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Derivation formulas of noncausal finite variation processes from the stochastic Fourier coefficients
Japan Journal of Industrial and Applied Mathematics ( IF 0.9 ) Pub Date : 2020-02-05 , DOI: 10.1007/s13160-020-00404-4
Kiyoiki Hoshino

Let $$(B_t)_{t\in [0,\infty )}$$ ( B t ) t ∈ [ 0 , ∞ ) be a real Brownian motion on a probability space $$(\varOmega ,{\mathcal {F}},P)$$ ( Ω , F , P ) . Our concern is whether and how a noncausal type stochastic differential $$dX_t=a(t,\omega )\,dB_t+b(t,\omega )\,dt$$ d X t = a ( t , ω ) d B t + b ( t , ω ) d t is determined from its stochastic Fourier coefficients (SFCs for short) $$(e_n,dX):$$ ( e n , d X ) : $$=\int _{0}^L\overline{e_n(t)}\,dX_t$$ = ∫ 0 L e n ( t ) ¯ d X t with respect to a CONS $$(e_n)_{n\in {\mathbb {N}}}$$ ( e n ) n ∈ N of $$L^2([0,L];{\mathbb {C}})$$ L 2 ( [ 0 , L ] ; C ) . This problem was proposed by Ogawa (Stochastics (85)(2), 286–294, 2013) and has been studied by Ogawa and Uemura (Ogawa in Ind J Stat 77-A(1):30–45, 2014, Ind J Stat 80-A:267–279, 2018; Ogawa and Uemura in J Theor Probab 27:370–382, 2014, Bull Sci Math 138:147–163, 2014, RIMS K $${\hat{\mathrm{o}}}$$ o ^ ky $${\hat{\mathrm{u}}}$$ u ^ roku 1952:128–134, 2015, J Ind Appl Math 35-1:373–390, 2018). In this paper we give several results on the problem for each of stochastic differentials of Ogawa type and Skorokhod type when [0, L ] is a finite or an infinite interval. Specifically, we first give a condition for a random function to be determined from the SFCs and apply it to obtain affirmative answers to the question with several concrete derivation formulas of the random functions.

中文翻译:

从随机傅立叶系数推导非因果有限变分过程的公式

令 $$(B_t)_{t\in [0,\infty )}$$ ( B t ) t ∈ [ 0 , ∞ ) 是概率空间 $$(\varOmega ,{\mathcal { F}},P)$$ ( Ω , F , P ) 。我们关心的是非因果类型的随机微分 $$dX_t=a(t,\omega )\,dB_t+b(t,\omega )\,dt$$ d X t = a ( t , ω ) d B t + b ( t , ω ) dt 由其随机傅立叶系数(简称 SFC)确定 $$(e_n,dX):$$ ( en , d X ) : $$=\int _{0}^L\ overline{e_n(t)}\,dX_t$$ = ∫ 0 L en ( t ) ¯ d X t 关于 CONS $$(e_n)_{n\in {\mathbb {N}}}$$ ( en ) n ∈ N of $$L^2([0,L];{\mathbb {C}})$$ L 2 ( [ 0 , L ] ; C ) 。这个问题是由 Ogawa (Stochastics (85)(2), 286–294, 2013) 提出的,并已被 Ogawa 和 Uemura 研究过(Ogawa in Ind J Stat 77-A(1):30–45, 2014, Ind J Stat 80-A:267–279, 2018; Ogawa and Uemura in J Theor Probab 27:370–382, 2014, Bull Sci Math 138:147–163, 2014, RIMS K $${\hat{\mathrm{o}}}$$ o ^ ky $${\hat{\mathrm{u}}}$$ u ^ roku 1952 :128–134, 2015, J Ind Appl Math 35-1:373–390, 2018)。在本文中,当 [0, L ] 是有限或无限区间时,我们针对 Ogawa 型和 Skorokhod 型的每个随机微分问题给出了几个结果。具体来说,我们首先给出了从 SFC 中确定随机函数的条件,并应用它来获得对随机函数的几个具体推导公式的问题的肯定答案。L ] 是有限或无限区间。具体来说,我们首先给出了从 SFC 中确定随机函数的条件,并应用它来获得对随机函数的几个具体推导公式的问题的肯定答案。L ] 是有限或无限区间。具体来说,我们首先给出了从 SFC 中确定随机函数的条件,并应用它来获得对随机函数的几个具体推导公式的问题的肯定答案。
更新日期:2020-02-05
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