Table of Contents
What is quadratic variation in stochastic process?
Definition 4 (Quadratic Variation) The quadratic variation of a stochastic process, Xt, is equal to the limit of Qn(T) as ∆t := maxi(ti − ti−1) → 0. Theorem 1 The quadratic variation of a Brownian motion is equal to T with probability 1.
Is quadratic variation variance?
Quadratic variation and variance are two different concepts. Let X be an Ito process and t≥0. Variance of Xt is a deterministic quantity where as quadratic variation at time t that you denoted by [X,X]t is a random variable.
What is the quadratic variation equation?
The quadratic variation is alternatively given by [X]=[X,X] [ X ] = [ X , X ] , and the covariation can be written in terms of the quadratic variation by the polarization identity, [X,Y]=([X+Y]−[X−Y])/4.
What is meant by the term stochastic process?
A stochastic process means that one has a system for which there are observations at certain times, and that the outcome, that is, the observed value at each time is a random variable.
What is variation function?
A variation function is a function in which the variables are related by how they change in relation to each other. For instance, in this function, if x increases or decreases then D does the same. There are two types of variation functions, direct and inverse variation functions.
What are the different stochastic processes?
Based on their mathematical properties, stochastic processes can be grouped into various categories, which include random walks, martingales, Markov processes, Lévy processes, Gaussian processes, random fields, renewal processes, and branching processes.
What is stochastic process example?
Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. Examples include the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule.
How is stochastic calculus different?
The fundamental difference between stochastic calculus and ordinary calculus is that stochastic calculus allows the derivative to have a random component determined by a Brownian motion. The derivative of a random variable has both a deterministic component and a random component, which is normally distributed.