Recursive representation of finite product in infinite sum

This may be a little off topic but if someone has some advice I would be very grateful.

I would like to find a recursive representation of the following expression (in latex syntax):

\kappa_t = \sum_{n=0}^\infty ( a^n * \prod_{k=0}^n x_{t+k} )

Is that actually possible? Thank you very much for you thoughts.

Best, Christian

Please check the attached calculation

Dear Johannes, thank you very much for your time!

I looked at your solution and I think it is not entirely correct. Nevertheless I was able to solve it following your approach. The solution (in latex syntax) is \kappa_t=x_t(1+a\kappa_{t+1}) which I believe is correct (see pdf).

Unfortunately, my model does not collapse to the problem I have initially stated and things get more complicated and now I am again not able to derive a recursive representation (if it even exists). The equation reads

\kappa_t = \sum_{n=0}^\infty ( a^n \frac{ \text{E}t \prod{k=0}^n R_{t+k}^{-1}C_{t+n}]} {\text{E}t \prod{k=0}^n R_{t+k}C_{t+n}^{-1}]} )

where \text{E}_t is the expectation operator. I have also created a pdf attached.

Again I would be very grateful if someone was willing to have a look at the problem. Thank you so much!

Best, Christian
recursive.pdf (12.8 KB)

Try to split the problem by defining separate auxilary variables for the numerate and denominator and finding recursive laws of motion. I don’t know if this is possible but that would be the way to go.