称号及び氏名

博士（理学）

星野 清生

学位授与の日付 令和 2 年 3 月 31 日
論文

名

論文審査委員Stochastic differentiability, Ogawa integrability and identificationfrom SFCs of random functions
乱関数の確率微分可能性, Ogawa 積分可能性, 及び SFC による同定
主査
副査
副査
副査

數見 哲也
壁谷 喜継
丸田 辰哉
山口 睦

Stochastic diﬀerentiability, Ogawa integrability andidentification from SFCs of random functions
Kiyoiki Hoshino (Osaka Prefecture University)

Abstract
We take up three topics in noncausal calculus, which is stochastic analysis for noncausal functions, and give results concerning each topic as follows:1. Stochastic diﬀerentiability: we introduce the notion of a stochastic diﬀerential of a noncausal function and give a result on the stochastic diﬀerentiability. Webriefly summarize the result. Let V : [0, ∞) × Ω → C be a right-continuous with leftlimits process with right-continuous quadratic variation process [V ] and with the the following property: for any interval [s, t] ⊂ [0, L] we have [V ]s = [V ]t a.s. implies ( Vs =Vu a.s. ) for any u ∈ (s, t]. We introduce the L0 (Ω)-module D(V ) of stochastic processes X with diﬀerential DV X with respect to V . Roughly speaking, the stochastic
diﬀerentiation DV , defined as a L0 (Ω)-module homomorphism, becomes the inverse operation of the stochastic integral and the space of integrands in D(V ) of a stochastic
integral forms the class of integrands of a version of the stochastic integral. We define
the subset QV of D(V ) and prove QV becomes a sub L0 (Ω)-module of D(V ). As aconsequence we have(2) for any X, Y ∈QV and α, β ∈ L0 (Ω):∫ · the following (1) and∫·(1) [αX + βY ]· = 0 |α DV X + β DV Y |2 d[V ]. (2) ⟨X, Y ⟩· = 0 DV X DV Y d[V ]. Here⟨, ⟩ and [ ] stand for the cross variation and quadratic variation, respectively.
2. Ogawa integrability: we give results on the Ogawa integrability, which has
been studied by S. Ogawa and J. Rosinski. The Ogawa integral is one of the integrals
defined for noncausal functions. Here, studying integrability means examining (necessary and) suﬃcient conditions for a function to be integrable and investigating how
representation the integral has. We compare our results on the Ogawa integrability
with previous results. Ogawa obtained theorems on the Ogawa integrability of quasimartingales. As extensions of those theorems, we obtained teorems on the integrabilityof Skorokhod integral processes. Moreover we got theorem on the integrability of S-type
Itô processes or more general Wiener functionals, as an extension of those theorems weobtained. On the other, Ogawa showed an integration by parts formula for the Ogawa
integral. We proved this type of formula as its extension.
3. Identification problem from stochastic Fourier coeﬃcients: we give
results on the identification problem from stochastic Fourier coeﬃcients (SFCs in abbr.),1

which is another topic posed and studied by Ogawa in noncausal calculus. We explain
the problem of identification. Let (en )n∈N be a complete orthonormal system (CONSin abbr.) of L2 ([0, L] ; C), (Bt )t∈[0,∞) a real Brownian motion on a probability space(Ω, F, P ) and a, b measurable functions on the product measure space [0, L] × Ω =([0, L]×Ω, B([0, L])⊗F, λ|B([0,L]) ⊗P ) taking values in C, which we call random functions.
Here, the symbol [0, L] is regarded as the interval [0, ∞) if L = ∞. We consider the∫L∫LSFC (en , dY ) := 0 en (t)a(t) dBt + 0 en (t)b(t) dt, where en (t) denotes the complex
conjugate of en (t), of the stochastic diﬀerential dYt = a(t) dBt + b(t) dt with respect to(en )n∈N , which is introduced by Ogawa. We note that the SFC (en , dY ) does not make∫Lsense unless the stochastic integral 0 dB is specified, en a is stochastic integrable anden b is Lebesgue integrable on [0, L]. Specifically, the SFC is called of Skorokhod type
∫L(SFC-S) if the stochastic integral 0 dB is the Skorokhod integral and of Ogawa type
∫L(SFC-O) if the stochastic integral 0 dB is the Ogawa integral. The question is asfollows: Letting Λ = N or Λ ⊂ N, is the map which associates a pair (a, b) of randomfunctions a and b with the sequence of SFCs ((en , dY ))n∈Λ injective? If yes, how is theinverse of the map ?
The following are the results we obtained: to begin with we introduce the notion of constructive identification in an assigned first-order language and introduce Bdependent (resp. B-independent) identification, which can be called identification ”inneed of” (resp. ”in no need of”) the condition that the underlying Brownian motion is(Bt )t∈[0,∞) . We obtained aﬃrmative answers to the question about the determinabilityof random functions from SFCs. The results are given in the following way. Here, theresults starting with ’Derivation’ give concrete derivation formulas of random functions.
(S) Identification from SFC-Ss(S-I) B-independent identification(S-I-1) Theorem: Derivation of |a| from SFC-Ss of a stochastic diﬀerential whose
diﬀusion coeﬃcient a(t) is a locally absolutely continuous Wiener functional and drift term b(t) is square integrable almost surely(S-D) B-dependent identification(S-D-1) Theorem (extension of a theorem by Ogawa, Uemura (2014)): Determination of a square integrable Wiener functional from its SFC-Ss(S-D-2) Theorem (extension of Theorem (S-D-1) and another theorem by Ogawa,Uemura (2014)): Determination of a stochastic diﬀerential whose coeﬃcients are square integrable Wiener functionals, from its SFC-Ss(S-D-3) Theorem: Derivation of a(t) from SFC-Ss of a stochastic diﬀerential whose
diﬀusion coeﬃcient a(t) is a locally absolutely continuous Wiener functional and drift term b(t) is square integrable almost surely(O) Identification from SFC-Os2

