# Generating time-correlated random noise with same start point

I'm trying to find an algorithm to generate time correlated random noise $\eta(t)$ (where $t$ is time, which I discretize for practical purposes) such that

$$\eta(0) = 0$$ $$\langle \eta(t), \eta(t+\Delta t)\rangle = \exp\left(-\frac{\Delta t}{\tau}\right)$$

for some correlation time $\tau$.

Attempt: I've had success generating the correlated random noise without the condition $\eta_i(0) = 0$ (i.e. starting at the same intensity) using Cholesky decomposition (i.e. cholcov in Matlab). I naïvely tried to set the (1,1) entry of the covariance matrix to zero to impose the condition, but that resulted in a not-positive-semi-definite matrix.

Take the autoregressive process $X_t = \rho X_{t-1} + \epsilon_t$ where $\epsilon_t$ is white noise with mean zero and variance $\sigma^2$. We will also assume $\rho < 1$. This process will be wide sense stationary (i.e. constant mean, covariance only depends on time differences).
Assuming $X_0 = 0$, then the process has mean zero for all time. The covariance $E[X_t X_{t+n}] = \frac{\sigma^2}{1-\rho^2} \rho^{|n|}$. You can set $\sigma^2, \rho$ to match the required covariance function as you need above.
• Thanks. To make the $\eta(t)$ normally distributed, would it suffice to make the $\epsilon_t$ normally distributed? – Dwagg Jul 3 '17 at 21:14
• If $\epsilon_t$ is i.i.d. Gaussian, then $X_t$ will also be a Gaussian process. – Batman Jul 3 '17 at 22:19