# Wiener integral

An abstract integral of Lebesgue type over sets in an infinite-dimensional function space of functionals defined on these sets. Introduced by N. Wiener in the nineteentwenties in connection with problems on Brownian motion [1], [2].

Let $C _ {0}$ be the vector space of continuous real-valued functions $x$ defined on $[ 0, 1]$ such that $x( 0) = 0$, with norm

$$\| x \| = \max _ {t \in [ 0, 1] } | x ( t) |.$$

The set

$$Q = \{ {x \in C _ {0} } : { a _ {i} < x ( t _ {i} ) \leq b _ {i} ,\ 0 = t _ {0} < t _ {1} < \dots < t _ {n} = 1 } \}$$

is called a quasi-interval of this space. Here, $a _ {i}$ and $b _ {i}$ may be equal to $- \infty$ and $+ \infty$, respectively, but then the symbol $<$ must replace $\leq$. The whole space $C _ {0} = \{ {x ( t) } : {- \infty < x ( 1) < + \infty } \}$ is an example of a quasi-interval.

The Wiener measure of a quasi-interval $Q$ is the number

$$\mu _ {W} ( Q) = \ { \frac{1}{\sqrt {\pi ^ {n} \prod _ { i = 1 } ^ { n } ( t _ {i} - t _ {i-} 1 ) }} } \int\limits _ { a _ 1 } ^ {b _ 1 } \dots \int\limits _ { a _ n } ^ {b _ n} e ^ {- L _ {n} } dx _ {n} \dots dx _ {1} ,$$

where

$$L _ {n} = \sum _ {j = 1 } ^ { n } \frac{( x _ {j} - x _ {j-} 1 ) ^ {2} }{t _ {j} - t _ {j-} 1 }$$

and $x _ {j} = x ( t _ {j} )$. This measure extends to a $\sigma$- additive measure on the Borel field of sets generated by the quasi-intervals, known also as Wiener measure.

Let $F$ be a functional defined on $C _ {0}$ that is measurable with respect to the measure $\mu _ {W}$. The Lebesgue-type integral

$$\int\limits _ { C _ 0 } F ( x) d \mu _ {W} ( x)$$

is known as the Wiener integral, or as the integral with respect to the Wiener measure, of the functional $F$. If $E \subset C _ {0}$ is measurable, then

$$\int\limits _ { E } F ( x) d \mu _ {W} ( x) = \ \int\limits _ { C _ 0 } F ( x) \chi _ {E} ( x) d \mu _ {W} ( x) ,$$

where $\chi _ {E}$ is the characteristic function of the set $E$.

Wiener's integral displays several properties of the ordinary Lebesgue integral. In particular, a functional which is bounded and measurable on a set $E$ is integrable with respect to the Wiener measure on this set and if, in addition, the functional $F$ is continuous and non-negative, then

$$\int\limits _ { C _ 0 } F ( x) d \mu _ {W} ( x) =$$

$$= \ \lim\limits _ {n \rightarrow \infty } \frac{1}{\sqrt { {\pi ^ {n} \prod _ {i = 1 } ^ { n } ( t _ {i} - t _ {i-} 1 ) } }} \int\limits _ { \mathbf R } ^ {n} \frac{F _ {n} ( x _ {1} \dots x _ {n} ) }{e ^ {L} _ {n} } dx _ {1} \dots dx _ {n} ,$$

where $F _ {n} ( x _ {1} \dots x _ {n} )$ is the value of $F$ at linear interpolation of $x( t)$ between points $( t _ {i} , x _ {i} \equiv x( t _ {i} ))$.

The computation of a Wiener integral presents considerable difficulties, even for the simplest functionals. The task may sometimes be reduced to solving a single differential equation [1].

There is a method by which Wiener's integral may be approximately computed through approximating it by finite-dimensional Stieltjes integrals of a high multiplicity (cf. Stieltjes integral).

