We link SVEs to SPDEs and derive Kolmogorov equations for singular kernels.
problem Solving stochastic Volterra equations with singular kernels.
method Establishing connections between SVEs and SPDEs, using stochastic calculus in Hilbert spaces.
result Solutions of SVEs can be expressed in terms of backward Kolmogorov equations.
The usual derivation of the Fokker-Planck partial differential eqn. assumes the Chapman-Kolmogorov equation for a Markov process. Starting instead with an Ito stochastic differential equation we argue that finitely many states of memory are allowed in Kolmogorov's two pdes, K1 (the backward time pde) and K2 (the Fokker…
The Accardi-Boukas quantum Black-Scholes equation can be used as an alternative to the classical approach to finance, and has been found to have a number of useful benefits. The quantum Kolmogorov backward equations, and associated quantum Fokker-Planck equations, that arise from this general framework, are derived usi…
Uniform diffusion approximation for SGD in non-convex settings.
problem Finite-time diffusion approximation for SGD.
method Establishing uniform-in-time diffusion approximation with strong convexity and mild conditions.
result Uniform-in-time diffusion approximation of SGD without convexity of each loss function.
The paper develops a computational method for efficient online filtering of diffusion processes.
problem Online filtering of discretely observed nonlinear diffusion processes.
method The approach involves Doob's h-transforms approximated by solving backward Kolmogorov equations using nonlinear Feynman-Kac formulas and neural networks. result The proposed method can be orders of magnitude more efficient than state-of-the-art particle filters.
High order splitting schemes with complex timesteps are applied to Kolmogorov backward equations stemming from stochastic differential equations in Stratonovich form. In the setting of weighted spaces, the necessary analyticity of the split semigroups can be easily proved. A numerical example from interest rate theory,…
A framework solves parametric families of MFGs efficiently.
problem Efficiently solving MFG systems with varying initial distributions and terminal costs.
method Operator learning framework for parametric families of MFGs.
result Accurate approximation for cybersecurity and quadratic MFGs.
Neural operators solve families of 2BSDEs efficiently.
problem Solving infinite families of 2BSDEs on bounded domains.
method Introduces a mild generative neural operator model to approximate solutions.
result Solution operators can be approximated by neural operators with polynomial parameters.
This paper deals with the exact calibration of semidiscretized stochastic local volatility (SLV) models to their underlying semidiscretized local volatility (LV) models. Under an SLV model, it is common to approximate the fair value of European-style options by semidiscretizing the backward Kolmogorov equation using fi…
Deriving option prices from operational-time Markov lattices
problem Option pricing
method Operational-time Markov lattice
result Derives option-pricing equations from an operational-time Markov lattice
Develops deep learning for fast, accurate option pricing models.
problem Computational efficiency and accuracy in option pricing models.
method Neural network generators solving backward Kolmogorov equations for TPDFs.
result Ultra-fast, highly accurate option pricing models for various asset models.
We study the solution to Kolmogorov-Feller equation and by using it provide pricing formulas of well known some options under jump-diffusion model.
During the last few years, significant attention has been paid to the stochastic training of artificial neural networks, which is known as an effective regularization approach that helps improve the generalization capability of trained models. In this work, the method of modified equations is applied to show that the r…
Stochastic differential equations (SDEs) and the Kolmogorov partial differential equations (PDEs) associated to them have been widely used in models from engineering, finance, and the natural sciences. In particular, SDEs and Kolmogorov PDEs, respectively, are highly employed in models for the approximative pricing of …
The goal of this article is to describe the concepts of system dynamics and its applications to the simulation modeling of financial institutions daily activity. The hybrid method of the re-engineering of banking business processes based upon combination of system dynamics, queuing theory and tools of ordinary differen…
Study finds conjugate points in geodesics of Kolmogorov flows on torus.
problem Characterizing pairs of integers (m,n) for which geodesics have conjugate points.
method Analysis of geodesics in the group of volume-preserving diffeomorphisms of a torus using stream functions.
result Existence of conjugate points for all pairs of strictly positive integers (m,n).
