State-price density and local deflator distinction

A local martingale deflator only forces deflated asset prices to be local martingales. True expectation pricing requires additional martingale and integrability properties. In a Brownian one-factor market, continuity of the coefficients ensures a pathwise finite market-price-of-risk square integral, but not expectation-one of its stochastic exponential. The reciprocal three-dimensional Bessel strict local martingale gives a counterexample to promoting this local conclusion automatically.