View source for Stochastic integration via the Fock space of white noise

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Consider the [[Probability space|probability space]] of (commutative) [[White noise|white noise]] $( \mathcal{S} ^ { \prime } ( \mathbf{R} ) , \mathcal{B} , d \mu )$, where $\mathcal{B}$ is the topological $\sigma$-algebra of $\mathcal{S} ^ { \prime } ( \mathbf{R} )$ and $d \mu$ is the measure determined by

\begin{equation} \tag{a1} \int _ { {\cal S} ^ { \prime } ( {\bf R} ) } e ^ { i \langle x , \xi \rangle } d \mu ( x ) = e ^ { - \| \xi \| _ { 2 } ^ { 2 } / 2 } , \xi \in {\cal S} ( {\bf R} ) \end{equation}

$\| . \| _ { 2 }$ being the norm of $L ^ { 2 } ( \mathbf{R} , d t )$ and $\langle \, .\, ,\,  . \, \rangle$ denoting the dual pairing. $( L ^ { 2 } ) \equiv L ^ { 2 } ( \mathcal{S} ^ { \prime } ( \mathbf{R} ) , d \mu )$ is unitary to the symmetric [[Fock space|Fock space]]

\begin{equation} \tag{a2} \Gamma ( L ^ { 2 } ( \mathbf{R} ) ) = \bigoplus _ { n = 0 } ^ { \infty } \sqrt { n !} L ^ { 2 } ( \mathbf{R} )^{ \widehat { \bigotimes } n } \simeq \bigoplus _ { n = 0 } ^ { \infty } \sqrt { n !} \widehat{ L ^ { 2 } ( \mathbf{R} ^ { n } ) }. \end{equation}

One can identify the last two spaces and denotes the unitary mapping from $( L ^ { 2 } )$ onto $\Gamma ( L ^ { 2 } ( \mathbf{R} ^ { n } ) )$ by $\mathcal{S}$.

Informally, one is looking for a pair $D _ { t }$, $D _ { t } ^ { * }$ of operators acting on the Fock space which implement the canonical commutation relations (cf. also [[Commutation and anti-commutation relationships, representation of|Commutation and anti-commutation relationships, representation of]])

\begin{equation} \tag{a3} [ D _ { t } , D _ { s } ^ { * } ] = \delta ( t - s ) , [ D _ { t } , D _ { s } ] = [ D _ { t } ^ { * } , D _ { s } ^ { * } ] = 0. \end{equation}

Still informally, this can be achieved as follows. If $f \in \Gamma ( L ^ { 2 } ( \mathbf{R} ) )$, $f = ( f ^ { ( n ) } ) _ { n \in \mathbf{N} _ { 0 } }$, $f ^ { ( n ) } \in L ^ { 2 } \widehat { ( {\bf R} ^ { n } ) }$, set

\begin{equation} \tag{a4} D _ { t }\, f = \left( ( n + 1 )\, f ^ { ( n + 1 ) } ( t , \cdot ) \right) _ { n \in \mathbf{N} _ { 0 } } \end{equation}

and let $D _ { t } ^ { * }$ be the informal adjoint, i.e.

\begin{equation} \tag{a5} D_t^*f = ( 0 , \delta _ { t } \widehat { \bigotimes } f ^ { n } ) _ { n \in \bf N }. \end{equation}

