Conditional Yeh-Wiener integrals with vector-valued conditioning functions

Park, Chull; Skoug, David

doi:10.1090/S0002-9939-1989-0960650-6

Conditional Yeh-Wiener integrals with vector-valued conditioning functions
HTML articles powered by AMS MathViewer

by Chull Park and David Skoug PDF

Proc. Amer. Math. Soc. 105 (1989), 450-461 Request permission

Abstract:

In this paper we establish various results involving conditional Yeh-Wiener integrals with vector-valued conditioning functions. We first develop a very simple formula for converting conditional Yeh-Wiener integrals into ordinary (i.e., nonconditional) Yeh-Wiener integrals. We then use this simple formula to evaluate the conditional Yeh-Wiener integral of various functionals, including functionals in which the sectional average of an ${L_2}$-function occurs naturally. Finally, we use these results to obtain a general Cameron-Martin type translation theorem for conditional Yeh-Wiener integrals.

References

Leo Breiman, Probability, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1968. MR 0229267
Kun Soo Chang and Jae Moon Ahn, Translation theorem for conditional Yeh-Wiener integrals, J. Korean Math. Soc. 20 (1983), no. 2, 163–172. MR 745352
Kun Soo Chang, Jae Moon Ahn, and Joo Sup Chang, An evaluation of the conditional Yeh-Wiener integral, Pacific J. Math. 124 (1986), no. 1, 107–117. MR 850669
Dong M. Chung and Jae Moon Ahn, Conditional Yeh-Wiener integrals, J. Korean Math. Soc. 20 (1983), no. 2, 209–221. MR 745356
J. L. Doob, Stochastic processes, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1990. Reprint of the 1953 original; A Wiley-Interscience Publication. MR 1038526
G. W. Johnson and D. L. Skoug, A stochastic integration formula for two-parameter Wiener $\times$ two-parameter Wiener space, SIAM J. Math. Anal. 18 (1987), no. 4, 919–932. MR 892480, DOI 10.1137/0518070
Chull Park, On Fredholm transformations in Yeh-Wiener space, Pacific J. Math. 40 (1972), 173–195. MR 304604
Chull Park and David Skoug, A simple formula for conditional Wiener integrals with applications, Pacific J. Math. 135 (1988), no. 2, 381–394. MR 968620
Howard G. Tucker, A graduate course in probability, Probability and Mathematical Statistics, Vol. 2, Academic Press, Inc., New York-London, 1967. MR 0221541
J. Yeh, Wiener measure in a space of functions of two variables, Trans. Amer. Math. Soc. 95 (1960), 433–450. MR 125433, DOI 10.1090/S0002-9947-1960-0125433-1
J. Yeh, Cameron-Martin translation theorems in the Wiener space of functions of two variables, Trans. Amer. Math. Soc. 107 (1963), 409–420. MR 189138, DOI 10.1090/S0002-9947-1963-0189138-6
J. Yeh, Stochastic processes and the Wiener integral, Pure and Applied Mathematics, Vol. 13, Marcel Dekker, Inc., New York, 1973. MR 0474528
J. Yeh, Inversion of conditional Wiener integrals, Pacific J. Math. 59 (1975), no. 2, 623–638. MR 390162
J. Yeh, Transformation of conditional Wiener integrals under translation and the Cameron-Martin translation theorem, Tohoku Math. J. (2) 30 (1978), no. 4, 505–515. MR 516883, DOI 10.2748/tmj/1178229910