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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.

An urn with $j$ red balls among $n$ is sampled independently $n$ times with replacement, and the next generation's red count is the number of red draws. Its proportion is a bounded discrete-time martingale. Accelerating generation time by $n$ gives a Wright–Fisher diffusion limit: the one-step variance of the proportion is $x(1-x)/n$, and the higher moments make Taylor remainders negligible.

For an independent exponential distribution default time of rate $\lambda$ and a continuous nonnegative martingale $S$, this process is a martingale. The compensating pre-default factor $e^{\lambda t}$ balances the survival probability $e^{-\lambda t}$. The strict survival inequality gives a càdlàg price process.