Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 29K / a

Between times $n-1$ and $n$, the investor holds ${\theta }_n$ shares worth ${\theta }_n{S}_{n-1}$ and places the remaining wealth $X_{n-1}-{\theta }_n{S}_{n-1}$ in the risk-free asset. The self-financing condition therefore gives