Time invarient pdf but nonstationary process

In summary, a general stochastic differential equation (SDE) does not always have a time independent probability density function (pdf) while the stochastic process Xt is nonstationary. However, it is possible to have a stochastic process with a constant pdf but a correlation function that is dependent on both values of the independent variable (time). An example of this is a Gaussian process with a correlation function of f(s,t)=1/(1+|s2-t2|). The correlation can be computed by plugging in values for s and t into the equation.
  • #1
shifo79
3
0
I am wondering if there exist some solution to the general stochastic differential equation (SDE) such that I get a time independent pdf(x) while the stochastic process Xt is nonstationary.. I really need some help with that..
 
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  • #2
I am not sure what you mean by a general stochastic diff. eq. However, it is possible to have a stochastic process with a constant distribution function, but where the correlation function is dependent on both values of the independent (time) variable, and not just the difference - therefore not stationary.
 
  • #3
OK, forget about the SDE .. can u give me example of a stochastic process such that the pdf (dosen't depend on time == > dp/dt=0) but the correlation has a time variable (nonstationary)? this will helpso much..
 
  • #4
Gaussian process (mean=0, s.d=1) with a correl. dep. on both variables. For example f(s,t)=1/(1+|s2-t2|).
 
  • #5
what's f(s,t)..
can u please tell me how to compute this correlation? if the pdf has not time in it, how come time appears in the correlation function?
 

1. What is a time-invariant PDF but nonstationary process?

A time-invariant PDF but nonstationary process is a type of stochastic process where the probability distribution function (PDF) stays the same over time, but the process itself changes and is not stationary. This means that the mean, variance, and autocorrelation of the process can vary over time, making it nonstationary.

2. How is a time-invariant PDF but nonstationary process different from a stationary process?

A stationary process has a constant PDF and its statistical properties, such as mean, variance, and autocorrelation, do not change over time. In contrast, a time-invariant PDF but nonstationary process has a constant PDF, but its statistical properties vary over time.

3. What are some examples of time-invariant PDF but nonstationary processes?

Some examples of time-invariant PDF but nonstationary processes include stock prices, weather patterns, and human heart rate. These processes have a constant PDF, but their statistical properties can change over time due to various factors such as market trends, climate change, or physical exertion.

4. How is the concept of time-invariant PDF but nonstationary process important in science?

The concept of time-invariant PDF but nonstationary process is important in science because it helps us understand and analyze complex systems that exhibit randomness and variability over time. By studying these processes, scientists can make predictions and identify patterns in various fields such as economics, physics, and biology.

5. Can a time-invariant PDF but nonstationary process be transformed into a stationary process?

Yes, a time-invariant PDF but nonstationary process can be transformed into a stationary process through a process known as differencing. Differencing involves taking the difference between consecutive observations of a nonstationary process, which can make the process stationary and easier to analyze using traditional statistical methods.

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