the integral against a semimartingale
The semimartingale is the most general integrator for which a satisfactory stochastic integral exists, and integrating against it is the culmination of the construction. A semimartingale is a process that can be written as a (local) martingale plus a finite-variation process; the integral against it is built by integrating against each piece separately and adding. The reason this class is exactly right is a deep theorem (Bichteler-Dellacherie): semimartingales are precisely the processes against which a reasonable, dominated stochastic integral with the right continuity properties can be defined at all.
Concretely, a semimartingale X has a decomposition X = X_0 + M + A, with M a local martingale (M_0 = 0) and A a process of finite variation (A_0 = 0). For a predictable, locally bounded integrand H, the stochastic integral is defined by integral_0^t H dX = integral_0^t H dM + integral_0^t H dA, where the first integral is the stochastic integral against the local martingale (the previous construction) and the second is an ordinary pathwise Lebesgue-Stieltjes integral against the finite-variation A. In the continuous case the decomposition M + A is unique (the canonical decomposition), so the integral is unambiguous; in the general cadlag case one uses the special-semimartingale decomposition with A predictable. The integral inherits the expected properties: it is again a semimartingale, it is linear and respects the dominated convergence theorem for stochastic integrals, and its quadratic variation is [integral H dX] = integral H^2 d[X]. For continuous semimartingales Ito's formula, the product rule, and the covariation bracket all hold with X in this generality, which is exactly why the entire calculus is phrased for semimartingales.
This is the natural home of stochastic calculus. Solutions of SDEs, prices of traded assets, transformed diffusions, and stopped or time-changed processes are all semimartingales, and the class is stable under the operations one performs: C^2 functions of a semimartingale are semimartingales (by Ito), stochastic integrals against a semimartingale are semimartingales, stopping preserves the class. The honest caveats: (1) The integrand must be predictable; for left-continuous adapted (e.g. caglad) integrands this is automatic, and predictability is what keeps the martingale part a local martingale. (2) The finite-variation part is integrated pathwise, so no isometry governs it; only the martingale part carries the L^2 / quadratic-variation machinery. (3) A subtle but important point is that under a change of measure (Girsanov) a process stays a semimartingale but its decomposition changes (the drift shifts), so 'semimartingale' is a measure-robust notion while 'martingale' is not.
Geometric Brownian motion S_t = S_0 exp(sigma B_t + (mu - sigma^2/2) t) is a continuous semimartingale with canonical decomposition S = S_0 + (local martingale integral sigma S dB) + (finite-variation drift integral mu S ds). Integrating a predictable holding strategy H against S gives the gain integral H dS = integral H sigma S dB + integral H mu S ds — the first piece a local martingale, the second a pathwise time integral; the value process of a trading strategy is exactly an integral against a semimartingale.
A semimartingale integral splits into a martingale (dB) part with L^2 machinery and a finite-variation (ds) part integrated pathwise.
Semimartingale is the largest class admitting a well-behaved integral (Bichteler-Dellacherie) and is preserved under C^2 maps, integration, stopping, and change of measure — but the martingale/finite-variation SPLIT is not measure-invariant; only the semimartingale property itself is.