Abstract
In view of rapid developments on nonparametric estimation of the drift and volatility functions in scalar diffusion models in financial econometrics, from discretetime observations, we provide, in this paper, a survey of its stateoftheart with new insights into current practices, as well as elaborating on our own recent contributions. In particular, in presenting the main principles of estimation for both stationary and nonstationary cases, we show the possibility to estimate nonparametrically the drift and volatility functions without distinguishing these two frameworks.
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Notes
 1.
Bachelier is credited for the first construction of the Brownian motion.
 2.
 3.
With electronic markets stock prices are now available at ultra high frequency, within the millisecond, or even less, but are not appropriate for the analysis of a diffusion model. Indeed they are submitted to microstructure noises due to the fact that prices are multiple of ticks, that they are different for buy and sell and depend on the type of posted order.
 4.
We will not discuss the estimation of the drift and volatility functions from observations of continuous time paths of the process, (see e.g. Bosq 2000 for general presentations of such an approach and Dalalyan and Kutoyants 2002 for constant volatility) or from point process sampling of the observations (Nguyen and Pham 1981; Nguyen 1985), or assuming that the frequency of observations increases. Even if this facilitates the estimation problem, this does not correspond to the need for Finance (see e.g. FlorensZmirou 1993; Bosq 1998; Hoffmann 1999; Bandi and Phillips 2003). For a recent work on risk analysis in diffusion models, see Nguyen et al. (2012). See also, Sriboonchitta et al. (2010).
 5.
This expression implies
$$\begin{aligned} \dfrac{d}{dy} \left[ f(y) \sigma ^2 (y)\right] = 2 \mu (y) f(y). \end{aligned}$$This equation allows to estimate the drift \(\mu \) given the volatility and the stationary density (see AitSahalia 1996b for an application).
 6.
Since the function \(g(y)=y\) is not in general an eigenfunction of the prediction operator (see Sect. 2.1).
 7.
 8.
See Meyn and Tweedie (1993) for the different definitions of recurrence for Markov processes.
References
AitSahalia, Y. (1996). Nonparametric pricing of interest rate derivative securities. Econometrica, 64, 527–560.
AitSahalia, Y. (1996). Testing continuoustime models of the spot interest rate. The Review of Financial Studies, 9(2), 385–426.
AitSahalia, Y. (2002). Maximum likelihood estimation of discretely sampled diffusions: A closedform approximation approach. Econometrica, 70(1), 223–262.
Akonom, J. (1993). Comportment asymptotique du temps d’occupation du processus des sommes partielles. Annales de l’Institut Henri Poincaré, 29, 57–81.
Bachelier, L. (1900). Théorie de la spé culation, Ph.D. Thesis, University Paris Sorbonne. Louis Bachelier’s Theory of Speculation: The Origin of Modern Finance (M. Davis & A. Etheridge, Trans.). Princeton University Press, 2006.
Bandi, F., & Phillips, P. (2003). Fully nonparametric estimation of scalar diffusion models. Econometrica, 71, 243–283.
Bandi, F., & Phillips, P. (2009). Nonstationary continuoustime processes. In Y. Ait Sahalia & L. Hansen (Eds.), Handbook of financial econometrics (pp. 139–202).
Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–639.
Bosq, D. (1998). Nonparametric statistics for stochastic processes. Lecture notes in statistics. New york: Springer.
Bosq, D. (2000). Linear processes in functions spaces: Theory and applications. Lecture notes in statistics. New York: Springer.
Bosq, D., & Davydov, Y. (1998). Local time and density estimation in continuous time. Lille: Publ. IRMA Univ. Lille 1.
Chen, X. (2007). Large sample sieve estimation of seminonparametric models. In Handbook of econometrics (vol. 5, pp. 5546–5632). Elsevier.
Chen, S. X., Gao, J., & Tang, C. Y. (2008). A test for model specification of diffusion processes. The Annals of Statistics, 36(1), 167–198.
Chung, K. L., & Williams, R. J. (1983). Introduction to stochastic integration. Boston: Birkhauser.
Cohen, A. (2000). Wavelet methods in numerical analysis. In P. Ciarlet (Ed.), Handbook of numerical analysis (7th ed., pp. 417–711). Amsterdam: NorthHolland.
Dalalyan, A., & Kutoyants, Y. (2002). Asymptotically efficient trend coefficient estimation for ergodic diffusion. Mathematical Methods of Statistics, 11, 402–421.
Darolles, S., Florens, J. P., & Gourieroux, C. (2004). Kernel based nonlinear canonical analysis and time reversibility. Journal of Econometrics, 119, 323–353.
Darolles, S., & Gourieroux, C. (2001). Truncated dynamics and estimation of diffusion equations. Journal of Econometrics, 102, 1–22.
Darolles, S. & Serot, I. (2000). Empirical local time for processes observed on a grid, CREST DP.