A new machine learning method solves high-dimensional Kolmogorov PDEs efficiently.
problem Solving high-dimensional Kolmogorov PDEs and SDEs.
method Stochastic weighted minimization and stochastic gradient descent with Malliavin weights.
result Accurate approximation of high-dimensional Kolmogorov PDEs and SDEs without curse of dimensionality.
Deep learning solves high-dimensional PDEs efficiently.
problem Solving high-dimensional Kolmogorov PDEs numerically.
method Reformulating as a statistical learning problem using Feynman-Kac formula.
result Single neural network learns entire family of PDEs.
New method speeds up SDE inference by matching moments to FPK equation.
problem Efficiency of sampling schemes in high-dimensional SDEs.
method Direct approximation of Fokker-Planck-Kolmogorov equation by matching moments.
result Fast, scalable inference in high-dimensional latent spaces.
We develop a new method to price SOFR futures contracts considering convexity, skew, and smile.
problem Analyzing and pricing SOFR futures contracts with convexity, skew, and smile adjustments.
method A perturbative formalism based on a time-ordered exponential series to solve the backward-Kolmogorov diffusion PDE.
result An analytic pricing formula for SOFR futures contracts that incorporates convexity, skew, and smile adjustments.
Novel method for SDE calibration from sparse data using neural flows.
problem Calibrating SDEs from sparse, noisy observations.
method Characterization of posterior SDE using neural networks trained to solve a PDE with multiplicative updates.
result Significant improvement in scalability and accuracy compared to classical methods.
Kernel method approximates dynamical operators from data.
problem Estimating eigenfunctions of dynamical operators from data.
method Kernel-based approach in reproducing kernel Hilbert spaces.
result Eigenfunctions estimated via matrix eigenvalue problems.
A new method for pricing exchange options under stochastic volatility and jumps.
problem Pricing European and American exchange options with stochastic volatility and jumps.
method Equivalent martingale measure, numeraire choice, integral transforms, Kolmogorov backward equation, integral equations.
result Reduced exchange option pricing to a one-dimensional problem of a call option.
Projects Markovian processes from Itô semimartingales with jumps.
problem Modeling Itô semimartingales with jumps using Markovian projections.
method Construct Markovian projections for Itô semimartingales with jumps using non-local FPKEs.
result Markovian projections match the marginal laws of the original process.
Survey and new results link hydrodynamics, molecular physics, and financial engineering.
problem Understanding financial engineering topics like Asian options and volatility swaps.
method Linking Kevin waves, Klein-Kramers, and Kolmogorov equations to financial models.
result Corrected the original solution of the Kolmogorov equation.
Paper approximates backward heat equation using wave equations and Ricci flow.
problem Solving backward heat equation on manifolds using wave equations.
method Approximates solutions of a wave equation on a larger manifold with Ricci flow to solve the backward heat equation.
result The approximation provides solutions to the backward heat equation on manifolds.
This work extends set-valued risk measures to discrete time, using difference inclusions and equations.
problem Defining set-valued dynamic risk measures in discrete time.
method Investigates discrete time setting with difference inclusions and difference equations.
result Provides insights for continuous time representations of set-valued dynamic risk measures.
The development of new classification and regression algorithms based on empirical risk minimization (ERM) over deep neural network hypothesis classes, coined deep learning, revolutionized the area of artificial intelligence, machine learning, and data analysis. In particular, these methods have been applied to the num…
It is an approach to introduce the forward Kolmogorov equation as an interesting natural ingredient in studying the evolution of the market stock prices.
Paper proves stability of complex equations under various conditions.
problem Stability of backward stochastic differential equations with jumps.
method General framework for convergent sequences of data and solutions.
result Convergent sequence of solutions for associated data.
Study BSΔE on lattices for asset price analysis.
problem Optimal investment and market equilibrium analysis in asset price models.
method Backward stochastic difference equations on lattices.
result Applications to optimal investment and market equilibrium analysis.
New Harnack inequality for heat equation on compact manifolds.
problem Developing a new Harnack inequality for heat equations.
method Gradient estimates by Hamilton combined with backward time comparison.
result Discovered a backward in time Harnack inequality for positive solutions.