This is made rigorous by introducing a suitable (complete) subspace $\Gamma ^ { + }$ of $\Gamma ( L ^ { 2 } ( \mathbf{R} ) )$ with dual $\Gamma^-$, so that one has a Gel'fand triple $\Gamma ^ { - } \supset \Gamma ( L ^ { 2 } ( \mathbf{R} ) ) \supset \Gamma ^ { + }$ (cf. also [[Gel'fand representation|Gel'fand representation]]) whose isomorphic pre-image gives the triple $( L ^ { 2 } ) ^ { - } \supset ( L ^ { 2 } ) \supset ( L ^ { 2 } ) ^ { + }$. For choices of $\Gamma ^ { \pm }$, see e.g. [[#References|[a5]]], [[#References|[a6]]], [[#References|[a7]]], [[#References|[a8]]], [[#References|[a9]]], [[#References|[a12]]]. Then $D _ { t } : \Gamma ^ { + } \rightarrow ( L ^ { 2 } )$, $D _ { t } ^ { * } : ( L ^ { 2 } ) \rightarrow \Gamma ^ { - }$. Denote the corresponding operators on $( L ^ { 2 } ) ^ { + }$ and $( L ^ { 2 } )$ by $\partial_t$ and $\partial _ { t } ^ { * }$, respectively. It turns out that multiplication by white noise is well-defined as an operator from $( L ^ { 2 } ) ^ { + }$ into $( L ^ { 2 } ) ^ { - }$ by $\partial _ { t } ^ { * } + \partial _ { t }$ [[#References|[a7]]], [[#References|[a8]]], [[#References|[a9]]]. In particular, [[Brownian motion|Brownian motion]] may be defined as

\begin{equation*} t \rightarrow \int _ { 0 } ^ { t } ( \partial _ { s } ^ { * } + \partial _ { s } ) 1 d s = \mathcal{S} ^ { - 1 } \left( \int _ { 0 } ^ { t } ( D _ { s } ^ { * } + D _ { s } ) \Omega d s \right), \end{equation*}

$\Omega$ being the Fock space vacuum $\Omega = ( 1,0 , \ldots )$.

Consider a process $\phi : [ 0,1 ] \rightarrow ( L ^ { 2 } )$ and assume for simplicity that this mapping is continuous. If one wishes to define the [[Stochastic integral|stochastic integral]] of $\phi$ with respect to Brownian motion $B ( t )$, $t \in {\bf R}_ +$, then one may set for $\phi$ taking values in $( L ^ { 2 } ) ^ { + }$,

\begin{equation} \tag{a6} \int _ { 0 } ^ { t } \phi ( s ) d B ( s ) : = \int _ { 0 } ^ { t } ( \partial _ { s } ^ { * } + \partial _ { s } ) \phi ( s ) d s, \end{equation}

following the heuristic idea that the  "time derivative of Brownian motion is white noise" . However, for most of the processes $\phi$ of interest (e.g. Brownian motion itself), one does not have $\phi ( s ) \in ( L ^ { 2 } ) ^ { + }$ and therefore the second term on the right-hand side of (a6) would be ill-defined. Moreover, heuristic calculations show [[#References|[a9]]], [[#References|[a11]]] that one should replace the term $\partial _ { s } \phi ( s )$ in (a6) by a proper version of

\begin{equation*} \partial _ { s + } \phi ( s ) = \operatorname { lim } _ { \epsilon \downarrow 0 } \partial _ { s + \epsilon } \phi ( s ) \end{equation*}

in order to reproduce the standard Itô integral (cf. also [[Itô formula|Itô formula]]). This extension of the operator $\partial _ { s }$ can be defined using a subspace of $L ^ { 2 } ( [ 0,1 ] ; ( L ^ { 2 } ) )$, constructed by means of the trace theorem of Sobolev spaces [[#References|[a9]]], [[#References|[a3]]]. So, put

\begin{equation} \tag{a7} \int _ { 0 } ^ { t } \phi ( s ) d B ( s + ) : = \int _ { 0 } ^ { t } ( \partial _ { s } ^ { * } + \partial _ { s + } ) \phi ( s ) d s. \end{equation}

It can be shown [[#References|[a9]]] that for processes $\phi$ adapted to the filtration generated by Brownian motion, $\partial _ { s + } \phi ( s ) = 0$ for all $s \in [ 0,1 ]$ and that the resulting stochastic integral (a7) coincides with the Itô-integral of $\phi$. Thus, (a7) is an extension of Itô's integral to anticipating processes. Clearly, also the first term on the right-hand side of (a7) alone is an extension of the Itô-integral to non-adapted processes and it is the white noise formulation of the [[Skorokhod integral|Skorokhod integral]], cf. e.g. [[#References|[a10]]]. Also, using instead of $\partial _ { s +}$ an analogous operator $\partial _ { s- }$, one obtains (an extension of) the Itô backward integral and the mean of both is (an extension of) the [[Stratonovich integral|Stratonovich integral]] [[#References|[a9]]], [[#References|[a3]]], [[#References|[a12]]].

