A
stochastic process
that can be represented as the sum of a local
martingale
and a process of locally bounded variation. For the
formal definition of a semi-martingale one starts from a
stochastic basis
,
where
(cf.
Stochastic processes, filtering of).
A stochastic process
is called a
semi-martingale
if its trajectories are right-continuous and have left limits,
and if it can be represented in the form
,
where
is a local martingale and
is a process of locally bounded variation, that is,
In general this representation is non-unique. But in
the class of representations with predictable processes
,
the representation is unique (up to stochastic equivalence). The following belong
to the class of semi-martingales (apart from local martingales
and processes of locally bounded variation):
local super-martingales and submartingales, processes
with independent increments for which
is a function of locally bounded variation for any
(and so all processes with stationary independent increments), Itô
processes, diffusion-type processes, and others. The class of
semi-martingales is invariant under an equivalent change of measure. If
is a semi-martingale and
is twice continuously differentiable, then
is also a semi-martingale. Here
(
Itô's formula)
or, equivalently,
where
is the
quadratic variation of the semi-martingale
,
that is,
is the continuous part of the quadratic variation
,
,
and the integrals are understood as stochastic
integrals with respect to a semi-martingale (cf.
Stochastic integral).
If
is a semi-martingale, then the process
with
has bounded jumps,
,
and so can be uniquely represented as
where
is a
predictable random process
of locally bounded variation and
is a local martingale. This martingale can be uniquely represented as
,
where
is a continuous local martingale (a continuous martingale forming the semi-martingale
)
and
is a purely-discontinuous local martingale that can be written in the form
where
is the
random jump measure
of
,
that is,
and
is its
compensator.
Since
each semi-martingale
admits a representation
called the
canonical representation
(decomposition).
The set of (predictable) characteristics
,
where
is the quadratic characteristic of
,
that is, a predictable increasing process such that
is a local martingale, is called a
triplet
of local (predictable) characteristics of
.