Lecture 3 Pseudo-random number generation

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Cholesky decomposition Algorithm for <strong>generation</strong> of N(µ, Σ) distributed pseudo-<strong>random</strong> variables: 1. Calculate the Cholesky decomposition AA T = Σ. 2. Calculate Z ∼ N(0,I) componentwise by Zi ∼ N(0, 1), i = 1,...,n. 3. µ + AZ has the desired distribution ∼ N(µ, Σ). Computational Finance – p. 38
PCA construction Principal Component Analysis (PCA) is the method of analyzing spectrum of the covariance matrix. Since Σ is symmetric positive definite matrix, it has d positive eigenvalues and d eigenvectors which span the space R d . In addition the following formula holds Σ = ΓΛΓ T , where Γ is the matrix of d eigenvectors of Σ and Λ is the diagonal matrix of eigenvalues of Σ. Computational Finance – p. 39
Page 1 and 2: Lecture 3 Pseudo-random number gene
Page 3 and 4: The numbers si have the following p
Page 5 and 6: Are the numbers Ui obtained with th
Page 7 and 8: Good congruential generator with a
Page 9 and 10: Apparently not so good congruential
Page 11 and 12: Fibonacci generators The original F
Page 13 and 14: ”Minimal Standard” generators W
Page 15 and 16: The Mersenne Twister The Mersenne T
Page 17 and 18: The state of the random number gene
Page 19 and 20: Generation of non-uniformly distrib
Page 21 and 22: Idea 2: Inversion of the distributi
Page 23 and 24: Idea 3: Transformation of random va
Page 25 and 26: If X ∼ N(µ, Σ), then X = (X1,X2
Page 27 and 28: We compute the determinant of the d
Page 29 and 30: Box-Muller with congruential genera
Page 31 and 32: Marsaglia algorithm The Box-Muller
Page 33 and 34: How to generate a sample from a mul
Page 35 and 36: Algorithm for generation of N(µ,
Page 37: Simulation For Monte Carlo simulati
Page 41 and 42: In general Cholesky decomposition a

Lecture 3 Pseudo-random number generation

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