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Extra info for Applied Quantitative Finance Free Printing
Suppose we know µ ˜ = E[g(zi )]. 50) N i=1 to reduce the unknown error 1 N N Wi − µ. 51) i=1 The controlled estimator has the form 1 N N Wi − β( i=1 1 N N ˜i − µ W ˜). 52) provides an unbiased estimator of µ as long as β is independent. 5 Variance Reduction Techniques in Monte-Carlo Simulation 27 if the function g(zi ) provides a close approximation of f (zi ), we usually set β = 1 to simplify the calculation. 3. , n, denote an independent standard normal random vector used to drive a simulation.
E. C1 (u, v) = Π(u, v) = u v. For θ → ∞ one finds for the Gumbel-Hougaard copula θ→∞ def Cθ (u, v) −→ min(u, v) = M (u, v). It can be shown that M is also a copula. Furthermore, for any given copula C one has C(u, v) ≤ M (u, v), and M is called the Fr´echet-Hoeffding upper bound. def The two-dimensional function W (u, v) = max(u+v−1, 0) defines a copula with W (u, v) ≤ C(u, v) for any other copula C. W is called the Fr´echet-Hoeffding lower bound. 4 Further Important Properties of Copulas In this section we focus on the properties of copulas.
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