Table of Contents
Introduction to Stochastic Calculus
Random variable (a refresh) and classification of stochastic processes
- Definition of probability space and random variable;
- joint probability; conditional probability; Bayes theorem;
- classification of stochastic process:Markov processes; Gaussian processes; process with independent increments; Martingales
- backward and forward Chapman-Kolmogorov equation;
Stochastic processes in discrete time and discrete space
- Markov chains;
- recurrent and transient states;
- generating function;
- ergodicity and invariant distribution;
- infinite number of states
Stochastic processes in continuous time and discrete space
- Poisson processes;
- generating function
- superposition of Poisson processes; non-linear processes;
- birth and death process;
Stochastic processes in continous time and continous space
- Wiener process;
- Ito and Stratonovih integral (derived from Steltjes integral);
- Langevin equation and the relevance of boundary conditions;
- Fokker-Plank equation;
- Pawula theorem and diffusive limit (if time remains)