#### References

 [1] I.M. Koval'chik, "The Wiener integral" Russian Math. Surveys , 18 : 1 (1963) pp. 97–134 Uspekhi Mat. Nauk , 18 : 1 (1963) pp. 97–134 [2] G.E. Shilov, "Integration in infinite dimensional spaces and the Wiener integral" Russ. Math. Surveys , 18 : 2 (1963) pp. 99–120 Uspekhi Mat. Nauk , 2 (1963) pp. 99–120

Further references on the computation of Wiener integrals in the sense described above are [a1] and [a2]. In the Western literature, the term "Wiener integral" normally refers to the stochastic integral of a deterministic function $f$ such that $f \in L _ {2} [ 0, t]$ for each $t \in \mathbf R _ {+}$, with respect to the Wiener process $X( t)$ defined on a probability space $( \Omega , {\mathcal F} , P)$. This is denoted by

$$I _ {t} ( f ) = \int\limits _ { 0 } ^ { t } f( s) dX( s) ,$$

and is defined as follows. If $f$ is a simple function, i.e. $f( s) = a _ {i}$ for $s \in [ t _ {t-} 1 , t _ {i} )$, where $a _ {i} \in \mathbf R$ and $0 = t _ {0} < t _ {1} < \dots < t _ {n} = t$, then

$$I _ {t} ( f ) = \sum _ { n= } 1 ^ { n } a _ {i} ( X( t _ {i} ) - X( t _ {i-} 1 )) .$$

Let $S$ denote the set of simple functions. For $f , g \in S$, a computation shows that ${\mathsf E} I _ {t} ( f ) = 0$, ${\mathsf E} ( I _ {t} ( f ) I _ {t} ( g)) = \int _ {0} ^ {t} f( s) g( s) ds$, i.e. $f \mapsto I _ {t} ( f )$ is an inner-product preserving mapping from $L _ {2} [ 0, t]$ to $L _ {2} ( \Omega , {\mathcal F} , P )$. For any $f \in L _ {2} [ 0, t]$ there exists a sequence $f _ {n} \in S$ such that $f _ {n} \rightarrow f$. $\{ I _ {t} ( f _ {n} ) \}$ is then a Cauchy sequence in $L _ {2} ( \Omega , {\mathcal F} , P)$, and one defines

$$\int\limits _ { 0 } ^ { t } f( s) dX( s) = \lim\limits _ {n \rightarrow \infty } I _ {t} ( f _ {n} ).$$

Notable features of this construction are as follows.

It is possible to define $I _ {t} ( f )$ simultaneously for all $t \geq 0$ and to obtain a version which is a Gaussian martingale with continuous sample paths

$$\mathop{\rm sp} \{ {X ( s) } : {0 \leq s \leq t } \} = \ \{ {I _ {t} ( f ) } : {f \in L _ {2} [ 0, t ] } \} ,$$

where "sp" denotes the closed linear span in $L _ {2} ( \Omega , {\mathcal F} , P )$. Information on the Wiener integral in this sense is given in [a3], [a4].

#### References

 [a1] A.J. Chorin, "Accurate evaluation of Wiener integrals" Math. Comp. , 27 (1973) pp. 1–15 [a2] G.L. Blankenschip, J.S. Baras, "Accurate evaluation of stochastic Wiener integrals with applications to scattering in random media and to nonlinear filtering" SIAM J. Appl. Math. , 41 (1981) pp. 518–552 [a3] M.H.A. Davis, "Linear estimation and stochastic control" , Chapman & Hall (1977) [a4] R.S. Liptser, A.N. Shiryaev, "Statistics of random processes" , I , Springer (1977) (Translated from Russian) [a5] J. Yeh, "Stochastic processes and the Wiener integral" , M. Dekker (1973) [a6] B. Simon, "Functional integration and quantum physics" , Acad. Press (1979) pp. 4–6 [a7] L.C.G. Rogers, D. Williams, "Diffusions, Markov processes, and martingales" , 2. Itô calculus , Wiley (1987) [a8] H. Bauer, "Probability theory and elements of measure theory" , Holt, Rinehart & Winston (1972) pp. Chapt. 11 (Translated from German)
How to Cite This Entry:
Wiener integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wiener_integral&oldid=49219
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article