Maximal regularity for nonuniformly parabolic problems with normal degeneration.
problem Nonuniformly parabolic boundary value problems with degeneration in normal direction.
method Theory of linear parabolic differential equations on noncompact Riemannian manifolds.
result Optimal solution theory for natural degeneration case.
We present an analytic approach to solve a degenerate parabolic problem associated to the Heston model, which is widely used in mathematical finance to derive the price of an European option on an risky asset with stochastic volatility. We give a variational formulation, involving weighted Sobolev spaces, of the second…
Study symplectic invariants of parabolic orbits and cuspidal tori in integrable systems.
problem Understanding symplectic invariants of degenerate singularities in integrable systems.
method Normal forms and new techniques for studying symplectic invariants.
result New insights into symplectic invariants of degenerate singularities.
The paper proves Hölder continuity for solutions of degenerate parabolic equations in any dimension.
problem Proving Hölder continuity for solutions of degenerate parabolic equations in arbitrary dimensions.
method Establishing Alexandroff-Bakelman-Pucci estimate, Harnack inequality, Hölder regularity, and Schauder estimates for a class of degenerate parabolic equations.
result The paper proves Hölder continuity for solutions of degenerate parabolic equations in all dimensions.
Smooth solutions up to evolving free boundaries for degenerate equations.
problem Degenerate parabolic equations with evolving free boundaries.
method Smooth short-time existence using linear degenerate equations on a fixed domain.
result Smoothness up to the free boundary for the p-Laplacian evolution equation and α-Gauss curvature flow. Unified diffusive bounds for non-linear parabolic equations.
problem Proving diffusive upper bounds for parabolic equations.
method Simple exponential deformation argument.
result Unified diffusive upper bounds for a wide class of non-linear parabolic equations.
Non-unique option pricing in Heston model analyzed mathematically.
problem Non-uniqueness of call option prices in the Heston model.
method Analysis of degenerate parabolic equations in the context of option pricing.
result Construction of a new example demonstrating the accuracy of a uniqueness theorem.
We prove the Bers' density conjecture for singly degenerate Kleinian surfaces groups without parabolics.
Study on solutions of degenerate equations on manifolds, linking behavior to geometry and decay rates.
problem Behavior of solutions to degenerate parabolic equations on manifolds with inhomogeneous density.
method Analysis of Cauchy problem on Riemannian manifolds, considering weight function as capacitary coefficient.
result Estimates of vanishing rate and finite speed of propagation in subcritical ranges, universal bounds and blow-up in supercritical ranges.
We show that Cannon-Thurston maps exist for degenerate free groups without parabolics, i.e. for handlebody groups. Combining these techniques with earlier work proving the existence of Cannon-Thurston maps for surface groups, we show that Cannon-Thurston maps exist for arbitrary finitely generated Kleinian groups witho…
The paper connects hyperpolygon spaces to Higgs bundle moduli spaces via degenerations.
problem Modeling hyperkähler 4-manifolds and their degenerations.
method Using parabolic SL(2,C)-Higgs bundles and Nakajima quiver varieties.
result ALG-D4 spaces degenerate to ALE-D4 spaces under a limit. Using results from our companion article [arXiv:1112.4824v2] on a Schauder approach to existence of solutions to a degenerate-parabolic partial differential equation, we solve three intertwined problems, motivated by probability theory and mathematical finance, concerning degenerate diffusion processes. We show that th…
This work is focused on the solvability of initial-boundary value problems for degenerate parabolic partial differential equations that arise in the pricing of Asian options, and on the investigation of differential and certain qualitative properties of solutions of such equations. The generalized solvability for such …
Motivated by applications to probability and mathematical finance, we consider a parabolic partial differential equation on a half-space whose coefficients are suitably Holder continuous and allowed to grow linearly in the spatial variable and which become degenerate along the boundary of the half-space. We establish e…
Develops theory for Kähler-Ricci flow on singular varieties.
problem Analyzing Kähler-Ricci flow on varieties with log terminal singularities.
method Parabolic pluripotential theory and complex Monge-Ampère equations.
result Establishes a parabolic theory analogous to Bedford-Taylor's.
Study on Higgs bundles and hyperpolygon spaces using Hitchin metrics.
problem Investigating the Hitchin metric on moduli spaces of Higgs bundles.
method Using Hitchin hyperkähler metric and parabolic Deligne-Hitchin moduli space.
result Rescaled Hitchin metric converges to hyperpolygon space's hyperkähler metric in the semiclassical limit.
