# Auto-correlation of a stochastic equation

• Tyler_D
In summary: W(t)>=0 and <W(t)W(t)>=t.In summary, the conversation discusses the calculation of the correlation function for two Wiener integrals, or Brownian motions, at different times. The conversation also mentions the periodic nature of the Wiener integral with a period of t, as well as the useful relations for calculating the expectation values of these integrals. Further steps for solving the equation involve Taylor expanding the exponentials and finding a nice expression for the expectation value.
Tyler_D
Homework Statement
Calculate the auto-correlation of ##x(t)=e^{-\beta W(t)}W(t)##
Relevant Equations
##\langle x(t)x(t+\tau)\rangle##, ##W(t)=\int_0^t dW(s)##
Here is my attempt:

\begin{align} \langle x(t)x(t+\tau) \rangle &= \langle (e^{-\beta W(t)}W(t))(e^{-\beta W(t+\tau)}W(t+\tau))\rangle \\ &= \langle (e^{-\beta W(t)}e^{-\beta W(t+\tau)})(W(t)W(t+\tau))\rangle \\ &= \langle (e^{-\beta W(t)}e^{-\beta W(t+\tau)})(\int_0^t dW(t)\int_0^{t+\tau}dW(t))\rangle \\ &= \langle (e^{-\beta W(t)}e^{-\beta W(t+\tau)})\int_0^t dW(t)(\int_0^{t}dW(t)+\int_t^{\tau}dW(t))\rangle \end{align}

From here on I am quite lost on what to do with the exponentials. Do I Taylor-expand them, or what is the best thing to do?

Is W(t) some special function? The Lambert W function maybe? That would make nice integrals.

It is the Wiener-integral/Brownian motion (my bad, should've written it in the OP):

##
W(t) = \int_0^t dW(s)
##

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I don't understand something, you wrote in Eq. (4) ##\int_0^{t+\tau}dW(s) = \int_0^t dW(s)+\int_t^{\tau}dW(s)##, but obviously this means that: ##\int_0^{t+\tau}dW(s)=\int_0^\tau dW(s)##.
Does this mean that Weiner integral is periodic with period ##t##, since then ##W(t+\tau)=W(\tau)##?

Last edited:
Tyler_D
That's also a typo, thanks for spotting it. The upper limit in the last integral should naturally be ##\tau+t##, not just ##\tau##.

## 1. What is auto-correlation and how is it related to a stochastic equation?

Auto-correlation refers to the relationship between a variable and its own past values. In the context of a stochastic equation, it measures the degree to which the current value of the variable is related to its previous values. This is important in understanding the behavior and predictability of the stochastic process.

## 2. How is auto-correlation calculated for a stochastic equation?

The auto-correlation for a stochastic equation is typically calculated using the autocovariance function, which is a measure of the covariance between a variable and its lagged values. This can be done using mathematical formulas or statistical software.

## 3. What factors can affect the auto-correlation of a stochastic equation?

The degree of randomness or volatility in the stochastic process can greatly affect its auto-correlation. Other factors such as the time interval between observations, the sample size, and the distribution of the data can also impact the auto-correlation.

## 4. How is auto-correlation useful in analyzing a stochastic equation?

Auto-correlation can provide insights into the patterns and trends of a stochastic process, allowing for better understanding and prediction of future values. It can also help identify if the process is stationary or if there are any underlying relationships between different variables in the equation.

## 5. Can auto-correlation be negative in a stochastic equation?

Yes, auto-correlation values can range from -1 to 1, with negative values indicating a negative relationship between the variable and its past values. This can occur in cases where the variable experiences oscillations or when there is a negative trend in the data.

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