Stochastic Differential Equations

General characterization

Prerequisites

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Bibliography

1. Hui-Hsiung Kuo, Introduction to Stochastic Integration. Springer. 2006
2. Bernt Oksendal, Stochastic Differential Equations. Sringer. 1998
3. Paul Malliavin,  Integration and Probability. Springer-Verlag. 1995
4. G. Prato, J. Zabczyk, Stochastic Equations in Infnite Dimensions. Cambridge University Press, Second Edition (2014).
5. M. Capinski. E. Kopp, Measure, Integral and Probability. Springer-Verlag (2004)
6. H.-H., Kuo, Gaussian Measures in Banach Spaces. Springer-Verlag (1975)
7. A. Ichikawa, Stability of Semilinear Stochastic Evolution Equations. Journal of Mathematical Analysis and Applications 90, 12-44 (1982)

Teaching method

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Evaluation method

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Subject matter

1) Gaussian measures on a Banach space.

2) Brownian Motion- Wiener integral- Conditional expectation- Martingales 

3) Stochastic integrals 

4) Itô formula 

5) Applications of the Itô Formula- Exponential process- Transformation of probability measures-Girsanov theorem 

6) Stochastic Differential equations-Existence and uniqueness-Markov property-Diffusion processes- Semigroups and Kolmogorov equations

7) Reproducing kernel of a Gaussian measure on a Banach space.

8) Construction of Gaussian random variables with values in a Banach space.

9) Mean and covariance operator of a Gaussian measure on a Hilbert space.

10) Q - Wiener process with values in a Hilbert space.

11) Stochastic integral with respect to a Q - Wiener process.

12) Itô''''s formula.

13) Linear stochastic differential equations in a Hilbert space.

Programs

Programs where the course is taught: