Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 211 / 1 / g / Solution

Assume condition (1). Given an adapted cash-flow process $X_1,\dots ,X_T$, construct the holdings backwards. Set $H_{T+1}=0$. Once $H_{t+1}$ is known, the random variableis ${\mathcal{F}}_t$-measurable. Condition (1) supplies an ${\mathcal{F}}_{t-1}$-measurable $H_t$ satisfyingHence ${\xi }_t^H=X_t$ for every $t\le T$, and setting later holdings to zero proves condition (2).

Conversely, let $X_T$ be any ${\mathcal{F}}_T$-measurable random variable and apply condition (2) to the adapted process with cash flow $X_T$ at $T$ and zero cash flow earlier. Since $H_{T+1}=0$,where $H_T$ is ${\mathcal{F}}_{T-1}$-measurable. This is condition (1). The conditions are therefore equivalent and describe market completeness.