Let

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

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

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

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

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

${\displaystyle \int _{0}^{T}X_{t}\,d_{\varphi }W_{t}:=\sum _{n=1}^{\infty }\left(\int _{0}^{T}X_{t}\varphi _{n}(t)\,dt\right)\int _{0}^{T}\varphi _{n}(t)\,dW_{t}}$ 

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

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

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

${\displaystyle \int _{0}^{T}X_{t}\,d_{\varphi }Z_{t}:=\sum _{n=1}^{\infty }\left(\int _{0}^{T}X_{t}\varphi _{n}(t)\,dt\right)\int _{0}^{T}\varphi _{n}(t)\,dZ_{t}}$ 

as long as the integrals

${\displaystyle \int _{0}^{T}\varphi _{n}(t)\,dZ_{t}}$ 

are well-defined.^[2]

• 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 (^{[2]: 239–241}). One way makes use of the Itô–Nisio theorem.

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

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

holds.

The following results on regularity are known:

• Every semimartingale (causal or not) is ${\displaystyle \varphi }$ -integrable if and only if ${\displaystyle \{\varphi _{n}\}}$  is regular.^{[2]: 242–243}
• It was proven that there exist a non-regular basis for ${\displaystyle L^{2}([0,1],dt)}$ .^[3]

• There exist a non-causal Itô formula,^[2]: 250 a non-causal integration by parts formula and a non-causal Girsanov theorem.^[4]

• Stratonovich integral: let ${\displaystyle X}$  be a continuous ${\displaystyle \mathbf {F} ^{W}}$ -adapted semimartingale that is universal Ogawa integrable with respect to the Wiener process, then the Stratonovich integral exist and coincides with the Ogawa integral.^[5]
• Skorokhod integral: the relationship between the Ogawa integral and the Skorokhod integral was studied in (^[6]).

• Ogawa, Shigeyoshi (2017). Noncausal Stochastic Calculus. Tokyo: Springer. doi:10.1007/978-4-431-56576-5. ISBN 978-4-431-56574-1.