Stochastic differentiability, Ogawa integrability and identification from SFCs of random functions

概要

To study uncertain phenomena such as movements of fluid particles and fluctuations in stock prices, the analysis of mathematical model of the nondeterministic phenomena called stochastic analysis is widely used nowadays. The classical theory of the stochastic analysis is (measure theoretic) causal calculus, where the causality on functions is imposed. Roughly speaking causality on a function f, which represents trajectory of a particle or reflects decision-making, means that the value f(t) of f is determined by the behaviour or information of a given basic moving object X up to the time t. Let me briefly explain the basic concept of causal calculus mathematically. Given a sequence X = (Xt)t∈[0,∞) of measurable functions on a probability space, we define and treat stochastic integrals with respect to X of a function f with causality for X, namely, the condition that f(t) is measurable with respect to the smallest σ-field which makes all Xs for s ≤ t measurable, for each t: ∫ t 0 f dX.

Itˆo integral and Fisk-Stratonovich integral are well used among these integrals.

On the other hand, this natural assumption on the integrand sometimes becomes constraints on development of the theory in itself, hence there is an extensive theory of (measure theoretic) noncausal calculus which is free from causality. Skorokhod integral, symmetric integral and Ogawa integral are integrals for noncausal, that is, causality-free functions. The Skorokhod integrability is well clarified, whereas the Ogawa integrability is not, and it has been studied by S. Ogawa and J. Rosinski. Here, studying integrability means examining (necessary and) sufficient conditions for a function to be integrable and investigating how representation the integral has. Another topic posed and studied by Ogawa in noncausal calculus is identification problem from stochastic Fourier coefficients (SFCs in abbr.).

We overview the essence of the problem of identification. Hereafter, refer to main sections after the introduction for the precise definition of each undefined term. Let (en)n∈N be a complete orthonormal system (CONS in abbr.) of L 2 ([0, L] ; C), (Bt)t∈[0,∞) a real Brownian motion (Definition 2.1) 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. We consider the SFC
(en, dY ) := ∫ L 0 en(t)a(t) dBt + ∫ L 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 S.Ogawa [29]-[31]. We note that the SFC (en, dY ) does not make sense unless the stochastic integral ∫ L 0 dB is specified, ena is stochastic integrable and enb is Lebesgue integrable on [0, L]. Specifically, the SFC is called of Skorokhod type (SFC-S) if the stochastic integral ∫ L 0 dB is the Skorokhod integral ([43], see also Definition 3.8) and of Ogawa type (SFCO) if the stochastic integral ∫ L 0 dB is the Ogawa integral ([25], see also Definition 4.1). The question is as follows: Letting Λ = N or Λ ⊂ N, is the map which associates a pair (a, b) of random functions a and b with the sequence of SFCs ((en, dY ))n∈Λ injective? If yes, how is the inverse of the map ? The question was originally posed by Ogawa [32] after a series of studies [29]-[31] of a stochastic integral equation (SIE for short) of Fredholm type, for the question is closely connected with the existence and uniqueness of solutions for the SIEs ([29]-[31], [33]).

In this note, we first review the theories of causal and noncausal calculus in Sections 2 through 4.1. After that, we give results in noncalsal calculus as follows: in Subsection 4.3, we introduce the notion of a stochastic differential of a noncausal function and give a result, Theorem 4.3 on the stochastic differentiability. Additionally, we also give in Proposition 4.1 in Subsection 4.2 a chain rule regarding the symmetric integral as a variation of the result in [20, Theorem 4.1.7]. In Section 5, we give results on the Ogawa integrability. Finally in Section 6, we give results on the identification problem. Here, the result, Theorem 4.3 is used to obtain the result, Theorem 6.13 on the identification problem.

We briefly summarize the result in Subsection 4.3. Let V : [0, ∞) × Ω → C be a r.c.l.l. process with right-continuous quadratic variation process 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 L 0 (Ω)-module D(V ) of stochastic processes differentiable with respect to V by
D(V ) = { X ∈ L 0 (Ω)[0,L] ∃ Xˆ = DV X ∈ L1 L (V ) ⟨V, X⟩· = ∫ · 0 X d ˆ [V ] } .

Roughly speaking, the stochastic differentiation DV , defined as a L 0 (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. The result, Theorem 4.3 is as follows:
QV = { X ∈ L 0 (Ω)[0,L] Xˆ = DV X ∈ L2 L (V ) and ∀t, s ∈ [0, L] [X]t = ∫ t 0 |Xˆ| 2 d[V ],⟨V·∧s, X⟩t = ∫ t∧s 0 X d ˆ [V ] }
becomes a sub L 0 (Ω)-module of D(V ). As a consequence we have the following: For any X, Y ∈ QV and α, β ∈ L 0 (Ω) we have
(1) [αX + βY ]· = |αXˆ + βYˆ | 2 L2([0,·], µ[V ] ) .
(2) ⟨X, Y ⟩· = ⟨X, ˆ Yˆ ⟩L2([0,·], µ[V ] ) .

We compare our results on the Ogawa integrability with previous results. Ogawa obtained Theorems 4.2 and 4.1 in Subsection 4.1 on the integrability. As extensions of those theorems, we obtained Theorems 5.1 and 5.2 in Subsection 5.1, respectively, where Theorem 5.3 is also given as a variation of these theorems we gave. Moreover we got Theorem 5.5 as an extension of Theorems from 5.1 to 5.3. On the other, Ogawa showed in [27, Example 1] an integration by parts formula for the Ogawa integral. We proved Lemma 5.4 as its extension.

