Maximal solution of a PDE shows boundary smoothness for certain domains.
problem Boundary behavior of solutions to a specific PDE.
method Reduction to a nonlinear Fuchsian elliptic PDE.
result Hyperbolic radius is smooth up to the boundary.
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48 results for “PDE arguments”
Proves a conjecture about Riemann surfaces using PDEs.
problem Griffiths' conjecture on holomorphic vector bundles on compact Riemann surfaces.
method Combines techniques from Uhlenbeck-Yau and Pingali's reduction to prove a system of PDEs.
result Analytic proof of Griffiths' conjecture on compact Riemann surfaces.
Improved flatness in annuli using PDE methods.
problem Flatness improvement in annuli.
method PDE-based approach adapted to exterior domains.
result Alternative proof of minimal surface end-structure and asymptotics.
Kernel for Lévy rough paths derived from PDE system.
problem Computing similarity measures for Lévy rough paths.
method Developed a PDE system for the expected signature of inhomogeneous Lévy processes.
result Gaussian martingales' expected signature kernel satisfies a Goursat PDE.
Sacks-Uhlenbeck's result on metric spaces expanded.
problem Existence of non-trivial harmonic 2-spheres in metric spaces.
method Developed a metric approach to generalize Sacks-Uhlenbeck's result.
result Generalized Sacks-Uhlenbeck's result to a broader class of compact metric spaces.
Of all real Lagrangian--Grassmannians , only admits a distinguished (Lorentzian) conformal structure and hence is identified with the indefinite M\"obius space . Using Cartan's method of moving frames, we study hyperbolic (timelike) surfaces in modulo the conformal symplectic gro…
Unified estimate for complex Monge-Ampère equations on Kähler manifolds.
problem Estimating solutions to complex Monge-Ampère equations on Kähler manifolds.
method Unified approach using PDE methods and entropy bounds to construct comparison metrics.
result Improves previous results on modulus of continuity, stability, and -estimates of Green's functions.
We show that the group of isometries (i.e., distance-preserving homeomorphisms) of an equiregular subRiemannian manifold is a finite-dimensional Lie group of smooth transformations. The proof is based on a new PDE argument, in the spirit of harmonic coordinates, establishing that in an arbitrary subRiemannian manifold …
In this article, we show how the scaling symmetry of the SABR model can be utilized to efficiently price European options. For special kinds of payoffs, the complexity of the problem is reduced by one dimension. For more generic payoffs, instead of solving the 1+2 dimensional SABR PDE, it is sufficient to solve u…
American option of stochastic volatility model with negative Fichera function on degenerate boundaryq-fin.PR
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 …
Cake wavelets minimize orientation score uncertainty.
problem Minimizing uncertainty in orientation scores.
method Axiomatically derived wavelets for orientation score lifting.
result Uncertainty gap of cake wavelets is less than 1.1.
Asians and cash dividends: Exploiting symmetries in pricing theorycond-mat.stat-mech
In this article we present new results for the pricing of arithmetic Asian options within a Black-Scholes context. To derive these results we make extensive use of the local scale invariance that exists in the theory of contingent claim pricing. This allows us to derive, in a natural way, a simple PDE for the price of …
We prove two theorems on the removal of singularities on the boundary of a pseudo-holomorphic curve. In one theorem, we need no apriori assumption on the area of the curve. The proof uses a doubling argument with the goal of converting curves with boundary to curves without boundary. Our method is new and geometric and…
The paper proves well-posedness of nonlocal PDEs related to stochastic control problems.
problem Characterizing equilibrium strategies and value functions for time-inconsistent stochastic control problems.
method Method of continuity and Banach's fixed point arguments, with Schauder prior estimates.
result Global well-posedness of nonlocal fully nonlinear PDEs with sharp a-priori estimates.
Study optimal consumption with drawdown limits over a fixed time frame.
problem Maximizing utility with consumption limits during a fixed period.
method Extended utility maximization problem with drawdown constraint, using PDE arguments and dual transform.
result Existence and uniqueness of classical solution to HJB variational inequality, with explicit free boundaries.
This paper establishes the existence of a unique nonnegative continuous viscosity solution to the HJB equation associated with a Markovian linear-quadratic control problems with singular terminal state constraint and possibly unbounded cost coefficients. The existence result is based on a novel comparison principle for…
Sharp inequalities and eigenvalue problems on Finsler manifolds with nonnegative Ricci curvature.
problem Establishing sharp Morrey-Sobolev inequalities and eigenvalue problems on Finsler manifolds.
method Combining sharp isoperimetric inequality and anisotropic symmetrization argument.
result Existence and multiplicity of solutions for eigenvalue problems and elliptic PDEs.
