Study dynamic asset allocation in incomplete markets using game theory and nonlocal BSDEs.
problem Dynamic mean-variance asset allocation in general incomplete markets with non-exponential discounting.
method Game-theoretic approach, decomposition into myopic and hedging strategies, nonlocal BSDEs, fixed-point theorem.
result Well-posedness of solutions to BSDEs, existence of equilibrium control policy.
Nonlocal neural networks have been proposed and shown to be effective in several computer vision tasks, where the nonlocal operations can directly capture long-range dependencies in the feature space. In this paper, we study the nature of diffusion and damping effect of nonlocal networks by doing spectrum analysis on t…
Investigates convexity of minimizers under mass constraint using nonlocal perimeter and potential.
problem Convexity of minimizers under mass constraint.
method Nonlocal free energy with nonlocal perimeter and convex potential.
result Quantitative stability theorem for nonlocal free energy assuming symmetry on the potential.
Approximates nonlocal curvature of curves using splines.
problem Approximating nonlocal curvature of planar curves.
method Extending nonlocal curvature definition, using incomplete beta function, and linear interpolating spline approximation.
result Nonlocal curvature of a planar curve can be approximated by a spline.
Paper connects Bäcklund transformations to nonlocal pseudosymmetries.
problem Finding Bäcklund transformations for differential equations.
method Factorization with nonlocal pseudosymmetries to determine Bäcklund transformations as C-morphisms. result Bäcklund transformations are determined by nonlocal pseudosymmetries' invariants.
The paper proves removable singularity for nonlocal minimal graphs.
problem Proving removable singularities for nonlocal minimal graphs.
method Analyzing (s,1)-capacity zero compact sets to ensure graphs are minimal in the entire domain. result Nonlocal minimal graphs are removable in the entire domain if they are minimal in a set of (s,1)-capacity zero. The paper studies nonlocal isoperimetric problems in hyperbolic space and finds unique minimizers for small volumes.
problem Nonlocal isoperimetric problem in hyperbolic space.
method Investigates minimization of a functional with perimeter and a nonlocal term derived from the negative power of distance.
result Geodesic balls are unique minimizers for small volumes in hyperbolic space.
The paper proves a nonlocal version of the Alexandrov Theorem for smooth boundaries.
problem Proving the nonlocal version of the Alexandrov Theorem for sets with smooth boundaries.
method Formulated a necessary and sufficient condition for the theorem to hold, used a specific formula for the tangential derivative of the nonlocal mean curvature, and applied the method of moving planes.
result The only set with smooth boundary and constant nonlocal mean curvature is an Euclidean ball.
Proposes a new nonlocal curvature tensor concept.
problem Various nonlocal curvature concepts in literature.
method Generalizes classical curvature tensor representation and uses fractional differential operator analogies.
result Introduces a new nonlocal curvature tensor.
New theorem for nonlocal minimal surfaces in any dimension.
problem Classical half-space theorems for minimal surfaces.
method Established a new half-space theorem.
result Result holds in any dimension.
Study uses G-BSDEs to decompose pricing kernels under robust G-expectation.
problem Long-term decomposition of robust pricing kernels under G-expectation.
method Proposes and analyzes three types of quadratic G-BSDEs to decompose pricing kernels.
result Pricing kernels decomposed into four components: discounting, transitory, symmetric martingale, and volatility uncertainty.
Nonlocal Bayesian modeling for continuous spatio-temporal dynamics
problem Handling irregular time points, sparse observations, and nonlocal interactions in spatio-temporal forecasting
method Hierarchical Bayesian framework with coordinate-based spatial basis expansion and continuous-time ODE
result Strong forecasting and uncertainty calibration
The paper tackles pricing vulnerable options via generalized BSDEs and penalization schemes.
problem Pricing options in a general hazard process setup.
method Establishes well-posedness and comparison theorems for generalized BSDEs and RBSDEs, studies penalization schemes.
result Well-posedness results and comparison theorems for generalized BSDEs and RBSDEs, extended penalization schemes.
Develops geometric BSDEs for modeling dynamic return risk measures.
problem Modeling continuous-time dynamic return risk measures.
method Introduces and develops Geometric Backward Stochastic Differential Equations (GBSDEs) and two-driver BSDEs.
result Establishes existence, regularity, uniqueness, and stability of solutions to GBSDEs.
Researchers solve a nonlocal parabolic equation on manifolds using source-to-solution maps.
problem Determine Riemannian manifolds up to isometry using local source-to-solution maps.
method Comprehensive spectrum analysis and semigroup theory for nonlocal parabolic operators.
result Can determine Riemannian manifold up to isometry using local source-to-solution maps in a small open cylinder.
