Stochastic Processes and Stochastic Calculus I

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Every fall semester

An introduction to the Ito stochastic calculus and stochastic differential equations through a development of continuous-time martingales and Markov processes. (1st of two courses in sequence)


MATH 6242 or equivalent

Course Text: 

At the level of Karatzas and Shreve, Brownian Motion and Stochastic Calculus

Topic Outline: 
  • Stochastic processes, filtrations and stopping times: some basics on these foundations
  • Continuous-parameter martingale theory: including basic properties and examples, the fundamental inequalities and convergence results and applications of these, optional sampling, decompositions, and square-integrable martingales
  • Weak convergence of processes
  • Continuous-parameter Markov processes
  • Brownian motion in one and several dimensions: include basic properties, functionals and sample path properties
  • Ito stochastic integration: include basic definitions, elementary properties, characterizations, Ito's rule (change-of-variable formulas), and applications of Ito's rule to results such as the martingale characterization of Brownian motion and to martingale moment inequalities