The One Step Malliavin scheme: new discretization of BSDEs implemented with deep learning regressions

Negyesi, Balint; Andersson, Kristoffer; Oosterlee, Cornelis W.

doi:10.48550/arxiv.2110.05421

Cited by 3 publications

(3 citation statements)

References 35 publications

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“…As for the LSMC method, these kinds of algorithms are not easily applied to coupled FBSDE. Algorithms of this type can be found in e.g., [42,41,43] and with error analysis [44]. For an overview of machine learning algorithms for approximation of PDE we refer to [45].…”

Section: Introductionmentioning

confidence: 99%

Convergence of a robust deep FBSDE method for stochastic control

Andersson¹,

Andersson²,

Oosterlee³

2022

Preprint

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In this paper we propose a deep learning based numerical scheme for strongly coupled FBSDE, stemming from stochastic control. It is a modification of the deep BSDE method in which the initial value to the backward equation is not a free parameter, and with a new loss function being the weighted sum of the cost of the control problem, and a variance term which coincides with the means square error in the terminal condition. We show by a numerical example that a direct extension of the classical deep BSDE method to FBSDE, fails for a simple linear-quadratic control problem, and motivate why the new method works. Under regularity and boundedness assumptions on the exact controls of time continuous and time discrete control problems we provide an error analysis for our method. We show empirically that the method converges for three different problems, one being the one that failed for a direct extension of the deep BSDE method.

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Section: Introductionmentioning

confidence: 99%

Convergence of a robust deep FBSDE method for stochastic control

Andersson¹,

Andersson²,

Oosterlee³

2022

Preprint

Self Cite

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“…Our second method, called Pathwise differential learning, considers furthermore the derivative of this martingale representation, computed by automatic differentiation, which gives another loss function to be minimized in order to train neural networks for approximating the value function and its first and second derivatives. Such differential representation has been also considered in the recent paper [22] for designing a deep learning scheme with one-step loss functions as in the deep backward approach in [16] for solving forward backward SDEs with new estimation and error control of the Z process. Actually, the addition of this derivative loss function permits a better approximation of the terminal condition of the PDE and improves the overall approximation of the PDE solution's value and derivatives on the entire domain.…”

Section: Introductionmentioning

confidence: 99%

Differential learning methods for solving fully nonlinear PDEs

Lefebvre¹,

Loeper²,

Pham³

2022

Preprint

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We propose machine learning methods for solving fully nonlinear partial differential equations (PDEs) with convex Hamiltonian. Our algorithms are conducted in two steps. First the PDE is rewritten in its dual stochastic control representation form, and the corresponding optimal feedback control is estimated using a neural network. Next, three different methods are presented to approximate the associated value function, i.e., the solution of the initial PDE, on the entire space-time domain of interest. The proposed deep learning algorithms rely on various loss functions obtained either from regression or pathwise versions of the martingale representation and its differential relation, and compute simultaneously the solution and its derivatives. Compared to existing methods, the addition of a differential loss function associated to the gradient, and augmented training sets with Malliavin derivatives of the forward process, yields a better estimation of the PDE's solution derivatives, in particular of the second derivative, which is usually difficult to approximate. Furthermore, we leverage our methods to design algorithms for solving families of PDEs when varying terminal condition (e.g. option payoff in the context of mathematical finance) by means of the class of DeepOnet neural networks aiming to approximate functional operators. Numerical tests illustrate the accuracy of our methods on the resolution of a fully nonlinear PDE associated to the pricing of options with linear market impact, and on the Merton portfolio selection problem.

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“…As for the LSMC method, these kinds of algorithms are not easily applied to coupled FBSDEs. Algorithms of this type can be found in e.g., [94,95,96] and with error analysis [97,98]. For an overview of machine learning algorithms for approximation of PDEs, we refer to [99].…”

Section: Introductionmentioning

confidence: 99%

Neural networks for stochastic control and decision making in mathematical finance

Andersson

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As the title suggests, this thesis is devoted to machine learning algorithms for stochastic control with applications in mathematical finance. In the first two chapters, we utilize neural networks to find the optimal exercise strategies of high-dimensional American options. Moreover, we compute option values along stochastic trajectories of the underlying assets, which is crucial in modern risk management. In Chapter 3, we approximate strongly coupled FBSDEs with a sequence of neural networks. More specifically, we are concerend with high-dimensional FBSDEs stemming from stochastic optimal control problems. These equations are inherently difficult to approximate due the coupling, and most classical methods as well as modern day machine learning algorithms break down. Our approach is novel in the sense that it combines mathematical structure from the initial control problem with the associate FBSDE which is shown to be both robust and accurate for a wide range of problems. In the final chapter, we consider time-inconsistent portfolio optimization problems. In general, for time-inconsistent problems the dynamic programming principle does not hold and in turn, the global problem cannot be divided into local sub-problems. Instead, we take on a fully data driven neural network-based approach to approximate the optimal allocation strategy directly. Beyond the classical instruments stocks and bonds, we also allow the investor to buy and sell options. In this way, we increase the flexibility in shaping the terminal wealth distribution.

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The One Step Malliavin scheme: new discretization of BSDEs implemented with deep learning regressions

Cited by 3 publications

References 35 publications

Convergence of a robust deep FBSDE method for stochastic control

Convergence of a robust deep FBSDE method for stochastic control

Differential learning methods for solving fully nonlinear PDEs

Neural networks for stochastic control and decision making in mathematical finance

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