Ito's formula is most useful when the right transformation reveals a linear equation, a martingale, or a deterministic drift. The examples below are reusable calculations rather than isolated tricks.
Stochastic exponentials
Theorem. If \[ X_t=\int_0^t\theta_s\,\mathrm{d}B_s, \qquad \int_0^t\theta_s^2\,\mathrm{d} s<\infty, \] then \[ Z_t = \exp\!\left\{ X_t-\frac12\int_0^t\theta_s^2\,\mathrm{d} s \right\} \] satisfies \[ \mathrm{d}Z_t=Z_t\theta_t\,\mathrm{d}B_t,\qquad Z_0=1. \] It is always a positive local martingale and is a true martingale under conditions such as Novikov's criterion.
Proof. Let \(Y_t=X_t-\frac12[X]_t\). Then \[ \mathrm{d}Y_t=\theta_t\,\mathrm{d}B_t-\frac12\theta_t^2\,\mathrm{d} t, \qquad \mathrm{d}[Y]_t=\theta_t^2\,\mathrm{d} t. \] Ito's formula for \(e^{Y_t}\) gives \[ \mathrm{d}Z_t = Z_t\,\mathrm{d}Y_t+\frac12Z_t\,\mathrm{d}[Y]_t =Z_t\theta_t\,\mathrm{d}B_t. \]
Geometric Brownian motion
Theorem. The SDE \[ \mathrm{d}X_t=\mu X_t\,\mathrm{d} t+\sigma X_t\,\mathrm{d}B_t, \qquad X_0>0, \] has the explicit solution \[ X_t = X_0\exp\!\left\{ \left(\mu-\frac12\sigma^2\right)t+\sigma B_t \right\}. \]
Proof. Before a possible hit of zero, apply Ito to \(f(x)=\log x\): \[ \mathrm{d}\log X_t = \frac{\mathrm{d}X_t}{X_t} -\frac12\frac{\mathrm{d}[X]_t}{X_t^2} = \left(\mu-\frac12\sigma^2\right)\mathrm{d} t +\sigma\,\mathrm{d}B_t. \] Integrate and exponentiate. The resulting explicit process is strictly positive and directly satisfies the SDE, so no hit of zero occurs.
Corollary. \[ \mathbb EX_t=X_0e^{\mu t}, \qquad \operatorname{Var}(X_t) = X_0^2e^{2\mu t}(e^{\sigma^2t}-1). \]
Ornstein-Uhlenbeck dynamics
Theorem. The linear SDE \[ \mathrm{d}X_t=-\theta X_t\,\mathrm{d} t+\sigma\,\mathrm{d}B_t, \qquad\theta>0, \] has solution \[ X_t=e^{-\theta t}X_0+ \sigma\int_0^te^{-\theta(t-s)}\,\mathrm{d}B_s. \] If \(X_0\sim N(0,\sigma^2/(2\theta))\) independently of future Brownian increments, the process is stationary with \[ \operatorname{Cov}(X_s,X_t) =\frac{\sigma^2}{2\theta}e^{-\theta|t-s|}. \]
Proof. Apply the product formula to the integrating factor: \[ \mathrm{d}(e^{\theta t}X_t) =\theta e^{\theta t}X_t\,\mathrm{d} t +e^{\theta t}\,\mathrm{d}X_t =\sigma e^{\theta t}\,\mathrm{d}B_t. \] Integrate and divide by \(e^{\theta t}\). The stochastic integral is Gaussian with variance \[ \sigma^2\int_0^te^{-2\theta(t-s)}\,\mathrm{d} s =\frac{\sigma^2}{2\theta}(1-e^{-2\theta t}). \] Combining it with the stationary initial variance gives constant marginal variance. Direct covariance calculation gives the formula.
Squared Bessel dynamics
Proposition. If \(B_t\) is \(d\)-dimensional Brownian motion and \(R_t^2=\|B_t\|^2\), then \[ \mathrm{d}(R_t^2) =2R_t\,\mathrm{d}\beta_t+d\,\mathrm{d} t \] for a one-dimensional Brownian motion \(\beta\) after the usual treatment at zero.
Proof. Multidimensional Ito gives \[ \mathrm{d}\|B_t\|^2 =2B_t^\top\mathrm{d}B_t+d\,\mathrm{d} t. \] The local martingale \(\int_0^t(B_s/\|B_s\|)^\top\mathrm{d}B_s\) has bracket \(t\) away from zero and is Brownian after localization, so \(B_t^\top\mathrm{d}B_t=R_t\,\mathrm{d}\beta_t\).
Exit-time equations
Theorem. Let \(X\) be a diffusion with generator \(L\), and let \(\tau_D\) be the exit time from a bounded domain. If a sufficiently regular function \(u\) solves \[ Lu=-1\quad\text{in }D,\qquad u=0\quad\text{on }\partial D, \] then \[ u(x)=\mathbb E_x\tau_D. \]
Proof. Ito's formula and the generator identity give \[ u(X_{t\wedge\tau_D}) = u(x)+M_{t\wedge\tau_D}-(t\wedge\tau_D). \] Take expectations. Boundary conditions and dominated or localized convergence let \(t\to\infty\), yielding \(0=u(x)-\mathbb E_x\tau_D\).
Backward equations
Proposition. If \(u\in C^{1,2}\) solves \[ \partial_tu+Lu=0, \] then \(u(t,X_t)\) is a local martingale. Under integrability, \[ u(t,x)=\mathbb E_{t,x}[g(X_T)] \] for terminal condition \(u(T,\cdot)=g\).
This drift-cancellation principle is the common source of dynamic programming, harmonic representations, and Feynman-Kac formulas. The next pages use the same stochastic exponentials for changes of measure and the same transformations for solving SDEs.