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Spot volatility is the instantaneous diffusion coefficient multiplying relative price increments in a stochastic differential equation for an asset. In a stochastic volatility model it is itself a stochastic process.

A squared Bessel process of dimension $\delta \ge 0$ is a nonnegative continuous solution of the displayed stochastic differential equation. It extends the squared radial part of integer-dimensional Brownian motion to arbitrary nonnegative dimensions. Before hitting zero its square root solves $dR=dB+(\delta -1)(2R{)}^{-1}dt$. At dimension zero, a process started at zero remains there. The coefficients are not Lipschitz at zero, so positivity or uniqueness must not be justified by a global Lipschitz theorem.

A stochastic differential equation has strong existence for specified coefficients and initial data if a strong stochastic solution can be constructed on every prescribed stochastic basis carrying the required initial variable and driving Brownian motion. This existence assertion is distinct from pathwise uniqueness and from a definition of the solution itself.

For bounded Borel $b:{\mathbb{R}}^d\to {\mathbb{R}}^d$, the stochastic differential equation $d{X}_t=b(X_t)dt+d{B}_t$ has strong existence and pathwise uniqueness. The identity diffusion matrix is nondegenerate. A Girsanov theorem argument alone establishes weak existence and uniqueness in law; the strong conclusion requires an additional theorem. This statement does not extend without further hypotheses to arbitrary path-dependent drift or arbitrary diffusion matrices. See the primary bounded-drift strong-solution theorem.

A Gaussian short rate model with Ornstein-Uhlenbeck process dynamics under the chosen pricing measure. For $a>0$, its unit zero-coupon bond value is $P(t,T)=A(\tau )e^{-B(\tau )r_t}$, where $\tau =T-t$, $B(\tau )=(1-e^{-a\tau })/a$ andSolve the linear stochastic differential equation, integrate over future time, and use the exponential moment of the resulting Gaussian integral to obtain this price. Its Gaussian law permits negative short rates.