Local Martingales, the Bessel Process, and Explosion

The stochastic integral $\int_0^t H_s \, dW_s$ is always a local martingale (see Extension to Local Martingales). But it is a true martingale only when $\mathbb{E}[\int_0^t H_s^2 \, ds] < \infty$. This gap between local martingale and true martingale has concrete consequences: when the diffusion coefficient of an SDE is large enough relative to the state, the solution process loses probability mass to infinity — it is a non-negative local martingale but not a true martingale.

The Bessel process is the canonical example. In dimension $d \ge 3$, the distance $R_t = |W_t|$ from the origin of a $d$-dimensional Brownian motion satisfies $dR_t = \frac{d-1}{2R_t} \, dt + dB_t$ — the Bessel-$d$ process. The process $1/R_t$ (the reciprocal) is a local martingale that is not a true martingale: $\mathbb{E}[1/R_t] < 1/R_0$ (probability "leaks to infinity"). In finance, this phenomenon appears in leveraged strategies where the variance grows faster than linearly — and explains why naive Black-Scholes delta-hedging can fail for strategies with unbounded leverage.

Learning objectives. By the end of this note you should be able to: