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**Inference for stochastic volatility models using time change transformations.**
*(English)*
Zbl 1189.91220

Diffusion processes provide a reasonable model for continuous time phenomena. Unfortunately, inference for diffusion processes is difficult. A review of results concerning inference for diffusion processes was compiled by H. Sørensen [Int. Stat. Rev. 72, 337–354 (2004)]. Likelihood-based methods usually approximate the likelihood so that the discretization error becomes arbitrarily small. These methods often depend on the Markov property.

Sequential Monte Carlo techniques can be used to implement parameter estimation for stochastic volatility models. Another approach to estimating parameters for these models arises from Bayesian inference using Markov chain Monte Carlo (MCMC) methods. Initial MCMC schemes were introduced by Jones [Review of Financial Studies 16, 793–843 (2003)]. Unfortunately, the algorithm’s convergence properties degenerate as the number of imputed points increases. G. O. Roberts and O. Stramer overcome this problem in the one-dimensional context with a reparametrization [Biometrika 88, No. 3, 603–621 (2001; Zbl 0985.62066)].

This paper introduces another reparametrization which operates on the time scale of the observed diffusion rather than on its path. This transformation alters the time axis of the diffusion. This allows for the construction of irreducible and efficient MCMC models to accommodate the time change of the diffusion path. This approach also relies upon the Markov property and can be coupled with the approaches presented in the paper by Roberts and Stramer [loc. cit.] and a submitted result by S. Chib, M. K. Pitt, and N. Shephard.

The authors discuss the need for reparametrization in section 2 and introduce the class of transformations based on changing the time scale of the diffusion process in section 3. Section 4 provides the details of the MCMC implementation. Numerical experiments of the method and an application to US Treasury bill rates are also presented. This method compares well with competing models and is not computationally expensive. The computing cost of the authors’ approach is \(O(m)\), compared to \(O(m^2)\) for competing models.

Sequential Monte Carlo techniques can be used to implement parameter estimation for stochastic volatility models. Another approach to estimating parameters for these models arises from Bayesian inference using Markov chain Monte Carlo (MCMC) methods. Initial MCMC schemes were introduced by Jones [Review of Financial Studies 16, 793–843 (2003)]. Unfortunately, the algorithm’s convergence properties degenerate as the number of imputed points increases. G. O. Roberts and O. Stramer overcome this problem in the one-dimensional context with a reparametrization [Biometrika 88, No. 3, 603–621 (2001; Zbl 0985.62066)].

This paper introduces another reparametrization which operates on the time scale of the observed diffusion rather than on its path. This transformation alters the time axis of the diffusion. This allows for the construction of irreducible and efficient MCMC models to accommodate the time change of the diffusion path. This approach also relies upon the Markov property and can be coupled with the approaches presented in the paper by Roberts and Stramer [loc. cit.] and a submitted result by S. Chib, M. K. Pitt, and N. Shephard.

The authors discuss the need for reparametrization in section 2 and introduce the class of transformations based on changing the time scale of the diffusion process in section 3. Section 4 provides the details of the MCMC implementation. Numerical experiments of the method and an application to US Treasury bill rates are also presented. This method compares well with competing models and is not computationally expensive. The computing cost of the authors’ approach is \(O(m)\), compared to \(O(m^2)\) for competing models.

Reviewer: Thomas Wakefield (Youngstown)

### MSC:

91G60 | Numerical methods (including Monte Carlo methods) |

65C05 | Monte Carlo methods |

65C60 | Computational problems in statistics (MSC2010) |

### Citations:

Zbl 0985.62066
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\textit{K. Kalogeropoulos} et al., Ann. Stat. 38, No. 2, 784--807 (2010; Zbl 1189.91220)

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