Let (d S_{t}=S_{t}left(r(t) d t+sigma d W_{t} ight)) where (r) is a deterministic function and let (h)

Let \(d S_{t}=S_{t}\left(r(t) d t+\sigma d W_{t}\right)\) where \(r\) is a deterministic function and let \(h\) be a convex function satisfying \(x h^{\prime}(x)-h(x) \geq 0\). Prove that \(\exp \left(-\int_{0}^{t} r(s) d s\right) h\left(S_{t}\right)=R_{t} h\left(S_{t}\right)\) is a local sub-martingale.

Apply the Itô-Tanaka formula to obtain that

\[\begin{aligned}R(t) h\left(S_{t}\right)= & h(x)+\int_{0}^{t} R(u) r(u)\left(S_{u} h^{\prime}\left(S_{u}\right)-h\left(S_{u}\right)\right) d u \\
& +\frac{1}{2} \int h^{\prime \prime}(d a) \int_{0}^{t} R(s) d_{s} L_{s}^{a}+\text { loc. mart. }\end{aligned}\]