By Marc Yor
The following notes symbolize nearly the second one half the lectures I gave within the Nachdiplomvorlesung, in ETH, Zurich, among October 1991 and February 1992, including the contents of six extra lectures I gave in ETH, in November and December 1993. half I, the elder brother of the current ebook [Part II], geared toward the computation, as explicitly as attainable, of a couple of attention-grabbing functionals of Brownian movement. it can be average that half II, the more youthful brother, appears to be like extra into the most process with which half i used to be "working", specifically: martingales and stochastic calculus. As F. Knight writes, in a assessment article on half I, within which examine on Brownian movement is in comparison to gold mining: "In the times of P. Levy, or even as overdue because the theorems of "Ray and Knight" (1963), it was once attainable for the practiced eye to select up important present with out the help of a lot expertise . . . Thereafter, in spite of the fact that, the rewards are more and more accomplished through the applying of excessive technology". even supposing one may possibly argue even if this golden age is admittedly foregone, and talk about the "height" of the know-how concerned, this citation is heavily with regards to the most motivations of half II: this know-how, inclusive of stochastic calculus for basic discontinuous semi-martingales, growth of filtrations, . . .
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Extra resources for Some Aspects of Brownian Motion: Part II: Some Recent Martingale Problems
Example text
1 "Zeta functions" and probability. : an Euler-product representation, or a functional equation, or ... 6. Some generalizations related to Bessel processes 23 we simply associate to a sequence A. = (An; n ~ 1) of strictly positive real numbers, the "zeta-function" : 00 1 (... (s) = AS' s > 0 . L n=l In the sequel, we shall assume that: (,"(1) n 1 L>: < 00. (t) = c)... nt with c)... (1). 25) PROOF: This is an immediate consequence of the equality: 1 00 r(s)a s = jdxxs-le- ax a a>O, s>O. 2 Some examples related to Bessel processes.
7. 39). 36). 3). By doing so, we obtain: E[X~-I] = 3 s(s-1) E [(~ N)S/2] , 7r and, changing s into: 2k + 2, we get: 1 k EX] = (k + 1)(2k + 1) [ (3"2 E[:E HI]) (k ~ 0). 36). 40) where, on the right-hand side, as t. ;>. ~)] == ( smh( . ) 1 f).. dx ="2 10 X3/2 (1- g(x)). 41) 11. 37). 41) follows, using integration by parts. 37). 00 J ds 1(R~9)' where (R~,s ~ 0) denotes the Bessel process o with index 1/, starting from 1, and define ~v-l = 0',,-1 + 0'~_1' where O'v-1 and 0'~_1 are two independent copies of the first hitting time of 1 by BESij-1, the Bessel process with index v - 1 starting from O.
1. 6 Example 4: We enlarge the Brownian filtration with the entire local time process (it, t ~ 0), and denote the enlarged filtration by (FlOC). 13) = inf{t > u: Bt = O}. We will not prove the above statement, but, in the sequel, a number of interesting consequences will be deduced from it. :1 = 00. 2 below. 14) PROOF: Remark that: Ou - u = inf{t > 0 : B t +u - Bu = -Bu}, so that, conditionally on Fu, and Bu = x, we have: law = law = O"x 2 def ' , = inf{t > 0: B t = X}, where B is a BM, X 0"1 x2 = N2' law .