# Stochastic integral

2010 Mathematics Subject Classification: Primary: 60H05 [MSN][ZBL]

An integral "∫ H dX" with respect to a 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).

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).

How to Cite This Entry:
Stochastic integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stochastic_integral&oldid=48851
This article was adapted from an original article by A.N. Shiryaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article