Sato, K. (1999) Levy Processes and In nitely Divisible Distributions. Cambridge University Press, Cambridge. Home/Search Document Not in Database Summary Related Articles Check |
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Regular Variation in the Mean and Stable Limits for.. - Klüppelberg, Mikosch, .. (2001) (Correct)
....the so called L evy representation of the logarithm of the characteristic function of an in nitely divisible distribution. A measure with the property (2.4) is called a L evy measure. The distribution of any in nitely divisible vector is uniquely determined by the triple ( Q) See Sato [38] for an encyclopedic treatment of in nitely divisible distributions and processes. 2.3. Weak limits of in nitely divisible distributions. It is well known that the weak limits of in nitely divisible distributions are in nitely divisible. Hence the weak limits of the nite dimensional ....
....sphere S . 2) b( 3) lim #0 lim 1 ( dx) Q( for all 2 R . In what follows, we use both symbols e x and x for x=jxj. 2.4. Multivariate stable distributions. Multivariate stable distributions are particular in nitely divisible distributions; see Sato [38] for the general case of in nitely divisible distributions and Samorodnitsky and Taqqu [37] for an encyclopedic treatment of stable distributions and processes. The characteristic function of a stable random vector X with values in R and index 2 (0; 2] is characterized by the triple ( ....
Sato, K. (1999) Levy Processes and In nitely Divisible Distributions. Cambridge University Press, Cambridge, UK.
Optimal Portfolios When Stock Prices Follow an Exponential.. - Emmer, Klüppelberg (2002) (1 citation) (Correct)
....optimization problems, when the price processes are governed by exponential L evy processes. The class of L evy processes includes the Brownian motion, but also processes with jumps. We explain some basic theory of L evy processes and refer to Bertoin [7] Protter [27] and, in particular, Sato [31] for relevant background. Each in nitely divisible distribution function F on R generates a d dimensional L evy process L by choosing F as distribution function of L(1) This can be seen immediately, since the characteristic function of L(t) is for each t 0 given by E exp(isL(t) exp(t ....
....denote by R the d dimensional Euclidean space. Its elements are column vectors and for x 2 R we denote by x the transposed vector; analogously, for a matrix we denote by its transposed matrix. We further denote by jxj = i=1 x i ) the Euclidean norm of x 2 R . According to Sato [31], Chapter 4, the following holds. For each in the probability space de ne L(t; L(t; L(t ; For each Borel set B [0; 1) R (R n f0g) set M(B; #f(t; L(t; 2 Bg : 1.3) L evy s theory says that M is a Poisson random measure with intensity m(dt; dx) dt (dx) 1.4) ....
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Sato, K-I. (1999) Levy Processes and In nitely Divisible Distributions. Cambridge University Press. Cambridge.
Risk Management with Extreme Value Theory - Klüppelberg (2002) (Correct)
....model analogously to the Black Scholes model in Section 5.1, but replace the Brownian motion by a general L evy process L. Before we specify this model further we summarize some results on L evy processes. For relevant background we refer to Bertoin [10] Protter [73] and, in particular, Sato [81]. A very interesting collection of research articles is Barndor Nielsen, Mikosch and Resnick [7] Each in nitely divisible df F on generates a L evy process L by choosing F as df of the d dimensional vector L(1) This can be seen immediately, since the characteristic function is for each t ....
.... (f0g) 0 and R d (jxj 1) dx) 1, called the L evy measure of the process L. The term corresponding to xI(jxj 1) represents a centering, without which the integral (5.23) may not converge. The characteristic triplet (a; characterizes the L evy process L. According to Sato [81], Chapter 4 (see Theorem 19.2) the following holds. For each in the probability space, de ne L(t; L 1 (t; L d (t; with L j (t; L j (t; L j (t ; for j = 1; d. For each Borel set B [0; 1) R n f0g) set M(B; cardft 0 : t; L(t; 2 ....
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Sato, K. I. (1999) Levy Processes and In nitely Divisible Distributions. Cambridge University Press. Cambridge.
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Sato, K-I. (1999) Levy Processes and In nitely Divisible Distributions. Cambridge University Press. Cambridge. 28
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