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Moment Evolution of Gaussian and Geometric Wiener Diffusions: Derived by Itô’s Lemma and Kolmogorov̂’s Forward Equation

In: The Elements and Dynamic Systems of Economic Growth and Trade Models

Author

Listed:
  • Bjarne S. Jensen

    (University of Southern Denmark (SDU))

Abstract

This chapter analyzes two basic stochastic models: The time-homogeneous Gaussian and the geometric Wiener diffusion of two-dimensional vector processes. Using the theory of stochastic processes and Ito lemma, the probability of distributions of the stochastic state vectors are described by the evolution of their moments (expectation and covariance as functions of time), as these moments satisfy certain systems of ordinary (deterministic) differential equations (ODE). By solving the latter ODE, the explicit solutions for the first-order and second-order moment functions are presented. The forward Kolmogorov equation is used partly to derive the results by alternative methods and partly to gain information on the probability distributions. The general closed form results on the moment evolutions (still unavailable) have many applications in the models of linear dynamics with uncertainty.

Suggested Citation

  • Bjarne S. Jensen, 2025. "Moment Evolution of Gaussian and Geometric Wiener Diffusions: Derived by Itô’s Lemma and Kolmogorov̂’s Forward Equation," Springer Books, in: The Elements and Dynamic Systems of Economic Growth and Trade Models, edition 0, chapter 0, pages 1197-1241, Springer.
    • Handle: RePEc:spr:sprchp:978-3-031-52493-6_33
    DOI: 10.1007/978-3-031-52493-6_33
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