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Set ${\theta }_t=(\mu -r)S_t/\sigma$. The Girsanov theorem uses the densityFor these nonexplosive linear stochastic differential equations, the linear drift change of measure ensures that $Z$ is a true martingale on each finite horizon. This matters: a stochastic exponential being a local martingale alone would not prove that its terminal expectation is one. Its strict positivity gives an equivalent risk-neutral measure $\mathbb{Q}$, under whichThe discounted asset is then a martingale. The Feynman-Kac formula, equivalently risk-neutral valuation of the replicable claim, givesHere the payoff must meet the usual integrability and replication assumptions; the bounded digital payoff below does.

For a non-dividend-paying traded asset $dS=\mu Sdt+\sigma dW$ and bank account $dB=rBdt$, delta hedging sets the share holding to $V_s$. Its bond value is $V-s{V}_s$. Itô's formula and the self-financing condition give the displayed pricing equation on the whole real line. The linear drift change of measure supplies risk-neutral dynamics $dS=rSdt+\sigma d{W}^{\mathbb{Q}}$, so the same price is the discounted conditional payoff expectation under the usual admissibility and integrability hypotheses.