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We propose a new algorithm for solving parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) in high dimension, by making an analogy between the BSDE and reinforcement learning with the…

We prove well-posedness results for backward stochastic differential equations (BSDEs) and reflected BSDEs with an optional obstacle process in the case of appropriately weighted $\mathbb{L}^2$-data when the generator is integrated with…

A novel discretization is presented for forward-backward stochastic differential equations (FBSDE) with differentiable coefficients, simultaneously solving the BSDE and its Malliavin sensitivity problem. The control process is estimated by…

Optimal control problems of forward-backward stochastic Volterra integral equations (FBSVIEs, in short) with closed control regions are formulated and studied. Instead of using spike variation method as one may imagine, here we turn to…

In this work, we extend deep learning-based numerical methods to fully coupled forward-backward stochastic differential equations (FBSDEs) within a non-Markovian framework. Error estimates and convergence are provided. In contrast to the…

We propose a novel numerical approach for nonlocal diffusion equations [8] with integrable kernels, based on the relationship between the backward Kolmogorov equation and backward stochastic differential equations (BSDEs) driven by L\`{e}vy…

In this paper, we present a backward deep BSDE method applied to Forward Backward Stochastic Differential Equations (FBSDE) with given terminal condition at maturity that time-steps the BSDE backwards. We present an application of this…

Deep learning methods based on backward stochastic differential equations (BSDEs) have emerged as competitive alternatives to physics-informed neural networks (PINNs) for solving high-dimensional partial differential equations (PDEs). By…

Bayesian methods have proved powerful in many applications for the inference of model parameters from data. These methods are based on Bayes' theorem, which itself is deceptively simple. However, in practice the computations required are…

Inverse problems involving partial differential equations (PDEs) are widely used in science and engineering. Although such problems are generally ill-posed, different regularisation approaches have been developed to ameliorate this problem.…

This article deals with the numerical approximation of Markovian backward stochastic differential equations (BSDEs) with generators of quadratic growth with respect to $z$ and bounded terminal conditions. We first study a slight…

We present a novel deep learning method for estimating time-dependent parameters in Markov processes through discrete sampling. Departing from conventional machine learning, our approach reframes parameter approximation as an optimization…

In this paper, we study the discrete-time approximation of multidimensional reflected BSDEs of the type of those presented by Hu and Tang [Probab. Theory Related Fields 147 (2010) 89-121] and generalized by Hamad\`ene and Zhang [Stochastic…

In this article, we introduce a novel backward method to model stochastic gene expression and protein level dynamics. The protein amount is regarded as a diffusion process and is described by a backward stochastic differential equation…

Backpropagation through (neural) SDE solvers is traditionally approached in two ways: discretise-then-optimise, which offers accurate gradients but incurs prohibitive memory costs; and optimise-then-discretise, which achieves constant…

Explicit solutions for a class of linear backward stochastic differential equations (BSDE) driven by Gaussian Volterra processes are given. These processes include the multifractional brownian motion and the multifractional…

The optimal stopping problem is one of the core problems in financial markets, with broad applications such as pricing American and Bermudan options. The deep BSDE method [Han, Jentzen and E, PNAS, 115(34):8505-8510, 2018] has shown great…

In this paper, we introduce a class of backward stochastic equations (BSEs) that extend classical BSDEs and include many interesting examples of generalized BSDEs as well as semimartingale backward equations. We show that a BSE can be…

Multivariate time series with missing values are common in areas such as healthcare and finance, and have grown in number and complexity over the years. This raises the question whether deep learning methodologies can outperform classical…

We are concerned with high-dimensional coupled FBSDE systems approximated by the deep BSDE method of Han et al. (2018). It was shown by Han and Long (2020) that the errors induced by the deep BSDE method admit a posteriori estimate…