arXiv research

Paper quantifies neural operators' efficiency for solving nonlinear parabolic PDEs.

Implicit schemes are popular methods for the integration of time dependent PDEs such as hyperbolic and parabolic PDEs. However the necessity to solve corresponding linear systems at each time step constitutes a complexity bottleneck in their application to PDEs with rough coefficients. We present a generalization of ga…

Meta-learning base distributions for efficient PDE solutions.

In this paper we introduce a numerical method for nonlinear parabolic PDEs that combines operator splitting with deep learning. It divides the PDE approximation problem into a sequence of separate learning problems. Since the computational graph for each of the subproblems is comparatively small, the approach can handl…

Paper establishes $L^{\infty}$ estimates for complex Monge-Ampere and Hessian equations.

New equivalences found linking parabolicity, comparison principle, and capacity on Riemannian manifolds.

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…

The comparison principle for scalar second order parabolic PDEs on functions $u(t,x)$ admits a topological interpretation: pairs of solutions, $u^1(t,\cdot)$ and $u^2(t,\cdot)$, evolve so as to not increase the intersection number of their graphs. We generalize to the case of multiple solutions $\{u^α(t,\cdot)\}_{α=1}^…

Tensor trains simplify solving complex PDEs efficiently.

New framework uses PDE for no-regret generative modeling.

The stability and robustness of compact schemes for parabolic PDEs are analyzed.

New method uses tensor trains for efficient PDE approximation.

Paper analyzes and proves convergence of a new method for solving complex PDEs.

Gradient estimates for a parabolic PDE under Ricci-Bourguignon flow on warped product manifolds.

Paper develops techniques to solve complex PDEs involving higher cohomology forms.

This paper includes a proof of well-posedness of an initial-boundary value problem involving a system of degenerate non-local parabolic PDE which naturally arises in the study of derivative pricing in a generalized market model. In a semi-Markov modulated GBM model the locally risk minimizing price function satisfies a…

New learning scheme solves high-dimensional semi-linear PDEs using sparse grids and Picard approximations.

This article provides a new representation for pricing adjustments in derivatives.

Quantum machine learning solves high-dimensional PDEs with lower variance and improved accuracy.

Study generalizes Picard iteration for nonlinear PDEs, deriving bounds on error.

Distance between evolving hypersurfaces is a PDE solution.

We provide an asymptotic expansion of the value function of a multidimensional utility maximization problem from consumption with small non-linear price impact. In our model cross-impacts between assets are allowed. In the limit for small price impact, we determine the asymptotic expansion of the value function around …

We give local descriptions of parabolic contact structures and show how their flat models yield explicit PDE having symmetry algebras isomorphic to all complex simple Lie algebras except $\mathfrak{sl}_2$. This yields a remarkably uniform generalization of the Cartan-Engel models from 1893 in the $G_2$ case. We give a …

D2SRM solves complex PDEs using deep learning.

Develops deep learning methods for non-linear PDEs in credit risk.

Extends XVA valuation under stochastic volatility, characterizing value processes via mild solutions.

We propose a new algorithm for solving parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) in high dimension, by making an analogy between the BSDE and reinforcement learning with the gradient of the solution playing the role of the policy function, and the loss functi…

We find normal forms for parabolic Monge-Ampere equations. Of these, the most general one holds for any equation admitting a complete integral. Moreover, we explicitly give the determining equation for such integrals; restricted to the analytic case, this equation is shown to have solutions. The other normal forms exha…

We develop a framework for estimating unknown partial differential equations from noisy data, using a deep learning approach. Given noisy samples of a solution to an unknown PDE, our method interpolates the samples using a neural network, and extracts the PDE by equating derivatives of the neural network approximation.…

Through the study of some elliptic and parabolic fully nonlinear PDEs, we establish conformal versions of quermassintegral inequality, the Sobolev inequality and the Moser-Trudinger inequality for the geometric quantities associated to the Schouten tensor on locally conformally flat manifolds.

New methods solve complex PDEs with mixed boundary conditions.

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 …

When the underlying stock price is a strict local martingale process under an equivalent local martingale measure, Black-Scholes PDE associated with an European option may have multiple solutions. In this paper, we study an approximation for the smallest hedging price of such an European option. Our results show that a…

We consider the problem of optimal portfolio selection under forward investment performance criteria in an incomplete market. Given multiple traded assets, the prices of which depend on multiple observable stochastic factors, we construct a large class of forward performance processes with power-utility initial data, a…

New theorems prove uniqueness of solutions to geometric PDEs.

We consider an integro-differential equation derived from a system of coupled parabolic PDE and an ODE which describes an European option pricing with liquidity shocks. We study the well-posedness and prove comparison principle for the corresponding initial value problem.

There was proposed the method of a factorization of PDE. The method is based on reduction of complicated systems to more easy ones (for example, due to dimension decrease). This concept is proposed in general case for the arbitrary PDE systems, and its concrete investigation is developing for the heat equation case. Th…

In this paper we consider a variation of the Merton's problem with added stochastic volatility and finite time horizon. It is known that the corresponding optimal control problem may be reduced to a linear parabolic boundary problem under some assumptions on the underlying process and the utility function. The resultin…

New boundary treatment improves accuracy for complex PDEs.

We study the asymptotic behavior of solutions to the second boundary value problem for a parabolic PDE of Monge-Ampère type arising from optimal mass transport. Our main result is an exponential rate of convergence for solutions of this evolution equation to the stationary solution of the optimal transport problem. We …

Deep learning model solves high-dimensional PDEs using Actor-Critic approach.

Survey on recent developments in isometric immersions using PDE techniques.

In the first half of the paper we construct a Morse-type theory on certain spaces of braid diagrams. We define a topological invariant of closed positive braids which is correlated with the existence of invariant sets of parabolic flows defined on discretized braid spaces. Parabolic flows, a type of one-dimensional lat…

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 …

Scalable solver reduces PDE uncertainty with active learning.

New method uses neural networks to solve complex PDEs from optimal control theory.

We use probabilistic methods to study classical solutions for systems of interacting semilinear parabolic partial differential equations. In a modeling framework for a financial market with interacting Ito and point processes, such PDEs are shown to provide a natural description for the solution of hedging and valuatio…

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 …