Let \(B\) be a BM and \(M_{t}^{B}:=\sup _{s \leq t} B_{s}\). Let \(f(t, x, y)\) be a \(C^{1,2,1}\left(\mathbb{R}^{+} \times \mathbb{R} \times \mathbb{R}^{+}\right)\)function such that

\[\begin{aligned}\frac{1}{2} f_{x x}+f_{t} & =0 \\f_{x}(t, 0, y)+f_{y}(t, 0, y) & =0 .\end{aligned}\]

Prove that \(f\left(t, M_{t}^{B}-B_{t}, M_{t}^{B}\right)\) is a local martingale. In particular, for \(h \in C^{1}\)

\[h\left(M_{t}^{B}\right)-h^{\prime}\left(M_{t}^{B}\right)\left(M_{t}^{B}-B_{t}\right)\]

is a local martingale.