Sequential Generalized Transforms on Function Space

Abstract

We define two sequential transforms on a function spaceCa,b[0,T]induced by generalized Brownian motion process. We then establish the existence of the sequential transforms for functionals in a Banach algebra of functionals onCa,b[0,T]. We also establish that any one of these transforms acts like an inverse transform of the other transform. Finally, we give some remarks about certain relations between our sequential transforms and other well-known transforms onCa,b[0,T].

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

“…In this section, we briefly list some of the preliminaries from [14,16,21] that we will need to establish our results in the next sections.…”

“…By [32,Theorem 14.2], the probability measure µ induced by Y , taking a separable version, is supported by We note that the coordinate process defined by e t (x) = x(t) on C a,b [0, T ] × [0, T ] is also the GBMP determined by a(t) and b(t). For more detailed studies about this function space C a,b [0, T ], see [14,15,16,21,31].…”

“…This integral is called the Paley-Wiener-Zygmund (PWZ) stochastic integral, see [21]. Our definition of the PWZ stochastic integral is different than the definition given in [14,16,23].…”

“…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].…”

Effect of drift of the generalized Brownian motion process: an example for the analytic Feynman integral

“…In this note, we set c = a(0) = b(0) = 0. Then the function space C a,b [0, T ] induced by the GBMP Y determined by the a(·) and b(·) can be considered as the space of continuous sample paths of Y , see [4][5][6][7][8][9][10][11][12][13][14][15][16]20], and one can see that for each t ∈ [0, T ],…”