Xiaotao Zheng

Xiaotao Zheng, Xingye Yue, Zhihong Xia et al.

Semilinear parabolic partial differential equations (PDEs) are fundamental to modeling complex dynamical systems across scientific domains. The Deep Backward Stochastic Differential Equation (BSDE) method is a promising approach for high-dimensional PDEs; however, existing convergence results apply only to globally Lipschitz generators, excluding important cases such as Allen--Cahn and Hamilton--Jacobi--Bellman (HJB) equations. This paper presents both a theoretical and a computational advance for Deep BSDE methods. Theoretically, we establish the convergence theory for non--Lipschitz generators--covering Allen--Cahn equations with cubic nonlinearity and HJB equations with quadratic gradient growth--based on a bounded double--well lemma and a truncated-BSDE analysis within the Bouchard--Touzi--Zhang theory. Computationally, we instantiate the framework with XNet, a shallow architecture with $\mathcal O(L)$ parameters that preserves strong approximation while substantially reducing optimization and computational cost. Numerical experiments on 100--dimensional PDEs corroborate the predicted convergence behavior and demonstrate significant efficiency gains over standard feedforward implementations.

Ziting Pei, Xingye Yue, Xiaotao Zheng

G-expectation, as a sublinear expectation, provides a powerful framework for modeling uncertainty in financial markets. Motivated by the need for robust valuation under model uncertainty, this work develops a unified risk-neutral valuation approach within the G-expectation environment, yielding a nonlinear generalization of the Black-Scholes model, termed the G-Black-Scholes equation. To enhance computational efficiency and reduce numerical cost, we introduce a logarithmic transformation of the asset price, which yields an alternative nonlinear PDE. Based on this transformed formulation, we design both explicit and implicit finite difference schemes that are rigorously demonstrated to be consistent, stable, monotone, and convergent to the viscosity solution. Numerical examples confirm that the proposed schemes achieve high accuracy, while the logarithmic transformation relaxes the stability constraints of explicit schemes and improves computational efficiency.

Ziting Pei, Shige Peng, Xingye Yue et al.

A $G$-normal random variable $X\sim \mathcal{N}(0,[\underlineσ^2,\overlineσ^2])$ does not admit a unique probability law due to volatility uncertainty. For a given test function $ϕ$, the $G$-expectation admits the stochastic control representation$$\mathbb{E}[ϕ(X)] = \sup_{σ\in[\underlineσ,\overlineσ]} {E}\!\left[ϕ(X_T^σ)\mid X_0^σ=0\right] ={E}\!\left[ϕ(X_T^\ast)\mid X_0^\ast=0\right].$$ This formulation interprets the nonlinear expectation as a linear expectation under the law induced by the optimally controlled diffusion $X^\ast$, namely, the terminal law of $X_T^\ast$. This observation motivates the notion of a \emph{responsive distribution}, a measurement-dependent probability density $f_ϕ$ such that, for a given test function $ϕ$, $$\mathbb{E}[ϕ(X)] = \int_{\mathbb{R}} ϕ(x)\,f_ϕ(x)\,dx.$$ Based on this viewpoint, we propose a coupled backward--forward trinomial tree framework for computing the $G$-expectation and constructing the corresponding responsive distribution. The backward trinomial tree discretizes the associated stochastic optimal control problem and yields approximations of the value function (i.e., the $G$-expectation) and the optimal feedback control, while the forward trinomial tree propagates the induced transition probabilities and produces a discrete approximation of the responsive distribution. We establish rigorous convergence results for both components of the method. Numerical results not only validate the theoretical convergence of the coupled schemes but also provide a powerful, practical sampling tool to visualize the complex responsive distributions under various measurements.

Xin Li, Xiaotao Zheng, Zhihong Xia

XNet is a single-layer neural network architecture that leverages Cauchy integral-based activation functions for high-order function approximation. Through theoretical analysis, we show that the Cauchy activation functions used in XNet can achieve arbitrary-order polynomial convergence, fundamentally outperforming traditional MLPs and Kolmogorov-Arnold Networks (KANs) that rely on increased depth or B-spline activations. Our extensive experiments on function approximation, PDE solving, and reinforcement learning demonstrate XNet's superior performance - reducing approximation error by up to 50000 times and accelerating training by up to 10 times compared to existing approaches. These results establish XNet as a highly efficient architecture for both scientific computing and AI applications.