By G. Letta, M. Pratelli
ISBN-10: 0387167870
ISBN-13: 9780387167879
Read Online or Download Probability and analysis: lectures given at the 1st 1985 session of the Centro internazionale matematico estivo PDF
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Extra resources for Probability and analysis: lectures given at the 1st 1985 session of the Centro internazionale matematico estivo
Example text
P(Yn ≤ yn )) = C(FY1 (y1 ), . . , FYn (yn )), where FYi (yi ), i = 1, . . , n denote the marginal distribution functions of the random variables Y i , i = 1, . . , n. 19 Mikosch ¨ (2006), Embrechts and Puccetti (2006), and Ruschendorf (2004) provide examples and further references for the application of copulas in risk management. 20 The importance of copulas in the modeling of the distribution of multivariate random variables is provided by Sklar’s theorem. The derivation was provided in Sklar (1959).
If for a given x, the gradient equals a vector of zeros, ∇f (x) = (0, . . , 0) then the function does not change in a small neighborhood of x ∈ Rn . It turns out that all points of local extrema of the objective function are characterized by a zero gradient. As a result, the points yielding the local extrema of the objective function are among the solutions of the system of equations, ∂f (x) =0 ∂x1 ... ∂f (x) = 0. 3) is often referred to as representing the first-order condition for the objective function extrema.
There is a connection between minimization and maximization. Maximizing the objective function is the same as minimizing the negative of the objective function and then changing the sign of the minimal value, maxn f (x) = − minn [−f (x)]. 1. As a consequence, problems for maximization can be stated in terms of function minimization and vice versa. 1 Minima and Maxima of a Differentiable Function If the second derivatives of the objective function exist, then its local maxima and minima, often called generically local extrema, can be characterized.
Probability and analysis: lectures given at the 1st 1985 session of the Centro internazionale matematico estivo by G. Letta, M. Pratelli
by David
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