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In this chapter we consider approximations of random walks in well-chosen spatial and time scales leading to the definition of Brownian motion, a famous random process indexed by continuous time. We describe its main properties and develop the study of continuous time martingales. Then we present the essential elements of stochastic calculus and applications to the study of stochastic differential equations.

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Brownian Motion and Diffusion Processes

  • 章节

  • 发表日期：

    2026年

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  • Sylvie Méléard

摘要

In this chapter we consider approximations of random walks in well-chosen spatial and time scales leading to the definition of Brownian motion, a famous random process indexed by continuous time. We describe its main properties and develop the study of continuous time martingales. Then we present the essential elements of stochastic calculus and applications to the study of stochastic differential equations.