Lecture 3 Pseudo-random number generation

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Congruential generator Define a sequence (si) recurrently Then si+1 = (asi + b) mod M. Ui = si M form an infinite sample from a uniform distribution on [0, 1]. Computational Finance – p. 2
The numbers si have the following properties: 1. si ∈ {0, 1, 2,... ,M − 1} 2. si are periodic with period < M. Indeed, since at least two values in {s0,s1,...,sM } must be identical, therefore si = si+p for some p ≤ M. In view of property 2 the number M should be as large as possible, because a small set of numbers make the outcome easier to predict. But still p can be small even is M is large! Computational Finance – p. 3
Page 1: Lecture 3 Pseudo-random number gene
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 and 38: Simulation For Monte Carlo simulati
Page 39 and 40: PCA construction Principal Componen
Page 41 and 42: In general Cholesky decomposition a

Lecture 3 Pseudo-random number generation

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