Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

AN EVALUATION FORMULA FOR A GENERALIZED CONDITIONAL EXPECTATION WITH TRANSLATION THEOREMS OVER PATHS

  • Received : 2019.02.11
  • Accepted : 2019.05.16
  • Published : 2020.03.01
Download PDF
⟨ Previous Next ⟩

Abstract

Let C[0, T] denote an analogue of Wiener space, the space of real-valued continuous functions on the interval [0, T]. For a partition 0 = t0 < t1 < ⋯ < tn < tn+1 = T of [0, T], define Xn : C[0, T] → ℝn+1 by Xn(x) = (x(t0), x(t1), …, x(tn)). In this paper we derive a simple evaluation formula for Radon-Nikodym derivatives similar to the conditional expectations of functions on C[0, T] with the conditioning function Xn which has a drift and does not contain the present position of paths. As applications of the formula with Xn, we evaluate the Radon-Nikodym derivatives of the functions ∫0T[x(t)]mdλ(t)(m∈ℕ) and [∫0Tx(t)dλ(t)]2 on C[0, T], where λ is a complex-valued Borel measure on [0, T]. Finally we derive two translation theorems for the Radon-Nikodym derivatives of the functions on C[0, T].

Keywords

Acknowledgement

Supported by : Kyonggi University

This work was supported by Kyonggi University Research Grant 2018.

References

  1. R. H. Cameron and W. T. Martin, Transformations of Wiener integrals under translations, Ann. of Math. (2) 45 (1944), 386-396. https://doi.org/10.2307/1969276
  2. D. H. Cho, A simple formula for an analogue of conditional Wiener integrals and its applications, Trans. Amer. Math. Soc. 360 (2008), no. 7, 3795-3811. https://doi.org/10.1090/S0002-9947-08-04380-8
  3. D. H. Cho, A simple formula for an analogue of conditional Wiener integrals and its applications. II, Czechoslovak Math. J. 59(134) (2009), no. 2, 431-452. https://doi.org/10.1007/s10587-009-0030-6
  4. D. H. Cho, Measurable functions similar to the Ito integral and the Paley-Wiener-Zygmund integral over continuous paths, Filomat 32 (2018), no. 18, 6441-6456. https://doi.org/10.2298/fil1818441c
  5. D. H. Cho, An evaluation formula for Radon-Nikodym derivatives similar to conditional expectations over paths, Mathematica Slovaca (2019), submitted.
  6. M. K. Im and K. S. Ryu, An analogue of Wiener measure and its applications, J. Korean Math. Soc. 39 (2002), no. 5, 801-819. https://doi.org/10.4134/JKMS.2002.39.5.801
  7. C. Park and D. Skoug, A simple formula for conditional Wiener integrals with applications, Pacific J. Math. 135 (1988), no. 2, 381-394. http://projecteuclid.org/euclid.pjm/1102688300 https://doi.org/10.2140/pjm.1988.135.381
  8. K. S. Ryu, The simple formula of conditional expectation on analogue of Wiener measure, Honam Math. J. 30 (2008), no. 4, 723-732. https://doi.org/10.5831/HMJ.2008.30.4.723
  9. K. S. Ryu, The generalized analogue of Wiener measure space and its properties, Honam Math. J. 32 (2010), no. 4, 633-642. https://doi.org/10.5831/HMJ.2010.32.4.633
  10. K. S. Ryu, The translation theorem on the generalized analogue of Wiener space and its applications, J. Chungcheong Math. Soc. 26 (2013), no. 4, 735-742. https://doi.org/10.14403/jcms.2013.26.4.735
  11. 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. https://doi.org/10.2748/tmj/1178229910