Reflection Principle of Brownian Motion

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If the path of a Brownian Motion reaches a value 'a' at time t=s, the reflected Brownian Motion 2a - W(t) from that point and so on, has the exact probability of occurring as the original process. This is known as the Reflection Principle and we can observe it below:

Animation of the zero mean and infinite variance of the Geometric Brownian Motion
Animation: Reflection Principle of the Brownian Motion

The source code related this visualization can be found in this Jupyter Notebook. For more information, please follow me on Twitter.

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