arXiv research

Sharp $L^\infty$ estimates proved for complex Monge-Ampère equations.

Paper studies solutions to a specific equation in conformal geometry with singular sets.

Sharp $L^\infty$ estimates for non-Kähler manifolds' PDEs are derived.

Study bi-Hermitian metrics on complex surfaces and solve geometric PDEs.

Uniform bounds for complex equations using Monge-Ampère method.

Paper proves uniform continuity bounds for complex Monge-Ampère solutions.

Solves a PDE for Landsberg surfaces using new Finsler surface insights.

New proof of $L^\infty$ estimates for Monge-Ampère and Hessian equations on nef classes.

Develops a new approach to study nonlinear PDEs and their singularities.

Develops deep learning methods for non-linear PDEs in credit risk.

In this work, we have presented a simple analytical approximation scheme for generic non-linear FBSDEs. By treating the interested system as the linear decoupled FBSDE perturbed with non-linear generator and feedback terms, we have shown that it is possible to carry out a recursive approximation to an arbitrarily highe…

Study of nonlinear PDEs using derived geometry and BV formalism.

This paper studies a limit order book (LOB) model, in which the order dynamics depend on both, the current best available prices and the current volume density functions. For the joint dynamics of the best bid price, the best ask price, and the standing volume densities on both sides of the LOB we derive a weak law of …

Uniform bounds derived for fully non-linear equations.

The paper examines solutions to a specific type of nonlinear equation in a disk, proving existence and uniqueness.

Paper develops techniques to solve complex PDEs involving higher cohomology forms.

In this paper, we study a semi-martingale optimal transport problem and its application to the calibration of Local-Stochastic Volatility (LSV) models. Rather than considering the classical constraints on marginal distributions at initial and final time, we optimise our cost function given the prices of a finite number…

High-dimensional partial differential equations (PDE) appear in a number of models from the financial industry, such as in derivative pricing models, credit valuation adjustment (CVA) models, or portfolio optimization models. The PDEs in such applications are high-dimensional as the dimension corresponds to the number …

New machine learning methods solve complex PDEs with improved accuracy.

We provide new exact Taylor's series with fixed coefficients and without the remainder. We demonstrate the usefulness of this contribution by using it to obtain very simple solutions to (non-linear) PDEs. We also apply the method to the portfolio model.

We present a PDE-based framework that generalizes Group equivariant Convolutional Neural Networks (G-CNNs). In this framework, a network layer is seen as a set of PDE-solvers where geometrically meaningful PDE-coefficients become the layer's trainable weights. Formulating our PDEs on homogeneous spaces allows these net…

In the neighborhood of a regular point, generalized Kahler geometry admits a description in terms of a single real function, the generalized Kahler potential. We study the local conditions for a generalized Kahler manifold to be a generalized Calabi-Yau manifold and we derive a non-linear PDE that the generalized Kahle…

We propose a numerical method for solving high dimensional fully nonlinear partial differential equations (PDEs). Our algorithm estimates simultaneously by backward time induction the solution and its gradient by multi-layer neural networks, while the Hessian is approximated by automatic differentiation of the gradient…

We present a stochastic numerical method for solving fully non-linear free boundary problems of parabolic type and provide a rate of convergence under reasonable conditions on the non-linearity.

Study shows long-term solutions for complex equations on curved spaces.

Introduces a new PDE involving differential forms for Kähler geometry.

Paper solves complex equations on noncompact manifolds.

We use Vessiot theory and exterior calculus to solve partial differential equations(PDEs) of the type uyy = F(x, y,u,ux,uy,uxx,uxy) and associated evolution equations. These equations are represented by the Vessiot distribution of vector fields. We develop and apply an algorithm to find the largest integrable sub-distr…

The paper establishes boundary estimates for solutions to elliptic equations on Hermitian manifolds.

Solves Dirichlet problem for elliptic equations on Hermitian manifolds.

We succeed in writing 2-dimensional conformally invariant non-linear elliptic PDE (harmonic map equation, prescribed mean curvature equations...etc) in divergence form. This divergence free quantities generalize to target manifolds without symmetries the well known conservation laws for harmonic maps into homogeneous s…

Study numerical methods for singular FBSDEs with degenerate forward component.

Paper generalizes sub-slope definition and solves complex equations on compact manifolds.

Anisotropic minimal graphs over half-spaces are flat.

The physics informed neural network (PINN) is evolving as a viable method to solve partial differential equations. In the recent past PINNs have been successfully tested and validated to find solutions to both linear and non-linear partial differential equations (PDEs). However, the literature lacks detailed investigat…

Paper establishes estimates for solutions on compact manifolds.

Geodesics in positive Lagrangian spaces map to special Lagrangian cylinders.

Sharp L∞ estimates for fully non-linear elliptic equations on compact complex manifolds.

We show that a properly convex projective structure $\mathfrak{p}$ on a closed oriented surface of negative Euler characteristic arises from a Weyl connection if and only if $\mathfrak{p}$ is hyperbolic. We phrase the problem as a non-linear PDE for a Beltrami differential by using that $\mathfrak{p}$ admits a compatib…

Extends parabolic study to flat hyperkähler manifolds.

We provide an asymptotic expansion of the value function of a multidimensional utility maximization problem from consumption with small non-linear price impact. In our model cross-impacts between assets are allowed. In the limit for small price impact, we determine the asymptotic expansion of the value function around …

In this work, we present a machine learning approach for reducing the error when numerically solving time-dependent partial differential equations (PDE). We use a fully convolutional LSTM network to exploit the spatiotemporal dynamics of PDEs. The neural network serves to enhance finite-difference and finite-volume met…

Calibration of stochastic local volatility (SLV) models to their underlying local volatility model is often performed by numerically solving a two-dimensional non-linear forward Kolmogorov equation. We propose a novel finite volume (FV) discretization in the numerical solution of general 1D and 2D forward Kolmogorov eq…

D2SRM solves complex PDEs using deep learning.

In this note we discuss some formal properties of universal linearization operator, relate this to brackets of non-linear differential operators and discuss application to the calculus of auxiliary integrals, used in compatibility reductions of PDEs.

We present a deep recurrent neural network architecture to solve a class of stochastic optimal control problems described by fully nonlinear Hamilton Jacobi Bellmanpartial differential equations. Such PDEs arise when one considers stochastic dynamics characterized by uncertainties that are additive and control multipli…

The paper solves curvature measure problem in hyperbolic space.

New complex-valued maps found on complex geometries.