Ftmeasurable

Ftmeasurable, typically written as F_t-measurable, is a term used in probability theory to describe random variables that are measurable with respect to a given filtration F_t. In stochastic processes, F_t represents the information available up to time t, encoded as a sigma-algebra.

Definition: Let (Ω, F, P) be a probability space and (F_t)_{t≥0} a filtration, meaning an increasing family of sub-sigma algebras of F. A random variable X: Ω → R is F_t-measurable if for every Borel set B ⊆ R, the preimage X^{-1}(B) belongs to F_t. Equivalently, X is determined by information available up to time t.

Relation to adaptedness: A stochastic process {X_t} is called adapted to the filtration (F_t) if, for every t, the random variable X_t is F_t-measurable. Adaptedness ensures that the value of the process at time t can be inferred from information up to that time, which is essential for defining conditional expectations and stochastic integration.

Examples: If F_t contains all information up to time t, then any function of this information is F_t-measurable. For instance, if X = g(Z_t) where Z_t is F_t-measurable and g is Borel, or X = h(Y_1, ..., Y_t) with each Y_k being F_k-measurable, then X is F_t-measurable.

Properties: If X is F_t-measurable and g is a Borel function, then g(X) is also F_t-measurable. Being F_t-measurable is not, in general, inherited by earlier times; X being F_t-measurable does not automatically imply X is F_s-measurable for s < t, unless X is additionally measurable with respect to F_s.

See also: Filtration, sigma-algebra, adapted process, conditional expectation, stochastic integration.