Derivation formulas of noncausal finite variation processes from the stochastic Fourier coefficients

Abstract

Let $(B_t)_{t\in [0,\infty )}$ be a real Brownian motion on a probability space $(\varOmega ,{\mathcal {F}},P)$. Our concern is whether and how a noncausal type stochastic differential $dX_t=a(t,\omega )\,dB_t+b(t,\omega )\,dt$ is determined from its stochastic Fourier coefficients (SFCs for short) $(e_n,dX):$$=\int _{0}^L\overline{e_n(t)}\,dX_t$ with respect to a CONS $(e_n)_{n\in {\mathbb {N}}}$ of $L^2([0,L];{\mathbb {C}})$. 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${\hat{\mathrm{o}}}$ky${\hat{\mathrm{u}}}$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.

Appendix

In Appendix, we follow the notation and terminology in Subsection 2.1. The following Propositions 3 and 4 are concerned with measurability and continuity of a stochastic process whose paths are of bounded variation.

Proposition 3

Let$a:[0,L]\times \varOmega \rightarrow \mathbb {R}$be a weak stochastic process, namely, $a_t$is a random variable for every$t\in [0,L]$. Suppose all paths ofa(t) are left-continuous and of bounded variation. Then, both$a_{+}(t)$and$a_{-}(t)$are weak stochastic processes, and their all paths are left-continuous.

Remark

In particular, both $a_{+}(t)$ and $a_{-}(t)$ become measurable stochastic process since any weak stochastic process whose all paths are left-continuous is measurable (see Proposition 1.13 in [6] for example).

Proof

First, the left-continuity of $a_{+}(t)$ and $a_{-}(t)$ follows from the left-continuity of a(t). Next, we show that $a_{-}(t)$ is a weak stochastic process. We can also show so is $a_+(t)$ in the same manner. Fix any $t\in (0,L]$. Put $V(t_0,...,t_n)=\textstyle \sum \limits \nolimits _{j=1}^{n}(a(t_j)-a(t_{j-1}))^{-}$ for $n\in \mathbb {N}$ and $t_0,\ldots ,t_n$ such that $0=t_0<t_1<\cdots <t_n=t$. All of them are random variables since a(t) is the weak stochastic process. While $a_{-}(t)$ is represented as

$$\begin{aligned} a_{-}(t)=\sup _{\begin{array}{c} n\in \mathbb {N} \\ 0=t_0<t_1<\cdots <t_n=t \end{array}}V(t_0,...,t_n), \end{aligned}$$

(35)

setting $A=\{\,(t_0,...,t_n)\,|\,n\in \mathbb {N},\, 0=t_0<t_1<\cdots <t_n=t,\, t_1,...,t_{n-1}\in \mathbb {Q}\,\}$, (35) is rephrased as

$$\begin{aligned} a_{-}(t)=\sup _{(t_0,...,t_n)\in A}V(t_0,...,t_n) \end{aligned}$$

by the denseness of $\mathbb {Q}$ and the left-continuity of a(t). Besides, the cardinality of A equals that of $\displaystyle {{\,{\cup }\,}}_{n=1}^{\infty }\mathbb {Q}^{n-1}$ ($\mathbb {Q}^{0}$ denotes some singleton), which is a countable set. Therefore, $a_{-}(t)$ is a random variable. $\square$

Proposition 4

Leta(t) be a noncausal finite variation process on [0, L]. Then, there exists a version$\widetilde{a}(t)$ofa(t) in$L^0([0,L]\times \varOmega )$such that

• 1. All paths of$\,\widetilde{a}(t)$are left-continuous and of bounded variation,

• 2. $\widetilde{a}_{+}(t)$and$\widetilde{a}_{-}(t)$are${\mathcal {L}}([0,L])\otimes {\mathcal {F}}$-measurable and all their paths are left-continuous and monotonically increasing.

