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Nonparametric estimation of a scalar diffusion model from discrete time data: a survey

Abstract

In view of rapid developments on nonparametric estimation of the drift and volatility functions in scalar diffusion models in financial econometrics, from discrete-time observations, we provide, in this paper, a survey of its state-of-the-art 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. 1.

    Bachelier is credited for the first construction of the Brownian motion.

  2. 2.

    See Merton (1990) and Duffie (1996) for classic textbooks for asset pricing theory written in continuous time.

  3. 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. 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. Florens-Zmirou 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. 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 Ait-Sahalia 1996b for an application).

  6. 6.

    Since the function \(g(y)=y\) is not in general an eigenfunction of the prediction operator (see Sect. 2.1).

  7. 7.

    They are derived for diffusions with reflecting barriers in Gobet et al. (2004), under much weaker assumptions in Karlsen and Tjostheim (2001).

  8. 8.

    See Meyn and Tweedie (1993) for the different definitions of recurrence for Markov processes.

References

  1. Ait-Sahalia, Y. (1996). Nonparametric pricing of interest rate derivative securities. Econometrica, 64, 527–560.

    Article  Google Scholar 

  2. Ait-Sahalia, Y. (1996). Testing continuous-time models of the spot interest rate. The Review of Financial Studies, 9(2), 385–426.

    Article  Google Scholar 

  3. Ait-Sahalia, Y. (2002). Maximum likelihood estimation of discretely sampled diffusions: A closed-form approximation approach. Econometrica, 70(1), 223–262.

    Article  Google Scholar 

  4. Akonom, J. (1993). Comportment asymptotique du temps d’occupation du processus des sommes partielles. Annales de l’Institut Henri Poincaré, 29, 57–81.

    Google Scholar 

  5. 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.

  6. Bandi, F., & Phillips, P. (2003). Fully nonparametric estimation of scalar diffusion models. Econometrica, 71, 243–283.

    Article  Google Scholar 

  7. Bandi, F., & Phillips, P. (2009). Nonstationary continuous-time processes. In Y. Ait Sahalia & L. Hansen (Eds.), Handbook of financial econometrics (pp. 139–202).

  8. Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–639.

    Article  Google Scholar 

  9. Bosq, D. (1998). Nonparametric statistics for stochastic processes. Lecture notes in statistics. New york: Springer.

    Book  Google Scholar 

  10. Bosq, D. (2000). Linear processes in functions spaces: Theory and applications. Lecture notes in statistics. New York: Springer.

    Book  Google Scholar 

  11. Bosq, D., & Davydov, Y. (1998). Local time and density estimation in continuous time. Lille: Publ. IRMA Univ. Lille 1.

    Google Scholar 

  12. Chen, X. (2007). Large sample sieve estimation of semi-nonparametric models. In Handbook of econometrics (vol. 5, pp. 5546–5632). Elsevier.

  13. 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.

    Article  Google Scholar 

  14. Chung, K. L., & Williams, R. J. (1983). Introduction to stochastic integration. Boston: Birkhauser.

    Book  Google Scholar 

  15. Cohen, A. (2000). Wavelet methods in numerical analysis. In P. Ciarlet (Ed.), Handbook of numerical analysis (7th ed., pp. 417–711). Amsterdam: North-Holland.

    Google Scholar 

  16. Dalalyan, A., & Kutoyants, Y. (2002). Asymptotically efficient trend coefficient estimation for ergodic diffusion. Mathematical Methods of Statistics, 11, 402–421.

    Google Scholar 

  17. Darolles, S., Florens, J. P., & Gourieroux, C. (2004). Kernel based nonlinear canonical analysis and time reversibility. Journal of Econometrics, 119, 323–353.

    Article  Google Scholar 

  18. Darolles, S., & Gourieroux, C. (2001). Truncated dynamics and estimation of diffusion equations. Journal of Econometrics, 102, 1–22.

    Article  Google Scholar 

  19. Darolles, S. & Serot, I. (2000). Empirical local time for processes observed on a grid, CREST DP.

  20. 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.

  21. Duffie, D. (1996). Dynamic asset pricing theory. Princeton: princeton University Press.

    Google Scholar 

  22. Dunford, N., & Schwartz, J. (1963). Linear operators (Vol. 2). New York: Wiley.

    Google Scholar 

  23. Fan, J., & Yao, Q. (2003). Nonlinear time series: Nonparametric and parametric methods. New York: Springer.

    Book  Google Scholar 

  24. 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.

    Article  Google Scholar 

  25. Florens-Zmirou, D. (1993). On estimating the diffusion coefficient from discrete observations. Journal of Applied Probability, 30, 790–804.

    Article  Google Scholar 

  26. Gao, J. (2007). Nonlinear time series: Semiparametric and nonparametric methods. London: Chapman & Hall.

    Book  Google Scholar 

  27. Gao, J., King, M., Lu, Z., & Tjostheim, D. (2009). Specification testing in nonlinear and nonstationary time series autoregression. Annals of Statistics, 37, 3893–3928.

    Article  Google Scholar 

  28. Geman, D., & Horowitz, J. (1980). Occupation densities. Annals of Probability, 8, 1–67.

    Article  Google Scholar 

  29. Gobet, E., Hoffman, M., & Reiss, M. (2004). Nonparametric estimation of scalar diffusions based on low frequency data. Annals of Statistics, 32, 2223–2253.

