Geodesic
A simple formula for an analogue of conditional Wiener integrals and its applications. II
Czechoslovak Mathematical Journal, Tome 59 (2009) no. 2, pp. 431-452 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

Let $C[0,T]$ denote the space of real-valued continuous functions on the interval $[0,T]$ with an analogue $w_\varphi $ of Wiener measure and for a partition $ 0=t_0 t_1 \cdots t_n
Let $C[0,T]$ denote the space of real-valued continuous functions on the interval $[0,T]$ with an analogue $w_\varphi $ of Wiener measure and for a partition $ 0=t_0 t_1 \cdots t_n $ of $[0, T]$, let $X_n\: C[0,T]\to \mathbb R^{n+1}$ and $X_{n+1} \: C [0, T]\to \mathbb R^{n+2}$ be given by $X_n(x) = ( x(t_0), x(t_1), \cdots , x(t_n))$ and $X_{n+1} (x) = ( x(t_0), x(t_1), \cdots , x(t_{n+1}))$, respectively. \endgraf In this paper, using a simple formula for the conditional $w_\varphi $-integral of functions on $C[0, T]$ with the conditioning function $X_{n+1}$, we derive a simple formula for the conditional $w_\varphi $-integral of the functions with the conditioning function $X_n$. As applications of the formula with the function $X_n$, we evaluate the conditional $w_\varphi $-integral of the functions of the form $F_m(x) = \int _0^T (x(t))^m d t$ for $x\in C[0, T]$ and for any positive integer $m$. Moreover, with the conditioning $X_n$, we evaluate the conditional $w_\varphi $-integral of the functions in a Banach algebra $\mathcal S_{w_\varphi }$ which is an analogue of the Cameron and Storvick's Banach algebra $\mathcal S$. Finally, we derive the conditional analytic Feynman $w_\varphi $-integrals of the functions in $\mathcal S_{w_\varphi }$.
Classification : 28C20, 60H05
Keywords: analogue of Wiener measure; Cameron-Martin translation theorem; conditional analytic Feynman $w_\varphi $-integral; conditional Wiener integral; Kac-Feynman formula; simple formula for conditional $w_\varphi $-integral 
@article{CMJ_2009_59_2_a10,
     author = {Cho, Dong Hyun},
     title = {A simple formula for an analogue of conditional {Wiener} integrals and its applications. {II}},
     journal = {Czechoslovak Mathematical Journal},
     pages = {431--452},
     year = {2009},
     volume = {59},
     number = {2},
     mrnumber = {2532375},
     zbl = {1224.28031},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2009_59_2_a10/}
}
TY  - JOUR
AU  - Cho, Dong Hyun
TI  - A simple formula for an analogue of conditional Wiener integrals and its applications. II
JO  - Czechoslovak Mathematical Journal
PY  - 2009
SP  - 431
EP  - 452
VL  - 59
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/CMJ_2009_59_2_a10/
LA  - en
ID  - CMJ_2009_59_2_a10
ER  - 
%0 Journal Article
%A Cho, Dong Hyun
%T A simple formula for an analogue of conditional Wiener integrals and its applications. II
%J Czechoslovak Mathematical Journal
%D 2009
%P 431-452
%V 59
%N 2
%U http://geodesic.mathdoc.fr/item/CMJ_2009_59_2_a10/
%G en
%F CMJ_2009_59_2_a10
Cho, Dong Hyun. A simple formula for an analogue of conditional Wiener integrals and its applications. II. Czechoslovak Mathematical Journal, Tome 59 (2009) no. 2, pp. 431-452. http://geodesic.mathdoc.fr/item/CMJ_2009_59_2_a10/

[1] Ash, R. B.: Real analysis and probability. Academic Press, New York-London (1972). | MR

[2] Cameron, R. H., Martin, W. T.: Transformations of Wiener integrals under translations. Ann. Math. 45 (1944), 386-396. | DOI | MR | Zbl

[3] Cameron, R. H., Storvick, D. A.: Some Banach algebras of analytic Feynman integrable functionals. Lecture Notes in Math. 798, Springer, Berlin-New York (1980). | DOI | MR | Zbl

[4] Chang, K. S., Chang, J. S.: Evaluation of some conditional Wiener integrals. Bull. Korean Math. Soc. 21 (1984), 99-106. | MR | Zbl

[5] Cho, D. H.: A simple formula for an analogue of conditional Wiener integrals and its applications. Trans. Amer. Math. Soc. 360 (2008), 3795-3811. | DOI | MR | Zbl

[6] Chung, D. M., Skoug, D.: Conditional analytic Feynman integrals and a related Schrödinger integral equation. SIAM J. Math. Anal. 20 (1989), 950-965. | DOI | MR | Zbl

[7] Im, M. K., Ryu, K. S.: An analogue of Wiener measure and its applications. J. Korean Math. Soc. 39 (2002), 801-819. | DOI | MR | Zbl

[8] Laha, R. G., Rohatgi, V. K.: Probability theory. John Wiley & Sons, New York-Chichester-Brisbane (1979). | MR | Zbl

[9] Park, C., Skoug, D.: A simple formula for conditional Wiener integrals with applications. Pacific J. Math. 135 (1988), 381-394. | DOI | MR | Zbl

[10] Ryu, K. S., Im, M. K.: A measure-valued analogue of Wiener measure and the measure-valued Feynman-Kac formula. Trans. Amer. Math. Soc. 354 (2002), 4921-4951. | DOI | MR | Zbl

[11] Yeh, J.: Transformation of conditional Wiener integrals under translation and the Cameron-Martin translation theorem. Tôhoku Math. J. 30 (1978), 505-515. | DOI | MR | Zbl

[12] Yeh, J.: Inversion of conditional Wiener integrals. Pacific J. Math. 59 (1975), 623-638. | DOI | MR | Zbl

[13] Yeh, J.: Inversion of conditional expectations. Pacific J. Math. 52 (1974), 631-640. | DOI | MR | Zbl

[14] Yeh, J.: Stochastic processes and the Wiener integral. Marcel Dekker, New York (1973). | MR | Zbl