The relation between Brownian Motion and Quadratic Variation in layman’s terms…

Brownian motion, a core concept in stochastic processes, paints a picture of a world in constant, random motion. It’s akin to a particle dancing erratically, with each step independent of the last, creating a path that is continuous but, intriguingly, non-differentiable at every point. 

If a continuous function is differentiable at a specific time point, t, it indicates that we could ascertain the derivative by observing how the function acts just prior to that point, at some moment slightly before t. But this notion conflicts with the fundamental nature of Brownian motion because it suggests that the value slightly before t can give us insights into the value at t, essentially predicting it. We know this isn’t possible due to Brownian motion’s inherent unpredictability and independence at each point in time.

One of the key aspects to understanding this enigmatic motion lies in the concept of quadratic variation, a mathematical tool that quantifies the “roughness” of Brownian paths over time.

The perpetual dance of a particle undergoing Brownian motion creates a path defined by continuous, yet infinitely complex movements. At every point, the path jitters, refusing to be pinned down to a single direction. This leads to a phenomenon where, despite being continuous, the path is non-differentiable everywhere. Such behavior is a stark deviation from the smooth and predictable paths charted by regular calculus, necessitating a fresh toolkit to analyze and understand it.

Here is where the quadratic variation (*) steps in. It doesn’t seek to smooth out the path but embraces and quantifies its “roughness.” In the mysterious world of Brownian motion, the quadratic variation is not just a static measure; it increases linearly with time. 

Every tick of the clock amplifies the erratic dance, making the path more complex and less predictable, echoing the intrinsic randomness of this motion.

In the context of Geometric Brownian Motion (GBM), a model often applied to predict stock prices, the linear relationship between quadratic variation and time becomes pivotal. 

Though it borrows characteristics from Brownian motion, the GBM adds a drift component, introducing a directional pull. Yet, the non-differentiable nature prevails. The quadratic variation mirrors the increasing complexity, offering insights into the volatility and behavior of financial markets.

(*)
Quadratic variation measures the cumulative squared increments of a stochastic process, quantifying its "roughness" over a given time interval.

For more information about quadratic variation : https://lnkd.in/ehydJi-v

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