Questions: Stochastic Integration for Semimartingales

Question 1

Which of the following is the correct definition of a semimartingale?

• A. A process with continuous paths and finite expectation
• B. A process that can be decomposed as X = M + A where M is a local martingale and A is a predictable process of finite variation
• C. Any process adapted to a filtration satisfying the usual conditions
• D. A process with independent and stationary increments

Answer: (B)

A semimartingale decomposes as a local martingale plus a predictable finite-variation process. This decomposition (the canonical decomposition for special semimartingales) separates the 'noise' (local martingale M) from the 'signal' (finite variation A). The class includes Brownian motion (M = W, A = 0), Poisson processes (M = N - λt, A = λt), Lévy processes, solutions to SDEs, and much more. The Bichteler-Dellacherie theorem proves that this class is exactly the set of processes for which stochastic integration is possible — no extension beyond semimartingales can produce a reasonable integral.

Question 2

The Burkholder-Davis-Gundy inequality states that for a continuous local martingale M, the maximal process sup_{s≤t}|M(s)| and the square root of the quadratic variation [M,M]_t^{1/2} have equivalent L^p norms for 1 ≤ p < ∞.

• T. True
• F. False

Answer: True

The BDG inequality provides universal constants c_p and C_p such that c_p E[[M,M]_t^{p/2}] ≤ E[sup_{s≤t}|M(s)|^p] ≤ C_p E[[M,M]_t^{p/2}] for all continuous local martingales M. This two-sided bound says that the 'size' of a martingale (measured by its maximum) is controlled by its quadratic variation and vice versa. For p = 2, the upper bound generalizes the Itô isometry E[M(t)²] = E[[M,M]_t] to a maximal inequality. The BDG inequality is the principal tool for proving L^p estimates on stochastic integrals — it reduces moment bounds on path suprema to bounds on quadratic variation, which is often computable.

Question 3

Explain why fractional Brownian motion with Hurst parameter H ≠ 1/2 is NOT a semimartingale, and what this implies about stochastic integration.

Model answer: Fractional Brownian motion (fBM) with H ≠ 1/2 has dependent increments — positively correlated for H > 1/2, negatively correlated for H < 1/2. A semimartingale must be decomposable as a local martingale plus a finite variation process. For H > 1/2, fBM paths are too smooth (Hölder continuous with exponent > 1/2) to be a martingale, and the 'drift' needed to compensate would have infinite variation. For H < 1/2, the paths are too rough. The Bichteler-Dellacherie theorem then implies that the standard stochastic integral ∫H dB^H is undefined — alternative integration theories (rough paths, Wick-Itô-Skorokhod) are needed.

This is a fundamental limitation result. The Bichteler-Dellacherie theorem says semimartingales are the LARGEST class of good integrators. Since fBM is not a semimartingale for H ≠ 1/2, one cannot define a pathwise stochastic integral using the standard theory. The rough paths theory of Lyons (1998) provides an alternative that extends stochastic calculus to processes with Hölder regularity > 1/3, covering fBM with H > 1/3. For H ≤ 1/3, even rough path theory requires additional structure.

Question 4

For a general semimartingale X = M + A (with M a local martingale and A finite variation), the quadratic variation [X,X]_t equals:

• A. [M,M]_t, since the finite variation part contributes nothing to quadratic variation
• B. [M,M]_t + 2[M,A]_t + [A,A]_t, the full bilinear expansion
• C. [M,M]_t + [A,A]_t, since [M,A]_t = 0 by orthogonality
• D. ∫₀ᵗ |dX(s)|², the pathwise squared total variation

Answer: (B)

The quadratic variation is bilinear: [X,X] = [M+A, M+A] = [M,M] + 2[M,A] + [A,A]. All three terms can be non-zero in general. However, if A is continuous and of finite variation, then [A,A]_t = 0 (continuous finite-variation processes have zero quadratic variation) and [M,A]_t = 0 (the cross-variation of a local martingale with a continuous finite-variation process vanishes). In this special case, [X,X]_t = [M,M]_t. But if A has jumps, [A,A]_t = Σ_{s≤t}(ΔA_s)² ≠ 0, and the full expansion applies.

Question 5

The Itô formula for a general semimartingale X and a C² function f states: 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]. What does the sum over jumps accomplish?

Model answer: The jump sum corrects for the fact that the continuous Itô formula (with only the ½f''d[X]^c term) is inaccurate at jump times. At a jump of size ΔX_s, the actual change in f is f(X_s) - f(X_{s-}), but the stochastic integral term contributes f'(X_{s-})ΔX_s (a linear approximation). The difference f(X_s) - f(X_{s-}) - f'(X_{s-})ΔX_s is the nonlinear correction — the 'second-order' effect of the jump that f' misses. For continuous semimartingales (no jumps), this sum vanishes and the formula reduces to the classical Itô formula. For pure jump processes, the continuous quadratic variation [X]^c vanishes and the sum captures all the nonlinearity.

The general Itô formula has three correction terms beyond the 'naive' chain rule ∫f'dX: the continuous quadratic variation term ½∫f''d[X]^c (the usual Itô correction), and the jump correction sum. Together they account for all second-order effects — continuous fluctuations via [X]^c and discontinuous jumps via the sum. This formula unifies Itô calculus for diffusions and the change-of-variable formula for jump processes into a single framework.