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{{MSC|60H05}}

[[Category:Stochastic analysis]]

An integral "∫ H dX" with respect to a [[Semi-martingale|semi-martingale]]  $  X $
on some stochastic basis  $  ( \Omega , {\mathcal F} , ( {\mathcal F} _ {t} ) _ {t} , {\mathsf P} ) $, 
defined for every locally bounded predictable process  $  H = ( H _ {t} , {\mathcal F} _ {t} ) $. 
One of the possible constructions of a stochastic integral is as follows. At first a stochastic integral is defined for simple predictable processes  $  H $, 
of the form

$$ 
H _ {t}  =  h( \omega ) I _ {( a,b] }  ( t),\  a < b,
$$

where  $  h $
is  $  {\mathcal F} _ {a} $-
measurable. In this case, by the stochastic integral  $  \int _ {0}  ^ {t} H _ {s}  dX _ {s} $(
or  $  ( H \cdot X) _ {t} $, 
or  $  \int _ {( t,0] }  H _ {s}  dX _ {s} $) 
one understands the variable

$$ 
h ( \omega ) ( X _ {b\wedge} t - X _ {a\wedge} t ).
$$

The mapping  $  H \mapsto H \cdot X $, 
where

$$ 
H \cdot X  =  ( H \cdot X) _ {t} ,\  t \geq  0,
$$

permits an extension (also denoted by  $  H \cdot X $) 
onto the set of all bounded predictable functions, which possesses the following properties:

a) the process  $  ( H \cdot X) _ {t} $, 
$  t \geq  0 $, 
is continuous from the right and has limits from the left;

b)  $  H \mapsto H \cdot X $
is linear, i.e.

$$ 
( cH _ {1} + H _ {2} ) \cdot X  =  c( H _ {1} \cdot X) + H _ {2} \cdot X;
$$

c) If  $  \{ H  ^ {n} \} $
is a sequence of uniformly-bounded predictable functions,  $  H $
is a predictable function and

$$ 
\sup _ { s\leq  } t  | H _ {s}  ^ {n} - H _ {s} |   \mathop \rightarrow \limits ^  {\mathsf P}    0,\  t >
0,
$$

then

$$ 
( H  ^ {n} \cdot X) _ {t}   \mathop \rightarrow \limits ^  {\mathsf P}    ( H \cdot X) _ {t} ,\  t > 0.
$$

The extension  $  H \cdot X $
is therefore unique in the sense that if  $  H \mapsto \alpha ( H) $
is another mapping with the properties a)–c), then  $  H \cdot X $
and  $  \alpha ( H) $
are stochastically indistinguishable (cf. [[Stochastic indistinguishability|Stochastic indistinguishability]]).

The definition

$$ 
( H \cdot X) _ {t}  =  h( \omega )( X _ {b\wedge} t - X _ {a\wedge} t ),
$$

given for functions  $  H _ {t} = h( \omega ) I _ {( a,b] }  ( t) $
holds for any process  $  X $, 
not only for semi-martingales. The extension  $  H \cdot X $
with properties a)–c) onto the class of bounded predictable processes is only possible for the case where  $  X $
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  $  X $
is a semi-martingale and  $  T = T( \omega ) $
is a Markov time (stopping time), then the "stopped" process  $  X  ^ {T} = ( X _ {t\wedge} T , {\mathcal F} _ {t} ) $
is also a semi-martingale and for every predictable bounded process  $  H $,

$$ 
( H \cdot X)  ^ {T}  =  H \cdot X  ^ {T}  = \ 
( HI _ {[[ 0,T ]] }  ) \cdot X .
$$

This property enables one to extend the definition of a stochastic integral to the case of locally-bounded predictable functions  $  H $. 
If  $  T _ {n} $
is a localizing (for  $  H $) 
sequence of Markov times, then the  $  H ^ {T _ {n} } $
are bounded. Hence, the  $  H \cdot I _ {[[ 0,T _ {n}  ]] } $
are bounded and

$$ 
[ ( HI _ {[[ 0, T _ {n+1} ]] }  ) \cdot X ] ^ {T _ {n} }
$$

is stochastically indistinguishable from  $  HI _ {[[ 0,T _ {n}  ]] } \cdot X $. 
A process  $  H \cdot X $, 
again called a stochastic integral, therefore exists, such that

$$ 
( H \cdot X) ^ {T _ {n} }  = \ 
HI _ {[[ 0,T _ {n}  ]] } \cdot X,\  n \geq  0.
$$

