Singularity of Gaussian measures on function spaces induced by Brownian motion processes with non-stationary increments

Yeh, J

doi:10.1215/ijm/1256052816

Cited by 45 publications

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“…The generalized Brownian motion process Y determined by a(t) and b(t) is a Gaussian process with mean function a(t) and covariance function r(s, t) = min{b(s), b(t)}. For more details, see [5,10,14,30,31]. By [31,Theorem 14.2], the probability measure μ induced by Y , taking a separable version, is supported by…”

Section: The Function Space C Ab [0 T ]mentioning

confidence: 99%

“…The function space C a,b [0, T ], induced by a GBMP, was introduced by Yeh in [30], and was used extensively in [4,8,[10][11][12][13]20]. There have also been several attempts to construct financial mathematical theories using this process, see [18,21,25].…”

mentioning

confidence: 99%

“…, Y tn (ω)) ∈ B is given by . For more details, see [30,31]. Note that when c = 0, a(t) ≡ 0, and b(t) = t on [0, T ], the GBMP reduces to a SBMP.…”

mentioning

confidence: 99%

“…However, the stochastic process used in this paper, as well as in [4][5][6][7][8][9][10][11][12][13][14][15][16]18,20,21,25,30], is non-stationary in time and is subject to a drift because In this note, we present an example of a bounded functional which is not generalized analytic Feynman integrable on the function space C a,b [0, T ].…”

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confidence: 99%

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

Chang¹,

Choi²

2016

Arch. Math.

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In the theory of the analytic Feynman integral, the integrand is a functional of the standard Brownian motion process. In this note, we present an example of a bounded functional which is not Feynman integrable. The bounded functionals discussed in this note are defined in sample paths of the generalized Brownian motion process.Mathematics Subject Classification. 28C20, 60J65, 46G12.1. Introduction. The purpose of this note is to illustrate an effect of drift of the generalized Brownian motion process (GBMP). To do this, we discuss the theory of analytic Feynman integrals. Frankly speaking, in order to emphasize an effect of drift of GBMPs, we present an example of a bounded functional which is not analytic Feynman integrable on the function space C a,b [0, T ]. The function space C a,b [0, T ] is a probability space induced by a GBMP. Let W ≡ C 0 [0, T ] denote one-parameter Wiener space; this is the space of all real-valued continuous functions x on [0, T ] with x(0) = 0. Let M denote the class of all Wiener measurable subsets of C 0 [0, T ] and let m be the Wiener measure. Then, as is well known, (C 0 [0, T ], M, m) is a complete measure space. The coordinate process W on C 0 [0, T ] × [0, T ] defined by (x, t)

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Section: The Function Space C Ab [0 T ]mentioning

confidence: 99%

mentioning

confidence: 99%

“…, Y tn (ω)) ∈ B is given by . For more details, see [30,31]. Note that when c = 0, a(t) ≡ 0, and b(t) = t on [0, T ], the GBMP reduces to a SBMP.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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

Chang¹,

Choi²

2016

Arch. Math.

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“…The function space C a,b [0, T ] induced by generalized Brownian motion was introduced by J. Yeh in [18] and was used extensively by Chang and Chung [6]. The Wiener process used in [1-5, 8, 11] is stationary in time and is free of drift while the stochastic process used in this paper as well as in [6,7,9], 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 [14].…”

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confidence: 99%

Integral Transforms of Functionals in L 2(C a,b [0,T])

Chang¹,

Chung²,

Skoug

2009

J Fourier Anal Appl

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Some relationships for the double modified generalized analytic function space Fourier–Feynman transform and its applications

Chang

Choi

Chung

2016

Mathematische Nachrichten

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In this paper we first introduce the concept of a double modified analytic function space Fourier–Feynman transform using the double modified analytic function space integral. We then proceed to establish the existence of the modified analytic function space Fourier–Feynman transform for all functionals in the Banach algebra. Finally we use this double modified analytic function space transform to explain various physical phenomenon.

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Singularity of Gaussian measures on function spaces induced by Brownian motion processes with non-stationary increments

Cited by 45 publications

References 0 publications

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

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

Integral Transforms of Functionals in L 2(C a,b [0,T])

Some relationships for the double modified generalized analytic function space Fourier–Feynman transform and its applications

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