Change of scale formulas for a generalized conditional Wiener integral
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Let $C[0,t]$ denote the space of real-valued continuous functions on $[0,t]$ and define a random vector $Z_n:C[0,t]\to\mathbb R^n$ by $Z_n(x)=(\int_0^{t_1}h(s) dx(s),\ldots,\int_0^{t_n}h(s) dx(s))$, where $0<t_1 < \cdots < t_n=t$ is a partition of $[0,t]$ and $h\in L_2[0,t]$ with $h\neq 0$ a.e. Using a simple formula for a conditional expectation on $C[0,t]$ with $Z_n$, we evaluate a generalized analytic conditional Wiener integral of the function $G_r(x) = F(x)\Psi(\int_0^t v_1 (s) dx(s),\ldots,\int_0^t v_r(s)dx(s))$ for $F$ in a Banach algebra and for $\Psi=f+\phi $ which need not be bounded or continuous, where $f\in L_p(\mathbb R^r)(1\le p\le\infty)$, $\{v_1,\ldots,v_r\}$ is an orthonormal subset of $L_2[0,t]$ and $\phi$ is the Fourier transform of a measure of bounded variation over $\mathbb R^r$. Finally we establish various change of scale transformations for the generalized analytic conditional Wiener integrals of $G_r$ with the conditioning function $Z_n$.
Abstract
Let $C[0,t]$ denote the space of real-valued continuous functions on $[0,t]$ and define a random vector $Z_n:C[0,t]\to\mathbb R^n$ by $Z_n(x)=(\int_0^{t_1}h(s) dx(s),\ldots,\int_0^{t_n}h(s) dx(s))$, where $0<t_1 < \cdots < t_n=t$ is a partition of $[0,t]$ and $h\in L_2[0,t]$ with $h\neq 0$ a.e. Using a simple formula for a conditional expectation on $C[0,t]$ with $Z_n$, we evaluate a generalized analytic conditional Wiener integral of the function $G_r(x) = F(x)\Psi(\int_0^t v_1 (s) dx(s),\ldots,\int_0^t v_r(s)dx(s))$ for $F$ in a Banach algebra and for $\Psi=f+\phi $ which need not be bounded or continuous, where $f\in L_p(\mathbb R^r)(1\le p\le\infty)$, $\{v_1,\ldots,v_r\}$ is an orthonormal subset of $L_2[0,t]$ and $\phi$ is the Fourier transform of a measure of bounded variation over $\mathbb R^r$. Finally we establish various change of scale transformations for the generalized analytic conditional Wiener integrals of $G_r$ with the conditioning function $Z_n$.
참고문헌 (15)
- R. H. Cameron, "The translation pathology of Wiener space", Duke Math. J, 21, pp.623~627 (1954)
- R. H. Cameron, "The behavior of measure and measurability under change of scale in Wiener space", Bull. Amer. Math. Soc, 53, pp.130~137 (1947)
- R. H. Cameron, "Change of scale formulas for Wiener integral", Rend. Circ. Mat. Palermo (2), (Suppl. 17), pp.105~115 (1987)
- R. H. Cameron, "Some Banach algebras of analytic Feynman integrable functionals", Analytic functions, Kozubnik 1979 (Proc. Seventh Conf.), pp.18~67 (1980), Springer
- K. S. Chang, "Functions in the Fresnel class", Proc. Amer. Math. Soc, 100(2), pp.309~318 (1987)
- 조동현, "Change of scale formulas for conditional Wiener integrals as integral transforms over Wiener paths in abstract Wiener space", 대한수학회논문집, 22(1), pp.91~109 (2007), 대한수학회
- D. H. Cho, "A simple formula for a generalized conditional Wiener integral and its appli-cations", Int. J. Math. Anal. (Ruse), 7(29-32), pp.1419~1431 (2013)
- D. H. Cho, "Analogues of conditional Wiener integrals and their change of scale transformations on a function space", J. Math. Anal. Appl, 359(2), pp.421~438 (2009)
- 임만규, "An analogue of Wiener measure and its applications", 대한수학회지, 39(5), pp.801~819 (2002), 대한수학회
- G. Kallianpur, "Stochastic analysis and applications", pp.217~267 (1984), Dekker
- 유일, "A Change of scale formula for conditional Wiener integrals on classical Wiener space", 대한수학회지, 44(4), pp.1025~1050 (2007), 대한수학회
- I. Yoo, "A change of scale formula for Wiener integrals on abstract Wiener spaces", Internat. J. Math. Math. Sci, 17(2), pp.239~247 (1994)
- I. Yoo, "A change of scale formula for Wiener integrals on abstract Wiener spaces II", J. Korean Math. Soc, 31(1), pp.115~129 (1994)
- I. Yoo, "A change of scale formula for Wiener integrals of unbounded functions", Rocky Mountain J. Math, 34(1), pp.371~389 (2004)
- I. Yoo, "Change of scale formulas for Yeh-Wiener integrals", Commun. Korean Math. Soc, 6(1), pp.19~26 (1991)