Actually, the identification problem has another background. The question stated above has been re-considered recently from the application viewpoint between Ogawa and Uemura. This is because P. Malliavin et al. proposed in [18], [19] an estimation method using the notion of the SFC in the study of volatility estimation problem in finance. In relation to this method, it arose that the problem of estimating the square of the diffusion coefficient a(t) from the SFCs of a stochastic differential dYt = a(t) dBt +b(t) dt. Here, it is required not to use the underlying Brownian motion (Bt)t∈[0,1] in the estimating process. Then, the following mathematical incentive appeared: distinguishing ”whether or not we use Brownian motion” to derive the random functions from the SFCs. However the interpretation and ways of thinking about the sentence in quotes differ among researchers and the criteria for the above distinction (nor the criteria for the classification based on the distinction) are not mathematically determined.

In this paper before treating the identification problem, as criteria for classifying derivations into those with and without the use of Brownian motion as described above, in Subsection 6.1 we introduce the following: First, introduce the notion of constructive identification in an assigned first-order language. The reason why we introduce the metamathematical notion of constructiveness in Appendix A is to give a framework to explain and evaluate derivations (or derivation formulas) or any other maps (or formulas of maps) by an organized notion based on the definite criteria. Second, introduce B-dependent (resp. B-independent) identification, which can be called identification ”in need of” (resp. ”in no need of”) the condition that the underlying Brownian motion is (Bt)t∈[0,∞) , from the purely mathematical interest. Here, B-dependent (resp. B-independent) is short for dependent on Brownian motion (resp. independent of Brownian motion). Note that for each derivation map of the random function, there could be various different formulas which represent the map, while these notions of B-dependent and B-independent identifications depend only on the derivation map in itself and does not depend on each derivation formula which represents the derivation map.

We obtained affirmative answers to the question about the determinability of random functions from SFCs. The results are to be stated in the following way. Here, the results starting with ’Derivation’ give concrete derivation formulas of random functions.

• Subsection 6.2: Identification from SFC-Ss
· B-independent identification
· Theorem 6.5, Corollary 6.1 (A) : 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
· B-dependent identification
· Theorem 6.3 (extension of Theorem 6.1 (Ogawa, Uemura (2014))) : Determination of a square integrable Wiener functional from its SFC-Ss
· Theorem 6.4 (extension of Theorem 6.2 (Ogawa, Uemura (2014)) and 6.3) : Determination of a stochastic differential whose coefficients are square integrable Wiener functionals, from its SFC-Ss
· Theorem 6.6, Corollary 6.1 (B) : 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
• Subsection 6.3: Identification from SFC-Os
· B-independent identification
· Theorem 6.9, Corollary 6.3 (A) (extension of Theorem 6.8 (Ogawa, Uemura (2018))): Derivation of the absolute value |a| from SFC-Os of any noncausal finite variation process a(t)
· Theorem 6.11, Corollary 6.4 (A) (extension of Theorem 6.9, Corollary 6.3 (A), respectively) : 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
· Assertion 2 (B) in Theorem 6.13 (extension of Theorem 6.11) : Derivations of Re a, Im a, Re a Im a and (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 (∗a) and a complex local martingale (∗b) and a Skorokhod integral process and Hilbert-Schmidt integral transforms of Wiener functionals, and whose drift term b(t) is square integrable almost surely
· Assertion 1 (B) in Theorem 6.13 (extension of Assertion 2 (B) in Theorem 6.13): Derivations of Re a, Im a, Re a Im a and (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.
0(∗a) The random function whose real and imaginary parts are noncausal finite variation processes 0(∗b) The random function whose real and imaginary parts are local martingales
· B-dependent identification
· Theorem 6.7 : 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
· Theorem 6.10, Corollary 6.3 (B) : Derivation of any noncausal finite variation process a(t) from its SFC-Os
· Theorem 6.12, Corollary 6.4 (B) (extension of Theorem 6.10, Corollary 6.3 (B), respectively) : Derivation of a(t) 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
· Assertion 2 (A) in Theorem 6.13 (extension of Theorems 6.7 and 6.12): Derivation of a(t) 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 and a Skorokhod integral process and Hilbert-Schmidt integral transforms of Wiener functionals, and whose drift term b(t) is square integrable almost surely
· Assertion 1 (A) in Theorem 6.13 (extension of Assertion 2 (A) in Theorem 6.13): Derivation of a(t) from SFC-Os of a stochastic differential whose diffusion coefficient a(t) is in L ∗,e and drift term b(t) is square integrable almost surely.
Here, in each statement above, a CONS which defines SFCs is taken generally. Besides, from the argument in the proof of Theorem 6.13, we can get the same result as in Theorem 6.13 on derivation from the SFC-Ss of a stochastic differential whose diffusion coefficient a(t) is a sum of a function in L 0 (Ω; L ∞[0, 1]) adapted to the Brownian filtration and a Wiener functional a ∈ L1,2 1 such that sup t∈[0,1] |a(t)|0,1,2 < ∞. Moreover, all the results listed above on B-independent (resp. B-dependent) identification assert at the same time constructive identification in L0 (resp. L0 with B). Additionally as an extension of Theorem 6.4, we obtained a result in [3] on the derivation from SFC-Ss of a stochastic differential whose diffusion coefficients are random fields on a locally compact group.