Study minimal supersolutions for BSDEs with infinite terminal values, solving portfolio liquidation problems.
problem Existence of minimal supersolutions for BSDEs with infinite terminal values.
method Generalized BSDEs on a general filtered probability space with singular terminal condition.
result Solved optimal portfolio liquidation problems using minimal supersolutions.
Paper aims to minimize ruin probability in insurance companies using Sparre Andersen model.
problem Minimizing ruin probability in insurance companies with Sparre Andersen surplus process.
method Markovization of the surplus process, investigation of value function's regularity, dynamic programming principle, and comparison of viscosity solutions.
result The value function is the unique constrained viscosity solution to the Hamilton-Jacobi-Bellman equation.
Improved bounds for online prediction with expert advice.
problem Online prediction with expert advice in finite-horizon games.
method Verification arguments from optimal control theory applied to PDEs to find sub- and supersolutions.
result Explicit bounds for any number of experts and horizon, improving upon previous results.
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.
Distance between evolving hypersurfaces is a PDE solution.
problem Tracking the distance between evolving hypersurfaces.
method Elliptic and parabolic PDEs, mean curvature flow.
result Local Harnack inequalities for the distance between evolving hypersurfaces.
Paper solves a Dirichlet problem using exit operator continuity.
problem Solving Dirichlet problems with fractional Laplacian.
method Continuity of exit operator under Skorokhod topology.
result Established sub and supersolutions for HJB equations.
Gradient estimate for nonlocal minimal graphs in higher dimensions.
problem Establishing a gradient bound for nonlocal minimal graphs in higher dimensions.
method Using a truncated fractional Jacobi operator and a weak Harnack inequality, the gradient bound is derived.
result Gradient of nonlocal minimal graphs is bounded by a power of their oscillation.
In an equity market model with "Knightian" uncertainty regarding the relative risk and covariance structure of its assets, we characterize in several ways the highest return relative to the market that can be achieved using nonanticipative investment rules over a given time horizon, and under any admissible configurati…
This paper is to study the conformal scalar curvature equation on complete noncompact Riemannian manifold of nonpositive curvature. We derive some estimates and properties of supersolutions of the scalar curvature equation, and obtain some nonexistence results for complete solutions of scalar curvature equation.
The paper solves two cases of the asymptotically flat scalar-flat Yamabe problem with boundary.
problem Finding asymptotically flat scalar-flat metrics on manifolds with boundary.
method Solving elliptic PDEs with sub- and supersolutions.
result Existence of conformally equivalent asymptotically flat scalar-flat metrics.
The paper studies equations on almost Hermitian manifolds with estimates and existence results.
problem Solving Monge-Ampère type equations on compact almost Hermitian manifolds.
method Derives C∞ a priori estimates and obtains existence results under admissible conditions. result Existence of solutions under admissible conditions for Monge-Ampère type equations.
Study optimal investment and consumption in a stochastic factor model.
problem Optimal investment and consumption decisions in a stochastic factor model.
method Characterization of well-posedness, numerical algorithm, and general theory of sub- and supersolutions for HJB equation.
result Proves existence and provides bounds for the solution to the HJB equation.
The paper proves the existence of finite-energy solutions to a specific elliptic equation on Riemannian manifolds.
problem Existence of finite-energy solutions to a singular elliptic equation on Riemannian manifolds.
method ε-regularized problems, mountain pass arguments, and limiting procedures.
result Existence of nonnegative finite-energy supersolutions under certain conditions.
In this paper, we introduce new methods for solving the vacuum Einstein constraints equations: the first one is based on Schaefer's fixed point theorem (known methods use Schauder's fixed point theorem) while the second one uses the concept of half-continuity coupled with the introduction of local supersolutions. These…
Paper proves solutions for deformed Hermitian-Yang-Mills equation on almost Hermitian manifolds.
problem Existence of solutions for deformed Hermitian-Yang-Mills equation on almost Hermitian manifolds.
method Derive a priori estimates under the existence of an admissible C-subsolution; prove existence of solutions under the condition of existence of a supersolution. result Proves existence of solutions for the deformed Hermitian-Yang-Mills equation.
Study pluri-subharmonic envelopes and solve complex Monge-Ampère equations.
problem Solving complex Monge-Ampère equations on compact Kähler manifolds and domains.
method Approximation process and lower envelopes of super-solutions.
result Quasi-psh envelope of a viscosity super-solution is a pluripotential super-solution.
