Stochastic integrals in Hilbert spaces

Let U and H be real separable Hilbert spaces, and let W be cylindrical Brownian motion on U relative to a filtration satisfying the usual completeness and right-continuity conditions. For a predictable integrand $\Phi_s$ taking values in the Hilbert–Schmidt operators from U to H, assume $\mathbb E\int_0^T\|\Phi_s\|_{\mathrm{HS}}^2ds<\infty$. The stochastic integral is the H-valued L² limit of sums of scalar Brownian integrals along an orthonormal input basis. Its value does not depend on that basis.

For simple predictable processes, coefficients are known at the beginning of each increment interval. Orthogonality of independent Brownian increments gives the isometry, which then completes the construction by density. Adaptedness alone needs suitable measurability before it supplies an admissible integrand; predictability is a clean sufficient formulation. A bounded operator can fail the Hilbert–Schmidt requirement in infinite dimension, as the identity does. This is the precise point where finite-dimensional intuition about arbitrary matrices stops applying.

If the input is a U-valued Q-Wiener process, then its eigen-directions already have amplitudes $\sqrt{q_k}$. For a bounded predictable operator G from U to H, the isometry therefore uses $\|GQ^{1/2}\|_{\mathrm{HS}(U,H)}^2=\sum_kq_k\|Ge_k\|_H^2$. Equivalently, one can formulate integrands on the covariance space $U_0=Q^{1/2}U$ with its canonical inverse-square-root norm, treating the kernel of Q appropriately. These are two descriptions of the same noise weighting.

An integrand defined only on the covariance space need not extend to a bounded operator on all of U, so the bounded-G formula is a convenient sufficient setup rather than the most general theorem. Conversely, applying Q again after using the covariance-space norm would weight the noise twice. In calculations, identify whether the symbol denotes covariance itself or a noise-amplitude operator. Some SPDE references call the amplitude Q and hence write QQ adjoint in the covariance; that notation differs from the covariance-Q convention here.

If integrands converge in their expected integrated Hilbert–Schmidt norm, the isometry gives convergence of the terminal integrals in mean square. Martingale inequalities also control the expected supremum norm of the differences, yielding continuous H-valued versions in this square-integrable setting. With only almost-sure local square integrability, localization constructs a local martingale instead; a global finite second moment cannot be asserted without the stronger expectation condition. These are distinct levels of integrability.

A stochastic integral is not an ordinary time integral of an H-valued derivative of W. Its increments retain quadratic variation, and their energy is governed by the isometry. This makes integration useful even when the noise has no pointwise state-space realization. Later, a semigroup can appear inside the integrand and provide the missing Hilbert–Schmidt smoothing. That time-dependent regularization must be integrable near the current time, where the semigroup approaches the identity and the high-frequency noise is least damped.