Wiener Process

A Wiener process, named after the mathematician Norbert Wiener, is a type of stochastic process that is fundamental to the theory of stochastic calculus and has significant applications in fields like mathematics, finance, and physics. It is also known as Brownian motion, after the botanist Robert Brown, who first observed the phenomenon in pollen grains in water.

Key characteristics of a Wiener process are:

1. Continuous Path: Unlike some other stochastic processes, a Wiener process has continuous paths. This means that its graphical representation is a continuous curve with no jumps or breaks.

2. Starts at Zero: By definition, a Wiener process starts at zero, $W(0) = 0$.

3. Independent Increments: The increments of the process are independent for non-overlapping time intervals. In other words, how the process moves in one time interval does not affect how it moves in another. Mathematically,
   for every $t>0, W_{t+u}-W_t, u \geq 0$, are independent of the past values $W_s, s<t$.

4. Normally Distributed Increments: The increments of the process over any time interval are normally distributed. The increment of the Wiener process over a time interval of length $(u)$ is normally distributed with mean $0$ and variance $(u)$. Mathematically, $(W_{t+u}-W_t) \sim N(0,u)$.

5. Scale Invariance: If $W(t)$ is a Wiener process, then for any positive constant $(c)$, $cW(t/c^2)$ is also a Wiener process. This property is known as self-similarity.

6. No Memory: Reflecting its independent increments, a Wiener process is a Markov process, which means that its future behavior is independent of its past, given its present state.

In finance, the Wiener process is used to model the random behavior of variables like stock prices and interest rates in the Black-Scholes model and other mathematical models. In physics, Brownian motion describes the random movement of particles suspended in a fluid.