arXiv research

A theorem connects integral of second-order derivatives to function rise.

A dynamical system on the total space of the fibre bundle of second order accelerations, $T^2M$, is defined as a third order vector field $S$ on $T^2M$, called semispray, which is mapped by the second order tangent structure into one of the Liouville vector field. For a regular Lagrangian of second order we prove that …

Paper derives estimates for complex Hessian equations on Hermitian manifolds.

Derives derivatives of risk measures for various types of portfolio losses.

Schwarzian derivative connects to Euler-Lagrange equations in variational calculus.

In the present paper we consider the problem of local equivalence of second order ODEs which are cubic in second derivative under the action of the pseudogroup of contact transformations. We show how it may be reduced to the equivalence problem of 2-webs in $\mathbb{R}^3$ under the action of finite-dimensional group, a…

Simplified argument for second order estimate in quaternionic Calabi-Yau problem.

Enhances gradient estimates for Hermitian Monge-Ampère equations.

In this paper we will discuss some new developments in the design of numerical methods for optimal control problems of Lagrangian systems on Lie groups. We will construct these geometric integrators using discrete variational calculus on Lie groups, deriving a discrete version of the second-order Euler-Lagrange equatio…

Paper introduces FoMoH for optimization without backpropagation.

Enhances SMC² with Hessian info for more efficient posterior approximation.

The quantification of diversification benefits due to risk aggregation plays a prominent role in the (regulatory) capital management of large firms within the financial industry. However, the complexity of today's risk landscape makes a quantifiable reduction of risk concentration a challenging task. In the present pap…

SOLBP extends efficient inference to uncertain Bayesian networks.

In this paper we investigate the relations between semispray, nonlinear connection, dynamical covariant derivative and Jacobi endomorphism on Lie algebroids. Using these geometric structures, we study the symmetries of second order differential equations in the general framework of Lie algebroids.

Risk management in financial derivative markets requires inevitably the calculation of the different price sensitivities. The literature contains an abundant amount of research works that have studied the computation of these important values. Most of these works consider the well-known Black and Scholes model where th…

Paper examines risk measure expansions under FGM dependence, improving accuracy at extreme levels.

New method for linear connections in ODEs with constraints.

Paper proposes a new method for efficient second-order neural network training.

Exact second-order optimization for deep learning reduces computational cost and improves performance.

Multiscale stochastic volatility models have been developed as an efficient way to capture the principle effects on derivative pricing and portfolio optimization of randomly varying volatility. The recent book Fouque, Papanicolaou, Sircar and Sølna (2011, CUP) analyzes models in which the volatility of the underlying i…

Paper generalizes connections between Lie groups and affine connections.

This paper presents a geometric-variational approach to continuous and discrete {\it second-order} field theories following the methodology of \cite{MPS}. Staying entirely in the Lagrangian framework and letting $Y$ denote the configuration fiber bundle, we show that both the multisymplectic structure on $J^3Y$ as well…

In this paper we derive a second order approximation for an infinite dimensional limit order book model, in which the dynamics of the incoming order flow is allowed to depend on the current market price as well as on a volume indicator (e.g.~the volume standing at the top of the book). We study the fluctuations of the …

We present high-order compact schemes for a linear second-order parabolic partial differential equation (PDE) with mixed second-order derivative terms in two spatial dimensions. The schemes are applied to option pricing PDE for a family of stochastic volatility models. We use a non-uniform grid with more grid-points ar…

Paper proposes a second-order method for faster SVI convergence.

A quasi-Lie scheme is a geometric structure that provides t-dependent changes of variables transforming members of an associated family of systems of first-order differential equations into members of the same family. In this note we introduce two quasi-Lie schemes for studying second-order Gambier equations in a geome…

Derives a new formula for optimal stopping problems with exploding derivatives.

New superintegrable systems derived from Frobenius structures.

Estimates for complex Hessian equations on Hermitian manifolds.

New Lagrangian approach for optimal control of second-order systems.

SOAR improves deep networks' robustness against adversarial examples.

In the present paper we establish the necessary and sufficient conditions for two ordinary differential equations of the form $y"{}^2+A(x,y,y') y"+B(x,y,y')=0$ to be equivalent under the action of the pseudogroup of contact transformations. These conditions are formulated in terms of integrals of some one-dimensional d…

Differential quantities, including normals, curvatures, principal directions, and associated matrices, play a fundamental role in geometric processing and physics-based modeling. Computing these differential quantities consistently on surface meshes is important and challenging, and some existing methods often produce …

The paper is mainly devoted to systematic developments and applications of geometric aspects of second-order variational analysis that are revolved around the concept of parabolic regularity of sets. This concept has been known in variational analysis for more than two decades while being largely underinvestigated. We …

New expressions for Nijenhuis tensor squares found.

The class of second order ODE's cubic with respect to the first order derivative is considered. Using geometric structures associated with these equations, the subclasses of umbilical equations, zero mean curvature equations, and zero Gaussian curvature equations are defined. Zero mean curvature equations are studied w…

We derive a priori second order estimates for solutions of a class of fully nonlinear elliptic equations on Riemannian manifolds under some very general structure conditions. We treat both equations on closed manifolds, and the Dirichlet problem on manifolds with boundary without any geometric restrictions to the bound…

New optimality conditions for sub-Riemannian geodesics derived.

New algorithm finds approximate stationary points in non-convex optimization.

We develop a second-order model for limit order books in a single scaling regime.

We prove comparison, uniqueness and existence results for viscosity solutions to a wide class of fully nonlinear second order partial differential equations $F(x, u, du, d^{2}u)=0$ defined on a finite-dimensional Riemannian manifold $M$. Finest results (with hypothesis that require the function $F$ to be degenerate ell…

As a difference with the positive-definite Riemannian case, in the Lorentzian case there exists proper second-order symmetric spacetimes, i.e., those with vanishing second covariant derivative of the Riemannian tensor ($R_{λμνρ;α;β}=0$) which are not locally symmetric ($R_{λμνρ;α}\neq 0$). In fact, they lie in the clas…

We derive a priori $C^2$ estimates for the $χ$-plurisubharmonic solutions of general complex Hessian equations with right-hand side depending on gradients.

Paper uses second-order differential geometry to study stochastic mechanics.

Lorentzian manifolds with vanishing second covariant derivative of the Riemann tensor are studied. Their existence, classification and explicit local expression are considered. Related issues and open questions are briefly commented.

Investigates second-order conditions for Cayley forms in eight dimensions.

We define a set of "second-order" L^(2)-signature invariants for any algebraically slice knot. These obstruct a knot's being a slice knot and generalize Casson-Gordon invariants, which we consider to be "first-order signatures". As one application we prove: If K is a genus one slice knot then, on any genus one Seifert …

Local search heuristics for non-convex optimizations are popular in applied machine learning. However, in general it is hard to guarantee that such algorithms even converge to a local minimum, due to the existence of complicated saddle point structures in high dimensions. Many functions have degenerate saddle points su…