Demoura, S. (1993). Theory and application of transition operator to the analysis of economic time series. Necessary and sufficient conditions for nonlinearities in economic dynamics, aliasing problem. Discussion papers, manuscripts.
Duffie, D. (1996). Dynamic asset pricing theory. Princeton: princeton University Press.
Dunford, N., & Schwartz, J. (1963). Linear operators (Vol. 2). New York: Wiley.
Fan, J., & Yao, Q. (2003). Nonlinear time series: Nonparametric and parametric methods. New York: Springer.
Fan, J., & Zhang, C. (2003). A reexamination of diffusion estimators with application to financial model validation. Journal of the American Statistical Association, 98(416), 118–134.
FlorensZmirou, D. (1993). On estimating the diffusion coefficient from discrete observations. Journal of Applied Probability, 30, 790–804.
Gao, J. (2007). Nonlinear time series: Semiparametric and nonparametric methods. London: Chapman & Hall.
Gao, J., King, M., Lu, Z., & Tjostheim, D. (2009). Specification testing in nonlinear and nonstationary time series autoregression. Annals of Statistics, 37, 3893–3928.
Geman, D., & Horowitz, J. (1980). Occupation densities. Annals of Probability, 8, 1–67.
Gobet, E., Hoffman, M., & Reiss, M. (2004). Nonparametric estimation of scalar diffusions based on low frequency data. Annals of Statistics, 32, 2223–2253.
Gourieroux, C., Monfort, A., & Renault, E. (1993). Indirect inference. Journal of Applied Econometrics, 8, 85–118.
Guerre, E. (2007). Design adaptive pointwise nonparametric regression estimation for recurrent Markov time series, DP.
Hansen, L., & Sargent, T. (1983). The dimensionality of the aliasing problem in models with rational spectral densities. Econometrica, 51, 377–387.
Hansen, L., & Scheinkman, J. (1995). Back to the future : Generating moment implications for continuoustime Markov processes. Econometrica, 63, 767–804.
Hansen, L., Scheinkman, J., & Touzi, N. (1998). Spectral methods for identifying scalar diffusions. Journal of Econometrics, 86, 1–32.
Hoffmann, M. (1999). Adaptive estimation in diffusion processes. Stochastic Processes and Their Applications, 79, 135–163.
Jiang, G., & Knight, J. (1997). A nonparametric approach to the estimation of diffusion processes with an application to a short term interest rate model. Econometric Theory, 13, 615–645.
Karlin, S., & Taylor, H. (1981). A second course in stochastic processes. New York: Academic Press.
Karlsen, H., & Tjostheim, D. (2001). Nonparametric estimation in null recurrent time series. Annals of Statistics, 29, 372–411.
Kutoyants, Yu A. (1999). Efficient density estimation for ergodic diffusion processes. Statistical Inference for Stochastic Processes, 1, 131–155.
Kutoyants, Y. (2013). Statistical inference for ergodic diffusion processes. New York: Springer.
Kessler, M., & Sorensen, M. (1999). Estimating equations based on eigenfunctions for directly observed diffusion process. Bernoulli, 5, 299–314.
Lindstrom, E. (2007). Estimating parameters in diffusion processes using an approximate maximum likelihood approach. Annals of Operations Research, 151, 269–288.
Levy, P. (1948). Processus Stochastiques et Mouvement Brownien. Paris: GauthierVillars.
Merton, R. (1973). Theory of rational option pricing. The Bell Journal of Economics and Management Science, 141–183.
Merton, R. (1990). Continuous time finance. Oxford: Blackwell.
Meyn, S., & Tweedie, R. (1993). Markov chains and stochastic stability. London: Springer.
Nguyen, H. T. (1985). On point process sampling in continuous time models. Publications de l’Institut de Statistique de l’Université de Paris, 30, 71–95.
Nguyen, H. T., & Pham, T. D. (1980). Sur l’utilisation du temps local en statistique des processus. Comptes rendus de l’Académie des sciences, 290, 165–168.
Nguyen, H. T., & Pham, T. D. (1981). Nonparametric estimation in diffusion models by discrete sampling. Publications de l’Institut de Statistique de l’Université de Paris, 26, 89–109.
Nguyen, H. T., & Pham, T. D. (1982). Identification of nonstationary diffusion model by the method of sieves. SIAM Journal on Control and Optimization, 20, 603–611.
Nguyen, H. T., Pham, U. H., & Tran, D. H. (2012). On some claims related to Choquet integral risk measures. Annals of Operations Research, 195(1), 5–31.
Park, J. & Phillips, P. (1998). Nonstationary density estimation and kernel autoregression, Cowles Foundation DP 1181, Yale Univ.
Revuz, D., & Yor, M. (1998). Continuous martingales and Brownian motion. NewYork: Springer.
Sriboonchitta, S., Wong, W. K., Dhompongsa, S., & Nguyen, H. T. (2010). Stochastic dominance and applications to finance, risk and economics. Boca Raton, FL: Chapman and Hall/CRC Press.
