For simplicial complexes, one such method is based on discrete Morse theory [114], which was adapted to filtrations of simplicial complexes in [115].
In the following, ({W t),tgeq0}) is a cylindrical Wiener process adapted to the filtration ({{mathcal{F}}_{t}}_{tgeq0}).
Let ((M_{n}(h))) be a locally square integrable real martingale adapted to a filtration (mathbb{F}=(mathcal{F}_{n})).
For all (tgeq0), the process ((X t), J t))) is adapted to the filtration (mathcal{F}_{t}).
Let ( M n ) be a locally square integrable real martingale adapted to a filtration F = ( F n ) with M 0 = 0.
The notion of vector fields adapted to this filtration and causal linear operators are introduced and geometric characterizations of these notions are derived.
Let ((M_{n}(h))) be a locally square integrable real martingale adapted to the filtration ((mathcal{F}_{n})) with (M_{0}=0).
Here (w(t)) is an m-dimensional Brownian motion defined on the probability space ((Omega,mathcal{F},P)) adapted to the filtration ((mathcal {F}_{t})_{tge{0}}).
We set (x t)) be adapted to the filtration (mathcal{F}_{t}) generated by Brownian motion ((W cdot),bar{W}(cdot))), and the control processes (c_{i}(t)) ((i=1,2)) be adapted to the observation filtration ({mathcal{G}_{t}subseteqmathcal{F}_{t}}).
In particular, the stochastic increment (dW_{t}) does not depend on (f(x_{t},t)) or (g(x_{t},t)) (i.e. (x_{t}) is adapted to the filtration generated by the noise).
It is shown in this article that we can define the stopping of noncommutative processes in Fock space in such a way that all the bounded martingales can be stopped at any stopping time T, are adapted to the filtration of the past before T and satisfy the optional stopping theorem.
