Machine learning approximation algorithms for highdimensional fully nonlinear partial differential equations and secondorder backward stochastic differential equations
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Machine learning approximation algorithms for highdimensional fully nonlinear partial differential equations and secondorder backward stochastic differential equations
Highdimensional partial differential equations (PDE) appear in a number of models from the financial industry, such as in derivative pricing models, credit valuation adjustment (CVA) models, or portfolio optimization models. The PDEs in such applications are highdimensional as the dimension corresponds to the number of financial assets in a portfolio. Moreover, such PDEs are often fully nonlinear due to the need to incorporate certain nonlinear phenomena in the model such as default risks, transaction costs, volatility uncertainty (Knightian uncertainty), or trading constraints in the model. Such highdimensional fully nonlinear PDEs are exceedingly difficult to solve as the computational effort for standard approximation methods grows exponentially with the dimension. In this work we propose a new method for solving highdimensional fully nonlinear secondorder PDEs. Our method can in particular be used to sample from highdimensional nonlinear expectations. The method is based on (i) a connection between fully nonlinear secondorder PDEs and secondorder backward stochastic differential equations (2BSDEs), (ii) a merged formulation of the PDE and the 2BSDE problem, (iii) a temporal forward discretization of the 2BSDE and a spatial approximation via deep neural nets, and (iv) a stochastic gradient descenttype optimization procedure. Numerical results obtained using ${rm T{small ENSOR}F{small LOW}}$ in ${rm P{small YTHON}}$ illustrate the efficiency and the accuracy of the method in the cases of a $100$dimensional BlackScholesBarenblatt equation, a $100$dimensional HamiltonJacobiBellman equation, and a nonlinear expectation of a $ 100 $dimensional $ G $Brownian motion.
Machine learning approximation algorithms for highdimensional fully nonlinear partial differential equations and secondorder backward stochastic differential equations
by Christian Beck, Weinan E, Arnulf Jentzen
https://arxiv.org/pdf/1709.05963v1.pdf
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