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Ogawa integral

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In stochastic calculus, the Ogawa integral, also called the non-causal stochastic integral, is a stochastic integral for non-adapted processes as integrands. The corresponding calculus is called non-causal calculus which distinguishes it from the anticipating calculus of the Skorokhod integral. The term causality refers to the adaptation to the natural filtration of the integrator.

The integral was introduced by the Japanese mathematician Shigeyoshi Ogawa in 1979.

Ogawa integral

Let

  • ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},P)} be a probability space,
  • W = ( W t ) t ∈ [ 0 , T ] {\displaystyle W=(W_{t})_{t\in }} be a one-dimensional standard Wiener process with T ∈ R + {\displaystyle T\in \mathbb {R} _{+}} ,
  • F t W = σ ( W s ; 0 ≤ s ≤ t ) ⊂ F {\displaystyle {\mathcal {F}}_{t}^{W}=\sigma (W_{s};0\leq s\leq t)\subset {\mathcal {F}}} and F W = { F t W , t ≥ 0 } {\displaystyle \mathbf {F} ^{W}=\{{\mathcal {F}}_{t}^{W},t\geq 0\}} be the natural filtration of the Wiener process,
  • B ( [ 0 , T ] ) {\displaystyle {\mathcal {B}}()} the Borel σ-algebra,
  • ∫ f d W t {\displaystyle \int f\;dW_{t}} be the Wiener integral,
  • d t {\displaystyle dt} be the Lebesgue measure.

Further let H {\displaystyle \mathbf {H} } be the set of real-valued processes X : [ 0 , T ] × Ω → R {\displaystyle X\colon \times \Omega \to \mathbb {R} } that are B ( [ 0 , T ] ) × F {\displaystyle {\mathcal {B}}()\times {\mathcal {F}}} -measurable and almost surely in L 2 ( [ 0 , T ] , d t ) {\displaystyle L^{2}(,dt)} , i.e.

P ( ∫ 0 T | X ( t , ω ) | 2 d t < ∞ ) = 1. {\displaystyle P\left(\int _{0}^{T}|X(t,\omega )|^{2}\,dt<\infty \right)=1.}

Ogawa integral

Let { φ n } n ∈ N {\displaystyle \{\varphi _{n}\}_{n\in \mathbb {N} }} be a complete orthonormal basis of the Hilbert space L 2 ( [ 0 , T ] , d t ) {\displaystyle L^{2}(,dt)} .

A process X ∈ H {\displaystyle X\in \mathbf {H} } is called φ {\displaystyle \varphi } -integrable if the random series

∫ 0 T X t d φ W t := ∑ n = 1 ∞ ( ∫ 0 T X t φ n ( t ) d t ) ∫ 0 T φ n ( t ) d W t {\displaystyle \int _{0}^{T}X_{t}\,d_{\varphi }W_{t}:=\sum _{n=1}^{\infty }\left(\int _{0}^{T}X_{t}\varphi _{n}(t)\,dt\right)\int _{0}^{T}\varphi _{n}(t)\,dW_{t}}

converges in probability and the corresponding sum is called the Ogawa integral with respect to the basis { φ n } {\displaystyle \{\varphi _{n}\}} .

If X {\displaystyle X} is φ {\displaystyle \varphi } -integrable for any complete orthonormal basis of L 2 ( [ 0 , T ] , d t ) {\displaystyle L^{2}(,dt)} and the corresponding integrals share the same value then X {\displaystyle X} is called universal Ogawa integrable (or u-integrable).

More generally, the Ogawa integral can be defined for any L 2 ( Ω , P ) {\displaystyle L^{2}(\Omega ,P)} -process Z t {\displaystyle Z_{t}} (such as the fractional Brownian motion) as integrators

∫ 0 T X t d φ Z t := ∑ n = 1 ∞ ( ∫ 0 T X t φ n ( t ) d t ) ∫ 0 T φ n ( t ) d Z t {\displaystyle \int _{0}^{T}X_{t}\,d_{\varphi }Z_{t}:=\sum _{n=1}^{\infty }\left(\int _{0}^{T}X_{t}\varphi _{n}(t)\,dt\right)\int _{0}^{T}\varphi _{n}(t)\,dZ_{t}}

as long as the integrals

∫ 0 T φ n ( t ) d Z t {\displaystyle \int _{0}^{T}\varphi _{n}(t)\,dZ_{t}}

are well-defined.

Remarks

  • The convergence of the series depends not only on the orthonormal basis but also on the ordering of that basis.
  • There exist various equivalent definitions for the Ogawa integral which can be found in (). One way makes use of the Itô–Nisio theorem.

Regularity of the orthonormal basis

An important concept for the Ogawa integral is the regularity of an orthonormal basis. An orthonormal basis { φ n } n ∈ N {\displaystyle \{\varphi _{n}\}_{n\in \mathbb {N} }} is called regular if

sup n ∫ 0 T ( ∑ i = 1 n φ i ( t ) ∫ 0 t φ i ( s ) d s ) 2 d t < ∞ {\displaystyle \sup _{n}\int _{0}^{T}\left(\sum _{i=1}^{n}\varphi _{i}(t)\int _{0}^{t}\varphi _{i}(s)\,ds\right)^{2}\,dt<\infty }

holds.

The following results on regularity are known:

  • Every semimartingale (causal or not) is φ {\displaystyle \varphi } -integrable if and only if { φ n } {\displaystyle \{\varphi _{n}\}} is regular.
  • It was proven that there exist a non-regular basis for L 2 ( [ 0 , 1 ] , d t ) {\displaystyle L^{2}(,dt)} .

Further topics

Relationship to other integrals

  • Stratonovich integral: let X {\displaystyle X} be a continuous F W {\displaystyle \mathbf {F} ^{W}} -adapted semimartingale that is universal Ogawa integrable with respect to the Wiener process, then the Stratonovich integral exist and coincides with the Ogawa integral.
  • Skorokhod integral: the relationship between the Ogawa integral and the Skorokhod integral was studied in ().

Literature

References

  1. Ogawa, Shigeyoshi (1979). "Sur le produit direct du bruit blanc par lui-même". C. R. Acad. Sci. Paris Sér. A. 288. Gauthier-Villars: 359–362.
  2. ^ Ogawa, Shigeyoshi (2007). "Noncausal stochastic calculus revisited – around the so-called Ogawa integral". Advances in Deterministic and Stochastic Analysis: 238. doi:10.1142/9789812770493_0016. ISBN 978-981-270-550-1.
  3. Majer, Pietro; Mancino, Maria Elvira (1997). "A counter-example concerning a condition of Ogawa integrability". Séminaire de probabilités de Strasbourg. 31: 198–206. Retrieved 26 June 2023.
  4. Ogawa, Shigeyoshi (2016). "BPE and a Noncausal Girsanov's Theorem". Sankhya A. 78 (2): 304–323. doi:10.1007/s13171-016-0087-x. S2CID 258705123.
  5. Nualart, David; Zakai, Moshe (1989). "On the Relation Between the Stratonovich and Ogawa Integrals". The Annals of Probability. 17 (4): 1536–1540. doi:10.1214/aop/1176991172. hdl:1808/17063.
  6. Nualart, David; Zakai, Moshe (1986). "Generalized stochastic integrals and the Malliavin calculus". Probability Theory and Related Fields. 73 (2): 255–280. doi:10.1007/BF00339940. S2CID 120687698.
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