Deep Learning Approach to Solve Cryptocurrency Lucas Tree Model

Abstract

Solving continuous-time macroeconomic finance models is very challenging. Especially when modeling the dynamics of multiple assets, state variables can easily lead to the curse of dimensionality once their number increases slightly. On the other hand, without appropriate boundary conditions, traditional methods for solving such models face significant challenges in convergence (for example, using the iterative method provided by Brunnermeier and Sannikov (2016)). This paper proposes a two Lucas trees model, consisting of a crypto tree. A key feature of this crypto tree is its dividends are not exogenous but tied to the value of the crypto tree itself. It's an abstraction of Bitcoin in reality: Bitcoin's total supply is tightly capped, with new coins periodically mined. The value of newly mined coins is ultimately determined by the market value of Bitcoin. Another tree produces dividends with follows OU process As a probabilistic method for solving PDEs, Deep BSDE has two key advantages. First, it bypasses the complexity of finite difference methods. Second, it is not highly sensitive to boundary conditions, allowing one to solve the PDE within a region of the state space that has a high probability(mass), which is helpful to solve this crypto tree model.