Introduction: Some problems, where SDE-s play an essential role in the solution.

Random variables, stochastic processes, Kolmogorov’s extension theorem

Brownian motion. Different constructions. Basic properties (e.g. in higher dimension, scaling)

Versions of processes: continuity. Nowhere differentiability

Ito integral. Mathematical interpretations of "noise" in equations. Preparation: Introduction of a special type  of measurability. Ito isometry on step function. Difficulties in the definition of an integral.

Ito integral. Stratonovich integral. Their relation. Some properties. Examles.

Important mathematical notion: Characteristic function of rv-s. Connection between different kind of measurabilities. Martingals, submartingals. Properties. Stopping time. Examples.

Quadratic process of the Brownian motion. Ito processes. 1-dimensional Ito formula. Applications, examples. Integration by parts.

The Multi-dimensional Ito formula. The martingal representation theorem. Exponential martingal

An application: the Tanaka formula

Stochastic DE. Examples and some solution methods .The law of iterated logarithm. Existence and uniqueness theorem

Strong solution, weak solution. Girsanov theorem  An example: the Orstein-Uhlenbeck equaton

Filtering problem, the general problem. Motivation: An elementary problem and its solution

The linear filtering problem Kalman-Bucy filter in continuous time. The Ricatti equation