    Article  Google Scholar 

  30. Gourieroux, C., Monfort, A., & Renault, E. (1993). Indirect inference. Journal of Applied Econometrics, 8, 85–118.

    Article  Google Scholar 

  31. Guerre, E. (2007). Design adaptive pointwise nonparametric regression estimation for recurrent Markov time series, DP.

  32. Hansen, L., & Sargent, T. (1983). The dimensionality of the aliasing problem in models with rational spectral densities. Econometrica, 51, 377–387.

    Article  Google Scholar 

  33. Hansen, L., & Scheinkman, J. (1995). Back to the future : Generating moment implications for continuous-time Markov processes. Econometrica, 63, 767–804.

    Article  Google Scholar 

  34. Hansen, L., Scheinkman, J., & Touzi, N. (1998). Spectral methods for identifying scalar diffusions. Journal of Econometrics, 86, 1–32.

    Article  Google Scholar 

  35. Hoffmann, M. (1999). Adaptive estimation in diffusion processes. Stochastic Processes and Their Applications, 79, 135–163.

    Article  Google Scholar 

  36. 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.

    Article  Google Scholar 

  37. Karlin, S., & Taylor, H. (1981). A second course in stochastic processes. New York: Academic Press.

    Google Scholar 

  38. Karlsen, H., & Tjostheim, D. (2001). Nonparametric estimation in null recurrent time series. Annals of Statistics, 29, 372–411.

    Article  Google Scholar 

  39. Kutoyants, Yu A. (1999). Efficient density estimation for ergodic diffusion processes. Statistical Inference for Stochastic Processes, 1, 131–155.

    Article  Google Scholar 

  40. Kutoyants, Y. (2013). Statistical inference for ergodic diffusion processes. New York: Springer.

    Google Scholar 

  41. Kessler, M., & Sorensen, M. (1999). Estimating equations based on eigenfunctions for directly observed diffusion process. Bernoulli, 5, 299–314.

    Article  Google Scholar 

  42. Lindstrom, E. (2007). Estimating parameters in diffusion processes using an approximate maximum likelihood approach. Annals of Operations Research, 151, 269–288.

    Article  Google Scholar 

  43. Levy, P. (1948). Processus Stochastiques et Mouvement Brownien. Paris: Gauthier-Villars.

    Google Scholar 

  44. Merton, R. (1973). Theory of rational option pricing. The Bell Journal of Economics and Management Science, 141–183.

  45. Merton, R. (1990). Continuous time finance. Oxford: Blackwell.

    Google Scholar 

  46. Meyn, S., & Tweedie, R. (1993). Markov chains and stochastic stability. London: Springer.

    Book  Google Scholar 

  47. 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.

    Google Scholar 

  48. 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.

    Google Scholar 

  49. 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.

    Google Scholar 

  50. 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.

    Article  Google Scholar 

  51. 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.

    Article  Google Scholar 

  52. Park, J. & Phillips, P. (1998). Nonstationary density estimation and kernel autoregression, Cowles Foundation DP 1181, Yale Univ.

  53. Revuz, D., & Yor, M. (1998). Continuous martingales and Brownian motion. New-York: Springer.

    Google Scholar 

  54. 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.

    Google Scholar 

  55. 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.

    Article  Google Scholar 

  56. Wang, Q., & Phillips, P. (2009). Asymptotic theory for local time density estimation and nonparametric cointegrating regression. Econometric Theory, 25, 710–738.

    Article  Google Scholar 

  57. Xu, K. L. (2009). Empirical likelihood-based inference for nonparametric recurrent diffusions. Journal of Econometrics, 153, 65–82.

    Article  Google Scholar 

  58. Xu, K. L. (2010). Reweighted functional estimation of diffusion models. Econometric Theory, 26, 541–563.

    Article  Google Scholar 

Download references

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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Correspondence to Hung T. Nguyen.

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 continuous-time 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

$$\begin{aligned} \displaystyle L(t,x)= & {} \lim _{\varepsilon \searrow 0}\dfrac{1}{2\varepsilon } \lambda \lbrace s\in [0,t]:W_{s}\in (x-\varepsilon ,x+\varepsilon )\rbrace \\ \displaystyle= & {} \lim _{\varepsilon \searrow 0}\dfrac{1}{2\varepsilon } \int _{0}^{t}1_{(x-\varepsilon ,x+\varepsilon )}(W_{s})ds, \end{aligned}$$

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(tx) 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

$$\begin{aligned} \displaystyle \int _{0}^{t}1_{B}(W_{s})ds=\int _{B}L(t,x)dx. \end{aligned}$$

In general, the empirical measure of a real-valued stochastic process \( (X_{t},t\ge 0)\) is defined on \({\mathcal {B}}({\mathbb {R}})\) as

$$\begin{aligned} \displaystyle \Gamma _{t}(B)=\dfrac{1}{t}\int _{0}^{t}1_{B}(X_{s})ds. \end{aligned}$$

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:

  1. (i)

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

  2. (ii)

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

  3. (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:

$$\begin{aligned} |W_{t}-x|=|x|+\int _{0}^{t}\text {sgn}(W_{s}-x)dW_{s}+L(t,x), \end{aligned}$$

where

$$\begin{aligned} sgn(y)=\left\{ \begin{array}{ll} -1&{}\quad \text {for}\;\;y\le 0, \\ 1&{}\quad \text {for}\;\;y>0. \end{array} \right. \end{aligned}$$

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/s10479-016-2273-6

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Keywords

  • Diffusion model
  • Local time
  • Low frequency data
  • Nonlinear canonical analysis
  • Prediction operator