arXiv research

Paper develops a new method for solving complex problems in generative modeling and mean-field games.

The paper generalizes Feynman-Kac formula for volatility uncertainty.

Paper proves existence and uniqueness of solutions to nonlocal systems, generalizing stochastic game theory.

The paper concerns singular solutions of nonlinear elliptic equations, which include removable singularities for viscosity solutions, a strengthening of the Hopf Lemma including parabolic equations, Strong maximum principle and Hopf Lemma for viscosity solutions including also parabolic equations.

This paper surveys various methods for dimensionality reduction and nearest neighbor search.

We present a deep recurrent neural network architecture to solve a class of stochastic optimal control problems described by fully nonlinear Hamilton Jacobi Bellmanpartial differential equations. Such PDEs arise when one considers stochastic dynamics characterized by uncertainties that are additive and control multipli…

Unified kernel framework extends to stochastic systems, improving numerical stability.

Novel filter uses deep BSDE for nonlinear density approximation.

This paper presents a novel approach to numerically solve stochastic differential games for nonlinear systems. The proposed approach relies on the nonlinear Feynman-Kac theorem that establishes a connection between parabolic deterministic partial differential equations and forward-backward stochastic differential equat…

This is an introduction to Wiener measure and the Feynman-Kac formula on general Riemannian manifolds for Riemannian geometers with little or no background in stochastics. We explain the construction of Wiener measure based on the heat kernel in full detail and we prove the Feynman-Kac formula for Schrödinger operators…

In this paper, we pursue the study of second order BSDEs with jumps (2BSDEJs for short) started in our accompanying paper [15]. We prove existence of these equations by a direct method, thus providing complete wellposedness for 2BSDEJs. These equations are a natural candidate for the probabilistic interpretation of som…

Deep density methods improve filtering in high-dimensional systems.

We prove Bismut-type formulae for the first and second derivatives of a Feynman-Kac semigroup on a complete Riemannian manifold. We derive local estimates and give bounds on the logarithmic derivatives of the integral kernel. Stationary solutions are also considered. The arguments are based on local martingales, althou…

A new numerical scheme approximates nonlinear filtering densities for noisy and partial measurements.

New methods solve SPDEs for financial derivative pricing.

Researchers prove a stability result for a 3-sphere inequality, extending previous work.

The paper develops a computational method for efficient online filtering of diffusion processes.

FKEE estimates expectations without samples, using diffusion bridges and PINNs.

New method recovers BSDE from financial data without ergodicity.

We prove a Feynman-Kac formula for differential forms satisfying absolute boundary conditions on Riemannian manifolds with boundary and of bounded geometry. We use this to construct $L^2$ harmonic forms out of bounded ones on the universal cover of a compact Riemannian manifold whose geometry displays a positivity prop…

The paper develops a Feynman-Kac formula for perturbations of order ≤ 1 in noncommutative geometry.

Proposes a transfer learning framework for sparse SIMs without raw source data.

Functional-analytic method for stochastic parallel transport in bundles.

New method trains partial Bayesian neural networks efficiently.

The usual Gromoll-Meyer's generalized Morse lemma near degenerate critical points on Hilbert spaces, so called splitting lemma, is stated for at least $C^2$-smooth functionals. In this paper we establish a splitting theorem and a shifting theorem for a class of continuously directional differentiable functionals (lower…

In this paper, we investigate an optimal investment and consumption problem for an investor who trades in a Black--Scholes financial market with stochastic coefficients driven by a non-Gaussian Ornstein--Uhlenbeck process. We assume that an agent makes investment and consumption decisions based on a power utility funct…

New method steers protein design towards desired properties.

Study analyzes derivative-free loss method for solving PDEs and fluid problems.

We study entire continuous viscosity solutions to fully nonlinear elliptic equations involving the conformal Hessian. We prove the strong comparison principle and Hopf Lemma for (non-uniformly) elliptic equations when one of the competitors is $C^{1,1}$. We obtain as a consequence a Liouville theorem for entire solutio…

This paper investigates sufficient conditions for a Feynman-Kac functional up to an exit time to be the generalized viscosity solution of a Dirichlet problem. The key ingredient is to find out the continuity of exit operator under Skorokhod topology, which reveals the intrinsic connection between overfitting Dirichlet …

Paper derives analytical formulas for NLD-CEV moments with regime switching.

Improved diffusion models using energy distillation and sequential Monte Carlo.

We propose a new method to solve eigenvalue problems for linear and semilinear second order differential operators in high dimensions based on deep neural networks. The eigenvalue problem is reformulated as a fixed point problem of the semigroup flow induced by the operator, whose solution can be represented by Feynman…

Paper improves robustness and sparsity in adversarially trained DNNs.

Study on PDEs in Heston model with unique solution and convergence proof.

The Gromoll-Meyer's generalized Morse lemma (so called splitting lemma) near degenerate critical points on Hilbert spaces, which is one of key results in infinite dimensional Morse theory, is usually stated for at least $C^2$-smooth functionals. It obstructs one using Morse theory to study most of variational problems …

We derive some consequences of the Liouville theorem for plurisubharmonic functions of L.-F. Tam and the author. The first result provides a nonlinear version of the complex splitting theorem (which splits off a factor of $\mathbb{C}$ isometrically from the simply-connected Kähler manifold with nonnegative bisectional …

We develop a new model for VIX derivatives with closed-form solutions.

We prove existence, regularity and a Feynman-Kač representation formula of the strong solution to the free boundary problem arising in the financial problem of the pricing of the American Asian option with arithmetic average.

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

Study heat profiles and eigenfunctions using Brownian motion.

Study of mean curvature flows with conical singularities using mathematical techniques.

Generalized tensor analysis in the sense of Colombeau's construction is employed to introduce a nonlinear distributional pseudo-Riemannian geometry. In particular, after deriving several characterizations of invertibility in the algebra of generalized functions we define the notions of generalized pseudo-Riemannian met…

The paper develops methods to price and hedge options in path-dependent stock models.

The paper provides gradient estimates for Neumann semigroups on manifolds with boundary under unbounded curvature conditions.

This paper relates parameter distance to gradient breakdown for a broad class of nonlinear compositional functions. The analysis leads to a new distance function called deep relative trust and a descent lemma for neural networks. Since the resulting learning rule seems to require little to no learning rate tuning, it m…

Stochastic delay differential equations (SDDE's) have been used for financial modeling. In this article, we study a SDDE obtained by the equation of a CIR process, with an additional fixed delay term in drift; in particular, we prove that there exists a unique strong solution (positive and integrable) which we call fix…

At first, we solve a problem of finding a risk-minimizing hedging strategy on a general market with ratings. Next, we find a solution to this problem on Markovian market with ratings on which prices are influenced by additional factors and rating, and behavior of this system is described by SDE driven by Wiener process…