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In a one-factor diffusion market, the market price of risk is the excess drift per unit spot volatility. It is the coefficient in the Brownian motion part of a local martingale deflator, $d{Y}_t=-Y_t(r_{t}dt+{\lambda }_{t}d{W}_t)$.

The natural Brownian filtration is the natural filtration generated by a Brownian motion. Its usual augmentation adds null events and makes it right-continuous; the Brownian martingale representation theorem holds for this augmented natural filtration. A larger Brownian filtration can preserve independence of future increments while containing additional randomness, so it need not have martingale representation with respect to the specified Brownian motion.

Almost surely, no restriction of a Brownian motion path to an interval of positive length is a monotone function. On a fixed interval, monotonicity forces the independent centered normal random variables given by every dyadic subdivision's increments to have the same sign. The probability is at most $2^{1-2^m}$ for a subdivision into $2^m$ pieces. A countable union over rational intervals gives the simultaneous pathwise assertion.

Two continuous local martingales are orthogonal when their quadratic covariation is zero. With zero initial values, the martingale product identity says that their product is a local martingale. For two Brownian motions in a common filtration, orthogonality and the Lévy characterization of multidimensional Brownian motion make the pair a two-dimensional Brownian motion and imply independence. Merely having the two marginal Brownian motion laws on the same space does not imply orthogonality.