Nnnconcavity and convexity pdf files

A point on the curve at which the curve exhibits on extra ordinary behavior is called a singular point. Pdf stochastic convexity and concavity of markov processes. Notes on online convex optimization columbia university. In fact, this function is most often a convex decreasing function. Just like dollar duration and duration, dollar convexities add, convexities average. Hence, quite often, investors see the convexity as a gain, and the realized increasedecrease in price over linear duration measure is referred to as convexity gain. Notes on online convex optimization karl stratos online convex optimization oco is a principled framework for online learning. It was written as an advanced text to serve as a basis for a.

Roughly speaking concavity of a function means that the graph is above chord. Hence, convexity of the curve increases the increase in price, and reduces the loss in price, compared to what is given by a linear measure of duration. R is concave convex if and only if its restriction to every line segment of rn is concave convex function of one variable. Preface the following notes were written before and during the course on convex geometry which was held at the university of karlsruhe in the winter term 20022003. Concavity, convexity and points of inflexion submitted by vikram kumar maths p. Increasing and decreasing functions, min and max, concavity studying properties of the function using derivatives typeset by foiltex 1. Increasing and decreasing functions, min and max, concavity. Concavity and convexity, inflection points of a function. Understanding duration and convexity of fixed income. A course on convex geometry university of ljubljana. Preface this book serves as an introduction to the expanding theory of online convex optimization. A second order taylor series expansion will capture the curvature in addition to the trend and will often well.