Quadratic Variation (Stochastic Processes and Brownian Motion)

[spenghali]

Homework Statement

No specific problem to solve, just looking for a better explanation of the implications of the quadratic variation not being zero in Brownian motion. Why is this so important in the study of stochastic calculus and Brownian motion? I understand that quadratic variation in usually zero in regular calculus. Also, what does the quadratic variation actually tell us in layman terms in both stochastic and regular calculus? I think the side-by-side comparison should help. Thanks for any responses/help.

Homework Equations

The Attempt at a Solution

P.s. does this lead to volatility in stock price models?

 

[lippyka]

In regular calculus, the quadratic variation of a function tells us how much the function changes over a period of time. In stochastic calculus and Brownian motion, the quadratic variation is not necessarily zero. This is important because it implies that the process is not a deterministic one, but rather a random one. In other words, the quadratic variation tells us how much the process changes over a period of time, but it also tells us that the changes are not predictable or consistent. This has implications in financial models, such as stock price models, where volatility is an important factor. The quadratic variation helps us to understand the magnitude of the volatility, and therefore gives us some insight into the behavior of the stock prices.