Let • (\Omega,\mathcal{F},P) be a
probability space, • W=(W_t)_{t\in[0,T]} be a one-dimensional standard
Wiener process with T\in\mathbb{R}_+, • \mathcal{F}_t^W=\sigma(W_s;0\leq s \leq t)\subset \mathcal{F} and \mathbf{F}^W=\{\mathcal{F}_t^W, t\geq 0\} be the natural filtration of the Wiener process, • \mathcal{B}([0,T]) the
Borel σ-algebra, • \int f\; dW_t be the Wiener integral, • dt be the
Lebesgue measure. Further let \mathbf{H} be the set of real-valued processes X\colon [0,T]\times \Omega \to\mathbb{R} that are \mathcal{B}([0,T])\times \mathcal{F}-measurable and
almost surely in L^2([0,T],dt), i.e. :P\left(\int_0^T |X(t,\omega)|^2 \, dt
Ogawa integral Let \{\varphi_n\}_{n\in \mathbb{N}} be a
complete orthonormal basis of the
Hilbert space L^2([0,T],dt). A process X\in\mathbf{H} is called \varphi-integrable if the random series :\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 \{\varphi_n\}. If X is \varphi-integrable for any complete orthonormal basis of L^2([0,T],dt) and the corresponding integrals share the same value then X is called
universal Ogawa integrable (or
u-integrable). More generally, the Ogawa integral can be defined for any L^2(\Omega,P)-process Z_t (such as the
fractional Brownian motion) as integrators :\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 :\int_0^T\varphi_n(t) \, dZ_t are well-defined. == Further topics ==