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.

Appendix

1.1 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(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

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

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:

• (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:

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

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.