A predictable process is a real-valued stochastic process whose values are known, in a sense, just in advance of time. Predictable processes are also called previsible.

predictable processes in discrete time

Suppose we have a filtration $(\mathcal{F}_n)_{n\in\mathbb{Z}_+}$ on a measurable space $(\Omega,\mathcal{F})$ . Then a stochastic process $X_n$ is predictable if $X_n$ is $\mathcal{F}_{n-1}$ -measurable for every $n\ge 1$ and $X_0$ is $\mathcal{F}_0$ -measurable. So, the value of $X_n$ is known at the previous time step. Compare with the definition of adapted processes for which $X_n$ is $\mathcal{F}_n$ -measurable.

predictable processes in continuous time

In continuous time, the definition of predictable processes is a little more subtle. Given a filtration $(\mathcal{F}_t)$ with time index $t$ ranging over the non-negative real numbers, the class of predictable processes forms the smallest set of real valued stochastic processes containing all left-continuous $\mathcal{F}_t$ -adapted processes and which is closed under taking limits of a sequence of processes.

Equivalently, a real-valued stochastic process

Alternatively, $\wp$ is generated by either of the following collections of subsets of $\mathbb{R}_+\times\Omega$

general predictable processes

The definition of predictable process given above can be extended to a filtration $(\mathcal{F}_t)$ with time index $t$ lying in an arbitrary subset $\mathbb{T}$ of the extended real numbers. In this case, the predictable sets form a $\sigma$ -algebra on $\mathbb{T}\times\Omega$ . If $\mathbb{T}$ has a minimum element $t_0$ then let $S$ be the collection of sets of the form $\{t_0\}\times A$ for $A\in\mathcal{F}_{t_0}$ , otherwise let $S$ be the empty set.Then, the predictable $\sigma$ -algebra is defined by \begin{equation*}\begin{split} \wp &=\sigma\left(\left\{(t,\infty]\times A:t\in\mathbb{T},A\in\mathcal{F}_t\right\}\cup S\right)\\ &= \sigma\left(\left\{(T,\infty]:T\colon\Omega\rightarrow\mathbb{T}\textrm{ is a stopping time}\right\}\cup S\right). \end{split}\end{equation*}Here, $(t,\infty]$ and $(T,\infty]$ are understood to be intervals containing only times in the index set $\mathbb{T}$ . If $\mathbb{T}$ is an interval of the real numbers then $\wp$ can be equivalently defined as the $\sigma$ -algebra generated by the class of left-continuous and adapted processes with time index ranging over $\mathbb{T}$ .

A stochastic process $X\colon\mathbb{T}\times\Omega\rightarrow\mathbb{R}$ is predictable if it is $\wp$ -measurable. It can be verified that in the cases where $\mathbb{T}=\mathbb{Z}_+$ or $\mathbb{T}=\mathbb{R}_+$ then this definition agrees with the ones given above.