arXiv research

Abstract: Survey on quadratic Hessian equations, their properties, and open problems.

Solutions to a quadratic matrix equation are linked to strongly regular graphs and multiplicative characters.

I use harmonic maps and minimal surfaces to study quadratic equations in groups.

We provide explicit solutions of certain forward-backward stochastic differential equations (FBSDEs) with quadratic growth. These particular FBSDEs are associated with quadratic term structure models of interest rates and characterize the zero-coupon bond price. The results of this paper are naturally related to simila…

The paper proves constant rank theorems for special Lagrangian equations.

Developed a theory of local convexity for second order differential equations on Lie algebroids.

Optimal contracts are found for agents with quadratic effort costs.

In this note, we derive a Liouville theorem for the complex Monge-Ampère equation. Our result states that if the global solution $u$ of the complex Monge-Ampère equation with constant right-hand side differs from a quadratic polynomial solution by $o(\abs{x}^2)$ at infinity, then $u$ is a quadratic polynomial.

In this note we find a 6-dimensional h-spaces of the $[(21...1)(21...1)...(1...1)]$ type and then determine quadratic first integrals of the geodesic equations of these h-spaces.

Solutions near infinity to special Lagrangian equations are asymptotic to quadratic polynomials with logarithmic terms.

Deep learning solves high-dimensional quadratic hedging problems.

Study solves HJB equations for time-inconsistent control problems.

We show that the linear trace Harnack quadratic on a steady gradient Ricci soliton satisfies the heat equation. Similar result holds for shrinkers. We also present an interpolation between Perelman's and Cao--Hamilton's Harnacks on a steady soliton.

On a complete non-compact gradient shrinking Ricci soliton, we prove the analyticity in time for smooth solutions of the heat equation with quadratic exponential growth in the space variable. This growth condition is sharp. As an application, we give a necessary and sufficient condition on the solvability of the backwa…

Building on previous results on the quadratic helicity in magnetohydrodynamics (MHD) we investigate particular minimum helicity states. Those are eigenfunctions of the curl operator and are shown to constitute solutions of the quasi-stationary incompressible ideal MHD equations. We then show that these states have inde…

Given a space it is easy to obtain the system of geodesic equations on it. In this paper the inverse problem of reconstructing the space from the geodesic equations is addressed. A procedure is developed for obtaining the metric tensor from the Christoffel symbols. The procedure is extended for determining if a second …

Paper derives estimates for Hessian equations under concavity assumptions.

We calculate explicitly the quadratic solution to the WDVV equations corresponds to the quasi-Coxeter conjugacy class $E_8(a_1)$ using the associated classical $W$-algebra.

Study proves existence of equilibrium in incomplete economies with discontinuous volatility.

A quadratic line complex is a three-parameter family of lines in projective space P^3 specified by a single quadratic relation in the Plucker coordinates. Fixing a point p in P^3 and taking all lines of the complex passing through p we obtain a quadratic cone with vertex at p. This family of cones supplies P^3 with a c…

This paper solves minimal surface equations near Hardt-Simon foliations.

We show that any global solution to the special Lagrangian equations with the phase larger than a critical value must be quadratic.

We develop algorithms for the numerical computation of the quadratic hedging strategy in incomplete markets modeled by pure jump Markov process. Using the Hamilton-Jacobi-Bellman approach, the value function of the quadratic hedging problem can be related to a triangular system of parabolic partial integro-differential…

New theory extends LQ control to non-exponential discount scenarios.

Study shows uniqueness of solutions on complex manifolds without requiring solution decay.

Study shows certainty equivalent policy minimizes regret in continuous-time systems.

New BDEs reveal singular surfaces from line congruences.

This research proves that quadratic regularized optimal transport can approximate the Laplace-Beltrami operator on smooth manifolds.

The paper classifies a specific type of quadratic variety with a small codimension.

Model liquidity premia using a risk-sharing economy with quadratic costs.

Analyzes how quadratic differential trajectories change with variation, proving a wall-crossing formula.

In this paper we study a continuous-time stochastic linear quadratic control problem arising from mathematical finance. We model the asset dynamics with random market coefficients and portfolio strategies with convex constraints. Following the convex duality approach, we show that the necessary and sufficient optimalit…

This article proposes a new approximation scheme for quadratic-growth BSDEs in a Markovian setting by connecting a series of semi-analytic asymptotic expansions applied to short-time intervals. Although there remains a condition which needs to be checked a posteriori, one can avoid altogether time-consuming Monte Carlo…

This paper optimizes portfolio selection for multivariate affine and quadratic Volterra models with rough volatilities.

We consider the optimal control problem for a linear conditional McKean-Vlasov equation with quadratic cost functional. The coefficients of the system and the weigh-ting matrices in the cost functional are allowed to be adapted processes with respect to the common noise filtration. Semi closed-loop strategies are intro…

This is a paper based on a talk given at the conference on Conformal Geometry which held at Roscoff in France in the 2008 summer. We study some aspects of the equation arising from the problem of the existence on a given closed Riemannian manifold of dimension at leat 4, of a conformal metric with constant $h_4$ curvat…

The authors prove that the logarithmic Monge-Ampère flow with uniformly bound and convex initial data satisfies uniform decay estimates away from time $t=0$. Then applying the decay estimates, we conclude that every entire classical strictly convex solution of the equation {equation*} \det D^{2}u=\exp\{n(-u+1/2\sum_{i=…

Analyticity of heat equation extended to Bakry-Émery Ricci curvature manifolds.

A Gaussian Process Ordinary Differential Equation framework for large continuous dynamical systems

Paper solves time-inconsistent control problems with BSDEs.

We derive a Bernstein type result for the special Lagrangian equation, namely, any global convex solution must be quadratic. In terms of minimal surfaces, the result says that any global minimal Lagrangian graph with convex potential must be a hyper-plane.

We deal with quadratic metric-affine gravity (QMAG), which is an alternative theory of gravity and present a new explicit representation of the field equations of this theory. In our previous work we found new explicit vacuum solutions of QMAG, namely generalised pp-waves of parallel Ricci curvature with purely tensor …

QENDy learns quadratic dynamics from nonlinear systems data.

We consider a general time-inconsistent stochastic linear-quadratic differential game. The time-inconsistency arises from the presence of quadratic terms of the expected state as well as state-dependent term in the objective functionals. We define an equilibrium strategy, which is different from the classical one, and …

Paper maps Hamiltonians and line elements in manifolds.

Optimal trading strategy with predictor and costs, derived equations and shape.

Minimax solutions are weak solutions to Cauchy problems involving Hamilton--Jacobi equations, constructed from generating families quadratic at infinity of their geometric solutions. We give a complete description of minimax solutions and we classify their generic singularities of codimension not greater than 2.

Study Einstein warped-product manifolds with specific curvature conditions.