Static Hedging of Barrier Options with a Smile: An Inverse Problem

Let L be a parabolic second order differential operator on the domain $ \bar{\Pi}=\left[ 0,T\right] \times {\mathbb R}.$ Given a function $\hat{u}: {\mathbb R\rightarrow R}$ and $\hat{x}>0$ such that the support of û is contained in $(-\infty ,-\hat{x}]$, we let $\hat{y}:\bar{\Pi}\rightarrow {\mathbb R}$ be the solution to the equation: \[ L\hat{y}=0,\text{\quad }\hat{y}|_{\{0\}\times {\mathbb R}}=\hat{u} . \] Given positive bounds $0