The constructed stochastic integral  $  H \cdot X $
possesses the following properties:  $  H \cdot X $
is a semi-martingale; the mapping  $  H \mapsto H \cdot X $
is linear; if  $  X $
is a process of locally bounded variation, then so is the integral  $  H \cdot X $, 
and  $  H \cdot X $
then coincides with the Stieltjes integral of  $  H $
with respect to  $  dX $; 
$  \Delta ( H \cdot X) = H \Delta X $; 
$  K \cdot ( H \cdot X) = ( KH) \cdot X $.

Depending on extra assumptions concerning  $  X $, 
the stochastic integral  $  H \cdot X $
can also be defined for broader classes of functions  $  H $. 
For example, if  $  X $
is a locally square-integrable martingale, then a stochastic integral  $  H \cdot X $(
with the properties a)–c)) can be defined for any predictable process  $  H $
that possesses the property that the process

$$ 
\left ( \int\limits _ { 0 } ^ { t }  H _ {s}  ^ {2}  d\langle  X\rangle _ {s} \right ) _ {t \geq  0 } 
$$

is locally integrable (here  $  \langle  X\rangle $
is the quadratic variation of  $  X $, 
i.e. the predictable increasing process such that  $  X  ^ {2} - \langle  X\rangle $
is a local martingale).

====References====
{|
|valign="top"|{{Ref|J}}|| J. Jacod, "Calcul stochastique et problèmes de martingales" , ''Lect. notes in math.'' , '''714''' , Springer (1979) {{MR|0542115}} {{ZBL|0414.60053}}
|-
|valign="top"|{{Ref|DM}}|| C. Dellacherie, P.A. Meyer, "Probabilities and potential" , '''A-C''' , North-Holland (1978–1988) (Translated from French) {{MR|0939365}} {{MR|0898005}} {{MR|0727641}} {{MR|0745449}} {{MR|0566768}} {{MR|0521810}} {{ZBL|0716.60001}} {{ZBL|0494.60002}} {{ZBL|0494.60001}}
|-
|valign="top"|{{Ref|LS}}|| R.Sh. Liptser, A.N. Shiryayev, "Theory of martingales" , Kluwer (1989) (Translated from Russian) {{MR|1022664}} {{ZBL|0728.60048}}
|}

====Comments====
The result alluded to above, that semi-martingales constitute the widest viable class of stochastic integrators, is the Bichteler–Dellacherie theorem {{Cite|B}}–{{Cite|D}}, and can be formulated as follows {{Cite|P}}, Thm. III.22. Call a process elementary predictable if it has a representation

$$ 
H _ {t}  =  H _ {0} I _ {\{ 0 \} }  ( t)+ \sum _ { i=1} ^ { n }  H _ {i} I _ {( T _ {i}  , T _ {i+1} ] } ( t) ,
$$

where  $  0 = T _ {0} \leq  T _ {1} \leq  \dots \leq  T _ {n+1} < \infty $
are stopping times and  $  H _ {i} $
is  $  {\mathcal F} _ {T _ {i}  } $-
measurable with  $  | H _ {i} | < \infty $
a.s.,  $  0< i< n $. 
Let  $  E $
be the set of elementary predictable processes, topologized by uniform convergence in  $  ( t, \omega ) $. 
Let  $  L  ^ {0} $
be the set of finite-valued random variables, topologized by convergence in probability. Fix a stochastic process  $  X $
and for each stopping time  $  T $
define a mapping  $  I _ {X}  ^ {T} :  E \rightarrow L  ^ {0} $
by

$$ 
I _ {X}  ^ {T} ( H)  =  H _ {0} X _ {0}  ^ {T} + \sum _ { i=1} ^ { n }  H _ {i} ( X _ {T _ {i+1} }   ^ {T} - X _ {T _ {i}  }  ^ {T} ),
$$

where  $  X  ^ {T} $
denotes the process  $  X _ {t}  ^ {T} = X _ {t\wedge T }  $. 
Say that "X has the property (C)" if  $  I _ {X}  ^ {T} $
is continuous for all stopping times.

