Related papers: Deep Operator BSDE: a Numerical Scheme to Approxim…

We present a parallel algorithm for solving backward stochastic differential equations (BSDEs in short) which are very useful theoretic tools to deal with many financial problems ranging from option pricing option to risk management. Our…

We study the numerical strong stability of explicit schemes for the numerical approximation of the solution to a BSDE where the driver has polynomial growth in the primary variable and satisfies a monotone decreasing condition, and we…

Operator learning has emerged as a promising paradigm for developing efficient surrogate models to solve partial differential equations (PDEs). However, existing approaches often overlook the domain knowledge inherent in the underlying PDEs…

In this paper, we present a numerical scheme to solve the initial-boundary value problem for backward stochastic partial differential equations of parabolic type. Based on the Galerkin method, we approximate the original equation by a…

We study the error arising in the numerical approximation of FBSDEs and related PIDEs by means of a deep learning-based method. Our results focus on decoupled FBSDEs with jumps and extend the seminal work of HAn and Long (2020) analyzing…

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…

We describe an approach for estimating the statistical risk of deep networks that contain a mix of learned and known operators. Building on the maximal training error bounds previously established for known operator learning, we derive a…

In this paper we propose an efficient third-order numerical scheme for backward stochastic differential equations(BSDEs). We use 3-point Gauss-Hermite quadrature rule for approximation of the conditional expectation and avoid spatial…

Pseudospectral approximation provides a means to approximate the dynamics of delay differential equations (DDE) by ordinary differential equations (ODE). This article develops a computer-aided algorithm to determine the distance between the…

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…

The aim of this work is to propose an extension of the deep solver by Han, Jentzen, E (2018) to the case of forward backward stochastic differential equations (FBSDEs) with jumps. As in the aforementioned solver, starting from a discretized…

We discuss a general dynamic replication approach to counterparty credit risk modeling. This leads to a fundamental jump-process backward stochastic differential equation (BSDE) for the credit risk adjusted portfolio value. We then reduce…

This paper proposes two efficient approximation methods to solve high-dimensional fully nonlinear partial differential equations (NPDEs) and second-order backward stochastic differential equations (2BSDEs), where such high-dimensional fully…

We propose a novel machine learning framework for solving optimization problems governed by large-scale partial differential equations (PDEs) with high-dimensional random parameters. Such optimization under uncertainty (OUU) problems may be…

Neural networks have been applied to control problems, typically by combining data, differential equation residuals, and objective costs in the training loss or by incorporating auxiliary architectural components. Instead, we propose a…

This work proposes a Bayesian inference method for the reduced-order modeling of time-dependent systems. Informed by the structure of the governing equations, the task of learning a reduced-order model from data is posed as a Bayesian…

In this paper, we revisit the backward Euler method for numerical approximations of random periodic solutions of semilinear SDEs with additive noise. Improved $L^{p}$-estimates of the random periodic solutions of the considered SDEs are…

We propose a new probabilistic scheme which combines deep learning techniques with high order schemes for backward stochastic differential equations belonging to the class of Runge-Kutta methods to solve high-dimensional semi-linear…

We construct the first rigorously justified probabilistic algorithm for recovering the solution operator of a hyperbolic partial differential equation (PDE) in two variables from input-output training pairs. The primary challenge of…

We analyze a natural extension of the backward Euler approximation for a class of BSDEs with Lipschitz generators and random (unbounded) time horizons. We derive strong error bounds in terms of the underlying stepsize; the distance between…