Almost Sure

23 December 18

Projection in Discrete Time

It has been some time since my last post, but I am continuing now with the stochastic calculus notes on optional and predictable projection. In this post, I will go through the ideas in the discrete-time situation. All of the main concepts involved in optional and predictable projection are still present in discrete time, but the theory is much simpler. It is only really in continuous time that the projection theorems really show their power, so the aim of this post is to motivate the concepts in a simple setting before generalising to the full, continuous-time situation. Ideally, this would have been published before the posts on optional and predictable projection in continuous time, so it is a bit out of sequence.

We consider time running through the discrete index set {{\mathbb Z}^+=\{0,1,2,\ldots\}}, and work with respect to a filtered probability space {(\Omega,\mathcal{F},\{\mathcal{F}_n\}_{n=0,1,\ldots},{\mathbb P})}. Then, {\mathcal{F}_n} is used to represent the collection of events observable up to and including time n. Stochastic processes will all be real-valued and defined up to almost-sure equivalence. That is, processes X and Y are considered to be the same if {X_n=Y_n} almost surely for each {n\in{\mathbb Z}^+}. The projections of a process X are defined as follows.

Definition 1 Let X be a measurable process. Then,

  1. the optional projection, {{}^{\rm o}\!X}, exists if and only if {{\mathbb E}[\lvert X_n\rvert\,\vert\mathcal{F}_n]} is almost surely finite for each n, in which case
    \displaystyle  {}^{\rm o}\!X_n={\mathbb E}[X_n\,\vert\mathcal{F}_n]. (1)
  2. the predictable projection, {{}^{\rm p}\!X}, exists if and only if {{\mathbb E}[\lvert X_n\rvert\,\vert\mathcal{F}_{n-1}]} is almost surely finite for each n, in which case
    \displaystyle  {}^{\rm p}\!X_n={\mathbb E}[X_n\,\vert\mathcal{F}_{n-1}]. (2)

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