Download PDF by Daniel Anderson: Factorization in Integral Domains
By Daniel Anderson
The contents during this paintings are taken from either the college of Iowa's convention on Factorization in vital domain names, and the 909th assembly of the yankee Mathematical Society's distinctive consultation in Commutative Ring concept held in Iowa urban. The textual content gathers present paintings on factorization in essential domain names and monoids, and the speculation of divisibility, emphasizing attainable various lengths of factorization into irreducible components.
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Extra resources for Factorization in Integral Domains
F ∈ Pa if 0 ≤ f < a is convex. f ∈ Pa ⇒ erτ Sτ f ∈ Pa . We are now going to introduce slightly larger classes of payoff functions than the classes Pa , a > 0. To this end, let a > 0 be given and suppose f : R m + → [0, a[ is a continuous function and set for fixed τ > 0, g = gτ = − 1 √ ln(1 − f /a) . σm τ (23) GEOMETRIC INEQUALITIES IN OPTION PRICING 49 We shall say that the the function f belongs to the class Pa,m if the function Φ−1 (P[gτ (x1 eσ1 √ τ c1 ,G , . . , xm eσm √ τ cm ,G ) ≤ s]) − s, s>0 is non-decreasing for every x ∈ R m + and τ > 0.
In fact, already the early paper  by Merton treats a variety of convexity properties of puts and calls, sometimes without any distributional assumptions on the underlying stock prices. Here, however, it will always be assumed that the price process X(t) = (X1 (t), . . , Xm (t)), t ≥ 0, of the underlying risky assets X1 , . . , Xm is governed by a so called joint geometric Brownian motion. Furthermore, all options will be of European type and so, from now on, option will always mean option of European type.
Then f ∈ C if and only if f > 0 and f (xeξ ) ≤ f (x)e ξ ∞ m x ∈ Rm +, ξ ∈ R . , (12) Moreover , f ∈ Ca if and only if a + f (xeξ ) ≤ (a + f (x))e ξ ∞ m x ∈ Rm +, ξ ∈ R . , In particular , any f ∈ Ca is a payoff function. Proof. Suppose first that f > 0 and set g(ξ) = ln f (eξ ), ξ ∈ Rm. Clearly, the inequality (12) just means that g ∈ Lip∞ (R m ; 1). Now let f ∈ C. e. e. and, hence, g ∈ Lip∞ (R m ; 1). Conversely, suppose g ∈ Lip∞ (R m ; 1). Then f ∈ Liploc (R m + ) and (13) holds. Accordingly, the inequality (11) must be true.
Factorization in Integral Domains by Daniel Anderson