Solution

Let

X_{s} = W_{s} - a s

and

M_{t} = sup_{0 \leq s \leq t} X_{s}

. Split the crossing event according to its endpoint:

P (M_{t} > b) = P (X_{t} > b) + P (M_{t} > b, X_{t} \leq b) .

X

relative to standard Wiener measure has endpoint density

exp (- a W_{t} - \frac{1}{2} a^{2} t) .

On paths that cross

b

and end at

x \leq b

, the Brownian reflection principle reflects the path after its first hit of

b

and changes the endpoint to

2 b - x \geq b

. Under this reflection the density becomes

e^{- 2 ab} exp (a W_{t} - \frac{1}{2} a^{2} t) .

The remaining exponential is the Cameron--Martin density for drift

+ a

. Therefore

P (M_{t} > b, X_{t} \leq b) = e^{- 2 ab} P (W_{t} + a t \geq b) = e^{- 2 ab} P (W_{t} - a t \leq - b),

where the last equality uses the symmetry of the normal distribution. Subtracting the crossing probability from one gives

P (0 \leq s \leq t sup (W_{s} - a s) \leq b) = P (W_{t} - a t \leq b) - e^{- 2 ab} P (W_{t} - a t \leq - b) .

Φ (\frac{b + a t}{t}) - e^{- 2 ab} Φ (\frac{a t - b}{t}) .

Solved by gpt-5.6-sol high.

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