(O-I) B-independent identification(O-I-1) Theorem (extension of a theorem by Ogawa, Uemura (2018)): Derivationof the absolute value |a| from SFC-Os of any noncausal finite variation
process a(t)(O-I-2) Theorem (extension of Theorem (O-I-1)): Derivation of |a| from SFC-Os
of a stochastic diﬀerential whose diﬀusion coeﬃcient a(t) is any noncausal
finite variation process and drift term b(t) is square integrable almost
surely(O-I-3) Theorem (extension of Theorem (O-I-2)): Derivations of Re a, Im a, Re a Im aand (sgn a)a from SFC-Os of a stochastic diﬀerential whose diﬀusion coeﬃcient a(t) is the sum of any complex noncausal finite variation process
and a complex local martingale, i.e. a process whose real and imaginaryparts are local martingales, and a Skorokhod integral process and HilbertSchmidt integral transforms of Wiener functionals, and whose drift termb(t) is square integrable almost surely(O-I-4) Theorem (extension of Theorem (O-I-3)): Derivations of Re a, Im a, Re a Im aand (sgn a)a from SFC-Os of a stochastic diﬀerential whose diﬀusion coefficient a(t) is in a certain class L ∗,e and drift term b(t) is square integrable
almost surely.
(O-D) B-dependent identification(O-D-1) Theorem: Determination of a stochastic diﬀerential whose coeﬃcients are
square integrable Wiener functionals and whose diﬀusion coeﬃcient is a
Skorokhod integral process, from its SFC-Os(O-D-2) Theorem: Derivation of any noncausal finite variation process a(t) fromits SFC-Os(O-D-3) Theorem (extension of Theorem (O-D-2)): Derivation of a(t) from SFCOs of a stochastic diﬀerential whose diﬀusion coeﬃcient a(t) is any noncausal finite variation process and drift term b(t) is square integrable
almost surely(O-D-4) Theorem (extension of Theorems (O-D-1) and (O-D-3)): Derivation ofa(t) from SFC-Os of a stochastic diﬀerential whose diﬀusion coeﬃcienta(t) is the sum of any complex noncausal finite variation process and acomplex local martingale and a Skorokhod integral process and HilbertSchmidt integral transforms of Wiener functionals, and whose drift termb(t) is square integrable almost surely(O-D-5) Theorem (extension of Theorem (O-D-4)): Derivation of a(t) from SFCOs of a stochastic diﬀerential whose diﬀusion coeﬃcient a(t) is in L ∗,eand drift term b(t) is square integrable almost surely.
3Here, in each statement above, the CONS (en )n∈N which defines SFCs is taken generally
and the above-mentioned result on the stochastic diﬀerentiability is used to obtain theresults (O-I-3), (O-I-4), (O-D-4) and (O-D-5). Besides, we can get the same result onderivation from the SFC-Ss of a stochastic diﬀerential whose diﬀusion coeﬃcient a(t)is the sum of a function in L0 (Ω; L∞ [0, 1]) adapted to the Brownian filtration and a
Wiener functional a ∈ L1,2
1 such that sup |a(t)|0,1,2 < ∞, as (O-I-3), (O-I-4), (O-D-4)
t∈[0,1]and (O-D-5) from the argument in their proofs. Moreover, all the results listed aboveon B-independent or B-dependent identification assert at the same time constructive
identification in a certain assigned first-order language L0 or LB
0 , respectively.
Journal:[1] On the identification of noncausal Wiener functionals from the stochastic Fouriercoeﬃcients, K. Hoshino, T. Kazumi, Journal of Theoretical Probability, vol. 32, pp
1973-1989, (2019).
[2] On the Ogawa integrability of noncausal Wiener functionals, K. Hoshino, T. Kazumi,
Stochastics: An International Journal of Probability and Stochastic Processes, vol. 91.
5, pp 773-796, (2019).
[3] Derivation formulas of noncausal finite variation processes from the stochastic Fouriercoeﬃcients, K. Hoshino, Japan Journal of Industrial and Applied Mathematics, published online: February 5, 2020. DOI: 10.1007/s13160-020-00404-4.
[4] Derivation of random functions from the stochastic Fourier coeﬃcients by the Ogawa
integral for regular CONSs, K. Hoshino, preprint submitted to Bull. Sci. Math.
Proceedings:[5] Identification of random functions from the SFCs defined by the Ogawa integral regarding regular CONSs, K. Hoshino, Probability Symposium, RIMS Kôkyûroku. 2116,
95-104 (RIMS, Kyoto university, 2019).

