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High-dimensional partial differential equations (PDE) appear in a number ofmodels from the financial industry, such as in derivative pricing models,credit valuation adjustment (CVA) models, or portfolio optimization models. ThePDEs in such applications are high-dimensional as the dimension corresponds tothe number of financial assets in a portfolio. Moreover, such PDEs are oftenfully nonlinear due to the need to incorporate certain nonlinear phenomena inthe model such as default risks, transaction costs, volatility uncertainty(Knightian uncertainty), or trading constraints in the model. Suchhigh-dimensional fully nonlinear PDEs are exceedingly difficult to solve as thecomputational effort for standard approximation methods grows exponentiallywith the dimension. In this work we propose a new method for solvinghigh-dimensional fully nonlinear second-order PDEs. Our method can inparticular be used to sample from high-dimensional nonlinear expectations. Themethod is based on (i) a connection between fully nonlinear second-order PDEsand second-order backward stochastic differential equations (2BSDEs), (ii) amerged formulation of the PDE and the 2BSDE problem, (iii) a temporal forwarddiscretization of the 2BSDE and a spatial approximation via deep neural nets,and (iv) a stochastic gradient descent-type optimization procedure. Numericalresults obtained using ${\rm T{\small ENSOR}F{\small LOW}}$ in ${\rm P{\smallYTHON}}$ illustrate the efficiency and the accuracy of the method in the casesof a $100$-dimensional Black-Scholes-Barenblatt equation, a $100$-dimensionalHamilton-Jacobi-Bellman equation, and a nonlinear expectation of a $ 100$-dimensional $ G $-Brownian motion.

journal of nonlinear science

Machine learning approximation algorithms for high-dimensional fully  nonlinear partial differential equations and second-order backward stochastic  differential equations