Stochastic integration via the Fock space of white noise

Consider the probability space of (commutative) white noise , where is the topological -algebra of and is the measure determined by

(a1)

being the norm of and denoting the dual pairing. is unitary to the symmetric Fock space

(a2)

One can identify the last two spaces and denotes the unitary mapping from onto by .

Informally, one is looking for a pair , of operators acting on the Fock space which implement the canonical commutation relations (cf. also Commutation and anti-commutation relationships, representation of)

(a3)

Still informally, this can be achieved as follows. If , , , set

(a4)

and let be the informal adjoint, i.e.

(a5)

This is made rigorous by introducing a suitable (complete) subspace of with dual , so that one has a Gel'fand triple (cf. also Gel'fand representation) whose isomorphic pre-image gives the triple . For choices of , see e.g. [a5], [a6], [a7], [a8], [a9], [a12]. Then , . Denote the corresponding operators on and by and , respectively. It turns out that multiplication by white noise is well-defined as an operator from into by [a7], [a8], [a9]. In particular, Brownian motion may be defined as

being the Fock space vacuum .

Consider a process and assume for simplicity that this mapping is continuous. If one wishes to define the stochastic integral of with respect to Brownian motion , , then one may set for taking values in ,

(a6)

following the heuristic idea that the "time derivative of Brownian motion is white noise" . However, for most of the processes of interest (e.g. Brownian motion itself), one does not have 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 in (a6) by a proper version of

in order to reproduce the standard Itô integral (cf. also Itô formula). This extension of the operator can be defined using a subspace of , constructed by means of the trace theorem of Sobolev spaces [a9], [a3]. So, put

(a7)

It can be shown [a9] that for processes adapted to the filtration generated by Brownian motion, for all and that the resulting stochastic integral (a7) coincides with the Itô-integral of . 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 an analogous operator , 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 (or any other suitable Hilbert space of functions) and use operators , which fulfil the canonical anti-commutation relations

(a8)

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 where is the Fock space vacuum .

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

with stochastic kernel .

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