Paper finds unique viscosity solution to complex control problems.
problem Complex stochastic control problems with singular terminal state constraints.
method Establishes existence of unique nonnegative continuous viscosity solution using novel comparison principle.
result Unique viscosity solution to HJB equation for linear-quadratic control problems.
Our goal is to resolve a problem proposed by Fernholz and Karatzas [On optimal arbitrage (2008) Columbia Univ.]: to characterize the minimum amount of initial capital with which an investor can beat the market portfolio with a certain probability, as a function of the market configuration and time to maturity. We show …
We develop classical globally supersymmetric theories. As much as possible, we treat various dimensions and various amounts of supersymmetry in a uniform manner. We discuss theories both in components and in superspace. Throughout we emphasize geometric aspects. The beginning chapters give a general discussion about su…
Set in Riemannian enviroment, the aim of this paper is to present and discuss some equivalent characterizations of the Liouville property relative to special operators, in some sense modeled after the p-Laplacian with potential. In particular, we discuss the equivalence between the Lioville property and the Khas'minski…
Study Kähler-Einstein potentials on stable varieties near singularities
problem Asymptotic behavior of Kähler-Einstein potentials on stable varieties near singularities
method Using iterated logarithmic functions and refined lower bounds
result Improved estimates for Kähler-Einstein potentials
Optimal dividends for a two-branch insurance company modelled by stochastic processes.
problem Maximizing dividends for an insurance company with two branches under ruin constraints.
method Solving a stochastic control problem using Hamilton-Jacobi-Bellman equations.
result The optimal strategy and value function are found.
Proves existence of solution to Lichnerowicz equation on non-CMC manifolds.
problem Existence of positive solution to Lichnerowicz equation on non-CMC closed manifolds with supercritical terms.
method Employed a fixed-point argument involving sub- and supersolutions, with conditions on coefficients to prevent classical solutions.
result Proves existence of a positive and essentially bounded solution.
New method decomposes submartingale systems for BSDEs with weak constraints.
problem Tackles decomposition of submartingale systems for BSDEs with weak constraints.
method Introduces Yg,ξ-submartingale systems and proves a Mertens decomposition using an original approach. result Proves a Mertens decomposition for Yg,ξ-submartingale systems. New decomposition for submartingales aids American option hedging in incomplete markets.
problem Hedging American options in incomplete markets with jumps.
method Introduced nonlinear optional decomposition for Yg,ξ-submartingales. result Infinitesimal characterization of buyer's superhedging price.
We give a sufficient condition, with no restrictions on the mean curvature, under which the conformal method can be used to generate solutions of the vacuum Einstein constraint equations on compact manifolds. The condition requires a so-called global supersolution but does not require a global subsolution. As a consequ…
Paper establishes estimates for nonlinear equations on compact manifolds.
problem Estimating solutions to fully nonlinear equations with gradient terms on compact almost Hermitian manifolds.
method Establishes second order estimates and proves existence of solutions for specific equations.
result Proves existence of solutions for various equations, including Monge-Ampère and Hessian equations.
We study viscosity solutions to complex hessian equations. In the local case, we consider Ω a bounded domain in Cn, β the standard Kähler form in Cn and 1≤m≤n. Under some suitable conditions on F,g, we prove that the equation $(dd^c \varphi)^m\wedgeβ^{n-m}=F(x,\varphi)β^n,\ \f=…
The study establishes conditions for quasiregular mappings from Heisenberg groups to link complements.
problem Conditions for nonconstant quasiregular mappings from Heisenberg groups to link complements.
method Translation of a growth condition on fundamental groups into the existence of a supersolution to the 4-harmonic equation.
result A link complement admits a nonconstant quasiregular mapping from the Heisenberg group only if the link is empty, an unknot, or a Hopf link.
Study optimal dividend policies for firms with random profitability.
problem Firms face a trade-off between bankruptcy and profit extraction.
method General cash flow drifts (Ornstein-Uhlenbeck, CIR) considered; rigorous proofs, numerical scheme provided.
result Optimal strategy includes barrier and band strategies, voluntary liquidation.
The paper proves a Harnack inequality for heat equations on Finsler metric measure manifolds.
problem Proving a Harnack inequality for positive solutions to heat equations on Finsler metric measure manifolds.
method Volume comparison theorem, weighted Poincaré inequality, local uniform Sobolev inequality, mean value inequalities.
result Derives a Harnack inequality for positive solutions to heat equations.
