preprints.org > business, economics and management > econometrics and statistics > doi: 10.20944/preprints202304.0369.v1

Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Version 1 : Received: 14 April 2023 / Approved: 14 April 2023 / Online: 14 April 2023 (10:44:44 CEST)

Many problems in the fields of finance and actuarial science can be transformed into the problem of solving backward stochastic differential equations (BSDE) and partial differential equations (PDE) with jumps, which are often difficult to solve in high-dimensional cases. To solve this problem, this paper applies the deep learning algorithm to solve a class of high-dimensional nonlinear partial differential equations with jump terms and their corresponding backward stochastic differential equations (BSDE) with jump terms. Using the nonlinear Feynman-Kac formula, the problem of solving this kind of PDE is transformed into the problem of solving the corresponding backward stochastic differential equations with jump terms, and the numerical solution problem is turned into a stochastic control problem. At the same time, the gradient and jump process of the unknown solution are separately regarded as the strategy function and they are approximated respectively by using two multilayer neural networks as function approximators. Thus, deep learning-based method is used to overcome the “curse of dimensionality” caused by high-dimensional PDE with jump, and the numerical solution is obtained. In addition, this paper proposes a new optimization algorithm based on the existing neural network random optimization algorithm, and compares the results with the traditional optimization algorithm, and achieves good results. Finally, the proposed method is applied to three practical high-dimensional problems: Hamilton-Jacobi-Bellman equation, bond pricing under the jump Vasicek model and high-dimensional option pricing model with default risk. The proposed numerical method has obtained satisfactory accuracy and efficiency. The method has important application value and practical significance in investment decision-making, option pricing, insurance and other fields.

Deep learning; Backward stochastic differential equation; Nonlinear Feynman-Kac formula; High dimensional PDE; Derivatives pricing; Neural network

Business, Economics and Management, Econometrics and Statistics

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