Mohamed Mnif: Deep Backward Schemes for High-Dimensional Portfolio Optimization Problem

Abstract: We consider the problem of portfolio choice in high dimension for an investor who wants to maximize the utility of his terminal wealth. For solving the Hamilton Jacobi Bellman Equation we relate our problem to a backward stochastic differential equation which is solved by two algorithms based on deep neural networks. At each time step from, the optimal portfolio is first estimated with a first neural network. Then we minimize a loss function defined recursively by backward induction estimating the solution and its gradient separately in the first algorithm and simultaneously in the second. We provide error estimates in terms of the universal approximation of neural networks and we compare the numerical results with a direct algorithm still using neural networks but estimating the control in a single optimization maximizing the expectation of the terminal wealth.