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**Statistical causality and adapted distribution.**
*(English)*
Zbl 1249.60063

Summary: D. N. Hoover and H. J. Keisler [Trans. Am. Math. Soc. 286, 159–201 (1984; Zbl 0548.60019)] introduced the notion of the adapted distribution of two stochastic processes, which in a way represents the notion of equivalence of those processes. This very important property is hard to prove directly, so we continue the work of Keisler and Hoover in finding sufficient conditions for two stochastic processes to have the same adapted distribution. For this purpose, we use the concept of causality between stochastic processes, which is based on Granger’s definition of causality. We also provide applications of our results to solutions of some stochastic differential equations.

### MSC:

60G07 | General theory of stochastic processes |

03C98 | Applications of model theory |

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

### Keywords:

filtration; causality; adapted distribution; weak solution of stochastic differential equation### Citations:

Zbl 0548.60019
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\textit{L. Petrović} and \textit{S. Dimitrijević}, Czech. Math. J. 61, No. 3, 827--843 (2011; Zbl 1249.60063)

### References:

[1] | D. Aldous: Weak convergence and the general theory of processes, preprint (1981). |

[2] | P. Bremaud, M. Yor: Changes of filtrations and of probability measures. Z. Wahrscheinlichkeitstheor. Verw. Geb. 45 (1978), 269–295. · Zbl 0415.60048 |

[3] | S. Fajardo, H. J. Keisler: Model Theory of Stochastic Processes, Lecture Notes in Logic vol. 14, Urbana, 2002, Association for Symbolic Logic. |

[4] | J.P. Florens, D. Fougères: Noncausality in continuous time. Econometrica 64 (1996), 1195–1212. · Zbl 0856.90020 |

[5] | J.B. Gill, L. Petrović: Causality and stochastic dynamic systems. SIAM J. Appl. Math. 47 (1987), 1361–1366. · Zbl 0636.93069 |

[6] | C.W. J. Granger: Investigating causal relations by econometric models and cross spectral methods. Econometrica 37 (1969), 424–438. · Zbl 1366.91115 |

[7] | D.N. Hoover: Adapted distribution, Probability theory and applications. Proc. World Congr. Bernoulli Soc., Tashkent/USSR 1986 1 (1987), 201–204. |

[8] | D.N. Hoover: Synonymity, generalized martingales, and subfiltrations. Ann. Probab. 12 (1984), 703–713. · Zbl 0545.60040 |

[9] | D.N. Hoover: A characterization of adapted distribution. Ann. Probab. 15 (1987), 1600–1611. · Zbl 0634.60033 |

[10] | D.N. Hoover: Extending probability spaces and adapted distribution, Séminare de probabilitités XXVI. Lect. Notes Math. 1526 (1992), 560–574. · Zbl 0776.60048 |

[11] | D.N. Hoover, H. J. Keisler: Adapted probability distributions. Trans. Am. Math. Soc. 286 (1984), 159–201. · Zbl 0548.60019 |

[12] | R. S. Lipster, A.N. Shiryaev: Statistics of Random Processes I, General theory. Applications of Mathematics vol. 5, Springer-Verlag, New York-Heidelberg-Berlin, 1977. |

[13] | P.A. Mykland: Statistical causality. Statistical Report no. 14, Dept. of Mathematics, University of Bergen, 1986. |

[14] | D. Nualart, Y. Ouknine: Regularization of differential equations by fractional noise. Stochastic Processes. Appl. 102 (2002), 103–116. · Zbl 1075.60536 |

[15] | L. Petrović: Causality and Markovian representations. Stat. Probab. Lett. 29 (1996), 221–227. · Zbl 0875.93483 |

[16] | L. Petrović: Causality and stochastic realization. Int. J. Math. Math. Sci. (2005), 349–356. · Zbl 1099.46051 |

[17] | L. Petrović, D. Stanojević: Statistical Causality, Extremal Measures and Weak Solutions of Stochastic Differential Equations with Driving Semimartingales, vol. 9. JMMA, 2010, pp. 113–128. · Zbl 1229.60055 |

[18] | L. Petrović, S. Dimitrijević: Some models of causality and stochastic differential equations driven by fractional Brownian motion. Facta Univ., Ser. Math. Inf. 20 (2005), 113–122. · Zbl 1104.60031 |

[19] | Y.A. Rozanov: Innovation Processes, Scripta Series in Mathematics. Washington, V.H. Winston and Sons, New York, 1977. |

[20] | Y.A. Rozanov: Markov Random Fields. Springer-Verlag, Berlin-New York-Heidelberg, 1982. |

[21] | D.W. Strook, M. Yor: On extremal solutions of martingale problems. Ann. Sci. éc. Norm. Supér. (4) 13 (1980), 95–164. |

[22] | M. Yor: Sur L’étude des Martingales Continues Extremales. Stochastics 2 (1979), 191–196. · Zbl 0409.60043 |

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