F11SS - Stochastic Simulation
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Aims
The aims of this module are:
- Develop understanding of continuous random variables and simulation
- Develop ideas of Monte-Carlo simulations and convergence
- Develop Brownian motion and properties
- Develop stochastic integrals and calculus
- Introduce SDEs examples such as Langevin equations, Geometric Brownian motion and Ornstein-Uhlenbeck process.
- Introduce numerical methods for SDEs
- Introduce different notions of convergence for numerical methods
- Convergence of Euler--Maruyama
Syllabus
1. Monte Carlo Methods (1.1 1. Random Number generation, 1.2 2. Inverse transform method, 1.3 3. Monte Carlo methods and applications)
2. Stochastic Integral (2.1 1. Brownian motion, 2.2 2. Definition of Ito and Stratonovich Integral, 2.3 3. Properties of Ito Integral)
3. Stochastic differential equations (3.1 1. Integral formulation, 3.2 2. Ito formula, 3.3 3. Use of Ito formula to solve SDE)
4. Numerical Methods for SDE (4.1 1. Derivation of numerical schemes, 4.2 2. Convergence analysis, 4.3 3. Weak error and multilevel Monte Carlo, 4.4 4. Implementation)
5. Fokker-Planck Equations (5.1 1. Forward Fokker-Planck Equation, 5.2 2. Backward Fokker-Planck Equation, 5.3 3. Applications)
Learning outcomes
By the end of the course, students should be able to do the following:
- Describe and use in practice standard methods for simulating random variables.
- Apply and analyse the Monte-Carlo method
- State the definitions of Ito and Stratonovich integrals and their basic properties
- Use integral formulation of a stochastic differential equation (SDE) to derive the Euler-Maruyama, Milstein and Heun numerical schemes.
- State the Ito formula and use it to solve standard one-dimensional SDE.
- Implement the Euler-Maruyama scheme and the higher-order Milstein scheme.
- Prove the weak convergence of Euler-Maruyama for geometric Brownian motion and perform key elements in the strong convergence analysis.
- Use the forward and backward Fokker-Planck equations to compute moments of the solution and mean exit times.
Further details
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SCQF Level: 11
Credits: 15