The paper studies dimensions of attractors for modified Leray-alpha equation on various surfaces.
problem Investigate attractor dimensions of the modified Leray-alpha equation.
method Existence and uniqueness of weak solutions, global attractor existence, estimates for vorticity scalar equations, Kolmogorov flows.
result Established upper and lower bounds for Hausdorff and fractal dimensions of global attractors on S2 and T2. Study proves existence of equilibrium in incomplete economies with discontinuous volatility.
problem Existence of incomplete Radner equilibrium with nondegenerate endogenous volatility.
method Established existence of solution for Markovian quadratic BSDEs with discontinuous generators using unique continuation and backward uniqueness.
result Existence of incomplete Radner equilibrium with nondegenerate endogenous volatility.
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…
In this paper, we derive a general evolution formula for possible Harnack quantities. As a consequence, we prove several differential Harnack inequalities for positive solutions of backward heat-type equations with potentials (including the conjugate heat equation) under the Ricci flow. We shall also derive Perelman's …
KaCGM models provide transparent causal inference from tabular data.
problem Limited auditability in deep causal models for tabular data.
method KaCGM uses Kolmogorov-Arnold Networks to parameterize structural equations, enabling direct inspection and visualization of causal mechanisms.
result KaCGM achieves competitive performance and interpretable causal effects in real-world applications.
We propose a new method for the numerical solution of backward stochastic differential equations (BSDEs) which finds its roots in Fourier analysis. The method consists of an Euler time discretization of the BSDE with certain conditional expectations expressed in terms of Fourier transforms and computed using the fast F…
Semi-analytical approach for optimal wealth management contributions.
problem Optimizing contributions to achieve a financial goal with uncertain returns.
method Controlled backward Kolmogorov equation and Schrodinger equation solution.
result Semi-analytical solutions for efficient frontiers in control space.
Kolmogorov-Arnold Networks offer interpretable models for energy applications.
problem Lack of interpretability in modern machine learning methods for sensitive industries.
method Symbolic regression with Kolmogorov-Arnold Networks compared to traditional feedforward neural networks.
result Kolmogorov-Arnold Networks yield perfectly interpretable models and learn real, physical relations.
In this paper, we introduce a large class of convergent numerical methods, based on (linear) basis function regression technique, to approximate the solution to a forward-backward stochastic differential equation with jumps (FBSDEJ hereafter). Numerical experiment shows good applicability of the proposed method.
The paper defines a frequency for mean curvature flow and proves its monotonicity.
problem Backwards uniqueness for solutions of mean curvature flow.
method Defining and proving monotonicity of a parabolic frequency for mean curvature flows.
result Frequency monotonicity implies backwards uniqueness for mean curvature flow solutions.
Develops a new trading strategy for renewable producers to manage price volatility.
problem Price volatility and imbalance risk in power markets due to renewable generation.
method Data-driven continuous-time stochastic optimal control framework using SDEs and diffusion models.
result Trading strategy outperforms benchmarks and reduces profit and loss.
New methods improve deep learning for solving linear PDEs.
problem Efficiently solving high-dimensional linear PDEs using deep learning.
method Rigorous investigation of gradient estimators for SDE-based variational formulations.
result Novel methods provide substantial performance improvements.
This paper formulates and studies a stochastic maximum principle for forward-backward stochastic Volterra integral equations (FBSVIEs in short), while the control area is assumed to be convex. Then a linear quadratic (LQ in short) problem for backward stochastic Volterra integral equations (BSVIEs in short) is present …
Backward SDEs help price XVA for OTC derivatives.
problem XVA valuation for OTC derivatives with default risk.
method Review and apply BSDEs with random horizon.
result Explicit formula for XVA correction terms.
New Hessian estimates for heat equations on manifolds.
problem Estimating Hessian matrices for heat-type equations on Riemannian manifolds.
method Using Bismut-Stroock Hessian formula, with explicit coefficients and delay/growth rate functions.
result Novel backward weak Harnack inequality and precise pointwise Hessian estimates for eigenfunctions.
We consider the heat equation associated with a class of hypoelliptic operators of Kolmogorov-Fokker-Planck type in dimension two. We explicitly compute the first meaningful coefficient of the small time asymptotic expansion of the heat kernel on the diagonal, and we interpret it in terms of curvature-like invariants o…