Extending probability spaces and adapted distribution.

*(English)*Zbl 0776.60048
Séminaire de probabilités XXVI, Lect. Notes Math. 1526, 560-574 (1992).

[For the entire collection see Zbl 0754.00008.]

Adapted distributions were originally inspired by stochastic differential equations which do not have strong solutions and for which it is therefore necessary to extend the space of filtrations. A first purpose of this article is to show that the equivalence of processes associated to the extension of the probability space corresponds to the notion of adapted distributions, the advantage of the latter being a more precise description of the fact that two processes have the same properties with respect to the filtrations associated with this extension. The second purpose is to show how the use of the method of extension of spaces instead of the notion of saturated spaces leads to much simpler and slightly more general versions of the theorems proved by the author and H. J. Keisler [Trans. Am. Math. Soc. 286, 159-201 (1984; Zbl 0548.60019)] and by the author [Ann. Probab. 15, 1600-1611 (1987; Zbl 0634.60033)].

Adapted distributions were originally inspired by stochastic differential equations which do not have strong solutions and for which it is therefore necessary to extend the space of filtrations. A first purpose of this article is to show that the equivalence of processes associated to the extension of the probability space corresponds to the notion of adapted distributions, the advantage of the latter being a more precise description of the fact that two processes have the same properties with respect to the filtrations associated with this extension. The second purpose is to show how the use of the method of extension of spaces instead of the notion of saturated spaces leads to much simpler and slightly more general versions of the theorems proved by the author and H. J. Keisler [Trans. Am. Math. Soc. 286, 159-201 (1984; Zbl 0548.60019)] and by the author [Ann. Probab. 15, 1600-1611 (1987; Zbl 0634.60033)].

Reviewer: M.Dozzi (Nancy)

##### MSC:

60G07 | General theory of stochastic processes |

60E05 | Probability distributions: general theory |

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

03B48 | Probability and inductive logic |