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**Diffusion-type models with given marginal distribution and autocorrelation function.**
*(English)*
Zbl 1066.60071

Summary: Flexible stationary diffusion-type models are developed that can fit both the marginal distribution and the correlation structure found in many time series from, for example, finance and turbulence. Diffusion models with linear drift and a known and prespecified marginal distribution are studied, and the diffusion coefficients corresponding to a large number of common probability distributions are found explicitly. An approximation to the diffusion coefficient based on saddlepoint techniques is developed for use in cases where there is no explicit expression for the diffusion coefficient. It is demonstrated theoretically as well as in a study of turbulence data that sums of diffusions with linear drift can fit complex correlation structures. Any infinitely divisible distribution satisfying a weak regularity condition can be obtained as a marginal distribution.

### MSC:

60J60 | Diffusion processes |

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

60H30 | Applications of stochastic analysis (to PDEs, etc.) |

### Keywords:

ergodic diffusions; generalized hyperbolic distributions; long-range dependence; saddlepoint approximation; stochastic differential equation; turbulence
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\textit{B. M. Bibby} et al., Bernoulli 11, No. 2, 191--220 (2005; Zbl 1066.60071)

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### References:

[1] | Aït-Sahalia, Y. (1996) Nonparametric pricing of interest rate derivative securities. Econometrica, 64, 527-560. · Zbl 0844.62094 |

[2] | Barndorff-Nielsen, O.E. (1998) Processes of normal inverse Gaussian type. Finance Stochastics, 2, 41-68. · Zbl 0894.90011 |

[3] | Barndorff-Nielsen, O.E. and Shephard, N. (2001) Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial econometrics (with discussion). J. Roy. Statist. Soc. Ser. B., 63, 167-241. JSTOR: · Zbl 0983.60028 |

[4] | Barndorff-Nielsen, O.E., Jensen, J.L. and Sørensen, M. (1990) Parametric modelling of turbulence. Philos. Trans. Roy. Soc. Lond. Ser. A., 332, 439-455. |

[5] | Barndorff-Nielsen, O.E., Jensen, J.L. and Sørensen, M. (1993) A statistical model for the streamwise component of a turbulent velocity field. Ann. Geophys., 11, 99-103. |

[6] | Barndorff-Nielsen, O.E., Jensen, J.L. and Sørensen, M. (1998) Some stationary processes in discrete and continuous time. Adv. in Appl. Probab., 30, 989-1007. · Zbl 0930.60026 |

[7] | Bibby, B.M. and Sørensen, M. (1997) A hyperbolic diffusion model for stock prices. Finance Stochastics, 1, 25-41. · Zbl 0883.90010 |

[8] | Bibby, B.M. and Sørensen, M. (2001) Simplified estimating functions for diffusion models with a high-dimensional parameter. Scand. J. Statist., 28, 99-112. · Zbl 0973.60071 |

[9] | Bibby, B.M. and Sørensen, M. (2003) Hyperbolic processes in finance. In S. Rachev (ed.), Handbook of Heavy Tailed Distributions in Finance, pp. 211-248. Amsterdam: Elsevier Science. |

[10] | Bibby, B.M. and Sørensen, M. (2004) Flexible stochastic volatility models of the diffusion type. In preparation |

[11] | Bleistein, N. (1966) Uniform asymptotic expansions of integrals with stationary point near algebraic singularity. Comm. Pure Appl. Math., 19, 353-370. · Zbl 0145.15801 |

[12] | Cox, D.R. (1984) Long-range dependence: A review. In H.A. David and H.T. David (eds), Statistics: An Appraisal. Ames: Iowa State University Press. |

[13] | Cox, J.C., Ingersoll Jr., J.E. and Ross, S.A. (1985) A theory of the term structure of interest rates. Econometrica, 53, 385-407. JSTOR: · Zbl 1274.91447 |

[14] | Daniels, H.E. (1954) Saddlepoint approximations in statistics. Ann. Math. Statist., 25, 631-650. · Zbl 0058.35404 |

[15] | Daniels, H.E. (1987) Tail probability approximations. Internat. Statist. Rev., 55, 37-48. JSTOR: · Zbl 0614.62016 |

[16] | De Jong, F., Drost, F.C. and Werker, B.J.M. (2001) A jump-diffusion model for exchange rates in a target zone. Statist. Neerlandica, 55, 270-300. · Zbl 1075.91533 |

[17] | Engelbert, H.J. and Schmidt, W. (1985) On solutions of one-dimensional stochastic differential equations without drift. Z. Wahrscheinlichkeitstheorie Verw. Geb., 68, 287-314. · Zbl 0535.60049 |

[18] | Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. II. New York: Wiley. · Zbl 0219.60003 |

[19] | Genon-Catalot, V., Jeantheau, T. and Larédo, C. (2000) Stochastic volatility models as hidden Markov models and statistical applications. Bernoulli, 6, 1051-1079. · Zbl 0966.62048 |

[20] | Karlin, S. and Taylor, M. (1981) A Second Course in Stochastic Processes. Orlando, FL: Academic Press. · Zbl 0469.60001 |

[21] | Larsen, K.S. and Sørensen, M. (2003) A diffusion model for exchange rates in a target zone. Preprint no. 6, Department of Applied Mathematics and Statistics, University of Copenhagen. |

[22] | Madan, D.B. and Seneta, E. (1990) The variance gamma (V.G.) model for share market returns. J. Business, 63, 511-524. · Zbl 0693.53038 |

[23] | Madan, D.B. and Yor, M. (2002) Making Markov martingales meet marginals: with explicit constructions. Bernoulli, 8, 509-536. · Zbl 1009.60037 |

[24] | Mikkelsen, H.E. (1988) Turbulence in the Wake Zone of a Vegetated 2-Dimensional Dune, Geoskrifter 30. Geology Institute, University of Aarhus. |

[25] | Mikkelsen, H.E. (1989) Wind Flow and Sediment Transport over a Low Coastal Dune, Geoskrifter 32. Geology Institute, University of Aarhus. |

[26] | Pedersen, A.R. (2000) Estimating the nitrous oxide emission rate from the soil surface by means of a diffusion model. Scand. J. Statist., 27, 385-404. · Zbl 0976.62111 |

[27] | Skorokhod, A.V. (1989) Asymptotic Methods in the Theory of Stochastic Differential Equation. Providence, RI: Americal Mathematical Society. · Zbl 0695.60055 |

[28] | Steutel, F.W. (1983) Infinite divisibility. In N.L. Johnson, S. Kotz and C.B. Read (eds), Encyclopedia of Statistical Sciences, Vol. 4, pp. 114-116. New York: Wiley. |

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