# C8.1 Stochastic Differential Equations - Material for the year 2019-2020

## Primary tabs

Integration theory: Riemann-Stieljes and Lebesgue integral and their basic properties

Probability and measure theory: $\sigma$-algebras, Fatou lemma, Borel-Cantelli, Radon-Nikodym, $L^p$-spaces, basic properties of random variables and conditional expectation,

Martingales in discrete and continuous time: construction and basic properties of Brownian motion, uniform integrability of stochastic processes, stopping times, filtrations, Doob's theorems (maximal and $L^p$-inequalities, optimal stopping, upcrossing, martingale decomposition), martingale (backward) convergence theorem, $L^2$-bounded martingales, Quadratic variation.

16 lectures

### Assessment type:

- Written Examination

Stochastic differential equations (SDEs) model quantities that evolve under the influence of noise and random perturbations. They have found many applications in diverse disciplines such as biology, physics, chemistry and the management of risk. Classic well-posedness theory for ordinary differential equations does not apply to SDEs. However, stochastic integration allows to develop a new calculus for such equations (Ito calculus). This leads to many new and interesting insights about quantities that evolve under randomness, that have found many real-world applications. This course is a introduction to stochastic differential equations.

The student will have learned about general existence and uniqueness results for stochastic differential equations, basic properties of such diffusive systems and how to calculate with them.

Recap on martingale theory in continuous time, quadratic variation, stochastic integration and Ito's calculus.

Levy's characterisation of Brownian motion, stochastic exponential, change of measure on pathspace, Burkholder-Davis-Gundy, Martingale represenation, Dambis-Dubins-Schwarz.

Strong and weak solutions of stochastic differential equations, existence and uniqueness.

Local times, Tanaka formula, Tanaka-Ito-Meyer formula.

- H. Oberhauser
*C8.1 Stochastic differential equations*online notes - M. Yor and D. Revaz,
*Continuous martingales and Brownian motion*(Springer). - R. Durrett,
*Stochastic Calculus*(CRC Press).

- N. Ikeda & S. Watanabe,
*Stochastic Differential Equations and Diffusion Processes*(North--Holland Publishing Company, 1989). - I. Karatzas and S. E. Shreve,
*Brownian Motion and Stochastic Calculus*, Graduate Texts in Mathematics 113 (Springer-Verlag, 1988). - L. C. G. Rogers & D. Williams,
*Diffusions, Markov Processes and Martingales Vol 1 (Foundations) and Vol 2 (Ito Calculus)*(Cambridge University Press, 1987 and 1994). - H. P. McKean,
*Stochastic Integrals*(Academic Press, New York and London, 1969). - B. Oksendal,
*Stochastic Differential Equations: An introduction with applications*(Universitext, Springer, 6th edition). Chapters II, III, IV, V, part of VI, Chapter VIII (F).