Quadratic variation

In mathematics, quadratic variation is a way to measure how much a process changes over time, especially when those changes are random. It is mainly used to study stochastic processes like Brownian motion, which is a model for random movement, and other martingales. These processes are important in fields like finance and physics because they help us understand and predict random behavior.

Quadratic variation is just one type of variation that can be calculated for a process. It looks at the squares of the changes that happen over small time intervals and adds them up. This helps mathematicians and scientists understand the overall variability and volatility of the process, even when the changes are very quick and unpredictable.

Because quadratic variation deals with random changes, it is a key tool in the study of randomness and uncertainty. It helps experts model real-world situations where things change in complex, unpredictable ways, such as stock market prices or the physical movement of tiny particles.

Definition

Quadratic variation is a way to measure how much a process, like Brownian motion, changes over time. Imagine you track the process at many points and look at the squares of the tiny changes between each point. If you add up these squares and make the time steps smaller and smaller, the total might approach a specific value. This value is the quadratic variation.

We can also measure how two different processes change together, called their covariation. This is done by multiplying the tiny changes of each process and adding them up, similar to quadratic variation. Quadratic variation helps us understand the behavior of processes that change in complex ways.

Finite variation processes

A process has finite variation if it changes by only small amounts over any short period of time. Many common processes, like smoothly drawn curves, fall into this category. For these processes, the quadratic variation — a way to measure how much the process jumps around — is always zero.

For processes that can have sudden jumps, the quadratic variation is simply the sum of the squares of all those jumps. This helps us understand how much the process changes in a more detailed way. Another related idea, called predictable quadratic variation, is used for certain types of processes that change in a way that can be forecasted, and it also helps describe these changes.

Main article: Doob–Meyer decomposition theorem