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

Comments

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)