Study on parabolic points and cylindrical surfaces in Euclidean 3-space.
problem Characterizing parabolic points and their geometric properties.
method Introducing contact cylindrical surfaces and analyzing their properties.
result Characterization of A-singularity through projections. In this work there is established an optimal existence and regularity theory for second order linear parabolic differential equations on a large class of noncompact Riemannian manifolds. Then it is shown that it provides a general unifying approach to problems with strong degeneracies in the interior or at the boundary…
Solves complex equation with singularities using transformations and numerical methods.
problem Solving a semilinear parabolic HJB equation with a singular initial condition.
method Transformed the equation to remove singularity, then constructed numerical schemes.
result Proved convergence of numerical schemes for the transformed equation.
Studying the (long-term) behavior of the Kähler-Ricci flow on mildly singular varieties, one is naturally lead to study weak solutions of degenerate parabolic complex Monge-Ampére equations. The purpose of this article, the second of a series on this subject, is to develop a viscosity theory for degenerate complex Mong…
In this paper we study a novel class of parabolic geometries which we call parabolic geometries of Monge type. These parabolic geometries are defined by special gradings of simple Lie algebras, namely, gradings with the property that their -1 component contains a nonzero co-dimension 1 abelian subspace whose bracket wi…
Non-uniqueness found in option valuation for certain α values.
problem Non-uniqueness in the value of call options for specific α values.
method Mathematical theory of degenerate parabolic equations and boundary conditions.
result Non-uniqueness explained by initial data outside Täcklind class and absence of boundary condition at infinity.
We prove long time existence and convergence results for the pluriclosed flow, which imply geometric and topological classification theorems for generalized Kähler structures. Our approach centers on the reduction of pluriclosed flow to a degenerate parabolic equation for a (1,0)-form, introduced in \cite{ST2}. We ob…
Study on Cauchy problem solutions and inverse problems for PDEs.
problem Analyzing the domain of solutions for a specific type of PDE.
method Examined a second-order quasi-linear PDE with parabolic degeneration, studied inverse problems, and provided conditions for solutions.
result The domain of the solution contains gaps under certain conditions.
A new spinorial heat flow framework studies geometric degeneration on 3-manifolds.
problem Analyzing geometric degeneration on 3-manifolds via spinor dynamics.
method Introducing a spinorial heat flow governed by the squared Dirac operator, where the metric is induced conformally by the spinor amplitude.
result Degeneration of the induced metric corresponds to nodal behavior of the spinor field.
Develops parabolic pluripotential theory for complex flows.
problem Complex Monge-Ampère equations in degenerate settings.
method Study of semi-concave envelopes and unique solutions.
result Shows semi-concave envelopes as unique solutions.
Study on totally geodesic planes in hyperbolic 3-manifolds with specific properties.
problem Characterizing totally geodesic planes in hyperbolic 3-manifolds with degenerate ends.
method Ratner-type phenomenon and finite surface count approach.
result Closed minimal invariant subsets are either immersed totally geodesic surfaces or the entire manifold.
We study the stochastic solution to a Cauchy problem for a degenerate parabolic equation arising from option pricing. When the diffusion coefficient of the underlying price process is locally Hölder continuous with exponent δ∈(0,1], the stochastic solution, which represents the price of a European option, is show…
For the system of second order quasilinear parabolic equations the problem of reducing them to the equations of diffusion type is considered. In non-degenerate case an effective algorithm for solving this problem is suggested.
Study bifurcations of curves on surfaces in Minkowski 3-space.
problem Understanding the behavior of curves on surfaces in Minkowski 3-space.
method Analyzing the degeneracy of induced pseudo metric, discriminant of principal curvatures, parabolic curve, and mean curvature vanishing points.
result Bifurcations of robust features on surfaces in Minkowski 3-space.
The paper studies Möbius inversion on surfaces in Minkowski 3-space.
problem Understanding transformations of surfaces in Minkowski space.
method Definition and properties of Möbius inversion on surfaces in Minkowski 3-space.
result Möbius inversion preserves lines of principal curvature and degenerate metric points but not parabolic sets.
IMEX schemes preserve positivity for European options with liquidity shocks.
problem Modeling European options with liquidity shocks.
method Two implicit-explicit (IMEX) schemes that preserve positivity are constructed and analyzed.
result The schemes confirm high accuracy and efficiency with Richardson extrapolation.
