What is a Wiener Process?

The Wiener Process, also known as Brownian Motion, is one of the most fundamental stochastic processes used in financial mathematics. 

It is a continuous-time stochastic process that describes random motion over time. It was originally developed to model the physical phenomenon of particle movement in a fluid, but it has become a cornerstone for modeling the randomness in stock prices, interest rates, and other financial variables. 


A stochastic process \( W(t) \), \( t \geq 0 \), is called a Wiener Process if it satisfies the following properties:


 1. Initial Value: \( W(0) = 0 \).

 2. \( W(t) \) has independent increments. 

For any \( 0 \leq t_1 < t_2 \):

The increment \( W(t_2) - W(t_1) \) is independent of the past values of \( W(s) \), \( s \leq t_1 \).

3. \( W(t) \) has normally distributed increments. 

For any \( 0 \leq t_1 < t_2 \):
 \( W(t_2) - W(t_1) \sim N(0, t_2 - t_1) \)

4. \( W(t) \) has continuous paths: the function \( t \mapsto W(t) \) is continuous almost surely.

Properties of the Wiener Process

  • Expected Value:

    E[W(t)]=0
    The average value of W(t) over many realizations is zero.

  • Variance Grows Linearly with Time:

    Var[W(t)]=t
    As time increases, the uncertainty (variance) grows proportionally.

  • Markov Property:

    The Markov Property is a fundamental characteristic of certain stochastic processes. It states that the future state of a process depends only on its current state and not on the sequence of states that preceded it. In other words, the process has no memory of the past—only the present matters for predicting the future.

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