A Fubini theorem on a function space and its applications

Abstract

In this paper we establish a Fubini theorem for functionals on a function space. We then establish some relationships as applications of our Fubini theorem. Finally, we present some historical remarks.

A Translation Theorem for the Generalized Fourier-Feynman Transform Associated With Gaussian Process on Function Space

“…Our definition of the PWZ stochastic integral is different than the definition given in [14,16,23]. But we will emphasize that the following fundamental facts are still true: ∼ exists and we have (w, x)…”

“…The function space C a,b [0, T ], induced by a GBMP, was introduced by Yeh in [31], and was used extensively in [14,15,16,21,23]. There have also been several recent attempts to construct financial mathematical theories using this process [22,24,26].…”

Some Expressions for the Inverse Integral Transform via the Translation Theorem on Function Space

“…However, when a(t) ≡ 0 and b(t) = t on [0, T ], the general function space C a,b [0, T ] reduces to the Wiener space C 0 [0, T ] and so most of the results in [6] follow immediately from the results in this paper. The Wiener process used in [4,6,11,13,14,15,16,19] is stationary in time and is free of drift while the stochastic process used in this paper as well as in [5,7,8,18], is nonstationary in time, is subject to a drift a(t), and can be used to explain the position of the Ornstein-Uhlenbeck process in an external force field [17].…”

“…The following lemma was established in [8] and used in [10]. The formula (3.2) is called the Fubini theorem with respect to the function space integrals.…”

“…The function space C a,b [0, T ] induced by a generalized Brownian motion was introduced by J. Yeh in [20] and was used extensively in [5,7,8,9,10]. In [5], the authors gave a necessary and sufficient condition that a functional F in L 2 (C a,b [0, T ]) has an integral transform F γ,β F also belonging to L 2 (C a,b [0, T ]).…”