Study proposes a nonlocal approximation of the Willmore functional using fractional Allen-Cahn energies.
problem Approximating the Willmore functional using nonlocal methods.
method Gamma-convergence and fractional Laplacian analysis in Fermi coordinates.
result Proves Γ-limsup estimate for the proposed nonlocal approximation. Study of nonlocal mean curvature and symmetry of surfaces.
problem Symmetry of surfaces with ordered nonlocal mean curvature.
method Generalization of Alexandrov's moving plane method.
result Similar result to classical setting proved in nonlocal setting.
We demonstrate that the use of asymptotic expansion as prior knowledge in the "deep BSDE solver", which is a deep learning method for high dimensional BSDEs proposed by Weinan E, Han & Jentzen (2017), drastically reduces the loss function and accelerates the speed of convergence. We illustrate the technique and its imp…
Study on BSDEs with random time horizon, focusing on existence and properties.
problem Existence of solutions to BSDEs and reflected BSDEs with a random time horizon.
method Method of reduction and examination of BSDEs with lahdlaug driver.
result Existence of solutions to BSDEs and reflected BSDEs with a random time horizon.
New methods solve complex financial equations.
problem Solving backward stochastic differential equations driven by continuous-time Markov chains.
method Multi-stage Euler-Maruyama methods and multilevel spatial discretization.
result Efficiently solved stiff Markov BSDEs.
Paper proves existence and uniqueness of solutions to nonlocal systems, generalizing stochastic game theory.
problem Time inconsistency in stochastic differential games.
method Proves existence and uniqueness of solutions to nonlocal fully-nonlinear parabolic systems.
result Generalizes stochastic game theory to include time-inconsistent preferences.
New nonlocal minimal surfaces on manifolds, proving Yau's conjecture.
problem Proving Yau's conjecture for nonlocal minimal surfaces.
method Introducing nonlocal minimal surfaces and applying min-max variational methods.
result Construction of infinitely many nonlocal s-minimal surfaces on closed manifolds. The paper characterizes dynamic return and star-shaped risk measures via BSDEs.
problem Characterizing dynamic return and star-shaped risk measures.
method Characterization of star-shaped functionals and BSDEs.
result Existence of convex BSDEs with non-empty set of supersolutions.
For surfaces without boundary, nonlocal notions of directional and mean curvatures have been recently given. Here, we develop alternative notions, special cases of which apply to surfaces with boundary. Our main tool is a new fractional or nonlocal area functional for compact surfaces.
(Working Paper) Using a purely probabilistic argument, we prove the global well-posedness of multidimensional superquadratic backward stochastic differential equations (BSDEs) without Markovian assumption. The key technique is the interplay between the local well-posedness of fully coupled path-dependent forward backwa…
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 the fundamental BSDE to a continuous BSDE. Depending on the close out value conve…
This paper is concerned with the determination of credit risk premia of defaultable contingent claims by means of indifference valuation principles. Assuming exponential utility preferences we derive representations of indifference premia of credit risk in terms of solutions of Backward Stochastic Differential Equation…
MetaNOR learns common nonlocal kernels for efficient metamaterial modeling.
problem Efficiently modeling wave propagation in new metamaterials.
method Meta-learns a common nonlocal kernel from existing tasks and transfers this knowledge to new tasks with minimal data.
result Substantial improvements in sampling efficiency for new metamaterials.
We are concerned with hypersurfaces of RN with constant nonlocal (or fractional) mean curvature. This is the equation associated to critical points of the fractional perimeter under a volume constraint. Our results are twofold. First we prove the nonlocal analogue of the Alexandrov result characterizing sph…
We study hypersurfaces of RN with constant nonlocal (or fractional) mean curvature. This is the equation associated to critical points of the fractional perimeter functional under a volume constraint. We establish the existence of a smooth branch of periodic cylinders in RN, N≥2, all of th…
Paper presents a new backward deep BSDE method for solving nonlinear FBSDE problems.
problem Nonlinear Forward Backward Stochastic Differential Equations (FBSDE) with terminal conditions.
method Backward deep BSDE method applied to FBSDE with nonlinear generators and random initial conditions.
result Derives exact and Taylor-based approximations for time-stepping nonlinear BSDEs.
We prove the existence of the analog of Lawson's minimal cones for a notion of nonlocal minimal surface introduced by Caffarelli, Roquejoffre and Savin, and establish their stability/instability in low dimensions. In particular we find that there are nonlocal stable minimal cones in dimension 7, in contrast with the ca…
A new image interpolation model using sparse representation and nonlocal linear regression.
problem Image interpolation without blurring and noise.
method Sparse representation, nonlocal self-similarity, nonlocal linear regression, adaptive sub-dictionary learning, weighted encoding.
result Our method outperforms state-of-the-art methods in quantitative measures and visual quality.
