It is well-known that decision-making problems from stochastic control can be formulated by means of a forward–backward stochastic differential equation (FBSDE). Recently, the authors of Ji et al. (2022) proposed an efficient deep learning algorithm based on the stochastic maximum principle (SMP). In this paper, we provide a convergence result for this deep SMP-BSDE algorithm and compare its performance with other existing methods. In particular, by adopting a strategy as in Han and Long (2020), we derive a-posteriori estimate, and show that the total approximation error can be bounded by the value of the loss functional and the discretization error. We present numerical examples for high-dimensional stochastic control problems, both in the cases of drift- and diffusion control, which showcase superior performance compared to existing algorithms. ... Read more

Backward stochastic differential equations (BSDE) are a fundamental tool in the mathematical modelling of financial problems. Through the famous nonlinear extensions to the Feynman-Kac formula, they do not merely provide a stochastic representation of the solution to large classes of partial differential equations such as pricing- or Hamilton- Jacobi-Bellman equations, but also include sensitivities, corresponding to derivatives of the solution, which are crucial in many financial mathematical applications. Henceforth, they simultaneously represent option pricing and hedging problems, and form a natural framework for the numerical treatment of stochastic optimal control.

The main challenge in the numerical approximation of such equations is the computation of conditional expectations over potentially high-dimensional spaces. In classical settings, where the dimensionality of the underlying randomness is moderate, many approaches have been proposed in the literature. However, for high-dimensional problems, one has to resort toMonte Carlo methods. In recent years, a new class of regression Monte Carlo methods has arisen in the literature, so called deep BSDE methods, which practically approximate the solution of BSDEs in a neural network regression Monte Carlo framework, after forming a suitable loss function motivated either by stochastic optimal control or the martingale representation theorem. These classes of methods can roughly be divided into two main categories. Forward methods, where the solution of the associated backward SDE is simultaneously optimized in a global optimization, minimizing a loss function stemming from a stochastic target problem reformulation. Alternatively, backward methods have been investigated, where the numerical resolution of the equation is decomposed into smaller sub-optimizations corresponding to a discrete set of points in a suitable time discretization. These methods enabled the numerical treatment of longstanding open challenges, such as the pricing and deltahedging of multi-asset financial options up to d = 100 risk factors and beyond.

The goal of this thesis is to analyze such modern machine learning based numerical methods, and apply them in the financial mathematical context. We propose numerical extensions of these methods in high-dimensional frameworks, analyze their convergence properties in discrete time, and investigate their robustness and accuracy in practical applications such as hedging and stochastic optimal control. Ourmain contributions in each chapter can be summarized as follows…
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A novel discretization is presented for decoupled forward–backward stochastic differential equations (FBSDE) with differentiable coefficients, simultaneously solving the BSDE and its Malliavin sensitivity problem. The control process is estimated by the corresponding linear BSDE driving the trajectories of the Malliavin derivatives of the solution pair, which implies the need to provide accurate Γ estimates. The approximation is based on a merged formulation given by the Feynman–Kac formulae and the Malliavin chain rule. The continuous time dynamics is discretized with a theta-scheme. In order to allow for an efficient numerical solution of the arising semidiscrete conditional expectations in possibly high dimensions, it is fundamental that the chosen approach admits to differentiable estimates. Two fully-implementable schemes are considered: the BCOS method as a reference in the one-dimensional framework and neural network Monte Carlo regressions in case of high-dimensional problems, similarly to the recently emerging class of Deep BSDE methods (Han et al. (2018 Solving high-dimensional partial differential equations using deep learning. Proc. Natl. Acad. Sci., 115, 8505–8510); Huré et al. (2020 Deep backward schemes for high-dimensional nonlinear PDEs. Math. Comp., 89, 1547–1579)). An error analysis is carried out to show L ² convergence of order 1/2, under standard Lipschitz assumptions and additive noise in the forward diffusion. Numerical experiments are provided for a range of different semilinear equations up to 50 dimensions, demonstrating that the proposed scheme yields a significant improvement in the control estimations. ... Read more