Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 3 / 29K / c

Condition (i) says that each discounted risky price $X^i$ is an integrable $\mathbb{Q}$-martingale. For a bounded predictable risky holding process $\theta$, self-financing gives the discounted wealth equation in discrete timeEach increment is integrable andHence $V$ is a $\mathbb{Q}$-martingale, proving (i) $\Rightarrow$ (ii).

If $V$ is a $\mathbb{Q}$-martingale, thenbecause ${\mathcal{F}}_0$ is $\mathbb{P}$-trivial and equivalence makes it $\mathbb{Q}$-trivial as well. Thus (ii) $\Rightarrow$ (iii).

Assume (iii). Fix $t\in \{1,\dots ,T\}$, a coordinate $i$, and an event $A\in {\mathcal{F}}_{t-1}$. Choose the bounded predictable risky strategyand choose the numéraire holding to make the strategy self-financing with $V_0=0$. Its terminal discounted value isCondition (iii) yieldsfor every $A\in {\mathcal{F}}_{t-1}$. The defining property of conditional expectation impliesEvery coordinate is therefore a $\mathbb{Q}$-martingale, so (iii) $\Rightarrow$ (i). This proves the martingale characterization by bounded self-financing strategies.