Proof

By the assumption on a(t), there exists $\varOmega ^V\in {\mathcal {F}}$ such that $P(\varOmega ^V)=1$ and a(t) is of bounded variation on $\varOmega ^V$. Set $a^{(0)}=a\mathsf {1}_{\varOmega ^V}$, then $a^{(0)}(t)$ is a version of a(t) in $L^0([0,L]\times \varOmega )$ and all paths of $a^{(0)}(t)$ are of bounded variation. Furthermore, set $\widetilde{a}(t,\omega )=\textstyle \lim \limits _{s\nearrow t}a^{(0)}(s,\omega )$, which is ${\mathcal {L}}([0,L])\otimes {\mathcal {F}}$-measurable. Since discontinuous points of a function of bounded variation are at most countable, $\widetilde{a}(t)$ is a version of $a^{(0)}(t)$ in $L^0([0,L]\times \varOmega )$ and all paths of $\,\widetilde{a}(t)$ are left-continuous and of bounded variation. Therefore by Proposition 3, $\widetilde{a}_{+}(t)$ and $\widetilde{a}_{-}(t)$ are measurable stochastic processes whose all paths are left-continuous and monotonically increasing. $\square$

In what follows, using Proposition 3, we show that

$$\begin{aligned} S=\{(t,\omega )\in [0,L]\times \varOmega \,|\,a_{+}(\cdot ,\omega ) \text{ and } a_{-}(\cdot ,\omega ) \text{ are } \text{ differentiable } \text{ in } \text{t}\} \end{aligned}$$

is a measurable set of $[0,L]\times \varOmega$ for a (noncausal) stochastic process $a:[0,L]\times \varOmega \rightarrow \mathbb {R}$ whose all paths are left-continuous and of bounded variation.

Proposition 5

Let$a:[0,L]\times \varOmega \rightarrow \mathbb {R}$ be ${\mathcal {L}}([0,L])\otimes {\mathcal {F}}$-measurable. Assume all paths ofa(t) are bounded and monotonically increasing. Then, all Dini derivatives ofa(t)

$$\begin{aligned} D_{\pm }a(t,\omega )=\underset{h\rightarrow \pm 0}{\varliminf }\frac{a(t+h,\omega )-a(t,\omega )}{h} ,\quad D^{\pm }a(t,\omega )=\underset{h\rightarrow \pm 0}{\varlimsup }\frac{a(t+h,\omega )-a(t,\omega )}{h} \end{aligned}$$

are${\mathcal {L}}([0,L])\otimes {\mathcal {F}}$-measurable.

Proof

Put $g_h(t,\omega )=\textstyle \inf \limits _{0<|\delta |<|h|}\frac{a(t+\delta ,\omega )-a(t,\omega )}{\delta }$ for each $h\in (-1,1)$. We show the measurability of $D_{+}a(t,\omega )$ as the others are shown by similar arguments. Since $D_{+}a(t,\omega ) =\textstyle \lim \limits _{n\rightarrow \infty }g_{\frac{1}{n}}(t,\omega ),$ we have only to show for any $h>0\,$$g_h$ is ${\mathcal {L}}([0,L])\otimes {\mathcal {F}}$-measurable, in other words, $g_h^{-1}(\{-\infty \}\cup (-\infty ,c))\in {\mathcal {L}}([0,L])\otimes {\mathcal {F}}$ for any real number c. We first note that

$$\begin{aligned} g_h^{-1}(\{-\infty \}\cup (-\infty ,c))&=\{\,(t,\omega )\in [0,L]\times \varOmega \,|\,\exists \delta \in (0,h)\,\,\frac{a(t+\delta ,\omega )-a(t,\omega )}{\delta }<c\}. \end{aligned}$$

Now, we show that for any $(t,\omega )\in [0,L]\times \varOmega$

$$\begin{aligned} \exists \delta \in (0,h)\,\,\frac{a(t+\delta ,\omega )-a(t,\omega )}{\delta }<c\quad \Rightarrow \quad \exists r\in (0,h)\cap \mathbb {Q}\,\,\frac{a(t+r,\omega )-a(t,\omega )}{r}<c. \end{aligned}$$

(36)

