Martingale problem

For a differential operator $L$, the martingale problem asks for a process $X$ such thatis a local martingale for every test function $f$ in a suitable domain.

A continuous adapted process started at a fixed point is Brownian motion exactly when $f(X_t)-f(X_0)-\frac{1}{2}{\int }_0^t\Delta f(X_s)ds$ is a martingale for every $f\in C_c^{\infty }({\mathbb{R}}^d)$. Cutoff coordinate and coordinate-product tests give continuous local martingales with quadratic covariations $[X^i,X^j{]}_t={\delta }_{ij}t$. The Itô formula makes $\exp (iu\cdot (X_t-X_0)+|u{|}^{2}t/2)$ a local martingale, whose deterministic modulus bound on bounded horizons makes it a true martingale. Its conditional expectation identifies every increment as centered Gaussian with covariance $(t-s)I$, independent of the past. This proves the converse via the Lévy characterization of multidimensional Brownian motion. The forward implication is the Itô formula.

For discrete-time Markov chains run at time step $1/n$, suppose the displayed conditional moments converge uniformly on compact sets to continuous coefficients, the local conditional large-jump second moments vanish, initial laws converge, and compact containment holds on each finite horizon. If the limiting diffusion martingale problem is well-posed and nonexplosive, the chain laws converge to its solution. Vanishing jumps give a continuous limit; linearly interpolated paths converge in the uniform path topology and step paths in Skorokhod J1 topology. Local moment bounds give tightness after exit-time localization. Taylor expansion identifies the limiting test-function martingales, and uniqueness of the martingale problem makes every subsequential limit the same. The diffusion approximation of a birth-death process is an important example.

A martingale problem is well-posed for a specified test domain and class of paths if it admits a solution for every specified starting point and that solution is unique in law. Existence on one fixed probability space is not required. Test-domain and true-versus-local conventions must be stated. For bounded diffusion coefficients and $C_b^2$ test functions, boundedness on every finite horizon upgrades the local identities to true martingale identities.

For bounded measurable $a,b$, with $a$ symmetric positive semidefinite, an L-diffusion is a continuous adapted process $X$ such that $g(X_t)-g(X_0)-{\int }_0^{t}Lg(X_s)ds$ is a true martingale for every $g\in C_b^2$. The general local martingale problem may instead use compactly supported test functions and require only a local martingale. Specify the test class and the true or local convention when coefficients are unbounded.

For a bounded-domain diffusion with almost surely finite exit time ${\tau }_D$, bounded $L$-harmonic functions continuous up to the boundary satisfy $u(x)={\mathbb{E}}_{x}u(X_{{\tau }_D})$. Stop the test-function martingale and use bounded convergence. Identical boundary values therefore imply identical harmonic functions. Uniform ellipticity and bounded SDE coefficients ensure finite exit by the projection argument.

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For an SDE with bounded drift and diffusion coefficients and ${\sum }_k⟨\xi ,{\sigma }_k(x){⟩}^2\ge \lambda |\xi {|}^2$, each fixed nonzero scalar projection of a solution is unbounded in absolute value almost surely. Rescale a slab to $(-1,1)$; its projected drift is controlled by its uniformly positive bracket density. The exit with drift controlled by quadratic variation then forces exit from every slab in a countable exhaustion.

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For an L-diffusion with bounded coefficients and $f\in C_b^{1,2}$, $M^f$ is a continuous martingale. Freeze the time argument along a deterministic partition, apply the spatial martingale problem on each interval, and pass to the limit by the dominated convergence theorem. The same proof works if the absolute drift coefficients and the trace of the diffusivity have integrable time integrals along the path on every finite horizon. Local hypotheses alone give a local martingale conclusion.