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Started at radius one, planar Brownian motion avoids the origin, and its logarithmic radius is a continuous local martingale because the logarithmic potential is harmonic away from zero. It is a strict local martingale: at deterministic $t>0$, the angular average of a logarithmic potential gives $\mathbb{E}\log |B_t|={\int }_1^{\infty }{e}^{-r^2/(2t)}dr/r>0$, whereas the initial value is zero. Absolute integrability follows from integrability of $r|\log r|$ near zero and the Gaussian tail. Its positive expectation does not contradict nonnegative-local-martingale bounds, since this process takes both signs.

Let $U_t=(1,0,0)+(B_t^1,B_t^2,B_t^3)$, with three independent standard Brownian motions, and set $R_t=|U_t|$ and $M_t=R_t^{-1}$. The radial stochastic process $R$ is a three-dimensional Bessel process, which never hits zero by the Hitting-zero classification for a Bessel process. In three dimensions $u\mapsto |u{|}^{-1}$ is a harmonic function away from zero: direct differentiation gives $\Delta (|u{|}^{-1})=0$.

For the stopping times $T_n=\inf \{t:R_t\notin (1/n,n)\}$, $n\ge 2$, the Itô formula givesThe coefficients are bounded by $n^2$, so each stopped Itô integral is a true martingale on every finite horizon. The stopping times form a localizing sequence; as permitted, its divergence need not be proved here. Thus $M$ is a nonnegative continuous local martingale.

To see that $M$ is not a true martingale, integrate the three-dimensional multivariate normal density of $U_t$ in spherical coordinates. The angular integral yieldswhere $\Phi$ is the standard normal distribution function. Therefore the reciprocal of a three-dimensional Bessel process is a strict local martingale.

For a three-dimensional Bessel process started at one, $R$ stays strictly positive and satisfies $dR=dW+R^{-1}dt$. The Itô formula makes its reciprocal a local martingale, but $\mathbb{E}[R_t^{-1}]=2\Phi (1/\sqrt{t})-1<1$ for $t>0$. Thus the reciprocal is a strict local martingale. It can deflate the market with constant bank account and stock price $R$ without yielding a true pricing density for the bank account.