From an SDE to Its Generator
A weak solution of an SDE consists of a process and a Brownian motion on a filtered probability space; the probability space need not be fixed in advance. Itô’s formula converts the SDE into an evolution formula for a smooth function of the process. The drift in that formula is the generator applied to the function, while the remaining stochastic integral supplies the martingale term. The key point is to justify that this term is a true martingale for the test functions used in the martingale problem.
Let \((\Omega,\mathcal{F},(\mathcal{F}_t),\mathbb{P})\) be a filtered probability space carrying an \(m\)-dimensional Brownian motion \(\mathbf{B}_t\) relative to this filtration. Let \(X_t\) take values in \(\mathbb{R}^d\), and consider coefficients \(b:\mathbb{R}^d\to\mathbb{R}^d\) and \(\sigma:\mathbb{R}^d\to\mathbb{R}^{d\times m}\). Assume both coefficients are locally bounded. A weak solution with initial value \(X_0\), which is \(\mathcal{F}_0\)-measurable, is a continuous adapted process satisfying, almost surely, for every \(t\ge 0\),
The drift integral is understood pathwise and is assumed finite on each bounded time interval. The stochastic integral is an Itô integral, locally square-integrable: after stopping when \(X\) exits a bounded region, its progressively measurable matrix integrand has finite expected time-integrated squared Frobenius norm. When the integrand itself satisfies \(\mathbb{E}\int_0^T\|\sigma(X_s)\|^2\,ds<\infty\), the unstopped integral is in the usual \(L^2\) class. These assumptions give meaning to the SDE without requiring global moment bounds on \(X\).
Set \(a(x)=\sigma(x)\sigma(x)^{\mathsf T}\). For a twice continuously differentiable function \(f:\mathbb{R}^d\to\mathbb{R}\), the diffusion generator is
Equivalently, the second-order term is \(\tfrac12\operatorname{tr}(a(x)D^2f(x))\). The factor \(\tfrac12\) is the Itô correction: it comes from quadratic covariation, not from the ordinary chain rule.
The Weak-SDE-to-Martingale-Problem Result
Proof. Fix \(f\in C_c^2(\mathbb{R}^d)\) and a finite time horizon \(T\). Itô’s formula gives, up to localization,
In particular, the Hessian contribution is explicitly the \(\tfrac12\partial_{ij}f\,a_{ij}\) term; it arises from \(dB_t^k\,dB_t^\ell=\delta_{k\ell}\,dt\), while \(dB_t^k\,dt=0\) and \(dt\,dt=0\). Combining the two drift terms gives \(\int_0^t\mathcal{L}f(X_s)\,ds\). The remaining term has integrand \(\sigma(X_s)^{\mathsf T}\nabla f(X_s)\) against \(\mathbf{B}\).
To justify the martingale claim, let \(\tau_n=\inf\{t\ge0:|X_t|\ge n\}\), with the infimum of the empty set defined as \(+\infty\). On \([0,T\wedge\tau_n]\), local boundedness of \(\sigma\) and boundedness of \(\nabla f\) imply that the stopped stochastic integrand is square-integrable in expectation. Indeed, on the event \(|X_0|<n\), its squared norm is bounded by a finite constant depending on \(n\) and \(f\), times \(T\); if \(|X_0|\ge n\), the stopped interval has length zero. The stopped Itô integral is therefore a square-integrable martingale by the Itô isometry and the martingale property of the Itô integral. It follows that \(M^f_{t\wedge\tau_n}\) is a martingale.
Compact support and local boundedness also imply that \(\mathcal{L}f\) is bounded on all of \(\mathbb{R}^d\): its derivatives vanish outside a compact set, and the coefficients are bounded on that set. Thus, for \(0\le t\le T\),
Every continuous path is bounded on \([0,T]\), so \(t\wedge\tau_n\to t\) almost surely for each \(t\le T\). For \(0\le s\le t\le T\), the stopped martingale property gives \(\mathbb{E}[M^f_{t\wedge\tau_n}\mid\mathcal{F}_s]=M^f_{s\wedge\tau_n}\). The displayed uniform bound permits dominated convergence in \(L^1\), including conditional expectations. Passing to the limit yields \(\mathbb{E}[M^f_t\mid\mathcal{F}_s]=M^f_s\). The same bound gives integrability. Since \(T\) was arbitrary, \(M^f\) is a true martingale on every bounded time interval. \(\square\)
The localization is important even though the final conclusion is stronger than a local martingale statement. In general, Itô’s formula first produces a local martingale. Here the compactly supported test function makes both \(f\) and \(\mathcal{L}f\) bounded, which gives the uniform bound needed to pass from stopped martingales to a true martingale. No uniqueness in law or pathwise uniqueness is asserted: the result only says that the law of a weak SDE solution satisfies the associated martingale problem.
Worked Examples
Worked Example: Brownian Motion Tested with Cosine
Let \(X_t=B_t\) be one-dimensional Brownian motion starting at \(0\), and use \(f(x)=\cos x\). This function is not compactly supported, so this calculation illustrates the same Itô argument for a bounded smooth test function; its bounded derivatives make the stochastic integral square-integrable on every finite horizon. For Brownian motion, \(b=0\), \(\sigma=1\), and \(\mathcal{L}f=\tfrac12 f''\). The derivatives are \(f'(x)=-\sin x\) and \(f''(x)=-\cos x\), so
Consequently \(M_t=\cos B_t+\tfrac12\int_0^t\cos B_s\,ds\). The stochastic integrand \(-\sin B_s\) is progressively measurable and has \(\mathbb{E}\int_0^T\sin^2(B_s)\,ds\le T\), so the Itô integral is a true square-integrable martingale. As an independent expectation check, \(B_s\sim N(0,s)\) gives \(\mathbb{E}[\cos B_s]=e^{-s/2}\). Therefore
This expectation check is consistent with the martingale property; the Itô-integral argument establishes the full conditional-expectation property, not just constancy of the mean.
