Specify a stochastic difference equation for solving?

Questions
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1

I’ve been trying to get full solutions to time series difference equations with a stochastic component. But my syntax has been off.

Solving a basic time series autoregressive order 1 function AR(1):

y_t = a_0 + a_1y_{t-1} + \epsilon_t

How could I go about solving this where the output would be the particular + all homogenous solutions? Such that for the above function I could specify initial conditions whereby the solution output is:

y_t = a_0 \sum_{i=0}^{t-1}{a_i^i} + a_1^t y_0 + \sum_{i=0}^{t-1}{a_i^i} \epsilon_{t-i}

Or then for the same function without initial conditions but a_1<1:

y_t = \frac{a_0}{1-a_1} + \sum_{i=0}^{\infty}{a_1^i} \epsilon_{t-1} + Aa_1^t

Even if I could get the notation to incorporate time correctly that’d be a great help. I’ve been trying to use DifferenceDelta in combination with with DSolve but can’t get it to work. Of course, moving forward the problems would extend to higher order ARMA process’. But I imagine once the functional form is entered correctly I could extend it a similar fashion?

Any help or suggestions appreciated.

2

I might be misunderstanding your problem, but I guess you need RSolve[] and friends. Have you tried:

RSolve[y[n] == a0 + a1 y[n - 1] + c[n], y[n], n]

Regards – Henrik