Quadratic variation is a measure of the accumulated variability of a stochastic process that takes into account the square of the increments. It is especially important in the analysis of financial markets, primarily for processes with continuous paths like Brownian motion, where it distinguishes between finite variances and increments that converge in quadratic mean.

Quadratic Variation

Quadratic variation

accumulated variability

A stochastic process is a collection of random variables representing the evolution of a system over time, where the future state depends on both the present state and inherent randomness. It is widely used in fields like finance, physics, and biology to model phenomena that evolve unpredictably over time.

stochastic process

square of the increments

Financial markets are platforms where individuals and institutions can trade financial securities, commodities, and other fungible items at prices that reflect supply and demand. They play a crucial role in the economy by facilitating liquidity, price discovery, and the efficient allocation of resources.

financial markets

continuous paths

Brownian motion is the random movement of particles suspended in a fluid, resulting from their collision with the fast-moving molecules in the fluid. It is a fundamental concept in statistical physics and serves as a mathematical model for stochastic processes, with applications in fields such as finance, biology, and physics.

Brownian motion

finite variances

increments that converge

The quadratic mean, also known as the root mean square (RMS), is a statistical measure of the magnitude of a set of numbers, calculated as the square root of the average of the squares of the numbers. It is particularly useful in contexts where values can be positive or negative, such as in physics and engineering, providing a measure of the average magnitude irrespective of direction.

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