Local limit theorem for random walks on hyperbolic groups with parabolic subgroups.
problem Analyzing the behavior of random walks on relatively hyperbolic groups.
method Study of convergent random walks with finite derivative of Green function at spectral radius.
result Proves a local limit theorem for the probability of returning to the origin.
Paper introduces a new metric for deforming surfaces with parabolics.
problem Deformation spaces of quasifuchsian groups with parabolics.
method Developed a mapping class group invariant pressure metric on QF(S).
result Hausdorff dimension of limit sets varies analytically over QF(S).
Study curve shortening flow on Riemann surfaces with conical singularities.
problem Evolution of curves on Riemann surfaces with singular points.
method Curve shortening flow governed by a degenerate quasilinear parabolic equation.
result Evolving curves stay fixed at singular points and show collapsing and convergence results.
The paper proves formulas for solving certain types of stochastic problems.
problem Solving boundary value and obstacle problems for degenerate elliptic and parabolic equations.
method Uses Feynman-Kac formulas for a general Markov diffusion process with degenerate elliptic generator.
result Provides unique solutions for smooth and non-smooth cases under Dirichlet boundary conditions.
This paper proves unique determination of certain non-quasi-Fuchsian manifolds by their end structure and bending lamination.
problem Unique determination of non-quasi-Fuchsian manifolds by their end structure and bending lamination.
method Analysis of the end structure (parabolic locus, ending laminations, conformal structures) and bending lamination.
result Non-quasi-Fuchsian manifolds are uniquely determined by their end structure and bending lamination.
Finite element method for SABR model pricing under various interest rates.
problem Pricing vanilla and barrier options under the SABR stochastic volatility model.
method Finite element discretization of non-symmetric Dirichlet forms for degenerate parabolic equations.
result Well-posedness of the variational formulation and error analysis for finite element discretization.
Backward stochastic partial differential equations of parabolic type in bounded domains are studied in the setting where the coercivity condition is not necessary satisfied and the equation can be degenerate. Some generalized solutions based on the representation theorem are suggested. In addition to problems with a st…
Study of light function singularities on surfaces.
problem Characterizing singularities of the slant function on surfaces.
method Analyzing the differential geometry of the parabolic set and its spherical image under the Gauss map.
result The type of singularities of the slant function is determined by the geometry of the parabolic set and its spherical image.
This thesis is concerned with the theory of invariant bilinear differential pairings on parabolic geometries. It introduces the concept formally with the help of the jet bundle formalism and provides a detailed analysis. More precisely, after introducing the most important notations and definitions, we first of all giv…
Paper proves well-posedness of PDE for option pricing in a complex market model.
problem Derivative pricing in a generalized market model with non-local parabolic PDE.
method Used Volterra integral equation and probabilistic approach with conditioning on stopping times.
result Proved well-posedness of the initial-boundary value problem for the PDE.
In this paper we study a general framework of American put option with stochastic volatility whose value function is associated with a 2-dimensional parabolic variational inequality with degenerate boundaries. We apply PDE methods to analyze the existences of the strong solution and the properties of the 2-dimensional …
According to an analogy to quasi-Fuchsian groups, we investigate topological and combinatorial structures of Lyubich and Minsky's affine and hyperbolic 3-laminations associated with the hyperbolic and parabolic quadratic maps. We begin by showing that hyperbolic rational maps in the same hyperbolic component have quasi…
Anisotropic obstacle problems and Stefan problem studied with evolving surfaces.
problem Anisotropic parabolic obstacle problems and Stefan problem.
method Cahn-Hoffman transform and anisotropic mean curvature flow.
result Optimal regularity of the solution and C1,α-regularity of the evolving free boundary. Study on singularities of Chern-Ricci flow on complex manifolds.
problem Understanding finite-time singularities of the Chern-Ricci flow.
method Extending Guedj-Lu's approach to establish uniform a priori estimates for degenerate complex Monge-Ampère equations, applied to Chern-Ricci flows on complex log terminal varieties.
result Showed solutions starting from positive currents are smooth outside some analytic subset.
The paper explains how to realize Kossowski metrics as wave front germs.
problem Realizing Kossowski metrics as wave front germs at non-parabolic singular points.
method Using coherent tangent bundles and Kossowski's realization theorem.
result Criteria for realizing Kossowski metrics as induced metrics of singularities.
We develop an option pricing model based on a tug-of-war game. This two-player zero-sum stochastic differential game is formulated in the context of a multi-dimensional financial market. The issuer and the holder try to manipulate asset price processes in order to minimize and maximize the expected discounted reward. W…