Worked Example: Brownian Motion and the Unbounded Test Function \(x^2\)
For \(f(x)=x^2\), the derivatives are \(f'(x)=2x\) and \(f''(x)=2\), so Brownian motion has \(\mathcal{L}f=1\). Itô’s formula, initially applied after stopping at \(\tau_n=\inf\{t:|B_t|\ge n\}\), gives \(B_{t\wedge\tau_n}^2-(t\wedge\tau_n)\) as a stopped local martingale. In this case the unstopped candidate is \(M_t=B_t^2-t\). It is in fact a true martingale: for \(s\le t\), write \(B_t=B_s+(B_t-B_s)\). The increment is independent of \(\mathcal{F}_s\), has mean \(0\), and has second moment \(t-s\). Thus
Integrability follows from \(\mathbb{E}[B_t^2]=t<\infty\). As a check at \(s=0\), \(\mathbb{E}[M_t]=t-t=0=M_0\). This example lies outside the compactly supported test domain in the definition; it shows that unbounded tests can also work when their martingale and integrability properties are verified separately.
Worked Example: An Ornstein–Uhlenbeck Generator
Consider the Ornstein–Uhlenbeck SDE \(dX_t=-X_t\,dt+2\,dB_t\), \(X_0=1\), and again take \(f(x)=x^2\). Here \(b(x)=-x\), \(\sigma(x)=2\), \(f'(x)=2x\), and \(f''(x)=2\). Substitution into the generator gives
Itô’s formula displays both the drift and correction explicitly:
The candidate martingale is \(M_t=X_t^2-1-\int_0^t(-2X_s^2+4)\,ds\). To check square-integrability of its stochastic-integral term on \([0,T]\), solve the linear SDE by multiplying by \(e^t\): \(X_t=e^{-t}+2\int_0^t e^{-(t-s)}\,dB_s\). The Itô isometry gives \(\mathbb{E}[X_t^2]=e^{-2t}+4\int_0^t e^{-2(t-s)}ds=2-e^{-2t}\). Hence \(\mathbb{E}\int_0^T(4X_s)^2ds=16\int_0^T(2-e^{-2s})ds<\infty\), and the stochastic integral is a true square-integrable martingale.
Finally, the mean of the candidate is zero. Indeed, \(\mathbb{E}[X_t^2]=2-e^{-2t}\), and \(\mathbb{E}[-2X_s^2+4]=2e^{-2s}\), so
The Itô-integral verification establishes the martingale property; the mean calculation is an independent check of the drift and the second-order correction. Since \(x^2\) is not compactly supported, this example is again an explicitly verified extension beyond the defining test domain.
Common Pitfall: A Local Martingale Is Not Automatically a Martingale
A frequent error is to apply Itô’s formula and stop at the phrase “the stochastic term is a local martingale,” then conclude that the martingale problem is solved. A local martingale need not be a true martingale: without additional integrability or boundedness, its conditional expectations need not be preserved. For compactly supported \(f\), the proof above supplies the missing argument: localization gives stopped martingales, and boundedness of \(f\) and \(\mathcal{L}f\) gives a uniform integrable bound so the stopping can be removed.
Another error is to omit the second-order term and use the ordinary chain rule. For \(f(x)=x^2\) and Brownian motion, that would give zero drift and incorrectly suggest that \(B_t^2\) is a martingale. The actual calculation has \(f''=2\) and correction \(\tfrac12 f''\,dt=dt\), so \(B_t^2-t\), not \(B_t^2\), is the martingale. The generator records precisely this correction.
| Object | Role in the calculation | What must be checked |
|---|---|---|
| \(\mathcal{L}f\) | Drift in Itô’s formula | Its integral is well-defined; for compactly supported \(f\), it is bounded under local boundedness of the coefficients. |
| Itô integral | Candidate martingale term | Square-integrability after stopping; then use boundedness or another valid integrability argument to obtain a true martingale. |
| \(M_t^f\) | Process required by the martingale problem | Adaptedness, integrability, and the conditional-expectation identity. |
Check Your Understanding
For each question, state the assumptions that make the claimed integral or conditional expectation meaningful.
- For \(dX_t=b(X_t)\,dt+\sigma(X_t)\,dB_t\), write the one-dimensional generator and the martingale candidate for \(f\in C_c^2(\mathbb{R})\).
- For \(f(x)=x^3\), compute the drift term in Itô’s formula for standard Brownian motion and identify the second-order correction.
- Why does local boundedness of \(b,\sigma\) suffice to make \(\mathcal{L}f\) bounded when \(f\in C_c^2(\mathbb{R}^d)\)?
- In the Ornstein–Uhlenbeck example, verify directly from the displayed second moment that \(\mathbb{E}\int_0^T(4X_s)^2ds\) is finite.
- Explain why constancy of \(\mathbb{E}[M_t]\) alone does not establish the martingale property.