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


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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  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.


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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.



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:

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


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

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).

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