Understanding the dual formulation for the hedging of path-dependent options with price impactmath.PR
We consider a general path-dependent version of the hedging problem with price impact of Bouchard et al. (2019), in which a dual formulation for the super-hedging price is obtained by means of PDE arguments, in a Markovian setting and under strong regularity conditions. Using only probabilistic arguments, we prove, in …
This work addresses the classic machine learning problem of online prediction with expert advice. We consider the finite-horizon version of this zero-sum, two-person game. Using verification arguments from optimal control theory, we view the task of finding better lower and upper bounds on the value of the game (regret…
This paper studies a class of nonMarkovian singular stochastic control problems, for which we provide a novel probabilistic representation. The solution of such control problem is proved to identify with the solution of a constrained BSDE, with dynamics associated to a non singular underlying forward process. Du…
Deep learning observables in computational fluid dynamicsphysics.comp-ph
Many large scale problems in computational fluid dynamics such as uncertainty quantification, Bayesian inversion, data assimilation and PDE constrained optimization are considered very challenging computationally as they require a large number of expensive (forward) numerical solutions of the corresponding PDEs. We pro…
We consider applications of the theory of balanced weight filtrations and iterated logarithms, initiated in arXiv:1706.01073, to PDEs. The main result is a complete description of the asymptotics of the Yang--Mills flow on the space of metrics on a holomorphic bundle over a Riemann surface. A key ingredient in the argu…
Letter analyzes training dynamics of a nonlinear contrastive learning model in high dimensions.
problem Understanding training dynamics of nonlinear contrastive learning models in high-dimensional settings.
method High-dimensional analysis using McKean-Vlasov PDEs and low-dimensional ODEs.
result The model's performance evolves according to specific ODEs, revealing features like feature learnability and noise effects.
Study on symmetries in wide neural networks' dynamics without bias.
problem Understanding symmetries in the dynamics of wide two-layer neural networks.
method Analyzing symmetries in gradient flow on population risk for infinitely wide networks.
result Symmetries can simplify the dynamics of predictors and reduce the dimensionality of the problem.
Study curvature flows on pinched Hadamard surfaces, proving convexity preservation and convergence.
problem Preserving convexity and convergence of curves under curvature flows on pinched Hadamard surfaces.
method Area- and length-preserving curvature flows, refined comparison arguments, delicate curvature estimates.
result Convexity is preserved and curves converge to a geodesic circle under certain conditions.
In this article we extend earlier work on the jump-diffusion risk-sensitive asset management problem [SIAM J. Fin. Math. (2011) 22-54] by allowing jumps in both the factor process and the asset prices, as well as stochastic volatility and investment constraints. In this case, the HJB equation is a partial integro-diffe…
We use commutator techniques and calculations in solvable Lie groups to investigate certain evolution Partial Differential Equations (PDEs for short) that arise in the study of stochastic volatility models for pricing contingent claims on risky assets. In particular, by restricting to domains of bounded volatility, we …
Study stabilizes translating solitons in hyperbolic space for MCF.
problem Stability of translating solitons in hyperbolic space.
method Developed theory, constructed rotationally invariant translators, used avoidance principle and maximum principle.
result Horospheres are dynamically stable as radial graphical solutions to MCF.
New proof shows all conformal fields are Killing on specific spaces.
problem Infinitesimal conformal rigidity on Damek-Ricci spaces.
method Formulated as PDEs, analyzed locally and directly.
result Constructive proof of rigidity without global methods.
We solve robust optimization problem and show the example of the market model for which the worst case measure is not a martingale measure. In our model the instantaneous interest rate is determined by the Hull-White model and the investor employs the HARA utility to measure his satisfaction.To protect against the mode…
Paper proves short-term existence of fractional mean curvature flow.
problem Existence of solutions for fractional mean curvature flow with capillary boundary conditions.
method Fixed point argument.
result Short time existence of solutions for fractional mean curvature flow.
Study finds existence of -curvature metrics on even-dimensional manifolds with conical singularities.
problem Existence of -curvature metrics on manifolds with conical singularities.
method Blow-up analysis of a th-order PDE and variational min-max argument.
result First existence result for supercritical conic manifolds (except spheres).
