Ito versus Stratonovich boundary

This block uses the left-endpoint Ito convention throughout: the integral ${\int }_0^{t}HdB$ of an elementary integrand Elementary predictable Brownian integrands is the finite sum with coefficients ${\xi }_k$ measurable at the left time $t_k$ of each interval $(t_k,t_{k+1}]$ Ito integral of an elementary predictable process. This describes the information available to the coefficient; with the left-open interval convention it does not assert ${\xi }_k=H_{t_k}$. The integral for a globally square-integrable predictable process is its $L^2$ extension Ito integral for square-integrable predictable processes, while the extension to locally square-integrable predictable processes is obtained by localization Localized Ito integral.

What is not defined here. Symmetric (Stratonovich) sums of the form ${\sum }_k\frac{1}{2}(H_{t_k}+H_{t_{k+1}})(B_{t_{k+1}}-B_{t_k})$, the Stratonovich integral, and any Ito--Stratonovich conversion rule are not defined or asserted on this page. None of the items above may be read as identifying a Stratonovich integral with an Ito integral plus a correction term; that identity would require its own definition, hypotheses and proof and belongs to a later stochastic-calculus development.

Finite-sum distinction. For any fixed finite partition and any specified real endpoint values H_k and B_k, subtraction gives the exact identity ${\sum }_k\frac{1}{2}(H_k+H_{k+1})(B_{k+1}-B_k)-{\sum }_k{H}_k(B_{k+1}-B_k)=\frac{1}{2}{\sum }_k(H_{k+1}-H_k)(B_{k+1}-B_k)\mathrm{.}$ Indeed each summand on the left simplifies to half the product of the two increments. These are cross-increment sums of the kind used in Quadratic covariation of Brownian Ito processes. They need not vanish merely because the mesh tends to zero; neither their convergence nor the convergence of either integral sum is asserted here for an arbitrary predictable integrand. For constant H the difference is exactly zero, whereas for H_k=B_k it is half the sum of squared increments.

An arbitrary predictable diffusion coefficient does not come with a covariation or a symmetric-integral conversion theorem. In particular this remark does not identify a correction for the complete stochastic integrand with just a Hessian term in an Ito formula. Such a claim needs its own hypotheses and proof. “Symmetric” above means the average of endpoint values, not evaluation at the time midpoint.

This finite algebraic comparison specifies a convention boundary; it defines no Stratonovich integral. No choices are made here, and the cited integral constructions retain their own declared AC and version assumptions.