##
**Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics. (With discussion).**
*(English)*
Zbl 0983.60028

The authors consider stochastic volatility asset price models following a stochastic differential equation of the form
\[
dX_t= (\mu+\beta\sigma^2(t)) dt+ \sigma(t) dB_t,
\]
where \(\mu\), \(\beta\) are constants, \(B_t\) is a Brownian motion and \(\sigma^2(t)\) is the instantaneous volatility process. The novel point of view is that \(\sigma^2\) is modelled using an Ornstein-Uhlenbeck type stochastic differential equation driven by a Lévy process \(Z_t\) which is in the simplest case
\[
d\sigma^2(t)= -\lambda\sigma^2(t) dt+ dZ_{\lambda t}.
\]
Notice that, although \(\sigma^2(t)\) has jumps, the price process \(X_t\) is continuous. Various authors have considered similar models but only with geometric Ornstein-Uhlenbeck volatility processes.

The Lévy process \(Z_t\) used here is a subordinator (i.e., \(Z_0= 0\) with increasing paths) which means that \(\sigma^2(t)\) jumps upwards and creeps (exponentially) downwards. The main theoretical part of the paper deals with the description of (distributional properties of) \(\sigma^2\). Particular attention is payed to the problem how to choose \(Z_t\) if \(\sigma^2\) has certain marginal distributions, and vice versa. An interesting approach is that the authors propose to model the driving Lévy process directly by using the (tail of its) Lévy measure which is particularly interesting for simulations [cf. L. Bondesson, Adv. Appl. Probab. 14, 855-869 (1982; Zbl 0494.60013)]. Using weighted averages of independent volatility processes of the type discussed above, the model can be changed to accommodate also long-range dependence effects in the price process.

The paper focusses, even in its theoretical aspects, on distributions of generalized inverse Gaussian type (a family that includes among others: inverse Gaussian, hyperbolic, inverse \(\chi^2\), Gamma, reciprocal Gamma distributions). The reasons for this choice are detailed in a long statistical section where on the basis of real-life data sets various estimation and fitting procedures are discussed. At the end of the paper the authors touch some generalizations of their model, mainly the vector-valued case.

A 34 page discussion (pp. 208–241 with references) concludes the paper where various mathematicians comment on the results. The authors are overwhelmingly commended upon their approach but some critical voices point out that even the most flexible and sophisticated models cannot capture everyday realities of the markets.

The Lévy process \(Z_t\) used here is a subordinator (i.e., \(Z_0= 0\) with increasing paths) which means that \(\sigma^2(t)\) jumps upwards and creeps (exponentially) downwards. The main theoretical part of the paper deals with the description of (distributional properties of) \(\sigma^2\). Particular attention is payed to the problem how to choose \(Z_t\) if \(\sigma^2\) has certain marginal distributions, and vice versa. An interesting approach is that the authors propose to model the driving Lévy process directly by using the (tail of its) Lévy measure which is particularly interesting for simulations [cf. L. Bondesson, Adv. Appl. Probab. 14, 855-869 (1982; Zbl 0494.60013)]. Using weighted averages of independent volatility processes of the type discussed above, the model can be changed to accommodate also long-range dependence effects in the price process.

The paper focusses, even in its theoretical aspects, on distributions of generalized inverse Gaussian type (a family that includes among others: inverse Gaussian, hyperbolic, inverse \(\chi^2\), Gamma, reciprocal Gamma distributions). The reasons for this choice are detailed in a long statistical section where on the basis of real-life data sets various estimation and fitting procedures are discussed. At the end of the paper the authors touch some generalizations of their model, mainly the vector-valued case.

A 34 page discussion (pp. 208–241 with references) concludes the paper where various mathematicians comment on the results. The authors are overwhelmingly commended upon their approach but some critical voices point out that even the most flexible and sophisticated models cannot capture everyday realities of the markets.

Reviewer: René L.Schilling (Brighton)

### MSC:

60G35 | Signal detection and filtering (aspects of stochastic processes) |

91B28 | Finance etc. (MSC2000) |

60J70 | Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) |

62P05 | Applications of statistics to actuarial sciences and financial mathematics |