In the present work, the integrable bi-Hamiltonian hierarchies related to compatible nonlocal Poisson brackets of hydrodynamic type are effectively constructed. For achieving this aim, first of all, the problem on the canonical form of a special type for compatible nonlocal Poisson brackets of hydrodynamic type is solv…
The study proves inequalities for complex operators on curved spaces.
problem Establishing inequalities for nonlocal operators on curved spaces.
method Defining and analyzing nonlocal Pucci operators on manifolds with nonnegative sectional curvatures, proving Harnack inequalities and Holder estimates.
result Harnack inequalities and Holder estimates for nonlocal operators on manifolds with nonnegative sectional curvatures.
Study on capillarity minimizers with nonlocal repulsion and gravity, proving existence and nonexistence.
problem Minimizing an energy functional with capillarity, nonlocal repulsion, and gravity.
method Quantitative isoperimetric inequalities applied to capillarity problem in a half-space.
result Existence and nonexistence of minimizers for various nonlocal kernels and masses.
BSDEs help in financial pricing and utility maximization.
problem Financial pricing and utility maximization in complex market models.
method Introduces and applies BSDEs to financial problems.
result Utilizes BSDEs for simple utility maximization solutions.
For any α>0, we study kα-type length-preserving and area-preserving nonlocal flow of convex closed plane curves and show that these two types of flow evolve such curves into round circles in C∞-norm. Other relevant kα-type nonlocal flow is also discussed when α≥1.
KANHedge improves hedging of high-dimensional options using learnable B-spline activation functions.
problem Challenges in high-dimensional option pricing and hedging due to the curse of dimensionality.
method Introduces KANHedge, a novel BSDE-based hedger leveraging Kolmogorov-Arnold Networks with learnable B-spline activation functions.
result KANHedge provides improved hedging performance, achieving significant reductions in hedging cost metrics.
We provide a probabilistic solution of a not necessarily Markovian control problem with a state constraint by means of a Backward Stochastic Differential Equation (BSDE). The novelty of our solution approach is that the BSDE possesses a singular terminal condition. We prove that a solution of the BSDE exists, thus part…
Proves uniqueness of solutions for a nonlocal Liouville equation with finite Q-curvature.
problem Proving uniqueness of solutions for a specific nonlocal Liouville equation.
method Connection to Calogero--Moser derivative NLS and ground state solitons.
result Uniqueness of solutions in the Gaussian case and general positive, symmetric-decreasing K. We study a doubly reflected backward stochastic differential equation (BSDE) with integrable parameters and the related Dynkin game. When the lower obstacle L and the upper obstacle U of the equation are completely separated, we construct a unique solution of the doubly reflected BSDE by pasting local solutions and…
In this paper, we study a nonlocal elliptic problem with the fractional Laplacian on Rn. We show that the problem has infinite positive solutions in Cτ(Rn)⋂Hlocα(Rn). Moreover each of these solutions tends to some positive constant limit at infinity. We extend Lin's result to the nonlocal problem on …
The paper proves well-posedness of nonlocal PDEs related to stochastic control problems.
problem Characterizing equilibrium strategies and value functions for time-inconsistent stochastic control problems.
method Method of continuity and Banach's fixed point arguments, with Schauder prior estimates.
result Global well-posedness of nonlocal fully nonlinear PDEs with sharp a-priori estimates.
A new deep generative model uses BSDEs for high-dimensional data generation.
problem Generating high-dimensional complex data, especially images.
method Combines BSDEs with deep neural networks for training with MMD loss.
result BSDE-Gen effectively generates high-dimensional data with stochasticity.
Paper introduces a new method to solve complex PDEs efficiently.
problem Solving high-dimensional semilinear PDEs and BSDEs.
method Decomposes PDEs into linear and nonlinear parts, uses Deep BSDE solver with control variate method.
result Errors of the new method are much smaller than those of the original Deep BSDE solver.
Develops a nonlocal PINN framework using PDDO for better solution of PDEs with sharp gradients.
problem Dealing with sharp gradients in solutions of PDEs using traditional PINN approaches.
method Integrates long-range interactions (nonlocality) into PINN using Peridynamic Differential Operator (PDDO).
result Nonlocal PINN approach improves solution accuracy and parameter inference for problems with sharp gradients.
Smoothness of graphs evolving by fractional mean curvature is proven.
problem Evolution of graphs by fractional mean curvature.
method Analytic semigroup approach to nonlocal quasilinear evolution equation.
result Short time existence, uniqueness, and optimal Hölder regularity of classical solutions.