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 |
Stochastic integration via the Fock space of white noise. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stochastic_integration_via_the_Fock_space_of_white_noise&oldid=12828