Probability and Stochastic Analysis

oral exam

Prerequisites

A good knowledge of the basic topics in mathematical analysis, linear algebra, and measure theory is required. Preliminary notions of probability theory are also useful, including random variables, distributions, probabilistic convergence, and conditional expectation.

Familiarity with function spaces and with some elements of ordinary or partial differential equations may help in understanding certain parts of the course, but is not strictly required.

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The course introduces the fundamental tools of modern probability theory and stochastic analysis, with particular emphasis on the rigorous construction of random processes and their analytical applications.

The first, introductory part reviews elements of measure theory, integration, random variables, probabilistic convergence, and conditioning. Products of probability spaces, Kolmogorov’s extension theorem, and Kolmogorov’s continuity criterion are then presented.

The second part is devoted to Brownian motion and Gaussian structures: Gaussian measures in finite and infinite dimensions, Cameron-Martin space, white noise, Wiener measure, quadratic variation, and the Wiener integral. Filtrations, stopping times, the Markov and strong Markov properties, the reflection principle, and some applications of Brownian motion to parabolic problems with boundary conditions are then introduced.

The final part presents the Itô integral, Itô’s formula, and finite-dimensional stochastic differential equations, including the existence and uniqueness theorem under standard regularity assumptions on the coefficients.

The objective is to provide a solid mathematical foundation for the study of continuous stochastic processes, stochastic equations, and the connections between probability, functional analysis, and the theory of partial differential equations.

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