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If $B_t$ is a Brownian motion then using Hermite polynomials one can find that $$1, B_t, B_t^2-t, B_t^3 - 3tB_t,...$$ are martingales. If $X_t$ is a diffusion $dX_t = \mu(X_t,t)dt + \sigma(X_t,t)...