4学位論文審査結果の要旨
学位論文題目

Stochastic diﬀerentiability, Ogawa integrability and identiﬁcation from SFCs of random functions（乱関数の確率微分可能性, Ogawa 積分可能性, 及び SFC による同定）
提出者氏名

星野浄生

本学大学院理学系研究科情報数理科学専攻博士後期課程４年に在籍の星野浄生氏は非因果的な乱関数についての確率解析を中心に研究を行っている。伝統的な確率解析の分野ではブラウン運動 B(t), 0 ≤ t ≤ 1が与えられて乱関数 f (t), 0 ≤ t ≤ 1 について因果的（causal）という条件が課される。ここで因果的とは

f (t) の値がブラウン運動の時刻 t までの値 B(s), 0 ≤ s ≤ t で決定されるということを意味する。f (t) が因∫ 1
果的である場合に確率積分
f (t) dB(t) としては Ito 積分と Stratonovichi 積分という２種類のものがあ0る。一方 f (t) が非因果的である場合には Skorokhod 積分と Ogawa 積分という２種類の確率積分があり、それぞれ Ito 積分と Stratonovichi 積分の拡張であるとみなされる。星野氏の研究は主に以下の２つの問題に関するものである：(A) Ogawa 積分の可積分性(B) 確率 Fourier 係数による乱関数の同定∫ t
ここで 乱関数 a(t), b(t), 0 ≤ t ≤ 1 に対して X(t) =∫ t

0

L2 (0, 1) の正規直交基底としたとき∫ 1(en , dX) :=0∫ 1
en (t)a(t) dB(t) +0b(s) ds と定め, (en (t))n∈N をa(s) dB(s) +en (t)b(t) dt0

を乱関数 X(t) の確率 Fourier 係数（stochastic Fourier coeﬃcient, SFC）と呼ぶ。SFC を定める確率積分が Skorokhod 積分か Ogawa 積分かに応じてそれぞれ SFC-S,SFC-O と表記する。SFC ((en , dX))n∈Nが与えられたとして, 乱関数 a(t), b(t) が SFC より復元されるかを問題 (B) では考察する。
星野氏は乱関数が Skorokhod 積分過程で表される Wiener 汎関数である場合に修士論文で得られた問題(A) に関する結果（定理 5.1, 定理 5.2, 定理 5.3）をさらに広い Wiener 汎関数である場合に対して問題 (A)に関する結果を得た（定理 5.5）
。また、乱関数が Wiener 汎関数である場合に修士論文で得られた問題 (B)の SFC-S に関する結果（定理 6.3, 定理 6.4）と SFC-O に関する結果（定理 6.7）を乱関数が局所絶対連続な Wiener 汎関数である場合の SFC-S に関する結果（定理 6.5, 定理 6.6）と非因果的有界変動過程である
場合の SFC-O に関する結果（定理 6.9, 定理 6.10, 定理 6.11, 定理 6.12）を得た。これらの結果は小川重義
氏と植村英明氏によって得られた問題 (B) に関する結果を適用される乱関数のクラスを有界変動過程の場合へと広げて一般化したもので単著の論文として専門雑誌に掲載が決定している。さらに正則な基底を用いて定義される SFC-O についての問題 (B) に関する結果（定理 6.13）を証明する際に用いられた X(t) から a(t) を導出する確率微分のアイデア（定理 6.4）は Ogawa 積分の提唱者である小川氏からも高く評価されている。
以上のことを踏まえて、本委員会は本論文を学位論文として十分な内容を持つものと判断した。
主査

數見哲也

壁谷喜継

丸田辰哉

山口睦