{"title":"AN EVALUATION FORMULA FOR A GENERALIZED CONDITIONAL EXPECTATION WITH TRANSLATION THEOREMS OVER PATHS","authors":"D. Cho","doi":"10.4134/JKMS.J190133","DOIUrl":null,"url":null,"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 = t 0 < t 1 < ··· < t n < t n +1 = T of [0 ,T ], define X n : C [0 ,T ] → R n +1 by X n ( x ) = ( x ( t 0 ) ,x ( t 1 ) ,...,x ( t n )). 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 X n which has a drift and does not contain the present position of paths. As applications of the formula with X n , we evaluate the Radon-Nikodym derivatives of the functions (cid:82) T 0 [ x ( t )] m dλ ( t )( m ∈ N ) and [ (cid:82) T 0 x ( 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 ].","PeriodicalId":49993,"journal":{"name":"Journal of the Korean Mathematical Society","volume":"45 1","pages":"451-470"},"PeriodicalIF":0.7000,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the Korean Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4134/JKMS.J190133","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 3
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 = t 0 < t 1 < ··· < t n < t n +1 = T of [0 ,T ], define X n : C [0 ,T ] → R n +1 by X n ( x ) = ( x ( t 0 ) ,x ( t 1 ) ,...,x ( t n )). 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 X n which has a drift and does not contain the present position of paths. As applications of the formula with X n , we evaluate the Radon-Nikodym derivatives of the functions (cid:82) T 0 [ x ( t )] m dλ ( t )( m ∈ N ) and [ (cid:82) T 0 x ( 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 ].
。设C [0,T]表示区间[0,T]上的实值连续函数空间的一个类似的Wiener空间。对于划分0 = t 0 < t 1 <···< t n < t n +1 = t ([0, t]),定义X n: C [0, t]→R n +1 × X n (X) = (X (t 0), X (t 1),…(x (t n))本文导出了一个简单的Radon-Nikodym导数的计算公式,类似于C [0,T]上函数的条件期望,条件函数X n具有漂移且不包含路径的当前位置。作为带X n的公式的应用,我们计算了函数(cid:82) t0 [X (T)] m dλ (T)(m∈n)和[(cid:82) t0 X (T) dλ (T)] 2在C [0,T]上的Radon-Nikodym导数,其中λ是[0,T]上的复值Borel测度。最后导出了C [0,T]上函数的Radon-Nikodym导数的两个平移定理。
期刊介绍:
This journal endeavors to publish significant research of broad interests in pure and applied mathematics. One volume is published each year, and each volume consists of six issues (January, March, May, July, September, November).