Stochastic Integration for Semimartingales

Core Idea

A semimartingale is a càdlàg adapted process X that decomposes as X = M + A, where M is a local martingale and A is an adapted process of finite variation. This is the most general class of "good integrators" — by the Bichteler-Dellacherie theorem, semimartingales are exactly the processes for which a reasonable stochastic integral ∫H dX can be defined. The Itô integral extends from Brownian motion to this full generality: the stochastic integral ∫₀ᵗ H(s) dX(s) is well-defined for predictable H, and the Burkholder-Davis-Gundy (BDG) inequality E[sup_{s≤t} |∫₀ˢ H dM|^p] ≤ C_p E[[∫H dM, ∫H dM]_t^p/2] controls the integral's moments through the quadratic variation.

Explainer

The semimartingale is the central object of modern stochastic calculus. A càdlàg adapted process X is a semimartingale if it admits a decomposition X = X_0 + M + A, where M is a local martingale (the "noise") and A is a predictable process of locally finite variation (the "drift"). This class is remarkably broad: it includes Brownian motion, Poisson processes, Lévy processes, diffusions, solutions to SDEs driven by any combination of continuous and jump noise, and much more. The Bichteler-Dellacherie theorem (1979) establishes that semimartingales are not merely a convenient class but the *largest* class of processes for which a reasonable stochastic integral can be defined — any attempt to extend integration beyond semimartingales violates either linearity or a minimal continuity requirement.

The construction of the stochastic integral ∫₀ᵗ H_s dX_s for a semimartingale X proceeds in two pieces corresponding to the decomposition X = M + A. The integral against A is a pathwise Lebesgue-Stieltjes integral (since A has finite variation), well-defined for any adapted H with ∫|H_s||dA_s| < ∞. The integral against M extends the Itô integral: first define it for simple predictable processes, then extend by an L² isometry (using the quadratic variation [M,M] as the "squared norm" of M), and finally localize to handle local martingales. The Itô isometry E[(∫₀ᵗ H dM)²] = E[∫₀ᵗ H² d[M,M]_s] generalizes from Brownian motion to any L²-martingale M, with the quadratic variation [M,M] playing the role that t plays for W.

The Burkholder-Davis-Gundy (BDG) inequality is the main analytic tool for controlling stochastic integrals. For a local martingale M and any p ≥ 1, it states c_p E[[M,M]_T^p/2] ≤ E[sup_{t ≤ T} |M_t|^p] ≤ C_p E[[M,M]_T^p/2], where c_p, C_p are universal constants depending only on p. This two-sided equivalence means that the "size" of a martingale (its maximal process) and the "size" of its randomness (the quadratic variation) are always comparable. For p = 2, the upper bound is the Doob maximal inequality upgraded with the quadratic variation. The BDG inequality is indispensable for proving existence and uniqueness of SDE solutions, for establishing convergence of numerical schemes, and for any argument that requires moment bounds on stochastic integrals.

The Itô formula for semimartingales unifies and extends the classical Itô formula to processes with jumps. For X a semimartingale and f ∈ C², it reads: f(X_t) = f(X_0) + ∫₀ᵗ f'(X_{s-}) dX_s + ½∫₀ᵗ f''(X_{s-}) d[X,X]_s^c + Σ_{0 < s ≤ t}[f(X_s) - f(X_{s-}) - f'(X_{s-})ΔX_s]. The three correction terms beyond the naive chain rule capture: (1) the stochastic integral (first-order), (2) the continuous quadratic variation correction (the familiar ½f''dt term for diffusions), and (3) a jump correction that accounts for the nonlinearity of f applied to finite-sized jumps. When X is continuous, the jump sum vanishes and the formula reduces to the standard Itô formula. When X is a pure jump process, the [X]^c term vanishes and the formula becomes a telescoping sum of discrete changes. This unified formula is the computational engine for pricing derivatives under jump-diffusion models, for deriving the generators of Markov processes, and for establishing the connections between SDEs and PDEs (Feynman-Kac type results) in the general semimartingale setting.