Consider a Lagrangian torus fibration a la SYZ over a non compact base. Using techniques from arXiv:1510.04265, I will discuss the construction of wrapped Floer theory in this setting. Note that this setting is generally not exact even near infinity. The construction allows the formulation of a version of the homological mirror symmetry conjecture for open manifolds which are not exact near infinity. According to time constraints, I will apply this to prove homological mirror symmetry in the case where the A-model is the complement of an anti-canonical divisor in a toric Calabi Yau manifold.