Filtrationsgränsen

Filtrationsgränsen, in Swedish typically translated as the filtration limit, is a term encountered in probability theory and the study of stochastic processes. A filtration is an increasing family of sigma-algebras {F_t}_{t≥0} that represents the information available up to time t. The notion of a filtration limit refers to the limiting behavior of such information as time progresses, or as a model is refined.

There are at least two common interpretations. First, for a fixed time horizon T, F_T denotes the terminal filtration, the sigma-algebra representing all information up to the terminal time. In many contexts one studies properties of processes relative to the terminal filtration, such as martingales with respect to F_T. Second, for a sequence of filtrations {F^n_t} one can define a limit filtration as n→∞. Several conventions exist; a typical one is F_t = σ(∪_n F^n_t), the sigma-algebra generated by events that eventually appear in some F^n_t. Another approach uses convergence of conditional expectations: E[X|F^n_t] → E[X|F_t] for all integrable X and all t.

In practice, the concept helps analyze how added information or longer horizons affect stochastic behavior, convergence of processes, and pricing in financial models with evolving information. The term is more common in Swedish-language texts; English literature more often speaks of terminal sigma-algebras, filtrations at a horizon, or convergence of filtrations without using the compound term "filtrationsgränsen."