optional process

Suppose we are given a filtration (http://planetmath.org/FiltrationOfSigmaAlgebras) ${(ℱ)}_{t\in 𝕋}$ on a measurable space $(\mathrm{\Omega },ℱ)$. A stochastic process is said to be adapted if $X_t$ is ${ℱ}_t$-measurable for every time $t$ in the index set $𝕋$. For an arbitrary, uncountable, index set $𝕋\subseteq ℝ$, this property is too restrictive to be useful. Instead, we can impose measurability conditions on $X$ considered as a map from $𝕋\times \mathrm{\Omega }$ to $ℝ$.For instance, we could require $X$ to be progressively measurable, but that is still too weak a condition for many purposes. A stronger condition is for $X$ to be optional. The index set $𝕋$ is assumed to be a closed subset of $ℝ$ in the following definition.

The class of optional processes forms the smallest set containing all adapted and right-continuous processes, and which is closed under taking limits of sequences of processes.

The $\sigma$-algebra, $𝒪$, on $𝕋\times \mathrm{\Omega }$ generated by the right-continuous and adapted processes is called the optional $\sigma$-algebra. Then, a process is optional if and only if it is $𝒪$-measurable.

Alternatively, the optional $\sigma$-algebra may be defined as

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

Here, $[T,\mathrm{\infty })$ is a stochastic interval, consisting of the pairs $(t,\omega )\in 𝕋\times \mathrm{\Omega }$ such that $T(\omega )\le t$.In continuous-time, the equivalence of these two definitions for $𝒪$ does require mild conditions on the filtration — it is enough for ${ℱ}_t$ to be universally complete.

In the discrete-time case where the index set $𝕋$ countable, then the definitions above imply that a process $X_t$ is optional if and only if it is adapted.