Authors improve accuracy analysis for portfolio optimization with multiple timescale factors.
problem Asymptotic accuracy of portfolio optimization approximations for general utility functions and two timescale factors.
method Construct sub- and super-solutions to fully nonlinear problem.
result Rigorous justification of accuracy for portfolio optimization with general utility functions and two timescale factors.
We find a simple strategy approximating optimal portfolio for short time horizons.
problem Optimizing portfolios in incomplete markets with general utility functions.
method Closed-form formula derived from HJB PDE, approximated by sub- and super-solutions.
result Approximation formula for optimal trading strategy is accurate for small time horizons.
Optimizes portfolio in volatile markets with jumps, providing accurate formulas.
problem Optimizing wealth in a volatile financial market with jumps.
method Analyzes an incomplete stochastic volatility model, derives closed-form portfolio formulas using HJB equation and super-solution/sub-solution.
result Proves accuracy of derived portfolio formulas for both small and finite time horizons.
The strong maximum principle is proved to hold for weak (in the sense of support functions) sub- and super-solutions to a class of quasi-linear elliptic equations that includes the mean curvature equation for C0 spacelike hypersurfaces in a Lorentzian manifold. As one application a Lorentzian warped product splittin…
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.
Optimizes investment portfolios with multiple correlated volatility factors.
problem Maximizing utility in a stochastic environment with multiple correlated volatility factors.
method Perturbation technique around perfectly correlated factors, reducing to single factor problem; numerical solution of linear equations.
result Approximation method reduces complexity of fully non-linear HJB equation to linear equations in lower dimension.
We extend the stochastic Perron method to analyze the framework of stochastic target games, in which one player tries to find a strategy such that the state process almost surely reaches a given target no matter which action is chosen by the other player. Within this framework, our method produces a viscosity sub-solut…
Optimizes trading in markets with unpredictable price impacts.
problem Optimizing trading strategies in markets with stochastic price impacts.
method Singular perturbation methods to approximate optimal control problem.
result Proves approximations are accurate to specified order using sub- and super-solutions.
In this paper, we prove the existence of classical solutions of the Dirichlet problem for a class of quasi-linear elliptic equations on unbounded domains like a cone or a U-type domain. This problem comes from the study of mean curvature flow and its generalization, the flow by powers of mean curvature. Our approach is…
In this paper, we adapt stochastic Perron's method to analyze a stochastic target problem with unbounded controls in a jump diffusion set-up. With this method, we construct a viscosity sub-solution and super-solution to the associated Hamiltonian-Jacobi-Bellman (HJB) equations. Under comparison principles, uniqueness o…
The paper tests hypotheses on two Lévy process-driven streams of observations.
problem Testing hypotheses on two Lévy process-driven streams of observations.
method Infinitesimal generators and super/sub-solutions are used to compute bounds and analyze the model.
result Bounds for infinitesimal generators are computed in terms of super/sub-solutions.
The paper presents a method for detecting jump sizes in crude oil prices.
problem Detecting jump sizes in crude oil price data.
method Sequential hypothesis testing using infinitesimal generators and super-solutions.
result The method improves the Barndorff-Nielsen and Shephard model for derivative and commodity market analysis.
Viscosity solutions are suitable notions in the study of nonlinear PDEs justified by estimates established via the maximum principle or the comparison principle. Here we prove that the isoperimetric profile functions of Riemannian manifolds with Ricci lower bound are viscosity super-solutions of some nonlinear differen…
We employ three different methods to prove the following result on prescribed scalar curvature plus mean curvature problem: Let (Mn,g0) be a n-dimensional smooth compact manifold with boundary, where n≥3, assume the conformal invariant Y(M,∂M)<0. Given any negative smooth functions f in M and…
Paper proves rigidity for compact manifolds with boundary.
problem Rigidity of compact connected locally conformally flat manifolds.
method Analyzes fully nonlinear elliptic conformally invariant equations.
result Compact connected locally conformally flat manifolds are isometric to closed geodesic balls.
In this paper we apply techniques from optimal transport to study the neckpinch examples of Angenent-Knopf which arise through the Ricci flow on Sn+1. In particular, we recover their proof of 'single-point pinching' along the flow. Using the methods of optimal transportation, we are able to remove the ass…
Solves boundary Yamabe problem with minimal boundary scenario.
problem Existence of solutions to a conformal metric equation with constant scalar curvature.
method Iteration schemes and perturbation methods to construct sub-solutions and super-solutions.
result Complete solution for three cases based on the sign of the first eigenvalue of the conformal Laplacian.
Study surfaces with nonnegative curvature in spectral sense, proving inequalities and bounds.
problem Closed orientable surfaces with nonnegative curvature in spectral sense.
method Spectral condition and associated conformal metrics to prove inequalities and bounds.
result Isoperimetric inequalities, area growth theorems, and diameter bounds for surfaces.
