Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 2 / 30L (Stochastic Financial Models)

An adapted integrable process is a supermartingale when $E[X_n\mid {\mathcal{F}}_{n-1}]\le X_{n-1}$ almost surely for every $n$.

$E[X_{n+1}\mid {\mathcal{F}}_n]=X_n[pc+(1-p)c^{-1}]$. Thus the condition is $pc+(1-p)/c\le 1$, or $p{c}^2-c+1-p\le 0$. Since the roots are $(1-p)/p$ and $1$, the range is $(1-p)/p\le c\le 1$.

Conditional Jensen and monotonicity give $E[U(X_n)\mid {\mathcal{F}}_{n-1}]\le U(E[X_n\mid {\mathcal{F}}_{n-1}])\le U(X_{n-1})$.

Predictability gives $E[H_k(X_k-X_{k-1})]=E[H_{k}E(X_k-X_{k-1}\mid {\mathcal{F}}_{k-1})]\le 0$. Sum over $k\le n$.

Fix $A\in {\mathcal{F}}_{n-1}$ and choose the nonnegative predictable strategy with $H_n=1_A$ and all other holdings zero. The assumed inequality yields $E[1_A(X_n-X_{n-1})]\le 0$ for every $A$, equivalent to the supermartingale conditional inequality.