It has been shown in [[#References|[a3]]] that Itô's lemma holds for the extended forward integral in its usual form (cf. also [[#References|[a11]]]). The proof is completely based on Fock space methods, i.e. (a3). For the calculus of the Skorokhod integral, cf. [[#References|[a2]]], [[#References|[a10]]], [[#References|[a12]]].

==Generalizations.==
Clearly one can define in the above way stochastic integrals (more precisely, stochastic differential forms) for processes with multi-dimensional time, or even with time parameter on manifolds, etc.

Also, instead of the symmetric Fock space one may work with the anti-symmetric Fock space over $L ^ { 2 } ( \mathbf{R} )$ (or any other suitable [[Hilbert space|Hilbert space]] of functions) and use operators $A _ { t }$, $A _ { t } ^ { * }$ which fulfil the canonical anti-commutation relations

\begin{equation} \tag{a8} \{ A_t , A _ { s } ^ { * } \} = \delta ( t - s ) , \{ A _ { t } , A _ { s } \} = \{ A _ { t } ^ { * } , A _ { s } ^ { * } \} = 0. \end{equation}

This way one arrives at the fermionic stochastic integration and its calculus, see e.g. [[#References|[a1]]], [[#References|[a4]]]. In particular, one may define a fermionic Brownian motion as $t \rightarrow \int _ { 0 } ^ { t } ( A _ { s } ^ { * } + A _ { s } ) \Omega d s$ where $\Omega$ is the Fock space vacuum $\Omega = ( 1,0,0 , \dots )$.

It is also possible to consider stochastic Volterra integral operators (cf. also [[Volterra equation|Volterra equation]])

\begin{equation*} \Phi ( t ) = \int _ { 0 } ^ { t } K ( t , s ) \phi ( s ) d B ( s + ) \end{equation*}

with stochastic kernel $K$.

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  D. Applebaum,   R.L. Hudson,   "Fermion diffusions"  ''J. Math. Phys.'' , '''25'''  (1984)  pp. 858–861</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  J. Asch,   J. Potthoff,   "A generalization of Itô's lemma"  ''Proc. Japan Acad.'' , '''63A'''  (1987)  pp. 289–291</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  J. Asch,   J. Potthoff,   "Itô's lemma without non-anticipatory conditions"  ''Probab. Th. Rel. Fields'' , '''88'''  (1991)  pp. 17–46</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  C. Barnett,   R.F. Streater,   I.F. Wilde,   "The Itô–Clifford integral"  ''J. Funct. Anal.'' , '''48'''  (1982)  pp. 172–212</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  T. Hida,   "Brownian motion" , Springer  (1980)</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  T. Hida,   H.-H. Kuo,   J. Potthoff,   L. Streit,   "White noise: An infinite dimensional calculus" , Kluwer Acad. Publ.  (1993)</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  I. Kubo,   S. Takenaka,   "Calculus on Gaussian white noise, I–IV"  ''Proc. Japan Acad.'' , '''56–58'''  (1980–1982)  pp. 376–380; 411–416; 433–437; 186–189</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  H.-H. Kuo,   "Brownian functionals and applications"  ''Acta Applic. Math.'' , '''1'''  (1983)  pp. 175–188</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  H.-H. Kuo,   A. Russek,   "White noise approach to stochastic integration"  ''J. Multivariate Anal.'' , '''24'''  (1988)  pp. 218–236</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  D. Nualart,   E. Pardoux,   "Stochastic calculus with anticipating integrands"  ''Th. Rel. Fields'' , '''78'''  (1988)  pp. 535–581</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  J. Potthoff,   "Stochastic integration in Hida's white noise calculus"  S. Albeverio (ed.)  D. Merlini (ed.) , ''Stochastic Processes, Physics and Geometry''  (1988)</td></tr><tr><td valign="top">[a12]</td> <td valign="top">  F. Russo,   P. Vallois,   "Forward, backward and symmetric stochastic integration"  ''Probab. Th. Rel. Fields'' , '''97'''  (1993)  pp. 403–421</td></tr></table>