Semimartingale integration combines two constructions: martingale integration for unpredictable noise and pathwise Lebesgue-Stieltjes integration for finite-variation drift. The resulting integral preserves jumps, brackets, stopping, and iterated integration in exactly the forms needed for stochastic differential equations.
Simple predictable integrands
Definition. For \[ H_t=H_01_{\{0\}}(t)+ \sum_{k=0}^{n-1}H_k1_{(t_k,t_{k+1}]}(t), \qquad H_k\in\mathcal F_{t_k}, \] and a cadlag adapted process \(X\), define \[ (H\mathbin{\cdot}X)_t = \sum_{k=0}^{n-1} H_k\{X_{t_{k+1}\wedge t}-X_{t_k\wedge t}\}. \] The left endpoint is essential: \(H_k\) is chosen before the increment occurs.
Integrability against the decomposition
Theorem. Let \(X=X_0+M+A\) be a special semimartingale. A predictable \(H\) is integrable with respect to \(X\) if it is locally integrable with respect to both components: \[ \int_0^tH_s^2\,\mathrm{d}\langle M^c\rangle_s +\sum_{s\le t}H_s^2(\Delta M_s)^2<\infty \quad\text{locally}, \] in the martingale sense, and \[ \int_0^t|H_s|\,\mathrm{d}\operatorname{Var}(A)_s<\infty \quad\text{almost surely}. \] Then \[ H\mathbin{\cdot}X = H\mathbin{\cdot}M+H\mathbin{\cdot}A, \] a semimartingale.
Proof outline. For bounded stopped martingales, complete simple predictable integrands under the bracket norm, exactly as for Brownian motion. Localization defines \(H\mathbin{\cdot}M\) in general. The \(A\)-integral is the pathwise Lebesgue-Stieltjes integral, controlled by \[ \left|\int_0^tH_s\,\mathrm{d}A_s\right| \le\int_0^t|H_s|\,\mathrm{d}\operatorname{Var}(A)_s. \] Simple-process approximations converge uniformly on compacts in probability, and the limits agree with the original sum definition.
Jumps and brackets of integrals
Proposition. If \(H\) is \(X\)-integrable, then \[ \Delta(H\mathbin{\cdot}X)_t=H_t\Delta X_t. \] If \(M,N\) are local martingales, then \[ [H\mathbin{\cdot}M,N]_t =\int_0^tH_s\,\mathrm{d}[M,N]_s, \] and in particular \[ [H\mathbin{\cdot}M]_t =\int_0^tH_s^2\,\mathrm{d}[M]_s. \]
Proof. For simple \(H\), a jump of the sum is exactly \(H_t\Delta X_t\). Partition products give \[ \sum\Delta(H\mathbin{\cdot}M)\Delta N =\sum H\,\Delta M\,\Delta N, \] which converges to the stated covariation integral. Approximation and localization extend both identities.
Associativity
Theorem. If \(H\) is integrable with respect to \(X\), \(Y=H\mathbin{\cdot}X\), and \(K\) is integrable with respect to \(Y\), then \(KH\) is integrable with respect to \(X\) and \[ K\mathbin{\cdot}Y=(KH)\mathbin{\cdot}X. \]
Proof. For simple integrands, refine both time partitions. Each increment of \(Y\) is \(H\) times the corresponding increment of \(X\), so summation gives the identity exactly. For general integrands, choose simple predictable approximations. Continuity of semimartingale integration in the ucp topology passes the identity to the limit.
Integration by parts and stochastic exponentials
Theorem. For semimartingales \(X,Y\), \[ X_tY_t-X_0Y_0 = X_{-}\mathbin{\cdot}Y_t +Y_{-}\mathbin{\cdot}X_t +[X,Y]_t. \] If \(X\) is continuous, the unique solution of \[ Z_t=1+\int_0^tZ_s\,\mathrm{d}X_s \] is the stochastic exponential \[ \mathcal E(X)_t =\exp\{X_t-\tfrac12[X]_t\}. \]
Proof. The product formula was obtained from the exact increment identity. For the exponential, apply Ito's formula to \(e^{X_t-[X]_t/2}\). The first derivative gives \(Z_t\,\mathrm{d}X_t-\frac12Z_t\,\mathrm{d}[X]_t\), while the quadratic correction contributes \(\frac12Z_t\,\mathrm{d}[X]_t\); they cancel, leaving \(\mathrm{d}Z_t=Z_t\,\mathrm{d}X_t\). Linear SDE uniqueness identifies the solution.
Example (compensated Poisson integral). Let \(N_t\) be Poisson with rate \(\lambda\), and \(\widetilde N_t=N_t-\lambda t\). For bounded predictable \(H\), \[ \int_0^tH_s\,\mathrm{d}\widetilde N_s = \sum_{0<s\le t}H_s\Delta N_s -\lambda\int_0^tH_s\,\mathrm{d} s \] is a local martingale with jumps \(H_s\) at arrival times and predictable bracket \[ \left\langle H\mathbin{\cdot}\widetilde N\right\rangle_t =\lambda\int_0^tH_s^2\,\mathrm{d} s. \]