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A Deep Learning Scheme for Solving Fully Nonlinear Partial Differential Equation

In: Peter Carr Gedenkschrift Research Advances in Mathematical Finance

Author

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  • Maxim Bichuch
  • Ke Chen

Abstract

We study the convergence of a deep learning algorithm applied to a general class of fully nonlinear second-order partial differential equations. By using a suitable finite difference approximation to the loss function of the deep learning scheme, we show the convergence of the numerical solution to the unique viscosity solution. We apply our results and illustrate this convergence to the finite horizon optimal investment problem with proportional transaction costs in single and multi-asset settings.

Suggested Citation

  • Maxim Bichuch & Ke Chen, 2023. "A Deep Learning Scheme for Solving Fully Nonlinear Partial Differential Equation," World Scientific Book Chapters, in: Robert A Jarrow & Dilip B Madan (ed.), Peter Carr Gedenkschrift Research Advances in Mathematical Finance, chapter 4, pages 101-140, World Scientific Publishing Co. Pte. Ltd..
    • Handle: RePEc:wsi:wschap:9789811280306_0004
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    Keywords

    Mathematical Finance; Quantitative Finance; Option Pricing; Derivatives; No Arbitrage; Asset Price Bubbles; Asset Pricing; Equilibrium; Volatility; Diffusion Processes; Jump Processes; Stochastic Integration; Trading Strategies; Portfolio Theory; Optimization; Securities; Bonds; Commodities; Futures;
    All these keywords.

    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • C6 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling

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