An
integral
"∫ H dX"
with respect to a
semi-martingale
on some
stochastic basis
,
defined for every locally bounded predictable process
.
One of the possible constructions of a stochastic integral is as
follows. At first a stochastic integral is defined for simple predictable processes
,
of the form
where
is
-measurable.
In this case, by the stochastic integral
(or
,
or
)
one understands the variable
The mapping
,
where
permits an extension (also denoted by
)
onto the set of all bounded predictable
functions, which possesses the following properties:
a) the process
,
,
is continuous from the right and has limits from the left;
b)
is linear, i.e.
c) If
is a sequence of uniformly-bounded predictable functions,
is a predictable function and
then
The extension
is therefore unique in the sense that if
is another mapping with the properties a)–c), then
and
are stochastically indistinguishable (cf.
Stochastic indistinguishability).
The definition
given for functions
holds for any process
,
not only for semi-martingales. The extension
with properties a)–c) onto the class of
bounded predictable processes is only possible for the case where
is a semi-martingale. In this sense, the class of semi-martingales
is the maximal class for which a stochastic integral
with the natural properties a)–c) is defined.
If
is a semi-martingale and
is a Markov time (stopping time), then the
"stopped"
process
is also a semi-martingale and for every predictable bounded process
,
This property enables one to extend the definition of a
stochastic integral to the case of locally-bounded predictable functions
.
If
is a localizing (for
)
sequence of Markov times, then the
are bounded. Hence, the
are bounded and
is stochastically indistinguishable from
.
A process
,
again called a stochastic integral, therefore exists, such that
The constructed stochastic integral
possesses the following properties:
is a semi-martingale; the mapping
is linear; if
is a process of locally bounded variation, then so is the integral
,
and
then coincides with the Stieltjes integral of
with respect to
;
;
.
Depending on extra assumptions concerning
,
the stochastic integral
can also be defined for broader classes of functions
.
For example, if
is a locally square-integrable martingale, then a stochastic integral
(with the properties a)–c)) can be defined for any predictable process
that possesses the property that the process
is locally integrable (here
is the
quadratic variation
of
,
i.e. the
predictable increasing process
such that
is a
local martingale).