- Simulate random numbers from a uniform distribution using techniques such as modular arithmetic and linear/mixed congruential generators.
* Assess the quality and performance of random number generators.
- Differentiate between random numbers and pseudo-random numbers.
- Simulate pseudo random numbers from various distributions using methods like the inverse transform method, rejection sampling, acceptance-rejection method, composition method, and ad-hoc methods.
- Simulate stochastic processes including random walks, Markov chains, Poisson processes, and Wiener processes and their extensions.
- Distinguish between exact simulation methods and numerical approximation methods for random variables and stochastic processes.
- Simulate simple stochastic differential equations (SDEs) ( learn/recall Ito's formula, simulate geometric Brownian motion, Wiener processes with drift, and the Ornstein-Uhlenbeck process).
- Apply numerical methods for solving SDEs, including the Euler-Maruyama method and the Milstein method.
- Compute the root mean square error (RMSE) to evaluate the simulations results.
- Conduct Monte Carlo simulations for various applications and implement variance reduction techniques (Analytical reduction, stratified sampling, importance sampling, use of covariates).
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Fundamental concepts of stochastic processes, techniques for random number generation, methods for simulating Markov chains and other stochastic models, Monte-Carlo simulation, practical applications of stochastic simulations in various fields, evaluation and interpretation of the results of stochastic simulations.
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