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This chapter introduces the basic probability concepts and stochastic processes. It includes random variables, probability density function (PDF), moments, Gaussian random variables, Wiener process, Markov jump process, and stochastic differential equations (SDEs). Essential tools for solving the path-wise solutions and the time evolution of the PDFs associated with the SDEs, namely Itô’s formula and Fokker-Planck equation, are also presented in this chapter. The exposition of the contents in this chapter is more towards the applied aspect and is therefore kept concise and non-rigorous. Simple examples are utilized to explain various concepts. The stochastic toolkits developed in this chapter will be used in the subsequent chapters of the book.

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Stochastic Toolkits

  • 章节

  • 发表日期：

    2025年

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  • Nan Chen

摘要

This chapter introduces the basic probability concepts and stochastic processes. It includes random variables, probability density function (PDF), moments, Gaussian random variables, Wiener process, Markov jump process, and stochastic differential equations (SDEs). Essential tools for solving the path-wise solutions and the time evolution of the PDFs associated with the SDEs, namely Itô’s formula and Fokker-Planck equation, are also presented in this chapter. The exposition of the contents in this chapter is more towards the applied aspect and is therefore kept concise and non-rigorous. Simple examples are utilized to explain various concepts. The stochastic toolkits developed in this chapter will be used in the subsequent chapters of the book.