# AN INTRODUCTION TO MALLIAVIN CALCULUS AND ITS APPLICATIONS

# AN INTRODUCTION TO MALLIAVIN CALCULUS AND ITS APPLICATIONS

28 Feb 2019

4 Mar 2019

7 Mar 2019

14 Mar 2019

Always at 11.00-13.00

Univ. of Barcelona

*Course description*

The Malliavin calculus extends the classical calculus of variations from deterministic functions to stochastic processes. It was introduced by Paul Malliavin in the 70’s to provide a probabilistic proof of Hormander hypoellipticity theorem. The main application of Malliavin calculus is to establish the regularity of the probability distributions of functionals of a Gaussian process. Basic examples are diffusion processes and solutions to stochastic partial differential equations. In addition to this main application and starting from the pioneering work by Ivan Nourdin and Giovanni Peccati, the Malliavin calculus, combined with Stein’s method for normal approximations, has proved to be a useful tool to derive quantitative central limit theorems.

The goal of this course is to introduce the basic elements of Malliavin calculus and discuss its applications to regularity of probability laws, normal approximations and mathematical finance.

*Motivation*

Malliavin calculus is an active research area in stochastic analysis, with a wide scope of applications in a number of fields including statistics, functional analysis and finance. The combination of Malliavin calculus and Stein’s method has proved to be a powerful theory leading to new results in areas as diverse as cosmology and statistical inference. In mathematical finance, the Malliavin calculus has played an important role in designing numerical computations of price sensitivities and finding hedging portfolios.

28 Feb 2019

4 Mar 2019

7 Mar 2019

14 Mar 2019

Always at 11.00-13.00

Univ. of Barcelona

*Course description*

The Malliavin calculus extends the classical calculus of variations from deterministic functions to stochastic processes. It was introduced by Paul Malliavin in the 70’s to provide a probabilistic proof of Hormander hypoellipticity theorem. The main application of Malliavin calculus is to establish the regularity of the probability distributions of functionals of a Gaussian process. Basic examples are diffusion processes and solutions to stochastic partial differential equations. In addition to this main application and starting from the pioneering work by Ivan Nourdin and Giovanni Peccati, the Malliavin calculus, combined with Stein’s method for normal approximations, has proved to be a useful tool to derive quantitative central limit theorems.

The goal of this course is to introduce the basic elements of Malliavin calculus and discuss its applications to regularity of probability laws, normal approximations and mathematical finance.

*Motivation*

Malliavin calculus is an active research area in stochastic analysis, with a wide scope of applications in a number of fields including statistics, functional analysis and finance. The combination of Malliavin calculus and Stein’s method has proved to be a powerful theory leading to new results in areas as diverse as cosmology and statistical inference. In mathematical finance, the Malliavin calculus has played an important role in designing numerical computations of price sensitivities and finding hedging portfolios.

David Nualart, University of Kansas (USA)

Email: nualart@ku.edu

David Nualart, University of Kansas (USA)

### Biosketch

David Nualart works in stochastic analysis. His research interests focus on the application of Malliavin calculus to a wide range of topics including regularity of probability laws, anticipating stochastic calculus, stochastic integral representations and central limit theorems for Gaussian functionals. His recent research deals with the stochastic calculus with respect to the fractional Brownian motion and related processes and central limit theorems. Other fields of interest are stochastic partial differential equations, rough path analysis and mathematical finance.

### Research Interests

- Malliavin calculus
- Anticipative stochastic calculus
- Central limit theorems
- Stochastic partial differential equations
- Fractional Brownian motion

##### Contents

The course will consist of five two-hours sessions devoted to the following topics:

1. One-dimensional Gaussian analysis. Brownian motion. Basic properties of the derivative and divergence operators on the Wiener space. The divergence as a stochastic integral.

2. Multiple stochastic integrals. Wiener chaos expansions. The Ornstein-Uhlenbeck semigroup. Meyer’s inequalities. Stochastic integral representations. Clark-Ocone formula.

3. Existence and regularity of the density for Wiener functionals. Density formulas. Application of Malliavin calculus to diffusion processes. Regularity of the density under Hormander’s conditions.

4. Stein’s method for normal approximations. Central limit theorems and Malliavin calculus.

5. Applications of Malliavin calculus in finance.

##### References

*Normal approximations with Malliavin calculus: form*

*Stein’s*

*method to universality*. Cambridge University Press, 2012.

D. Nualart: *The Malliavin calculus and **related **topics*. Second edition. Springer, 2006.

D. Nualart: *Malliavin** **calculus** **and** **its** **applications*. CBMS **110**, AMS, 2009.

E. Nualart and D. Nualart: *Introduction to Malliavin calculus*. Cambridge University Press, 2018.