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I am currently considering a simple life-cycle problem. We consider a market with equity risk only, which follows a geometric Brownian motion. We seek to maximize the terminal wealth of a CRRA utility individual over the investment strategies $\omega_t$ subject to the corresponding wealth dynamics. Of course, this has a known closed form solution, however I am trying to use this simple example to understand the method of dynamic programming, or, numerical backward induction.

Firstly, I understand we assume values for wealth $W(t-1)$ and then use Guassian quadrature to evaluate the expected value of wealth in this time period. I am confused in the next step. Somehow, interpolation is used and I fail to understand the intuition.

If there is someone out there who is knowledgeable on this subject, I would be ever-grateful if you can explain to me the procedure of numerical backward induction in this simple case, using both guassian quadrature and interpolation.

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