Some Examples of Peacocks in a Markovian Set-Up

Abstract

We give, in a Markovian setup , some examples of processes which are increasing in the convex order (we call them peacocks). We then establish some relation between the stochastic and convex orders.

An application of multivariate total positivity to peacocks

“…Then (α(λ)X, λ ≥ 0) is SCM. 4) Note that (SCM) implies the conditional monotonicity as defined in [18] and [5]. Indeed,…”

“…We introduce the notion of strong conditional monotonicity which strictly implies conditional monotonicity as defined in [5] and [18]. 1) A random vector (X 1 , · · · , X n ) with real components is said to be strongly conditionally monotone (SCM) if, for every i ∈ {1, · · · , n}, every continuous and strictly positive functions f k : R → R + , k = 1, · · · , n such that:…”

“…The converse is not true. For example, the Gamma subordinator is conditionally monotone (see [5], Section 2 or [18], Section 1.4), but not strongly conditionally monotone.…”

“…and we recover (CEX08) thanks to (4.2). In [5], further examples of conditionally monotone processes are presented. For example, the Gamma subordinator and "well-reversible" diffusions at a fixed time are conditionally monotone.…”

“…We shall introduce the notion of strong conditional monotonicity. This concept is strictly more restrictive than the conditional monotonicity as defined in [18] (see also [2], [5] and [36]). There are many random vectors which are strongly conditionally monotone (SCM).…”