234 Modeling and Inverse Problems in the Presence of Uncertainty
The above theorem implies that if a sa mple-continuous martingale has finite
variatio n, then it must be a constant. It is worth no ting that there exist some
martingales with finite variation. But by Theorem 7.1.10, we k now that such
martingales must be sample discontinuous . An example of such martingales
is {X(t) − λt : t ≥ 0}, where {X(t) : t ≥ 0} is a Poisson process with rate λ.
7.1.6.1 Examples of Sample-Continuous Martingales
Let F
t
be the σ-algebra generated by {W (s) : 0 ≤ s ≤ t}. Then by the
definition of the Wiener process, it can be shown that the Wiener process is
a martingale with r espe ct to {F
t
}. Specifically, by Jensen’s inequality (2 .67)
and W (t) ∼ N(0, t) we find that ...