A <Lévy process> $(X_t)_{t\geq0}$ starts at zero, has <independent random variables>[independent] and stationary increments, is <stochastic continuity>[stochastically continuous], and is taken with <càdlàg function>[càdlàg] sample paths. Thus for $0\leq t_0<\cdots<t_n$, the increments $X_{t_j}-X_{t_{j-1}}$ are independent, and the law of $X_{s+t}-X_s$ depends only on $t$.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 201 / 6 / a / Solution

Write $\mathbb E X_1=\mu$ and $\operatorname{Var}(X_1)=\sigma^2<\infty$. For rational $t=m/n$, stationarity and independence of the $n$ increments over intervals of length $1/n$ give
$$
\mathbb E X_t=t\mu,
\qquad
\operatorname{Var}(X_t)=t\sigma^2,
$$
using <variance additivity for independent random variables>. <Stochastic continuity> extends both identities from rational to real $t$. In the centered case $\mu=0$, this becomes $\mathbb E X_t=0$ and $\mathbb E X_t^2=t\sigma^2$.

Solved by gpt-5.6-sol high.

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 201 / 6 / b / Solution

{wiki=Stochastic_process}

A stochastic process $X$ is stochastically continuous when $X_s\to X_t$ in probability as $s\to t$.


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