We study the differentiability of the CIR process with respect to its parameters. We give a stochastic representation for these derivatives in terms of the paths of V .

Isidore Fleishcher, Centre de recherches mathematiques, Universite de Montreal, C.P. 6128, succ. Centre-ville, Montreal, QC H3C 3J7,Canada. Green’s Theorem with NO Differentiability. The result is established for a Jordan measurable region with locally connected rectifiable boundary. The integrand F for the new plane integral to be used is a function of axisparallel rectangles, finitely additiv...

This case study shows how ACL2 can be used to reason about the real and complex numbers, using non-standard analysis. It describes some modifications to ACL2 that include the irrational real and complex numbers in ACL2’s numeric system. It then shows how the modified ACL2 can prove classic theorems of analysis, such as the intermediate-value and mean-value theorems.

This paper is concerned with the relationships between L differentiability and Sobolev functions. It is shown that if / is a Sobolev function with weak derivatives up to order k in L , and 0 s / s k, then / has an L derivative of order / everywhere except for a set which is small in the sense of an appropriate capacity. It is also shown that if a function has an 2V derivative P everywhere excep...

We characterize some major algorithmic randomness notions via differentiability of effective functions. (1) As the main result we show that a real number z ∈ [0, 1] is computably random if and only if each nondecreasing computable function [0, 1] → R is differentiable at z. (2) We prove that a real number z ∈ [0, 1] is weakly 2-random if and only if each almost everywhere differentiable computa...

In this paper, we will provide some examples of Banach spaces that are Gâteaux differentiability spaces but not weak Asplund, weak Asplund but not in class(
̃), in class(
̃) but whose dual space is not weak∗ fragmentable. We begin with some definitions. A Banach space X is called a weak Asplund space [almost weak Asplund] (Gâteaux differentiability space) if each continuous convex function define...

A well-known open question is whether every countable collection of Lipschitz functions on a Banach space X with separable dual has a common point of Fréchet differentiability. We show that the answer is positive for some infinite-dimensional X. Previously, even for collections consisting of two functions this has been known for finite-dimensional X only (although for one function the answer is...

In this paper, we prove the existence and uniqueness of solution to the impulsive fuzzy functional differential equations under generalized Hukuhara differentiability via the principle of contraction mappings. Some examples are provided to illustrate the result.