Autocorrelation of a wiener process

In summary, the conversation discusses using a Brownian Walk or Wiener Process, represented by z(t)=e^{i\phi(t)}, to calculate E(z(t)z(t+\tau)). It is mentioned that E(e^{i\phi(t)})=e^{-\frac{1}{2}\sigma^{2}(t)}, where the mean is 0 and \sigma^{2}(t)=2Dt. The speaker is seeking assistance on how to proceed with this calculation.
  • #1
aimforclarity
33
0
Let [itex]\phi(t)[/itex] be a Brownian Walk (Wiener Process), where [itex]\phi\in[0,2\pi)[/itex]. As such we work with the variable [itex]z(t)=e^{i\phi(t)}[/itex]. I would like to calculate

[itex]E(z(t)z(t+\tau))[/itex]

This is equal to [itex]E(e^{i\phi(t)+i\phi(t+\tau)})[/itex] and I know that
[itex]E(e^{i\phi(t)})=e^{-\frac{1}{2}\sigma^{2}(t)}[/itex], where the mean is 0 and [itex]\sigma^{2}(t)=2Dt[/itex].
However, I have been stuck a week on how to proceed, any thoughts?

Thank you :)

Aim For Clarity
 
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  • #2
Hey aimforclarity.

Have you made an attempt to partition this whole thing in terms of independent processes so that you can calculate E[XY] = E[X]E[Y]?
 
  • #3
ok, so can I say [itex]<e^{\phi(t)+\phi(t+\tau)}>=<e^{\phi(t)+\phi(t)+\phi(\tau)}>[/itex] since [itex]W(t+\Delta t)=W(t)+W(\Delta t)[/itex]. Next we can say [itex]<e^{\phi(t)+\phi(t)+\phi(\tau)}>=<e^{2\phi(t)}><e^{\phi(\tau)}>[/itex] because [itex]<\phi(t)\phi(\tau)>=0[/itex]
 

FAQ: Autocorrelation of a wiener process

1. What is the definition of autocorrelation of a wiener process?

The autocorrelation of a wiener process is a measure of the linear relationship between the values of the process at different time points. It is a statistical concept commonly used in time series analysis to understand the degree of similarity or correlation between a process and its own past values.

2. How is autocorrelation calculated for a wiener process?

The autocorrelation function for a wiener process is calculated by taking the product of the values of the process at two different time points, and then averaging the results over all possible time intervals. This can be expressed as a mathematical formula using the covariance and variance of the process.

3. What does a high autocorrelation value indicate for a wiener process?

A high autocorrelation value for a wiener process indicates a strong linear relationship between the values at different time points. This means that the process is highly correlated with its own past values and is likely to exhibit similar behavior in the future. It can also suggest that the process is not random and may follow a specific pattern or trend.

4. How does autocorrelation affect the predictability of a wiener process?

The autocorrelation of a wiener process can greatly impact its predictability. A high autocorrelation value indicates that the process is not independent and its future values can be predicted based on its past values. On the other hand, a low autocorrelation value suggests that the process is more random and harder to predict.

5. Can autocorrelation be negative for a wiener process?

Yes, autocorrelation can be negative for a wiener process. This means that there is a negative linear relationship between the values at different time points. A negative autocorrelation value indicates that as the process increases, its past values tend to decrease, and vice versa. It is important to consider both positive and negative autocorrelation when analyzing a wiener process.

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