Peacocks Obtained by Normalisation: Strong and Very Strong Peacocks

Abstract

This paper contains two parts:Part I. Let $({V }_{t},t\,\geq \,0)$ be an integrable right-continuous process such that $\mathrm{e}\left [\vert {V }_{t}\vert \right ]\,<\,\infty$, for every t ≥ 0. Let us consider the three types of processes:

• 1. $\left ({C}_{t} := {V }_{t} -\mathrm{ e}\left [{V }_{t}\right ],t \geq 0\right )$,

• 2. $\left ({N}_{t} := \dfrac{{V }_{t}} {\mathrm{e}\left [{V }_{t}\right ]},t \geq 0\right )$, with $\mathrm{e}\left [{V }_{t}\right ] > 0$ for every $t \geq 0$,

• 3. $\left ({Q}_{t} := \dfrac{{V }_{t}} {\alpha (t)},t \geq 0\right )$, where $\mathrm{e}[{V }_{t}] = 0$ for every t ≥ 0 and, $\alpha : {\mathbb{R}}_{+} \rightarrow {\mathbb{R}}_{+}$ is a Borel function which is strictly positive.

We shall give some classes of processes $({V }_{t},t \geq 0)$ such that C, N or Q are peacocks, i.e.: whose one-dimensional marginals are increasing in the convex order. Part II. We introduce the notions of strong and very strong peacocks which lead to the study of new classes of processes.

AMS Classification: 60J25, 32F17, 60G44, 60E15