The Bichteler–Dellacherie theorem:  $  X $
has property (C) if and only if  $  X $
is a semi-martingale.

Since the topology on  $  E $
is very strong and that on  $  L  ^ {0} $
very weak, property (C) is a minimal requirement if the definition of  $  I _ {X}  ^ {T} $
is to be extended beyond  $  E $.

It is possible to use property (C) as the definition of a semi-martingale, and to develop the theory of stochastic integration from this point of view {{Cite|P}}. There are many excellent textbook expositions of stochastic integration from the conventional point of view; see, e.g., {{Cite|CW}}–{{Cite|RW}}.

====References====
{|
|valign="top"|{{Ref|B}}|| K. Bichteler, "Stochastic integrators" ''Bull. Amer. Math. Soc.'' , '''1''' (1979) pp. 761–765 {{MR|0537627}} {{ZBL|0416.60066}}
|-
|valign="top"|{{Ref|B2}}|| K. Bichteler, "Stochastic integrators and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013088.png" /> theory of semimartingales" ''Ann. Probab.'' , '''9''' (1981) pp. 49–89
|-
|valign="top"|{{Ref|D}}|| C. Dellacherie, "Un survol de la théorie de l'intégrale stochastique" ''Stoch. Proc. &amp; Appl.'' , '''10''' (1980) pp. 115–144 {{MR|0587420}} {{MR|0562680}} {{MR|0577985}} {{ZBL|0436.60043}} {{ZBL|0429.60053}} {{ZBL|0427.60055}}
|-
|valign="top"|{{Ref|P}}|| P. Protter, "Stochastic integration and differential equations" , Springer (1990) {{MR|1037262}} {{ZBL|0694.60047}}
|-
|valign="top"|{{Ref|CW}}|| K.L. Chung, R.J. Williams, "Introduction to stochastic integration" , Birkhäuser (1990) {{MR|1102676}} {{ZBL|0725.60050}}
|-
|valign="top"|{{Ref|E}}|| R.J. Elliott, "Stochastic calculus and applications" , Springer (1982) {{MR|0678919}} {{ZBL|0503.60062}}
|-
|valign="top"|{{Ref|KS}}|| I. Karatzas, S.E. Shreve, "Brownian motion and stochastic calculus" , Springer (1988) {{MR|0917065}} {{ZBL|0638.60065}}
|-
|valign="top"|{{Ref|RW}}|| L.C.G. Rogers, D. Williams, "Diffusions, Markov processes and martingales" , '''II. Ito calculus''' , Wiley (1987) {{MR|0921238}} {{ZBL|0627.60001}}
|-
|valign="top"|{{Ref|McK}}|| H.P. McKean jr., "Stochastic integrals" , Acad. Press (1969)
|-
|valign="top"|{{Ref|MP}}|| M. Metivier, J. Pellaumail, "Stochastic integration" , Acad. Press (1980) {{MR|0578177}} {{ZBL|0463.60004}}
|-
|valign="top"|{{Ref|McSh}}|| E.J. McShane, "Stochastic calculus and stochastic models" , Acad. Press (1974)
|-
|valign="top"|{{Ref|R}}|| M.M. Rao, "Stochastic processes and integration" , Sijthoff &amp; Noordhoff (1979) {{MR|0546709}} {{ZBL|0429.60001}}
|-
|valign="top"|{{Ref|SV}}|| D.W. Stroock, S.R.S. Varadhan, "Multidimensional diffusion processes" , Springer (1979) {{MR|0532498}} {{ZBL|0426.60069}}
|-
|valign="top"|{{Ref|K}}|| P.E. Kopp, "Martingales and stochastic integrals" , Cambridge Univ. Press (1984) {{MR|0774050}} {{ZBL|0537.60047}}
|-
|valign="top"|{{Ref|F}}|| M. Fukushima, "Dirichlet forms and Markov processes" , North-Holland (1980) {{MR|0569058}} {{ZBL|0422.31007}}
|-
|valign="top"|{{Ref|AFHL}}|| S. Albeverio, J.E. Fenstad, R. Høegh-Krohn, T. Lindstrøm, "Nonstandard methods in stochastic analysis and mathematical physics" , Acad. Press (1986) {{MR|0859372}} {{ZBL|0605.60005}}
|}