 Solution

A trading strategy chooses a vector $H_t$ of holdings for the period $(t-1,t]$, with $H_t$ measurable with respect to ${\mathcal{F}}_{t-1}$. Its end-of-period wealth is $X_t=H_t\cdot P_t$. Rebalancing at date $t$ is self-financing whennew holdings cost exactly the value released by the old holdings. With fixed initial capital $x$, the equivalent gains identity isA European contingent claim is a maturity-$T$ payoff $\xi$ measurable with respect to ${\mathcal{F}}_T$. It is attainable if there exists a predictable self-financing strategy and an initial capital $x$ for which $X_T=\xi$ almost surely. Such a strategy is a replicating strategy, and $x$ is its initial replication cost. Holdings are understood only up to the maturity being replicated.