New method for handling multi-dimensional singular controls with jump costs in mean-field problems.
problem Handling jump costs in multi-dimensional singular controls.
method Introducing two-layer parametrisations to interpolate jumps on both distributional and pathwise levels.
result Derivation of a DPP and characterisation of the value function as a minimal super-solution to a quasi-variational inequality.
Study compares H-type sub-Riemannian manifolds using uniform metrics.
problem Comparing H-type sub-Riemannian manifolds with Riemannian metrics.
method Establishes sub-Hessian and sub-Laplacian comparison theorems for a family of approximating Riemannian metrics.
result Proves a sharp sub-Riemannian Bonnet-Myers theorem.
Maps commuting with sub-Laplacians on Carnot groups are conformal.
problem Characterizing maps preserving sub-Laplacians on sub-Riemannian Lie groups.
method Analyzing smooth maps between sub-Riemannian Lie groups that commute with sub-Laplacians.
result Sub-Laplacian determines the sub-Riemannian structure in Carnot groups.
Study of intrinsic sub-Laplacian for hypersurfaces in contact sub-Riemannian manifolds.
problem Characterizing the intrinsic sub-Laplacian for hypersurfaces in contact sub-Riemannian manifolds.
method Construction and analysis of the intrinsic sub-Laplacian using Riemannian approximations and stochastic processes.
result The intrinsic sub-Laplacian is stochastically complete, ensuring the process does not hit characteristic points.
The authors first in this paper define a semi-symmetric metric non-holonomic connection (called in briefly a semi-sub-Riemannian connection) on sub-Riemannian manifolds, and study the relations between sub-Riemannian connections and semi-sub-Riemannian connections. An invariant under a connection transformation $\nabla…
Sub-Riemannian spectral distance defined using eigenfunctions of sub-Laplacian
problem Sub-Riemannian geometry and eigenvalues of sub-Laplacian
method Embedding manifold into Hilbert space using eigenfunctions
result Defined spectral distance between sub-Riemannian manifolds
Essential self-adjointness proven for sub-Laplacians on sub-Riemannian manifolds.
problem Proving essential self-adjointness for sub-Laplacians on sub-Riemannian manifolds.
method General essential self-adjointness criterion for sub-Laplacians on complete sub-Riemannian manifolds, defined with respect to singular measures.
result Intrinsic sub-Laplacian is essentially self-adjoint on equiregular connected components of a sub-Riemannian manifold under mild regularity assumptions.
Study of sub-Riemannian cubics in SU(2) for long-term dynamics.
problem Optimizing curves in a sub-Riemannian manifold with constraints.
method Analysis of sub-Riemannian Lie quadratics in SU(2).
result Characterization of sub-Riemannian Lie quadratics in SU(2).
Proves new concentration inequalities for sub-gaussian and sub-exponential variables.
problem Understanding functions of independent random variables better.
method Sub-gaussian and sub-exponential conditions, Rademacher complexities, Lipschitz function classes.
result Extension of Rademacher complexities to unbounded sub-exponential distributions.
Introduce sub-Randers metrics by adding a one-form to a sub-Riemannian metric
problem Define a new class of sub-Finsler metrics
method Derive equations for sub-Randers normal geodesics
result Prove a Hopf-Rinow type theorem for sub-Randers manifolds
The paper proves inequalities for optimal transport on sub-Finslerian manifolds.
problem Optimal transport inequalities on sub-Finslerian manifolds.
method Introduction of sub-Finslerian Jacobi fields and optimal transport theory.
result Characterization of generalized distortion coefficients and fundamental geometric inequalities.
Study compares sub-Riemannian curvature to optimal control variational problems.
problem Comparing sub-Riemannian curvature to optimal control variational problems.
method Introducing sub-Riemannian Bakry-Émery curvature and proving sub-Laplacian comparison theorems.
result Established sharp measure contraction property for 3-Sasakian manifolds.
Study on nonholonomic mechanics and sub-Finsler geometry.
problem Understanding nonholonomic mechanical systems and their geometric properties.
method Variational approach, sub-Finsler manifolds, nonholonomic sub-Finslerian structure.
result Existence and properties of extremals in nonholonomic mechanics.
Derives sub-Riemannian Ricci curvature for various manifolds.
problem Calculating Ricci curvature in sub-Riemannian geometry.
method Generalized Gamma z calculus and z--Bochner's formula. result Analytical bounds for sub-Riemannian curvature dimension and log-Sobolev inequalities.
