대한수학회지 (Journal of the Korean Mathematical Society)

  • 제57권2호
  • /
  • Pages.451-470
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  • 2020
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  • 0304-9914(pISSN)
  • /
  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
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AN EVALUATION FORMULA FOR A GENERALIZED CONDITIONAL EXPECTATION WITH TRANSLATION THEOREMS OVER PATHS

  • 투고 : 2019.02.11
  • 심사 : 2019.05.16
  • 발행 : 2020.03.01
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초록

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].

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과제정보

연구 과제 주관 기관 : Kyonggi University

This work was supported by Kyonggi University Research Grant 2018.

참고문헌

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