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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 iscontained 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 $0<x_{0}<x_{1},$ 
we seek a function u with supportin $\left[ x_{0},x_{1}\right] $ 
such that the corresponding solution ysatisfies: \[y(t,0)=\hat{y}(t,0)\quad \quad \forall t\in \left[ 0,T\right] .\] 
We prove in this article that, under some regularity conditions on thecoefficients of L, continuous solutions are unique and dense in the sensethat $\hat{y}|_{[0,T]\times \{0\}}$ 
can be C 0-approximated, but anexact solution does not exist in general. This result solves the problem of almost replicating a barrier option in thegeneralised Black–Scholes framework with a combination of European options,as stated by Carr et al. in [6].
