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For $a,b>0$,This follows from the Brownian reflection principle weighted by the Cameron-Martin theorem for a linear drift.

Apply Cameron-Martin theorem to the absolutely continuous path $h(t)=xt/T$ up to $T$ and $h(t)=x$ thereafter. Its derivative is $x/T$ on $[0,T]$ and zero later, so the Radon-Nikodym factor is $\exp ((x/T)W_T-x^2/(2T))$, yielding the formula for every bounded measurable functional of the path after $T$.

The Cameron-Martin theorem for a linear drift changes the density of Brownian paths through time $t$ byAt the driftless hitting time ${\tau }_{a,0}=t$, the endpoint is $B_t=a$. Multiplying its given density by the likelihood $e^{ab-b^{2}t/2}$ therefore yields

Setand define $Q$ on ${\mathcal{F}}_T$ byThe Cameron-Martin theorem for a linear drift says thatis a Brownian motion under $Q$. ThereforeThe discounted stock is an exponential Brownian martingale, so $e^{-rt}{S}_t$ is a $Q$-martingale. Thus $Q$ is a Risk-neutral measure for the Black-Scholes model.

Let $X_s=W_s-as$ and $M_t={\sup }_{0\le s\le t}{X}_s$. Split the crossing event according to its endpoint:Under the Cameron-Martin theorem for a linear drift, the law of $X$ relative to standard Wiener measure has endpoint densityOn paths that cross $b$ and end at $x\le b$, the Brownian reflection principle reflects the path after its first hit of $b$ and changes the endpoint to $2b-x\ge b$. Under this reflection the density becomesThe remaining exponential is the Cameron--Martin density for drift $+a$. Thereforewhere the last equality uses the symmetry of the normal distribution. Subtracting the crossing probability from one givesEquivalently, in terms of the standard normal distribution function,This is the finite-horizon maximum of Brownian motion with negative drift formula.

Under the Cameron-Martin theorem for a linear drift, the probability that Brownian motion with drift $-\mu$ reaches $x$ isUsing the supplied Laplace transform with $\lambda ={\mu }^2/2$ givesand henceThis is the survival function of the exponential distribution with rate $2\mu$.