predictable process

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.

1 predictable processes in discrete time

Suppose we have a filtration (http://planetmath.org/FiltrationOfSigmaAlgebras) ${({ℱ}_n)}_{n\in {\mathbb{Z}}_{+}}$ on a measurable space $(\mathrm{\Omega },ℱ)$. Then a stochastic process $X_n$ is predictable if $X_n$ is ${ℱ}_{n-1}$-measurable (http://planetmath.org/MeasurableFunctions) for every $n\ge 1$ and $X_0$ is ${ℱ}_0$-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 ${ℱ}_n$-measurable.

2 predictable processes in continuous time

In continuous time, the definition of predictable processes is a little more subtle. Given a filtration $({ℱ}_t)$ 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 ${ℱ}_t$-adapted processes and which is closed under taking limits of a sequence of processes.

Equivalently, a real-valued stochastic process

$X:{ℝ}_{+}\times \mathrm{\Omega }\to ℝ$

$(t,\omega )\mapsto X_t(\omega )$

is predictable if it is measurable with respect to the predictable sigma algebra $\mathrm{\wp }$. This is defined as the smallest $\sigma$-algebra on ${ℝ}_{+}\times \mathrm{\Omega }$ making all left-continuous and adapted processes measurable.

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

$\mathrm{\wp }$$=\sigma \left(\{(t,\mathrm{\infty })\times A:t\ge 0,A\in {ℱ}_t\}\cup \{\{0\}\times A:A\in {ℱ}_0\}\right)$

$=\sigma \left(\{(T,\mathrm{\infty }):T\text{ is a stopping time}\}\cup \{\{0\}\times A:A\in {ℱ}_0\}\right)$

$=\sigma \left(\{[T,\mathrm{\infty }):T\text{ is a predictable stopping time}\}\right)$

Note that in these definitions, the sets $(T,\mathrm{\infty })$ and $[T,\mathrm{\infty })$ are stochastic intervals, and subsets of ${ℝ}_{+}\times \mathrm{\Omega }$.

3 general predictable processes

The definition of predictable process given above can be extended to a filtration $({ℱ}_t)$ with time index $t$ lying in an arbitrary subset $𝕋$ of the extended real numbers. In this case, the predictable sets form a $\sigma$-algebra on $𝕋\times \mathrm{\Omega }$. If $𝕋$ has a minimum element $t_0$ then let $S$ be the collection of sets of the form $\{t_0\}\times A$ for $A\in {ℱ}_{t_0}$, otherwise let $S$ be the empty set.Then, the predictable $\sigma$-algebra is defined by

Here, $(t,\mathrm{\infty }]$ and $(T,\mathrm{\infty }]$ are understood to be intervals containing only times in the index set $𝕋$. If $𝕋$ is an interval of the real numbers then $\mathrm{\wp }$ can be equivalently defined as the $\sigma$-algebra generated by the class of left-continuous and adapted processes with time index ranging over $𝕋$.

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