Analysis on Wiener space and anticipating stochastic calculus.

*(English)*Zbl 0915.60062
Barlow, Martin T. et al., Lectures on probability theory and statistics. Ecole d’Eté de probabilités de Saint-Flour XXV - 1995. Lectures given at the summer school in Saint-Flour, France, July 10-26, 1995. Berlin: Springer. Lect. Notes Math. 1690, 123-237 (1998).

The author’s work is essentially based on lectures given in St. Flour in 1995, and covers, in greater detail, most of the courses given there. His lectures are a real work of reference in the domain of analysis on the Wiener space and anticipating stochastic calculus, and they are endowed with a very complete bibliography of works published in this subject.

The first three chapters are devoted to the stochastic calculus of variations on the Wiener space, introduced by Malliavin (1976), which is in particular applied to prove Meyer’s inequalities (second chapter) and to study regularity properties of probability laws (third chapter). In Chapter 4 the properties of the support of the probability distribution of a random element are studied in the underlying Gaussian space. For nondegenerate smooth random vectors, the points where the density is strictly positive are characterized by means of the notion of the skeleton. This characterization is used to deduce Varadhan-type estimates. The last two chapters are devoted to the anticipating stochastic calculus. In Chapter 5 the Skorokhod and the extended Stratonovich integrals are introduced, the relationship between the both is studied and a change-of-variable formula is established. The anticipating stochastic integrals allow to consider stochastic differential equations which have coefficients anticipating the driving Brownian motion, or on which some two-sided boundary condition is imposed. Existence and uniqueness results for such types of equations are presented in Chapter 6.

For the entire collection see [Zbl 0894.00045].

The first three chapters are devoted to the stochastic calculus of variations on the Wiener space, introduced by Malliavin (1976), which is in particular applied to prove Meyer’s inequalities (second chapter) and to study regularity properties of probability laws (third chapter). In Chapter 4 the properties of the support of the probability distribution of a random element are studied in the underlying Gaussian space. For nondegenerate smooth random vectors, the points where the density is strictly positive are characterized by means of the notion of the skeleton. This characterization is used to deduce Varadhan-type estimates. The last two chapters are devoted to the anticipating stochastic calculus. In Chapter 5 the Skorokhod and the extended Stratonovich integrals are introduced, the relationship between the both is studied and a change-of-variable formula is established. The anticipating stochastic integrals allow to consider stochastic differential equations which have coefficients anticipating the driving Brownian motion, or on which some two-sided boundary condition is imposed. Existence and uniqueness results for such types of equations are presented in Chapter 6.

For the entire collection see [Zbl 0894.00045].

Reviewer: R.Buckdahn (Brest)

##### MSC:

60H05 | Stochastic integrals |

60-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory |

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

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |

60J65 | Brownian motion |