Numerical methods for stochastic modelling

Additional Info

  • ECTS credits: 3
  • Semester: 3
  • University: University of L'Aquila
  • Objectives:

     

    The Aim of this course is to provide the student with the knowledge of numerical modeling for stochastic problems and the ability to analyze theoretical properties and design mathematical software based on the proposed schemes.

    On successful completion of this module, the student should
    - have profound knowledge and understanding of the most relevant numerical methods for the approximation of stochastic differential problems and the design of accurate and efficient mathematical software;
    - demonstrate skills in choosing the most suitable discretization in relation to the problem to be solved and ability to provide theoretical analysis and mathematical software based on the proposed schemes;
    - demonstrate capacity to read and understand other texts on the related topics.

  • Topics:

     

    - Discretized Brownian motion.
    - Ito and Stratonovich stochastic integrals.
    - Stochastic differential equations: motivation, modeling, existence and uniqueness, strong and weak solutions.
    - Ito formula.
    - Explicit methods: Euler-Maruyama and Milstein.
    - Implicit methods: stochastic theta-methods.
    - Strong and weak convergence.
    - Mean-square and asymptotic linear stability.
    - Nonlinear stability analysis.
    - Stochastic geometric numerical integration.

  • Prerequisites:

     

    Basic Numerical Analysis, differential equations and stochastic processes.

  • Books:

     

    An Introduction to the Numerical Simulation of Stochastic Differential Equations, D.J. Higham and P. E. Kloeden, SIAM, 2021.

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