# Stochastic integral

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2020 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).

#### References

 [J] J. Jacod, "Calcul stochastique et problèmes de martingales" , Lect. notes in math. , 714 , Springer (1979) MR0542115 Zbl 0414.60053 [DM] C. Dellacherie, P.A. Meyer, "Probabilities and potential" , A-C , North-Holland (1978–1988) (Translated from French) MR0939365 MR0898005 MR0727641 MR0745449 MR0566768 MR0521810 Zbl 0716.60001 Zbl 0494.60002 Zbl 0494.60001 [LS] R.Sh. Liptser, A.N. Shiryayev, "Theory of martingales" , Kluwer (1989) (Translated from Russian) MR1022664 Zbl 0728.60048

The result alluded to above, that semi-martingales constitute the widest viable class of stochastic integrators, is the Bichteler–Dellacherie theorem [B][D], and can be formulated as follows [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 [P]. There are many excellent textbook expositions of stochastic integration from the conventional point of view; see, e.g., [CW][RW].

#### References

 [B] K. Bichteler, "Stochastic integrators" Bull. Amer. Math. Soc. , 1 (1979) pp. 761–765 MR0537627 Zbl 0416.60066 [B2] K. Bichteler, "Stochastic integrators and the theory of semimartingales" Ann. Probab. , 9 (1981) pp. 49–89 [D] C. Dellacherie, "Un survol de la théorie de l'intégrale stochastique" Stoch. Proc. & Appl. , 10 (1980) pp. 115–144 MR0587420 MR0562680 MR0577985 Zbl 0436.60043 Zbl 0429.60053 Zbl 0427.60055 [P] P. Protter, "Stochastic integration and differential equations" , Springer (1990) MR1037262 Zbl 0694.60047 [CW] K.L. Chung, R.J. Williams, "Introduction to stochastic integration" , Birkhäuser (1990) MR1102676 Zbl 0725.60050 [E] R.J. Elliott, "Stochastic calculus and applications" , Springer (1982) MR0678919 Zbl 0503.60062 [KS] I. Karatzas, S.E. Shreve, "Brownian motion and stochastic calculus" , Springer (1988) MR0917065 Zbl 0638.60065 [RW] L.C.G. Rogers, D. Williams, "Diffusions, Markov processes and martingales" , II. Ito calculus , Wiley (1987) MR0921238 Zbl 0627.60001 [McK] H.P. McKean jr., "Stochastic integrals" , Acad. Press (1969) [MP] M. Metivier, J. Pellaumail, "Stochastic integration" , Acad. Press (1980) MR0578177 Zbl 0463.60004 [McSh] E.J. McShane, "Stochastic calculus and stochastic models" , Acad. Press (1974) [R] M.M. Rao, "Stochastic processes and integration" , Sijthoff & Noordhoff (1979) MR0546709 Zbl 0429.60001 [SV] D.W. Stroock, S.R.S. Varadhan, "Multidimensional diffusion processes" , Springer (1979) MR0532498 Zbl 0426.60069 [K] P.E. Kopp, "Martingales and stochastic integrals" , Cambridge Univ. Press (1984) MR0774050 Zbl 0537.60047 [F] M. Fukushima, "Dirichlet forms and Markov processes" , North-Holland (1980) MR0569058 Zbl 0422.31007 [AFHL] S. Albeverio, J.E. Fenstad, R. Høegh-Krohn, T. Lindstrøm, "Nonstandard methods in stochastic analysis and mathematical physics" , Acad. Press (1986) MR0859372 Zbl 0605.60005
How to Cite This Entry:
Stochastic integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stochastic_integral&oldid=54958
This article was adapted from an original article by A.N. Shiryaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article