On a dyadic approximation of predictable processes of finite variation

Pietro Siorpaes

doi:10.1214/ecp.v19-2972

Abstract

We show that any cadlag predictable process of finite variation is an a.s. limit of elementary predictable processes; it follows that predictable stopping times can be approximated from below by predictable stopping times which take finitely many values. We then obtain as corollaries two classical theorems: predictable stopping times are announceable, and an increasing process is predictable iff it is natural.

Highlights

We recall that a process S = (St)t is called of class D if the family of random variables (Sτ )τ, where τ ranges through all stopping times, is uniformly integrable
We notice how these results can be used to provide an alternate derivation of the following well known theorems: predictable stopping times are announceable, and an increasing process is predictable iff it is natural
We prove this classic result without using the deep debut and section theorems, by showing explicitly that the jumps times, and the ‘first-approach time’, of a cadlag adapted process are stopping times; our proofs are elementary, and hold even if the filtration does not satisfy the usual conditions

Summary

The main results

After introducing some definitions and conventions, we state our results on the approximation of predictable processes of integrable variation and of predictable stopping times, and we prove the second one. This fact is often derived as a consequence of the (difficult) section theorem for predictable sets; one can find in [RW00a, Chapter 6, Theorem 12.6] a direct proof which does not use the section theorems themselves, but does involve ideas from their proofs, which are essentially based on Choquet’s capacity theorem Another possibility is to proceed as [MP80] and give a proof of the Doob-Meyer decomposition which shows inter-alia that increasing predictable processes are natural; applying this to the predictable process A := 1⁄2[τ,1] shows that predictable stopping times are fair (in particular in this paper, instead of proving Theorem 2 directly, we could see it as an immediate corollary of Theorem 9). Another way is to proceed as in [Low13]; this proof, intuitive, uses the theory of integration with respect to general predictable bounded integrands, as well as the BichtelerDellacherie theorem and Jacod’s countable expansion theorem

Predictable stopping times

How to approximate the compensator

Consequences for special semimartingales