• Title, Summary, Keyword: generalized Brownian motion

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GENERALIZED ANALYTIC FEYNMAN INTEGRALS INVOLVING GENERALIZED ANALYTIC FOURIER-FEYNMAN TRANSFORMS AND GENERALIZED INTEGRAL TRANSFORMS

  • Chang, Seung Jun;Chung, Hyun Soo
    • Journal of the Chungcheong Mathematical Society
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    • v.21 no.2
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    • pp.231-246
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    • 2008
  • In this paper, we use a generalized Brownian motion process to define a generalized analytic Feynman integral. We then establish several integration formulas for generalized analytic Feynman integrals generalized analytic Fourier-Feynman transforms and generalized integral transforms of functionals in the class of functionals ${\mathbb{E}}_0$. Finally, we use these integration formulas to obtain several generalized Feynman integrals involving the generalized analytic Fourier-Feynman transform and the generalized integral transform of functionals in ${\mathbb{E}}_0$.

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CONDITIONAL GENERALIZED FOURIER-FEYNMAN TRANSFORM AND CONDITIONAL CONVOLUTION PRODUCT ON A BANACH ALGEBRA

  • Chang, Seung-Jun;Choi, Jae-Gil
    • Bulletin of the Korean Mathematical Society
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    • v.41 no.1
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    • pp.73-93
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    • 2004
  • In [10], Chang and Skoug used a generalized Brownian motion process to define a generalized analytic Feynman integral and a generalized analytic Fourier-Feynman transform. In this paper we define the conditional generalized Fourier-Feynman transform and conditional generalized convolution product on function space. We then establish some relationships between the conditional generalized Fourier-Feynman transform and conditional generalized convolution product for functionals on function space that belonging to a Banach algebra.

TRANSFORMS AND CONVOLUTIONS ON FUNCTION SPACE

  • Chang, Seung-Jun;Choi, Jae-Gil
    • Communications of the Korean Mathematical Society
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    • v.24 no.3
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    • pp.397-413
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    • 2009
  • In this paper, for functionals of a generalized Brownian motion process, we show that the generalized Fourier-Feynman transform of the convolution product is a product of multiple transforms and that the conditional generalized Fourier-Feynman transform of the conditional convolution product is a product of multiple conditional transforms. This allows us to compute the (conditional) transform of the (conditional) convolution product without computing the (conditional) convolution product.

TRANSLATION THEOREM ON FUNCTION SPACE

  • Choi, Jae Gil;Park, Young Seo
    • Korean Journal of Mathematics
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    • v.11 no.1
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    • pp.17-30
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    • 2003
  • In this paper, we use a generalized Brownian motion process to define a translation theorem. First we establish the translation theorem for function space integrals. We then obtain the general translation theorem for functionals on function space.

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A FRESNEL TYPE CLASS ON FUNCTION SPACE

  • Chang, Seung-Jun;Choi, Jae-Gil;Lee, Sang-Deok
    • The Pure and Applied Mathematics
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    • v.16 no.1
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    • pp.107-119
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    • 2009
  • In this paper we define a Banach algebra on very general function space induced by a generalized Brownian motion process rather than on Wiener space, but the Banach algebra can be considered as a generalization of Fresnel class defined on Wiener space. We then show that several interesting functions in quantum mechanic are elements of the class.

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MULTIPLE Lp ANALYTIC GENERALIZED FOURIER-FEYNMAN TRANSFORM ON THE BANACH ALGEBRA

  • Chang, Seung-Jun;Choi, Jae-Gil
    • Communications of the Korean Mathematical Society
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    • v.19 no.1
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    • pp.93-111
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    • 2004
  • In this paper, we use a generalized Brownian motion process to define a generalized Feynman integral and a generalized Fourier-Feynman transform. We also define the concepts of the multiple Lp analytic generalized Fourier-Feynman transform and the generalized convolution product of functional on function space $C_{a,\;b}[0,\;T]$. We then verify the existence of the multiple $L_{p}$ analytic generalized Fourier-Feynman transform for functional on function space that belong to a Banach algebra $S({L_{a,\;b}}^{2}[0, T])$. Finally we establish some relationships between the multiple $L_{p}$ analytic generalized Fourier-Feynman transform and the generalized convolution product for functionals in $S({L_{a,\;b}}^{2}[0, T])$.

A FUBINI THEOREM FOR GENERALIZED ANALYTIC FEYNMAN INTEGRALS AND FOURIER-FEYNMAN TRANSFORMS ON FUNCTION SPACE

  • Chang, Seung-Jun;Lee, Il-Yong
    • Bulletin of the Korean Mathematical Society
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    • v.40 no.3
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    • pp.437-456
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    • 2003
  • In this paper we use a generalized Brownian motion process to define a generalized analytic Feynman integral. We then establish a Fubini theorem for the function space integral and generalized analytic Feynman integral of a functional F belonging to Banach algebra $S(L^2_{a,b}[0,T])$ and we proceed to obtain several integration formulas. Finally, we use this Fubini theorem to obtain several Feynman integration formulas involving analytic generalized Fourier-Feynman transforms. These results subsume similar known results obtained by Huffman, Skoug and Storvick for the standard Wiener process.

Note on the generalized Fourier-Feynman transform on function space (함수공간에서의 일반화된 푸리에-파인만 변환에 관한 고찰)

  • Chang, Seung-Jun
    • Journal for History of Mathematics
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    • v.20 no.3
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    • pp.73-90
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    • 2007
  • In this paper, we define a generalized Feynman integral and a generalized Fourier-Feynman transform on function space induced by generalized Brownian motion process. We then give existence theorems and several properties for these concepts. Finally we investigate relationships of the Fourier transform and the generalized Fourier-Feynman transform.

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GENERALIZED BROWNIAN MOTIONS WITH APPLICATION TO FINANCE

  • Chung, Dong-Myung;Lee, Jeong-Hyun
    • Journal of the Korean Mathematical Society
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    • v.43 no.2
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    • pp.357-371
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    • 2006
  • Let $X\;=\;(X_t,\;t{\in}[0, T])$ be a generalized Brownian motion(gBm) determined by mean function a(t) and variance function b(t). Let $L^2({\mu})$ denote the Hilbert space of square integrable functionals of $X\;=\;(X_t - a(t),\; t {in} [0, T])$. In this paper we consider a class of nonlinear functionals of X of the form F(. + a) with $F{in}L^2({\mu})$ and discuss their analysis. Firstly, it is shown that such functionals do not enjoy, in general, the square integrability and Malliavin differentiability. Secondly, we establish regularity conditions on F for which F(.+ a) is in $L^2({\mu})$ and has its Malliavin derivative. Finally we apply these results to compute the price and the hedging portfolio of a contingent claim in our financial market model based on a gBm X.

GENERALIZED ANALYTIC FEYNMAN INTEGRAL VIA FUNCTION SPACE INTEGRAL OF BOUNDED CYLINDER FUNCTIONALS

  • Chang, Seung-Jun;Choi, Jae-Gil;Chung, Hyun-Soo
    • Bulletin of the Korean Mathematical Society
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    • v.48 no.3
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    • pp.475-489
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    • 2011
  • In this paper, we use a generalized Brownian motion to define a generalized analytic Feynman integral. We then obtain some results for the generalized analytic Feynman integral of bounded cylinder functionals of the form F(x) = $\hat{v}$(($g_1,x)^{\sim}$,..., $(g_n,x)^{\sim}$) defined on a very general function space $C_{a,b}$[0,T]. We also present a change of scale formula for function space integrals of such cylinder functionals.