arXiv research

The paper explores symmetry in solutions of semilinear PDEs on Riemannian domains.

Paper introduces a new method to solve complex PDEs efficiently.

Deep neural nets solve complex insurance math equations.

The aim of this paper is to suggest a new viewpoint to study qualitative properties of solutions of semilinear elliptic PDE's defined outside a compact set. The relevant tools come from spectral theory and from a combination of stochastic properties of the relevant differential operators. Possible links between spectra…

We extend the viscosity solution characterization proved in [5] for call/put American option prices to the case of a general payoff function in a multi-dimensional setting: the price satisfies a semilinear re-action/diffusion type equation. Based on this, we propose two new numerical schemes inspired by the branching p…

We study the semilinear partial differential equation (PDE) associated with the non-linear BSDE characterizing buyer's and seller's XVA in a framework that allows for asymmetries in funding, repo and collateral rates, as well as for early contract termination due to counterparty credit risk. We show the existence of a …

New learning scheme solves high-dimensional semi-linear PDEs using sparse grids and Picard approximations.

New approach simplifies proof of wave equations on black holes.

We use probabilistic methods to study classical solutions for systems of interacting semilinear parabolic partial differential equations. In a modeling framework for a financial market with interacting Ito and point processes, such PDEs are shown to provide a natural description for the solution of hedging and valuatio…

We study conditions for existence, uniqueness and invariance of the comprehensive nonlinear valuation equations first introduced in Pallavicini et al (2011). These equations take the form of semilinear PDEs and Forward-Backward Stochastic Differential Equations (FBSDEs). After summarizing the cash flows definitions all…

Paper tackles DOCTR-L with SciPhy RL, solving neural PDEs from data.

We take the holistic approach of computing an OTC claim value that incorporates credit and funding liquidity risks and their interplays, instead of forcing individual price adjustments: CVA, DVA, FVA, KVA. The resulting nonlinear mathematical problem features semilinear PDEs and FBSDEs. We show that for the benchmark v…

Study phase transitions with prescribed mean curvature in Riemannian manifolds.

Extends XVA valuation under stochastic volatility, characterizing value processes via mild solutions.

The article derives gradient estimations for semilinear equations on geometric flows.

Researchers solve an inverse problem for a semilinear elliptic equation on complex manifolds.

NO approximates non-Markovian BSDEs with polynomial scaling in 1/ε.

Derives matrix Harnack inequalities for semilinear heat equations on manifolds.

The paper proves Liouville theorems and nonexistence results for semilinear equations on pseudo-Hermitian manifolds.

Paper proves a Liouville theorem for a generalized elliptic equation on H-type groups.

The paper classifies solutions to semilinear equations on curved spaces.

On an affine flat manifold with coordinates x^j and convex local potential function f, we call the affine Kahler metric f_{ij} dx^i dx^j semi-flat Calabi-Yau if it satisfies det f_{ij} = 1. Recently Gross-Wilson have constructed many such metrics on S^2 minus 24 singularities, as degenerate limits of Calabi-Yau metrics…

Derives Li & Yau estimates for heat equations on manifolds.

Trivial solution proof for heat equation on certain manifolds.

Sharp conditions found for solving heat equation on Riemannian manifolds.

New Kelvin transform for anisotropic elliptic problems.

Paper proves no nontrivial solutions to certain elliptic equations on graphs.

Develops comparison methods for semilinear elliptic problems on Riemannian manifolds with Ricci lower bound.

We study bifurcation from a branch of trivial solutions of semilinear elliptic Dirichlet boundary value problems on a geodesic ball, whose radius is used as the bifurcation parameter. In the proof of our main theorem we obtain in addition a special case of an index theorem due to S. Smale.

Classifies self-similar solutions for heat equations with positive speed.

Measuring wave sources uniquely identifies manifold properties.

The paper classifies solutions to a specific elliptic equation in the Heisenberg group.

In this paper, we investigate the problem of blow up and sharp upper bound estimates of the lifespan for the solutions to the semilinear wave equations, posed on asymptotically Euclidean manifolds. Here the metric is assumed to be exponential perturbation of the spherical symmetric, long range asymptotically Euclidean …

Nonexistence results for semilinear parabolic and hyperbolic inequalities on metric graphs

We consider the Dirichlet problem for semilinear elliptic equations on a bounded domain which is diffeomorphic to a ball and investigate bifurcation from a given (trivial) branch of solutions, where the radius of the ball serves as bifurcation parameter. Our methods are based on well known results from variational bifu…

We show that a wide range of overdetermined boundary problems for semilinear equations with position-dependent nonlinearities admits nontrivial solutions. The result holds true both on the Euclidean space and on compact Riemannian manifolds. As a byproduct of the proofs we also obtain some rigidity, or partial symmetry…

Study semilinear equations on weighted manifolds to prove rigidity.

We develop a framework for computing the total valuation adjustment (XVA) of a European claim accounting for funding costs, counterparty credit risk, and collateralization. Based on no-arbitrage arguments, we derive backward stochastic differential equations (BSDEs) associated with the replicating portfolios of long an…

Assume that $f(s) = F'(s)$ where $F$ is a double-well potential. Under certain conditions on the Lipschitz constant of $f$ on $[-1,1]$, we prove that arbitrary bounded global solutions of the semilinear equation $Δu = f(u)$ on hyperbolic space $\HH^n$ must reduce to functions of one variable provided they admit asympto…

The paper extends Liouville theorems to sub-Riemannian manifolds.

We consider a semilinear parabolic degenerated Hamilton-Jacobi-Bellman (HJB) equation with singularity which is related to a stochastic control problem with fuel constraint. The fuel constraint translates into a singular initial condition for the HJB equation. We first propose a transformation based on a change of vari…

Study on ground states of semilinear elliptic equations with various potential wells.

We introduce a method for solving Calderón type inverse problems for semilinear equations with power type nonlinearities. The method is based on higher order linearizations, and it allows one to solve inverse problems for certain nonlinear equations in cases where the solution for a corresponding linear equation is not…

Consider a nontrivial solution to a semilinear elliptic system of first order with smooth coefficients defined over an $n$-dimensional manifold. Assume the operator has the strong unique continuation property. We show that the zero set of the solution is contained in a countable union of smooth $(n-2)$-dimensional subm…

In this paper, we generalize the Gauduchon metrics on a compact complex manifold and define the $γ_k$ functions on the space of its hermitian metrics.

The classical Pohozaev identity constrains potential solutions of certain semilinear PDE boundary value problems. The Kazdan-Warner identity is a similar necessary condition important for the Nirenberg problem of conformally prescribing scalar curvature on the sphere. For dimensions $n\geq 3$ both identities are captur…

Study on price formation in financial markets with a single default event.

Paper analyzes multidimensional PIDEs for financial modeling, proving existence and uniqueness in Bessel spaces.