Consider \(\left(B_{t}ight)_{t \in \mathbb{R}_{+}}\)a Brownian motion started at \(B_{0}=x \in[a, b]\) with \(a

\[\tau:=\inf \left\{t \in \mathbb{R}_{+}: B_{t}=a \text { or } B_{t}=bight\}\]

denote the first exit time of the interval \([a, b]\). Show that the solution \(f(x)\) of the differential equation \(f^{\prime \prime}(x)=-2\) with \(f(a)=f(b)=0\) satisfies \(f(x)=\mathbb{E}\left[\tau \mid B_{0}=xight]\).

Consider the process \(X_{t}:=f\left(B_{t}ight)-\frac{1}{2} \int_{0}^{t} f^{\prime \prime}\left(B_{s}ight) d s\), and apply the Doob Stopping Time Theorem 14.7.