Stanton, R. (1997). A nonparametric model of term structure dynamics and the market price of interest rate risk. The Journal of Finance, 52(5), 1973–2002.
Wang, Q., & Phillips, P. (2009). Asymptotic theory for local time density estimation and nonparametric cointegrating regression. Econometric Theory, 25, 710–738.
Xu, K. L. (2009). Empirical likelihoodbased inference for nonparametric recurrent diffusions. Journal of Econometrics, 153, 65–82.
Xu, K. L. (2010). Reweighted functional estimation of diffusion models. Econometric Theory, 26, 541–563.
Acknowledgments
We thank two referees for their valuable comments and suggestions which have led to a significant improvement of the paper. We acknowledge the partial support of the Center of Excellence in Econometrics, Faculty of Economics, Chiang Mai University, Thailand.
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Appendix
Appendix
Local time
Originally, in probability theory, the concept of local time was invented by Levy (1948) as a tool to investigate sample paths of a Brownian motion. Later, it became an important tool in the theory of stochastic integration. In statistics of continuoustime stochastic processes, such as diffusion processes, the existence of local times turns out to be a useful information for statistical inference, as first pointed out by Nguyen and Pham (1980). Nowadays, the use of local times for general stochastic processes (called occupation densities) seems to become a routine in statistics, especially in nonparametric density estimation, see e.g. Bosq and Davydov (1998), Kutoyants (1999) and Wang and Phillips (2009).
Let \((W_{t},t\ge 0)\) be a Brownian motion on \({\mathbb {R}}\). The amount of time it spends in the neighborhood of a point \(x\in {\mathbb {R}}\), up to time \( t>0\), can be measured as
where \(\lambda (dx)\), or simply dx, is the Lebesgue measure on \({\mathcal {B}} ({\mathbb {R}})\).
Of course, the existence of the above limit, as well as properties of the stochastic process \(t\rightarrow L(t,x)\) and of the random field \( (t,x)\rightarrow L(t,x)\) are major achievements in the studies of Brownian motion. The random variable L(t, x) is called the local time of the Brownian motion at x. Its existence implies that the occupation measure \( \mu _{t}\) on \({\mathcal {B}}({\mathbb {R}})\), defined as \(\displaystyle \mu _{t}(B)=\int _{0}^{t}1_{B}(W_{s})ds\), is absolutely continuous with respect to the Lebesgue measure, and the local time is precisely the Radon–Nikodym derivative \(\dfrac{d\mu _{t}}{d\lambda }\), so that
In general, the empirical measure of a realvalued stochastic process \( (X_{t},t\ge 0)\) is defined on \({\mathcal {B}}({\mathbb {R}})\) as
If \(\Gamma _{t}\) is absolutely continuous with respect to the Lebesgue measure on \({\mathcal {B}}({\mathbb {R}})\), then its density is \(\dfrac{L(t,x)}{t}\), i.e., its occupation density is \(L(t,x)=\dfrac{d(t\Gamma _{t})}{d\lambda } (x)\). For the existence of occupation densities, see Geman and Horowitz (1980).
The existence and properties of local time of Brownian motion and of diffusion processes are essential in the theory of stochastic integral where it is used to extend Ito’s formula. Basic properties of the local time of Brownian motion are:

(i)
For every \(x\in {\mathbb {R}},\) the stochastic process \(t\rightarrow L(t,x)\) is nonnegative and nondecreasing,

(ii)
For almost all \(\omega \), the function \((t,x)\rightarrow L(t,x)(\omega ) \) is jointly continuous,

(iii)
For every bounded measurable function \(g:{\mathbb {R}}\rightarrow \mathbb { R}\),
$$\begin{aligned} \displaystyle \int _{0}^{t}g(W_{s})ds=\int _{{\mathbb {R}}}g(x)L(t,x)dx. \end{aligned}$$
Local time is used to extend Ito’s rule for changing variables in stochastic integral. Specifically, if \((W_{t},t\ge 0)\) is a Brownian motion, then for \( x\in {\mathbb {R}}\), \(W_{t}x\) is a positive submartingale, and as such, by Doob–Meyer decomposition, it is the (unique) sum of a martingale and a continuous, increasing process, completely specified by Tanaka’s formula with the local time being precisely the continuous, increasing process:
where
The Tanaka’s formula provides the generalization of Ito’s formula to convex functions. See, e.g. Chung and Williams (1983) for details.
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Gourieroux, C., Nguyen, H.T. & Sriboonchitta, S. Nonparametric estimation of a scalar diffusion model from discrete time data: a survey. Ann Oper Res 256, 203–219 (2017). https://doi.org/10.1007/s1047901622736
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Keywords
 Diffusion model
 Local time
 Low frequency data
 Nonlinear canonical analysis
 Prediction operator