Why is OU process stationary?
The mean and variance of Ornstein–Uhlenbeck (OU) process have time dependence (exponentially decay in time). So they are not constant in time. How can it to be stationary? Stationary means that the process does not depend on a specific time instant, but only on a time interval.
What does geometric Brownian motion do?
Geometric Brownian motion, and other stochastic processes constructed from it, are often used to model population growth, financial processes (such as the price of a stock over time), subject to random noise .
What is the mean reverting process?
Mean reversion is the process that describes that when the short-rate r is high, it will tend to be pulled back towards the long-term average level; when the rate is low, it will have an upward drift towards the average level. In Vasicek’s model the short-rate is pulled to a mean level b at a rate of a.
Is Brownian motion an ITO process?
An Ito process is a type of stochastic process described by Japanese mathematician Kiyoshi Itô, which can be written as the sum of the integral of a process over time and of another process over a Brownian motion.
What is a stochastic process?
A stochastic process is defined as a collection of random variables X={Xt:t∈T} defined on a common probability space, taking values in a common set S (the state space), and indexed by a set T, often either N or [0, ∞) and thought of as time (discrete or continuous respectively) (Oliver, 2009).
What are stochastic differential equations used for?
A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as stock prices or physical systems subject to thermal fluctuations.
What is the difference between Brownian motion and geometric Brownian motion?
The key distinguishing point among different Brownian motions is the different types of drift. If the drift is 0, it is standard BM. If the drift is constant, it is BM with constant drift. If the drift is linear, it is geometric BM.