what is F-previsible process? And what would be F?

Question

What is F-previsible process? I tried to search in the Internet but I couldn't find it... Also what is F here?

context: http://en.m.wikipedia.org/wiki/Martingale_representation_theorem#section_2

Answer 1

I agree with @André Nicolas' comment. Predictable and previsible are synonyms in the context of stochastic processes.

Consider the following setup: $(\Omega,\mathcal{F},(\mathcal{F}_t),P)$ is a filtered probability space, i.e. $(\mathcal{F}_t)_{t\geq 0}$ is a filtration in $\mathcal{F}$. Let $\mathcal{P}$ denote the predictable/previsible sigma-algebra, i.e. the sigma-algebra on $\Omega\times\mathbb{R}_+$ generated by all adapted left-continuous processes. If a process $$ (\omega,t)\mapsto X_t(\omega) $$ is measurable with respect to $\mathcal{P}$, then $(X_t)_{t\geq 0}$ is said to be a predictable/previsible process.

But note that this definition depends on the filtration (because of $\mathcal{P}$ being generated by all adapted left-continuous processes) and hence it would be more correct to include the filtration:

• $(X_t)_{t\geq 0}$ is an $(\mathcal{F}_t)_{t\geq 0}$-previsible/predictable process or,
• $(X_t)_{t\geq 0}$ is an $\mathbf{F}$-previsible/predictable process, where $\mathbf{F}=(\mathcal{F}_t)_{t\geq 0}$.