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We analyze a fixed-point algorithm for reinforcement learning (RL) of optimalportfolio mean-variance preferences in the setting of multivariate generalizedautoregressive conditional-heteroskedasticity (MGARCH) with a small penalty ontrading. A numerical solution is obtained using a neural network (NN)architecture within a recursive RL loop. A fixed-point theorem proves that NNapproximation error has a big-oh bound that we can reduce by increasing thenumber of NN parameters. The functional form of the trading penalty has aparameter $\epsilon>0$ that controls the magnitude of transaction costs. When$\epsilon$ is small, we can implement an NN algorithm based on the expansion ofthe solution in powers of $\epsilon$. This expansion has a base term equal to amyopic solution with an explicit form, and a first-order correction term thatwe compute in the RL loop. Our expansion-based algorithm is stable, allows forfast computation, and outputs a solution that shows positive testingperformance.

A Deep Neural Network Algorithm for Linear-Quadratic Portfolio Optimization with MGARCH and Small Transaction Costs