An $\mathbb R^d$-valued process $(X_t)_{t\geq0}$ is a <Brownian motion> started at $x$ when:

* $X_0=x$ almost surely;
* its sample paths are almost surely continuous;
* increments over disjoint time intervals are independent; and
* for $0\leq s<t$, $X_t-X_s$ has the centered <multivariate normal distribution> with covariance matrix $(t-s)I_d$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 201 / 4 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-201/4/a/solution