Simulating random walk with "known" prediction Suppose a random walk that looks a bit like this
set.seed(420)
x=rnorm(1000)
y=rep(NA,length(x))
y[1]=x[1]
for (i in 2:length(x)) {
  y[i]=y[i-1]+x[i]*0.7
}


But it is not a real random walk like that, and it 
is just something which is very hard to predict (think weather). first 700 points are actual data, the other 300 are predictions.
The predictions are assumed to be "true" on average, they should follow the given path. However, we suspect that the path will be a lot more variable than predicted, there will be subintervals of higher and/or lower values, they shouldn't jump up or down too strongly.
How can we simulate some random walks, that will oscillate around the predictions (having a sort of similar mean) but with arbitrary variability. We suspect there will be subintervals with higher correlation and so it will take longer here to return to the mean. This to get a better idea of possible paths / scenarios during this time.
 A: The specification of the sort of data that you want to generate is very broad, and there are many ways to simulate sort of random walks with paths that have a tendency to 'return to the mean'.
For instance, you can change your code like:
set.seed(420)
a = 0.9
b = 0.7
n = 10^4
x=rnorm(n)
y=rep(NA,n)
y[1]=x[1]
for (i in 2:n) {
  y[i]=a*y[i-1]+b*x[i]
}

Which is a damped random walk. 
related question (and possibly duplicate): Creating auto-correlated random values in R
more similar types of random walks: https://en.wikipedia.org/wiki/Autoregressive-moving-average_model
A: A random walk process that you're using has a constant (or zero) drift and variance:
$$dW_t=\xi_t,\\\xi_t\sim \mathcal N(0,\sigma^2)$$ 
You want "arbitrary variablility", i.e. $\sigma^2$ is not only changing with time, but also in some arbitrary way, whatever you meant by this word. If you meant that it's stochastic, unpredictable, then maybe you need to look at stochastic variance processes where $\sigma^2_t$ is a random process itself. One such model is Heston model, which is popular in derivative pricing in finance. 
There are simpler models such as GARCH. 
In GARCH the variance is not stochastic in sense that the variance of next step is completely determined by information to date. However, since you constantly get new information, the future variance gets updated at every step. So the variance is also arbitrary albeit in a narrower sense compared to the stochastic volatility process such as Heston.
