Deep learning techniques in portfolio optimization under constraints

Abstract

We consider the constrained utility maximization problem and the corresponding dual problem with regard to theoretical results which allow the formulation of algorithmic solvers which make use of deep learning techniques. We place great emphasis on detailed proofs of the underlying theoretical results. At first, the deep controlled 2BSDE algorithm from [5] is derived in a Markovian setting. It combines the dynamic programming approach with the adjoint equation from the stochastic maximum principle (SMP). In the case of random coefficients, we prove stochastic maximum principles for the primal and the dual problem, respectively. Furthermore, we show that the strong duality property holds under additional assumptions. This leads to the formulation of the deep SMP algorithm as in [5]. Moreover, we use the aforementioned result for the primal problem for defining a new algorithm, which we call deep primal SMP algorithm. Numerical examples illustrate the effectiveness of the studied algorithms - in particular for higher-dimensional problems and problems with random coefficients, which are either path dependent or satisfy their own SDEs. Moreover, our numerical experiments for constrained problems show that the novel deep primal SMP algorithm overcomes the deep SMP algorithm's weakness of erroneously producing the value of the corresponding unconstrained problem. Furthermore, in contrast to the deep controlled 2BSDE algorithm, this algorithm is also applicable to problems with path dependent coefficients. As the deep primal SMP algorithm even yields the most accurate results in many of our studied problems, we can highly recommend its usage.