Attainable European contingent claim ($X_T=\xi$)

A maturity-$T$ claim is attainable if a predictable self-financing strategy with fixed initial capital has terminal wealth equal to the claim payoff almost surely. In discrete time its wealth is $X_t=x+{\sum }_{s=1}^t{H}_s\cdot (P_s-P_{s-1})$. The strategy replicates the claim; attainability is a statement about exact pathwise replication, not only equality of expectations.