Inference for Diffusion Processes using Combined Estimating Functions

A class of martingale estimating functions provides a convenient framework for studying inference for nonlinear time series models. Further, when information about higher order conditional moments of the observed process is available, the estimation based on combined estimating functions becomes more informative. In this paper, a general framework is developed for estimating parameters of diffusion processes with discretely sampled data using combined estimating functions. The approach is used to study parameter estimation for diffusion models for asset pricing including the Black Scholes model, the Vasicek model, and the Cox-Ingersoll-Ross (CIR) model. Closed form expressions for the gain in information are also discussed in some detail.


Introduction
For nonlinear time series models, Chandra and Taniguchi [1], Bera et al. [2], Merkouris [3], Ghahramani and Thavaneswaran [4], and more recently Liang et al. [5] among others have studied inference using estimating functions.For discretely sampled diffusion-type models, parameter estimation using estimating functions has been studied in Bibby and Sørensen [6], Sørensen [7], and Bibby et al. [8].However, additional assumptions were made and constraints were imposed to obtain the estimates.Moreover, information issues related to the estimating function approach have not been sufficiently addressed in the literature.In this paper, we study combined martingale estimating functions and show that the combined estimating functions are more informative when the conditional mean and variance of the observed process depend on the same parameter of interest.We then apply our approach to discretely sampled observations from diffusion models.
This paper is organized as follows.The rest of Section 1 presents the basics of estimating functions and information associated with estimating functions.Section 2 presents the general model framework for discretely sampled observations from a continuous process, and presents the form of the optimal combined estimating function.In Section 3, the theory is applied to three different diffusion models that are widely used in asset pricing.
Suppose that { ,  h h y y θ be specified q -dimensional vectors that are martingales.We consider the class M of zero mean and square integrable p -dimensional martingale estimating functions of the form g θ which maximizes, in the partial order of nonnegative definite matrices, the information matrix ( ) and the corresponding optimal information reduces to [9]).It follows from Lindsay ([10], page 916) that if we solve an unbiased estimating equation ( ) n = g θ 0 to get an estimator, then the asymptotic variance of the resulting estimator is the inverse of the information n g I .Hence the estimator obtained from a more informative estimating equation is asymptotically more efficient.

Estimating Function Approach for a Discretely Sampled Continuous Stochastic Process
Assume that a real-valued continuous-time process { } ( ) E ( ) | ,and ( ) E ( ) | , where ( For the general model in (2.1)- (2.4), in the class of all combined estimating .functions of the form ( ) (a) the optimal estimating function is given by ( ) , where , , 1 1 and ; (2.9) Proof.Proof of Theorem 1 is similar to that of Theorem 2.1 in Liang et al.

Examples
In the three examples provided in this section, we assume that t W is a Wiener process.

Geometric Brownian Motion with Volatility as a Function of Drift
Consider the Black and Scholes model (Black and Scholes [11]) of the form ( ) .
We estimate the unknown parameter θ appearing simultaneously in the conditional mean and variance.The first four conditional moments of , The optimal estimating function based on the martingale difference t m is given by ( ) which gives an estimator for θ of the form Similarly, the optimal estimating function based on the martingale difference t M is given by The corresponding information associated with * ( ) m θ g and * ( ) , and ( ) .
It follows from Theorem 1 that the optimal combined estimating function based on t m and t M has the form ( 1) ( ) 1 nh e e e e θ θ θ which also approaches ( ) ( ) / / ( )  , , , ,

Ornstein-Uhlenbeck Model
, the martingale differences are derived as 1 , ) The optimal estimating functions based on the martingale differences t m and t M are respectively given by Moreover, the information matrices associated with * ( ) m g θ and * ( )

I
It follows from Theorem 1 that the optimal combined estimating functions based on t m and t M for α , µ and 2 σ have the following forms: The above estimating functions are set equal to zero and can be solved simultaneously to obtain the estimators for α , µ and 2 σ as ( 1)
Based on t m and t M , the optimal combined estimating function is given by ( ) ) )