## Bernoulli

• Bernoulli
• Volume 4, Number 3 (1998), 283-303.

### Limit theorems for discretely observed stochastic volatility models

#### Abstract

A general set-up is proposed to study stochastic volatility models. We consider here a two-dimensional diffusion process $( Y t,V t)$ and assume that only $( Y t)$ is observed at $n$ discrete times with regular sampling interval $Δ$. The unobserved coordinate $( V t)$ is an ergodic diffusion which rules the diffusion coefficient (or volatility) of $( Y t)$. The following asymptotic framework is used: the sampling interval tends to $0$, while the number of observations and the length of the observation time tend to infinity. We study the empirical distribution associated with the observed increments of $( Y t)$. We prove that it converges in probability to a variance mixture of Gaussian laws and obtain a central limit theorem. Examples of models widely used in finance, and included in this framework, are given.

#### Article information

Source
Bernoulli, Volume 4, Number 3 (1998), 283-303.

Dates
First available in Project Euclid: 19 March 2007

https://projecteuclid.org/euclid.bj/1174324982

Mathematical Reviews number (MathSciNet)
MR1653264

Zentralblatt MATH identifier
0916.60075

#### Citation

Genon-Catalot, Valentine; Jeantheau, Thierry; Larédo, Catherine. Limit theorems for discretely observed stochastic volatility models. Bernoulli 4 (1998), no. 3, 283--303. https://projecteuclid.org/euclid.bj/1174324982

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