This paper concerns an optimal dividend distribution problem for an insurance company whose risk process evolves as a spectrally negative Lévy process (in the absence of dividend payments). The management of the company is assumed to control timing and size of dividend payments. The objective is to maximize the sum of …
The paper characterizes stochastic incompleteness in Riemannian manifolds.
problem Stochastic incompleteness of Riemannian manifolds and its characterization.
method Characterization through solutions to nonlinear parabolic equations.
result Stochastic incompleteness is equivalent to the nonuniqueness of bounded solutions to certain nonlinear parabolic equations.
Proves conditions for solving Einstein constraint equations on Euclidean manifolds.
problem Finding solutions to Einstein constraint equations on asymptotically Euclidean manifolds.
method Conformal method, Lichnerowicz equation, global supersolutions, limit equation criterion.
result Characterizes the Yamabe classes and solves the prescribed scalar curvature problem for nonpositive scalar curvatures.
Two insurance companies collaborate to maximize dividend payouts until ruin.
problem Maximizing dividends for two collaborating insurance companies with compound Poisson surplus processes.
method Solving a stochastic control problem using viscosity solutions and numerical approximation.
result Curve strategies identified as optimal, outperforming stand-alone companies.
Interdisciplinary study linking potential theory and elliptic PDEs.
problem Understanding solutions to nonlinear elliptic PDEs.
method Combining geometric and potential theory approaches.
result Validity of comparison principle and existence/uniqueness of solutions.
Model financial market with fundraiser and stock, derive option prices.
problem Derive option prices in a market with a fundraiser and multiple solutions to the Black-Scholes equation.
method Model financial market with two types of agents, use Pitman's theorem for Bessel process, derive option prices using numerical scheme.
result Derive option prices for European options and call options in a market with a bubble.
Study optimal dividend strategies for insurers with natural catastrophe claims.
problem Maximizing dividends for a catastrophe insurer over its lifetime.
method Two-dimensional stochastic control problem, viscosity solutions, numerical approximation.
result Optimal dividend strategies identified for natural catastrophe insurers.
We prove existence of solutions to boundary value problems and obstacle problems for degenerate-elliptic, linear, second-order partial differential operators with partial Dirichlet boundary conditions using a new version of the Perron method. The elliptic operators considered have a degeneracy along a portion of the do…
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.
It was recently proved that embedded solutions of Euclidean hypersurface flows with speeds given by concave (convex), degree one homogeneous functions of the Weingarten map are interior (exterior) non-collapsing. These results were subsequently extended to hypersurface flows in the sphere and hyperbolic space. In the f…
Proves no minimal hypersurfaces in regions bounded by minimal cones.
problem Existence of minimal hypersurfaces in bounded regions.
method Analyzes minimal hypersurfaces in Euclidean space.
result No minimal hypersurfaces in regions bounded by unstable minimal cones.
Proves cones over minimal embeddings of R-spaces are area-minimizing.
problem Finding area-minimizing properties of cones over minimal embeddings.
method Constructing area-nonincreasing retractions.
result Proves cones over minimal embeddings of R-spaces are area-minimizing.
Study shows area-minimizing submanifolds are mostly smooth except for specific types.
problem Understanding when area-minimizing submanifolds are smooth in mod 2 homology.
method Proved area-minimizing submanifolds are not generically smooth except for geodesics, minimal surfaces, and minimal hypersurfaces.
result Area-minimizing submanifolds are not generically smooth, except for geodesics, minimal surfaces, and minimal hypersurfaces.
Study shows area-minimizing submanifolds are mostly smooth except for geodesics, minimal surfaces, and hypersurfaces.
problem Understanding when area-minimizing submanifolds are smooth in mod 2 homology.
method Proving the mod 2 area-minimizing submanifolds are smooth in specific cases and establishing lower bounds on singular sets.
result Area-minimizing submanifolds are not generically smooth, except for geodesics, minimal surfaces, and minimal hypersurfaces.
Minimal elastic networks minimize energy and length at fixed angles.
problem Finding optimal network configurations under elastic constraints.
method Minimizing a combination of elastic energy and length.
result Existence and regularity of minimizers with prescribed angles.
Explains minimal surfaces and their properties.
problem Understanding minimal surfaces and their characteristics.
method Analyzes various minimal surfaces and their properties.
result Discusses the properties and stability of minimal surfaces.
Minimal Lagrangian submanifolds deform to J-minimal ones under small perturbations.
problem Understanding how minimal Lagrangian submanifolds behave under small perturbations in Kaehler-Einstein manifolds.
method Analyzing the deformation of minimal Lagrangian submanifolds under Kaehler-Einstein perturbations.
result Deformed submanifolds remain J-minimal under certain conditions.