In this course we will introduce the audience to the basic elements of Ito non anticipative stochastic calculus and Stochastic Differential Equations.
Synthetic programme : Brownian Motion, Martingales, Ito integral, Ito formula and martingale representation theorem, strong and weak solutions of a stochastic differential equation, diffusion theory, Feynman Kac formula, application to filtering and stochastic control theory.
- Probabilty spaces, random variables, stochastic processes and martingales
- Brownian motion
- Ito integral: construction, properties and extensions
- Ito formula and Martingale representation theorem
- Stochastic differential equations: examples of solution, existence and uniqueness of strong solution, weak solutions, Girsanov theorem and Cameron-Martin formula.
- Diffusion theory : Markov property, generators.
- PDEs problems associated to a diffusion : Dirichlet problem, Parabolic equations, Feynman-Kac formula.
- Application to filtering and stochastic control theory
Discussion of home works.
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