Malliavin Calculus and its Apllications
In 1976, P. Malliavin founded the Stochastic Calculus of Variation, nowadays called the Malliavin calculus, and gave a probabilistic proof of the famous Hörmander theorem for hypoelliptic operators. Since then, the Malliavin calculus has been one of the central topics in stochastic analysis and has many significant applications in mathematics and in other sciences.
The aim of this course of 48 hours is to give a complete introduction of the basic materials of the Malliavin calculus and then to present the probabilistic proofs of Hörmander's hypoellipticity theorem and the Atiyah-Singer index theorem for the Dirac operator, together with its applications in the ergodicity of 2D stochastic Navier-Stokes equations, numerical scheme to SDEs and financial mathematics.
The aim of this course of 48 hours is to give a complete introduction of the basic materials of the Malliavin calculus and then to present the probabilistic proofs of Hörmander's hypoellipticity theorem and the Atiyah-Singer index theorem for the Dirac operator, together with its applications in the ergodicity of 2D stochastic Navier-Stokes equations, numerical scheme to SDEs and financial mathematics.
Lecturer
Xiangdong Li
Date
15th March ~ 14th June, 2024
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Friday | 13:00 - 16:00 | A6-101 | ZOOM 07 | 559 700 6085 | BIMSA |
Syllabus
1. Brownian motion and Lévy-Doob's martingale theory
2. Itô's stochastic integrals and Itô's formula
3. Itô's stochastic differential equations and diffusion processes
4. Probabilistic approach to PDEs and Feynman-Kac formula
5. Cameron-Martin-Girsanov formula and Malliavin gradient operator on the Wiener space
6. Integration by parts formula and Sobolev spaces on the Wiener space
7. Martingale representation theorem and Malliavin-Bismut-Clark-Ocone formula with application in finance
8. Malliavin's probabilistic proof of Hörmander's hypoellipticity theorem
9. P.-L. Lions et al's work on Greeks formula in financial math using Malliavin calculus
10. M. Hairer and J. Mattingly's work on the ergodicity of 2D stochastic Navier-Stokes equations
11. Application of Malliavin calculus to short time asymptotic behavior of the heat kernel of hypoelliptic operators and numerical scheme of SDEs
12. Bismut's probabilistic proof of the Atiyah-Singer index theorem for Dirac operator
2. Itô's stochastic integrals and Itô's formula
3. Itô's stochastic differential equations and diffusion processes
4. Probabilistic approach to PDEs and Feynman-Kac formula
5. Cameron-Martin-Girsanov formula and Malliavin gradient operator on the Wiener space
6. Integration by parts formula and Sobolev spaces on the Wiener space
7. Martingale representation theorem and Malliavin-Bismut-Clark-Ocone formula with application in finance
8. Malliavin's probabilistic proof of Hörmander's hypoellipticity theorem
9. P.-L. Lions et al's work on Greeks formula in financial math using Malliavin calculus
10. M. Hairer and J. Mattingly's work on the ergodicity of 2D stochastic Navier-Stokes equations
11. Application of Malliavin calculus to short time asymptotic behavior of the heat kernel of hypoelliptic operators and numerical scheme of SDEs
12. Bismut's probabilistic proof of the Atiyah-Singer index theorem for Dirac operator
Reference
1. J.-M. Bismut, Large Deviations and the Malliavin Calculus, Birkhauser, 1984.
2. N. Ikeda, S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Publishing Company, Second Ed. 1989
3. K. Itô, Selected Papers, Edited by D.W. Stroock and S.R.S. Varadhan, Springer, 1987.
4. P. Malliavin, Stochastic Analysis, Springer, 1997.
5. Original papers by A.N. Kolmogorov, P. Lévy, N. Wiener, P. Malliavin, J.-M. Bismut, M. Hairer, J. Mattingly, P.-L. Lions et al., S. Kuosoka and D. W. Stroock etc.
2. N. Ikeda, S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Publishing Company, Second Ed. 1989
3. K. Itô, Selected Papers, Edited by D.W. Stroock and S.R.S. Varadhan, Springer, 1987.
4. P. Malliavin, Stochastic Analysis, Springer, 1997.
5. Original papers by A.N. Kolmogorov, P. Lévy, N. Wiener, P. Malliavin, J.-M. Bismut, M. Hairer, J. Mattingly, P.-L. Lions et al., S. Kuosoka and D. W. Stroock etc.
Audience
Undergraduate
, Graduate
Video Public
Yes
Notes Public
No
Language
Chinese
Lecturer Intro
李向东，中国科学院数学与系统科学研究院研究员，主要研究领域：随机分析、随机微分几何、随机矩阵、最优传输理论。1990年本科毕业于武汉大学，1994-1999年获中国科学院应用数学研究所及葡萄牙里斯本大学博士学位，1999-2003年先后在里斯本大学与牛津大学从事博士后研究，2003年获法国图卢兹大学Maitre de Conference终身职位，2007年获法国图卢兹大学“指导研究证书”(Habilitation a Diriger des Recherches )，2008-2009年任复旦大学数学科学学院教授。2009年至今，先后任中国科学院数学与系统科学研究院研究员、华罗庚应用数学首席研究员。2015年至今，兼任中国科学院大学岗位教授。