Assume the antecedent and fix one of the $\delta \in (0,h)$. For the denseness of $\mathbb {Q}$ there exists $r\in (0,\delta )\cap \mathbb {Q}$ such that $\frac{a(t+\delta ,\omega )-a(t,\omega )}{r}<c$. Here, because $a(\cdot ,\omega )$ monotonically increase we have $\frac{a(t+r,\omega )-a(t,\omega )}{r}\le \frac{a(t+\delta ,\omega )-a(t,\omega )}{r}<c$. Thus, the consequent holds. Also, the converse of (36) is trivial, so we have the following:

$$\begin{aligned} g_h^{-1}(\{-\infty \}\cup (-\infty ,c))&=\Bigl \{\,(t,\omega )\in [0,L]\times \varOmega \,|\,\exists r\in (0,h)\cap \mathbb {Q}\,\,\frac{a(t+r,\omega )-a(t,\omega )}{r}<c \Bigl \}\\&=\displaystyle {{\,{\cup }\,}}_{r\in (0,h)\cap \mathbb {Q}}\Bigl (\,\frac{a(\cdot +r)-a(\cdot )}{r}\,\Bigl )^{-1}((-\infty ,c)). \end{aligned}$$

$\frac{a(\cdot +r)-a(\cdot )}{r}$ is ${\mathcal {L}}([0,L])\otimes {\mathcal {F}}$-measurable function for r since $a(\cdot )$ is ${\mathcal {L}}([0,L])\otimes {\mathcal {F}}$-measurable, then $g_h^{-1}(\{-\infty \}\cup (-\infty ,c))\in {\mathcal {L}}([0,L])\otimes {\mathcal {F}}$ is proved. $\square$

Proposition 6

Let$a:[0,L]\times \varOmega \rightarrow \mathbb {R}$be a (noncausal) stochastic process whose all paths are left-continuous and of bounded variation. Then, $S=\{(t,\omega )\in [0,L]\times \varOmega \,|\,$$a_{+}(\cdot ,\omega ) \text{ and } a_{-}(\cdot ,\omega ) \text{ are } \text{ differentiable }$$\text{ in } t\}$is a measurable set of$[0,L]\times \varOmega$. In particular, the following are equivalent.

• (i) $a(\cdot ,\omega )$and$a_\mathrm{tv}(\cdot ,\omega ) \text{ are } \text{ differentiable } \text{ in } t \, \text{ a.a. } \,\, (t,\omega )\in [0,L]\times \varOmega$.

• (ii) $( a(\cdot ,\omega )$and$a_\mathrm{tv}(\cdot ,\omega ) \text{ are } \text{ differentiable } \text{ in } t \, \text{ a.a. } \,\, t\in [0,L] ) \, \text{ a.a. } \,\, \omega \in \varOmega$.

• (iii) $( a(\cdot ,\omega )$and$a_\mathrm{tv}(\cdot ,\omega ) \text{ are } \text{ differentiable } \text{ in } t \, \text{ a.a. } \,\, \omega \in \varOmega ) \, \text{ a.a. } \,\, t\in [0,L]$.

Proof

For convenience, we denote $a_{+}$ and $a_{-}$ by $a_1$ and $a_2$, respectively. For $i\in \{1,2\}$ set

$$\begin{aligned} D_{\pm }a_i(t,\omega )=\underset{h\rightarrow \pm 0}{\varliminf }\frac{a_i(t+h,\omega )-a_i(t,\omega )}{h} ,\quad D^{\pm }a_i(t,\omega )=\underset{h\rightarrow \pm 0}{\varlimsup }\frac{a_i(t+h,\omega )-a_i(t,\omega )}{h}. \end{aligned}$$

Then, by Proposition 3 each $a_i$ turns out to be ${\mathcal {L}}([0,L])\otimes {\mathcal {F}}$-measurable, thus by Proposition 5, $S=\displaystyle \displaystyle {{\,{\cap }\,}}_{i=1,2}\{\,(t,\omega )\in [0,L]\times \varOmega \,|\, D^{+}a_i(t,\omega )\le D_{-}a_i(t,\omega ) \text{ and } D^{-}a_i(t,\omega )\le D_{+}a_i(t,\omega ) \text{ and }$$|D^{+}a_i(t,\omega )|<\infty \,\}$ belongs to ${\mathcal {L}}([0,L])\otimes {\mathcal {F}}$. The latter statement holds for $S=\{(t,\omega )\in [0,L]\times \varOmega \,|\,a(\cdot ,\omega ) \text{ and }$$a_\mathrm{tv}(\cdot ,\omega ) \text{ are } \text{ differentiable } \text{ in } t \}.$$\square$