Study solves sub-Laplacian equivalence on a specific Heisenberg group.
problem Contact equivalence problem for sub-Laplacians on the second Heisenberg group.
method Solves the contact equivalence problem for generalised sub-Laplacians on $\He^2$.
result Parameterises sub-Laplacians on $\He^2$ by R+. We study sub-Riemannian and sub-Lorentzian geometry on the Lie group $\SU(1,1)$ and on its universal cover $\CSU(1,1)$. In the sub-Riemannian case we find the distance function and completely describe sub-Riemannian geodesics on both $\SU(1,1)$ and $\CSU(1,1)$, connecting two fixed points. In particular, we prove that …
Study on curvature exponent of sub-Finsler Heisenberg groups, proving N_min ≥ 5.
problem Determining the curvature exponent of sub-Finsler Heisenberg groups.
method Analyzing the measure contraction property and constructing sub-Finsler structures.
result Proved that curvature exponent N_min ≥ 5, with equality if sub-Riemannian.
Study optimal transport in 4D sub-Riemannian spaces with many singular geodesics.
problem Existence and uniqueness of optimal transport maps in sub-Riemannian structures.
method Analysis of Monge optimal transport problem in sub-Riemannian manifolds.
result Extension of previous results to sub-Riemannian structures of rank two in 4D.
SDSM extracts statistically significant sub-trajectories from large trajectory datasets.
problem Discerning moving patterns that are more characteristic of one group of trajectories than another.
method Statistically Discriminative Sub-trajectory Mining (SDSM) method using tree representation and permutation-based statistical inference.
result SDSM efficiently extracts statistically significant sub-trajectories from massive trajectory datasets.
We study eigenvalue problems for intrinsic sub-Laplacians on regular sub-Riemannian manifolds. We prove upper bounds for sub-Laplacian eigenvalues λk of conformal sub-Riemannian metrics that are asymptotically sharp as k→+∞. For Sasakian manifolds with a lower Ricci curvature bound, and more generally, for…
Sub-Riemannian Geometry is proved to play an important role in many applications, e.g., Mathematical Physics and Control Theory. The simplest example of sub-Riemannian structure is provided by the 3-D Heisenberg group. Sub-Riemannian Geometry enjoys major differences from the Riemannian being a generalisation of the la…
Study on extremals in sub-Lorentzian geometry defined by antinorm.
problem Characterizing extremals in sub-Lorentzian structures.
method Deriving Hamiltonian system and conditions for extremal trajectories.
result Conditions for normal extremal trajectories and properties of abnormal extremals.
Maps on Sasakian manifolds limit to sub-Riemannian distance bounds.
problem Understanding sub-Riemannian distances on Sasakian manifolds.
method Parallel and mirror maps along geodesics of a taming Riemannian metric.
result Limits of transport maps outside sub-Riemannian cut-locus provide bounds on sub-Riemannian distance.
We consider different sub-Laplacians on a sub-Riemannian manifold M. Namely, we compare different natural choices for such operators, and give conditions under which they coincide. One of these operators is a sub-Laplacian we constructed previously in \cite{GordinaLaetsch2014a}. This operator is canonical with respec…
Study calculates volume of small sub-Riemannian balls in 3D contact manifolds.
problem Computing the volume of small sub-Riemannian balls in 3D contact manifolds.
method Asymptotic expansion and geometric invariants of the sub-Riemannian structure.
result Expressed first geometric coefficients in terms of sub-Riemannian structure invariants.
Harmonic maps studied in sub-Riemannian geometry for Lie groups.
problem Defining and characterizing harmonic maps in sub-Riemannian settings.
method Generalization of Riemannian harmonic maps to sub-Riemannian manifolds and Lie groups.
result Conditions for sub-Riemannian harmonic maps and their classification.
Note shows Cheng-Yau estimate for Carnot groups and sub-Riemannian manifolds.
problem Proving Cheng-Yau gradient estimate for specific geometric structures.
method Utilizing previous results on sub-Riemannian manifolds and Carnot groups.
result Established Cheng-Yau estimate for Carnot groups and sub-Riemannian manifolds.
The Hopf-Rinow theorem is extended to sub-Finslerian geometry.
problem Extending the Hopf-Rinow theorem to sub-Finslerian manifolds.
method Investigation of sub-Finslerian bundle, exponential map, and Legendre transformation.
result Established a relation between completeness, geodesic completeness, and compactness in sub-Finslerian geometry.
Two flat sub-Lorentzian problems on Martinet distribution differ in attainable set intersections.
problem Flat sub-Lorentzian structures on Martinet distribution.
method Analysis of attainable sets, optimal trajectories, sub-Lorentzian distances and spheres.
result The attainable set for the first problem intersects with the Martinet plane, while for the second it does not.
Constant curvature models in sub-Riemannian geometry are explored.
problem Understanding constant curvature models in sub-Riemannian geometry.
method Cohomological description of principal invariants for parabolic geometries.
result Constant curvature models are discussed for specific sub-Riemannian geometries.