2Semi-martingales
III Stochastic Calculus and Applications
2 Semi-martingales
The title of the chapter is “semi-martingales”, but we are not going even meet
the definition of a semi-martingale till the end of the chapter. The reason is that
a semi-martingale is essentially defined to be the sum of a (local) martingale and
a finite variation process, and understanding semi-martingales mostly involves
understanding the two parts separately. Thus, for most of the chapter, we
will be studying local martingales (finite variation processes are rather more
boring), and at the end we will put them together to say a word or two about
semi-martingales.
From now on, (Ω
, F,
(
F
t
)
t≥0
, P
) will be a filtered probability space. Recall
the following definition:
Definition
(C`adl`ag adapted process)
.
A c`adl`ag adapted process is a map
X : Ω × [0, ∞) → R such that
(i) X is c`adl`ag, i.e. X(ω, ·) : [0, ∞) → R is c`adl`ag for all ω ∈ Ω.
(ii) X is adapted, i.e. X
t
= X( ·, t) is F
t
-measurable for every t ≥ 0.
Notation.
We will write
X ∈ G
to denote that a random variable
X
is measur-
able with respect to a σ-algebra G.