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What is the variance of a random walk process?
The variance of a random variable X is defined as var[X] = E[(X −E[X])2]. In other words, on average, what is the square of your distance to the expectation.
Is a random walk mean reverting?
The time it takes to revert is often referred to as the time to reversion. If the process is very persistent, it might take a long time to revert to the mean. The key difference between a mean-reverting process and a random-walk is that after the shock, the random-walk price process does not return to the old level.
Why does the variance of the random walk increase with time?
If we extend this example to the random walk, we can see that the variance increases with time, even though the mean stays at 0. In the random walk case, it seems strange that the mean stays at 0, even though you will intuitively know that it almost never ends up at the origin exactly.
What is the probability of the random walk?
For instance, each time step the random walk takes a step ± 1 with equal probability p = 0.5. This is equivalent to taking each two time steps a step ± 2 with equal probability p = 0.25, and staying in place with probability p = 0.5. Then The normal distribution with σ ∝ √t satisfies the above equation process.
How is mean reversion tested in a time series?
Mathematically, the ADF is based on the idea of testing for the presence of a unit root in an autoregressive time series sample. It makes use of the fact that if a price series possesses mean reversion, then the next price level will be proportional to the current price level.
Which is the stochastic equation for mean reversion?
A continuous mean-reverting time series can be represented by an Ornstein-Uhlenbeck stochastic differential equation: d x t = θ (μ − x t) d t + σ d W t Where θ is the rate of reversion to the mean, μ is the mean value of the process, σ is the variance of the process and W t is a Wiener Process or Brownian Motion.