process. In a general way, plays the role of an attractor at each point in the sense that, when the trend term and therefore decreases and when a similar argument establishes that grows.

To establish the relation between the mean reversion function and the expected value of the process , we can consider (1) written in integral form as

Taking the expected value, and thus obtain the ordinary differential equation:

The solution of this equation is .

To illustrate graphically the relation between and , we use as an example. In this case the solution to (3) is given bywhere .

Figure 1 shows the dynamic behavior of , and one path of for some specific values in the parameters.