Progressively measurable process

Definition

Let

• (\Omega, \mathcal{F}, \mathbb{P}) be a probability space;
• (\mathbb{X}, \mathcal{A}) be a measurable space, the state space;
• { \mathcal{F}_{t} \mid t \geq 0 } be a filtration of the sigma algebra \mathcal{F};
• X : 0, \infty) \times \Omega \to \mathbb{X} be a [stochastic process (the index set could be [0, T] or \mathbb{N}_{0} instead of [0, \infty));
• \mathrm{Borel}([0, t]) be the Borel sigma algebra on [0,t].

The process X is said to be progressively measurable (or simply progressive) if, for every time t, the map [0, t] \times \Omega \to \mathbb{X} defined by (s, \omega) \mapsto X_{s} (\omega) is \mathrm{Borel}([0, t]) \otimes \mathcal{F}{t}-measurable. This implies that X is \mathcal{F}{t} -adapted.

A subset P \subseteq 0, \infty) \times \Omega is said to be progressively measurable if the process X_{s} (\omega) := \chi_{P} (s, \omega) is progressively measurable in the sense defined above, where \chi_{P} is the [indicator function of P. The set of all such subsets P form a sigma algebra on [0, \infty) \times \Omega, denoted by \mathrm{Prog}, and a process X is progressively measurable in the sense of the previous paragraph if, and only if, it is \mathrm{Prog}-measurable.

Properties

• It can be shown that L^2 (B), the space of stochastic processes X : [0, T] \times \Omega \to \mathbb{R}^n for which the Itô integral :: \int_0^T X_t , \mathrm{d} B_t : with respect to Brownian motion B is defined, is the set of equivalence classes of \mathrm{Prog}-measurable processes in L^2 ([0, T] \times \Omega; \mathbb{R}^n).
• Every adapted process with left- or right-continuous paths is progressively measurable. Consequently, every adapted process with càdlàg paths is progressively measurable.
• Every measurable and adapted process has a progressively measurable modification.

References

References

1. (1991). "Brownian Motion and Stochastic Calculus". Springer.
2. Pascucci, Andrea. (2011). "PDE and Martingale Methods in Option Pricing". Springer.