Limit theorems for discretely observed stochastic volatility models
type: Item
instance of: scholarly article
title: Limit theorems for discretely observed stochastic volatility models
zbMATH Open document ID: 0916.60075
DOI: 10.2307/3318718
author: Thierry Jeantheau
author: V. Genon-Catalot
author: Catherine Larédo
published in: Bernoulli
publication date: 1999-07-12T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.bj/1174324982
review text: The authors consider the following two-dimensional diffusion model for stock prices with stochastic volatility: \(Y_0 = 0\), \(dY_t = \varphi(V_t) dt + V_t^{1/2} dB_t\) and \(V_0 = \eta\), \(dV_t = b(V_t) dt + a(V_t) dW_t\) where \(B\) and \(W\) are independent Wiener processes. \(Y\) is interpreted as the process of logarithmic stock prices. \(V\) is assumed to be a strictly positive, ergodic, stationary diffusion. As typical for the finance background, the volatility process \(V\) is unobservable, while one has a discrete series \(Y_{t_1}, \ldots, Y_{t_n}\) of observations for \(Y\). The authors' aim is to derive information about the stationary distribution of the hidden process \(V\) using the given discrete data for \(Y\). They assume that the observation times are regularly spaced, \(t_i = i \Delta\), and show that the empirical distribution \(\hat{P}\) of the renormalized increments \(X_i = (Y_{t_i}-Y_{t_{i-1}})/\Delta^{1/2}\) converges to a variance mixture of Gaussian laws \(P_\pi\) as the length of the sampling interval \(\Delta\) approaches zero and the total observation time \(t_n = n \Delta\) tends to infinity. Moreover, \(P_\pi\) is identified as the distribution of a random variable \(X = \varepsilon \eta^{1/2}\) where \(\varepsilon\) is standard normal distributed and independent of \(\eta=V_0\). If the asymptotic framework is such that \(n \Delta^2\) tends to zero, the authors also show that a central limit theorem holds.
Mathematics Subject Classification ID: 60J70
Mathematics Subject Classification ID: 91B28
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 1239892
zbMATH Keywords: diffusion processes
zbMATH Keywords: discrete time observations
zbMATH Keywords: empirical distributions
zbMATH Keywords: limit theorems
zbMATH Keywords: mathematical finance
zbMATH Keywords: stochastic volatility
reviewed by: Peter Bank
MaRDI profile type: Publication
OpenAlex ID: W2007383562
description: scientific article; zbMATH DE number 1239892