Suppose a variable 𝑧𝑧 follows a Wiener process if it has the following twoproperties

The Wiener process, also known as Brownian motion, is a mathematical model that describes the random movement of particles suspended in a fluid. It possesses two key properties that define its behavior:

1. Continuity: The sample paths of the Wiener process are continuous. This means that the value of the process at any point in time can be approached as closely as desired by the values at nearby points in time. Mathematically, if $z(t)$ is the Wiener process, then for any $t$, as $h$ approaches $0$, $z(t+h)$ approaches $z(t)$.

2. Independent Increments: The increments of the process over non-overlapping intervals are independent of each other. This implies that for $0 \leq s < t$, the difference $z(t) - z(s)$ is independent of the past values of the process up to time $s$. Furthermore, the increments $z(t_2) - z(t_1), z(t_3) - z(t_2), \ldots$ are independent for times $t_1 < t_2 < t_3 < \ldots$.

These properties enable the Wiener process to model various phenomena in fields such as physics, finance, and other areas requiring stochastic processes.