We propose a new, unified approach to solving jump-diffusion partial integro-differential equations (PIDEs) that often appear in mathematical finance. Our method consists of the following steps. First, a second-order operator splitting on financial processes (diffusion and jumps) is applied to these PIDEs. To solve the diffusion equation, we use standard finite-difference methods, which for multi-dimensional problems could also include splitting on various dimensions. For the jump part, we transform the jump integral into a pseudo-differential operator. Then for various jump models we show how to construct an appropriate first and second order approximation on a grid which supersets the grid that we used for the diffusion part. These approximations make the scheme to be unconditionally stable in time and preserve positivity of the solution which is computed either via a matrix exponential, or via P{á}de approximation of the matrix exponent. Various numerical experiments are provided to justify these results.
INEUS: Iterative Neural Solver for High-Dimensional PIDEs
Department of Mathematics and RiskLab, ETH Zurich（数学系和风险实验室，苏黎世联邦理工学院）；Applied Mathematics: Institute for Analysis and Numerics, University of Münster（应用数学：分析与数值计算研究所，蒙斯特大学）
