# Natural filtration

In the theory of stochastic processes in mathematics and statistics, the **generated filtration** or **natural filtration** associated to a stochastic process is a filtration associated to the process which records its "past behaviour" at each time. It is in a sense the simplest filtration available for studying the given process: all information concerning the process, and only that information, is available in the natural filtration.

More formally, let (Ω, *F*, **P**) be a probability space; let (*I*, ≤) be a totally ordered index set; let (*S*, Σ) be a measurable space; let *X* : *I* × Ω → *S* be a stochastic process. Then the **natural filtration of** *F* **with respect to** *X* is defined to be the filtration *F*_{•}^{X} = (*F*_{i}^{X})_{i∈I} given by

i.e., the smallest *σ*-algebra on Ω that contains all pre-images of Σ-measurable subsets of *S* for "times" *j* up to *i*.

In many examples, the index set *I* is the natural numbers **N** (possibly including 0) or an interval [0, *T*] or [0, +∞); the state space *S* is often the real line **R** or Euclidean space **R**^{n}.

Any stochastic process *X* is an adapted process with respect to its natural filtration.

## References

- Delia Coculescu; Ashkan Nikeghbali (2010), "Filtrations",
*Encyclopedia of Quantitative Finance*