We prove a scaling limit theorem for the super-replication cost of options in a Cox--Ross--Rubinstein binomial model with transient price impact. The correct scaling turns out to keep the market depth parameter constant while resilience over fixed periods of time grows in inverse proportion with the duration between tr…
Develops PAC-Bayesian framework for physics-informed machine learning.
problem Lack of statistical generalisation understanding for PIML models.
method PAC-Bayesian framework with multi-task perspective, incorporating physical structure.
result High-probability generalisation guarantees with unbounded losses.
Deep ResNets can achieve zero loss with enough layers and weights.
problem Finding parameters in deep ResNets that fit training data perfectly.
method Mean-field analysis and gradient descent flow to a PDE.
result Gradient descent for ResNet training converges to a zero-loss solution in the large-NN limit.
Paper tackles stochastic control with mean and higher-order moments, finding Nash equilibria.
problem Time-inconsistent stochastic control problems with mean and higher-order moments.
method Developed closed-loop and open-loop Nash equilibrium controls using PDEs and maximum principles.
result Identical closed-loop and open-loop Nash equilibria controls, independent of state value and random path.
Parameter Inference based on Gaussian Processes Informed by Nonlinear Partial Differential Equationsmath.NA
A new method infers parameters from PDEs using Gaussian processes.
problem Estimating unknown parameters in PDEs from noisy data.
method PDE-Informed Gaussian Process (PIGP) method.
result The method bypasses numerical solvers for PDEs and provides uncertainty quantification.
Study optimal consumption with relaxed benchmarks and drawdown constraints.
problem Optimal consumption under relaxed benchmark tracking and consumption drawdown constraint.
method Transformed stochastic control problem into regular control problem with state-control constraints, then solved using dual transform and optimal consumption behavior.
result Closed-form solution for optimal investment and consumption in feedback form.
Neural-PDE learns PDEs from data using LSTM, outperforming traditional methods.
problem Solving time-dependent PDEs numerically is challenging.
method Bidirectional LSTM encoder to learn governing rules from data.
result Neural-PDE efficiently predicts PDE dynamics with minimal parameters.
We give a new proof for the local existence of a smooth isometric embedding of a smooth -dimensional Riemannian manifold with nonzero Riemannian curvature tensor into -dimensional Euclidean space. Our proof avoids the sophisticated arguments via microlocal analysis used in earlier proofs. In Part 1, we introduce …
The paper proves unboundedness of a functional on G2 forms and describes manifold limits.
problem Proving unboundedness of a functional on G2 forms and describing manifold limits.
method Scaling arguments, geometric estimates, collapsing theorem for orbifolds.
result Explicit descriptions of large volume limits of two G2 manifolds.
Study optimal investment strategies with entropy regularization in volatile markets.
problem Optimal portfolio selection under stochastic volatility with constraints.
method Entropy-regularized relaxed controls, dynamic programming, nonlinear PDEs.
result Existence of classical solutions to nonlinear HJB equation for value function.
Neural Q-learning tackles high-dimensional PDEs.
problem Solving high-dimensional PDEs is computationally challenging.
method Adapting Q-learning from reinforcement learning to solve PDEs.
result The neural network approximator converges to the PDE solution as the network width increases.
Partial differential equations (PDEs) are commonly derived based on empirical observations. However, recent advances of technology enable us to collect and store massive amount of data, which offers new opportunities for data-driven discovery of PDEs. In this paper, we propose a new deep neural network, called PDE-Net …
We determine bubble tree convergence for a sequence of harmonic maps, with uniform energy bounds, from a compact Riemann surface into a compact locally CAT(1) space. In particular, we demonstrate energy quantization and the no-neck property for such a sequence. In the smooth setting, Jost and Parker respectively establ…
PDE-DKL combines NNs and GPs for high-dimensional PDE problems.
problem High-dimensional PDE problems with scarce data.
method PDE-constrained Deep Kernel Learning (PDE-DKL) framework.
result High accuracy with reduced data requirements.
We consider two cases of the asymptotically flat scalar-flat Yamabe problem on a non-compact manifold with boundary, in dimension . First, following arguments of Cantor and Brill in the compact case, we show that given an asymptotically flat metric , there is a conformally equivalent asymptotically flat scal…
Solves second-order PDEs using quotients and differential invariants.
problem Solving second-order PDEs with first-order quotients.
method Solve the quotient PDE using differential invariants, then add new constraints to solve the original PDE.
result New method for solving second-order scalar PDEs with infinite-dimensional symmetry algebras.