Recursive representation of finite product in infinite sum

[schoderch]

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

[jpfeifer]

Please check the attached calculation

[schoderch]

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

[jpfeifer]

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.