Stochastic integration via the Fock space of white noise

Consider the probability space of (commutative) 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

\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)

\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) whose isomorphic pre-image gives the triple $( L ^ { 2 } ) ^ { - } \supset ( L ^ { 2 } ) \supset ( L ^ { 2 } ) ^ { + }$. For choices of $\Gamma ^ { \pm }$, see e.g. [a5], [a6], [a7], [a8], [a9], [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 $\partial _ { t } ^ { * } + \partial _ { t }$ [a7], [a8], [a9]. In particular, 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 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 [a9], [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). 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 [a9], [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 [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, cf. e.g. [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 [a9], [a3], [a12].

It has been shown in [a3] that Itô's lemma holds for the extended forward integral in its usual form (cf. also [a11]). The proof is completely based on Fock space methods, i.e. (a3). For the calculus of the Skorokhod integral, cf. [a2], [a10], [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 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. [a1], [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)

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

with stochastic kernel $K$.

References

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