Ects : 10
Volume horaire : 78

Description du contenu de l'enseignement :

• Part I. Continuous-Time Markov Chains (15-18 hours)
  • Càdlàg and increasing processes: definition. Poisson process (on RR) and counting processes.
  • Continuous-time Markov chains (countable state space): embedded Markov chain, jump times, Kolmogorov equations, generator, recurrence vs. transience, invariant measures.
  • Possible applications (in lectures and/or tutorials): M/M/s queues, birth and death chains, branching processes.
• Part II. Brownian Motion and Diffusions (21-24 hours)
  • Continuous processes: definition. Random continuous functions. Brownian motion: construction of Brownian motion (as a Gaussian process), existence of a continuous version (via Kolmogorov’s criterion or admitted). Fundamental properties.
  • Introduction to stochastic calculus (continuous martingales, Itô integral and Itô’s formula) and stochastic differential equations. One may restrict to the one-dimensional case for simplicity.
  • Application in physics: Langevin equation (in lectures or tutorials).
  • Application in finance and asset pricing: Black-Merton-Scholes model.

Compétence à acquérir :

Continuous-time Markov chains and their applications, Brownian motion, stochastic calculus, stochastic differential equations, and applications to physics and finance.