Reinhold Schneider: “Solving Backward Stochastic Differential Equation & HJB equations with Tree...“

Tensor Methods and Emerging Applications to the Physical and Data Sciences 2021 Workshop I: Tensor Methods and their Applications in the Physical and Data Sciences “Solving Backward Stochastic Differential Equation and Hamilton Jacobi Bellmann (HJB) equations with Tree Based Tensor Networks (HT/TT)“ Reinhold Schneider - Technische Universität Berlin, Institut für Mathematik, FG Modellierung, Simulation & Optimierung Abstract: For many high dimensional PDEs of practical interest, e.g. Backward Kolmogorov equations etc. the PDE operator cannot be easily expanded in matrix product operator form. In this case, we propose a variational Monte Carlo approach confined to the manifold of tree based tensors with fixed multi-rank (i.e. bond dimensions). In particular the (stochastic) HJB can be reformulated by an (un-coupled) Forward Backward SDE system. The forward dynamics can be computed easily by standard Euler-Mayurana scheme. For the backward equation for the value function, we use variational interpolation (Ben