Semimartingales combine local-martingale noise with finite-variation motion. They include diffusions, counting processes, jump processes, and deterministic finite-variation signals, and they form the maximal natural class stable under predictable stochastic integration.
Decomposition and special semimartingales
Definition. An adapted cadlag process \(X\) is a semimartingale if locally in time it admits a decomposition \[ X_t=X_0+M_t+A_t, \] where \(M\) is a local martingale starting at zero and \(A\) is adapted with finite variation on compact intervals.
Definition. A semimartingale is special if the finite-variation term can be chosen predictable. Its canonical decomposition \[ X=X_0+M+A \] with \(A_0=0\), \(A\) predictable, is unique.
Proof of uniqueness. If \(M+A=M'+A'\), then \(N=M-M'=A'-A\) is both a local martingale and predictable finite variation. After localization, its total variation is integrable. A finite-variation martingale has zero conditional expected signed increments and zero expected total variation only when its increments vanish; hence \(N\) is indistinguishable from zero.
Canonical examples
Examples.
- If \(B\) is Brownian motion and \(b,\sigma\) are adapted with suitable integrability, then \[ X_t=X_0+\int_0^tb_s\,\mathrm{d} s +\int_0^t\sigma_s\,\mathrm{d}B_s \] is a continuous semimartingale.
- If \(N_t\) is a Poisson process of rate \(\lambda\), then \[ N_t=(N_t-\lambda t)+\lambda t \] is its martingale-plus-compensator decomposition.
- Every adapted finite-variation process is a semimartingale with zero local-martingale part.
Proposition. If \(X=M+A\) is a cadlag semimartingale, then \[ [X]_t=[M]^c_t+\sum_{0<s\le t}(\Delta X_s)^2, \] where \([M]^c\) is the continuous part of the local-martingale quadratic variation.
Proof outline. The continuous finite-variation part has zero quadratic variation and zero covariation with continuous local martingales. At a jump time, the partition square converges to the full squared jump \((\Delta M+\Delta A)^2=(\Delta X)^2\). Isolating finitely many large jumps and controlling the remaining small jumps yields the formula.
Stability properties
Theorem. The class of semimartingales is closed under:
- addition and scalar multiplication;
- stopping at stopping times;
- equivalent changes of probability measure;
- stochastic integration by predictable integrands;
- \(C^2\) transformations, through Ito's formula.
Proof. The first two claims follow by applying the operation separately to the local-martingale and finite-variation parts. Equivalent measure changes preserve the semimartingale class by the density-process decomposition. Integration maps \(M+A\) to \(H\mathbin{\cdot}M+H\mathbin{\cdot}A\), again a local martingale plus finite variation. Ito's formula explicitly decomposes \(f(X)\) into stochastic-integral, finite-variation, and jump-correction terms.
Good integrators
Definition. For bounded predictable simple \[ H_t=\sum_{k=0}^{m-1}H_k1_{(t_k,t_{k+1}]}(t), \qquad H_k\in\mathcal F_{t_k}, \] define \[ (H\mathbin{\cdot}X)_T =\sum_kH_k(X_{t_{k+1}}-X_{t_k}). \] A cadlag adapted \(X\) is a good integrator if \(H^n\mathbin{\cdot}X\to0\) in probability whenever \(\|H^n\|_\infty\to0\).
Theorem (Bichteler-Dellacherie). A cadlag adapted process is a good integrator if and only if it is a semimartingale.
Proof outline. If \(X=M+A\), localization and martingale inequalities give \(H^n\mathbin{\cdot}M\to0\) in probability, while \[ |H^n\mathbin{\cdot}A|_T \le\|H^n\|_\infty\operatorname{Var}_T(A)\to0. \] Conversely, the good-integrator property bounds all predictable trading gains. A separation argument in the space of adapted processes constructs a finite-variation compensator whose subtraction leaves a local martingale. This direction is deep and is the structural reason semimartingales, rather than arbitrary adapted paths, support a stable stochastic calculus.
Processes outside the class
Remark. Fractional Brownian motion with Hurst parameter \(H\ne1/2\) is generally not a semimartingale. For \(H>1/2\), paths are regular enough for Young or rough-path integration but do not have martingale quadratic variation; for \(H<1/2\), rougher integration theories are needed. The integrator determines the calculus.