Abstract
In this chapter, we study some of the basic properties of Markov stochastic processes, and in particular, the properties of diffusion processes. In Sect. 2.1, we present various examples of Markov processes in discrete and continuous time. In Sect. 2.2, we give the precise definition of a Markov process and we derive the fundamental equation in the theory of Markov processes, the Chapman–Kolmogorov equation. In Sect. 2.3, we introduce the concept of the generator of a Markov process. In Sect. 2.4, we study ergodic Markov processes. In Sect. 2.5, we introduce diffusion processes, and we derive the forward and backward Kolmogorov equations. Discussion and bibliographical remarks are presented in Sect. 2.6, and exercises can be found in Sect. 2.7.
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Notes
- 1.
In later chapters, we will also consider Markov processes with state space a subset of \(\mathbb{R}^{d}\), for example the unit torus.
- 2.
In fact, all we need is that \(u \in C^{2,1}(\mathbb{R} \times \mathbb{R}_{+})\). This can be proved using our assumptions on the transition function, on f, and on the drift and diffusion coefficients.
- 3.
The backward Kolmogorov equation can also be derived using Itô’s formula. See Chap. 3.
- 4.
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Pavliotis, G.A. (2014). Diffusion Processes. In: Stochastic Processes and Applications. Texts in Applied Mathematics, vol 60. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1323-7_2
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