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By Leopoldo Nachbin

Trans. shape the Portuguese through Richard M. Aron

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Extra resources for Introduction to Functional Analysis: Banach Spaces and Differential Calculus

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B) Using pseudocode, describe how you could use n simulations of the price S(T ) to estimate pK . (c) For T = 1 and K = 110, compare an approximate value for pK (based on approximations for Φ(x) such as those mentioned in Prob. 10) with the one obtained using simulation as in (b), with n = 1000. Does the 95% confidence interval based on these n simulations contain the approximate value? (d) 38 1 The Monte Carlo Method Describe two different functions f1 and f2 defined over [0, 1)s for some s (s does not need to be the same for f1 and f2 ) whose integral is equal to E(S(T )) for T = 1.

For i = 1, . . , B: a. Randomly and uniformly choose n indices li,1 , . . , li,n from {1, . . , n} with replacement. b. Compute θˆi = θ(xli,1 , . . , xli,n ). 2. Use the obtained sample θˆ1 , . . , θˆB to infer on the desired property. For instance, if the goal is to estimate Var(θ), then we can use the estimator σ ˆθ2 = 1 B−1 where 1 θ¯ = B B ¯ 2, (θˆi − θ) i=1 B θˆi . i=1 Problems 39 To get a confidence interval for E(θˆi ), we can either use a percentile approach and construct the 100(1 − α)% confidence interval θˆ(Bα/2) , θˆ(B(1−α/2)) or a central limit theorem approach with σ ˆθ σ ˆθ θ¯ − zα/2 √ , θ¯ + zα/2 √ B B .

33, . 05, . . 33, . ) = 36. j=1 u1 u2 u3 u4 u5 . . 05 . . 33 → a1 → s1 → a2 → s2 → a3 . . 1 . . 7 . . 0 . . 5 Fig. 75) as a function evaluation. Times are in seconds. C5 is updated each time a new waiting time wj is computed. Aj = a1 + . . + aj is the arrival time of the jth client. 12). 11) based on the input vector of uniform numbers ui = (ui1 , ui2 , . ) required to generate the random observations a1 , s1 , a2 . . for the ith simulation. d. d. 12) using the Monte Carlo integration method.

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