Toolbox for stochastic Euler equations using Ebin-Marsden theory.
problem Applying geometric framework to stochastic PDEs.
method Combining infinite-dimensional geometry and stochastic analysis.
result Local well-posedness of maximal solutions for incompressible Euler equation with noise.
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.
Derives new equations for stochastic volatility models.
problem Modeling local-stochastic-volatility models and their derivatives.
method Conditional forward equation, Dupire stochastic PDE, rolling expiry vanilla option SPDE.
result New equations for LSV models and their derivatives.
Derives new equations for volatility models and option pricing.
problem Modeling and pricing options in local-stochastic-volatility models.
method Develops conditional forward equations and Dupire stochastic PDEs.
result Derives new SPDE for vanilla options.
Study shows how market firm capitalization models converge to stochastic PDE solutions.
problem Understanding convergence of rank-based models with common noise to stochastic PDE solutions.
method Analysis of mean field limit, martingale problem, and pathwise entropy solutions.
result Empirical cumulative distribution function converges to solution of a stochastic PDE under certain conditions.
Flow Annealing Posterior Sampling unifies stochastic-process regression and PDE inverse problems.
problem Function-space posterior sampling for stochastic processes and inverse problems.
method Flow Annealing Posterior Sampling (FAPS) using pretrained function-space flow-matching priors.
result Coherent posterior samples with accurate uncertainty quantification.
FM4PDE learns PDE solutions from sparse data.
problem Reconstructing PDE solutions from limited observations.
method Flow-matching generative framework that learns PDE coefficients and solutions.
result Error guarantees for guided procedures, including deterministic and stochastic samplers.
Derives Black-Scholes model without stochastic calculus or PDEs.
problem Deriving the Black-Scholes model without advanced math.
method Continuum limit of Binomial tree approach.
result Derives Black-Scholes model and exchange-option generalization.
Paper solves PDEs for optimal investment strategies in volatile markets.
problem Finding optimal investment strategies in volatile markets.
method Numerical methods using time-changed Bessel bridges.
result Solves PDEs for relative arbitrage opportunities in volatility-stabilized markets.
Path-dependent PDEs model VIX and Realised Variance options.
problem Modeling volatility derivatives with path-dependence.
method Continuous stochastic volatility model with Gaussian Volterra process, proving well-posedness of PDEs.
result Formulae for greeks and implied volatility provided, finite-dimensional pricing PDEs obtained in Markovian models.
The aim of this paper is to suggest a new viewpoint to study qualitative properties of solutions of semilinear elliptic PDE's defined outside a compact set. The relevant tools come from spectral theory and from a combination of stochastic properties of the relevant differential operators. Possible links between spectra…
PLoM learns stochastic solutions to PDEs with limited data.
problem Synthesizing solutions to nonlinear PDEs with scarce data.
method Probabilistic Learning on Manifolds constrained by PDEs.
result Learned stochastic solutions minimize PDE residuals.
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.
In this paper, we propose the uncertain volatility models with stochastic bounds. Like the regular uncertain volatility models, we know only that the true model lies in a family of progressively measurable and bounded processes, but instead of using two deterministic bounds, the uncertain volatility fluctuates between …
Deep learning model solves high-dimensional PDEs using Actor-Critic approach.
problem Solving high-dimensional nonlinear PDEs efficiently.
method Reformulated PDE into BSDE system, inspired by Actor-Critic algorithm for deep RL.
result Improved model with fewer parameters, faster convergence, and less hyperparameter tuning.
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 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.
The numerical solution of large-scale PDEs, such as those occurring in data-driven applications, unavoidably require powerful parallel computers and tailored parallel algorithms to make the best possible use of them. In fact, considerations about the parallelization and scalability of realistic problems are often criti…
Tensor trains simplify solving complex PDEs efficiently.
problem Solving high-dimensional parabolic PDEs using traditional methods is computationally infeasible.
method Reformulate PDEs as backward stochastic differential equations and use tensor train format for compression and efficient computation.
result Tensor train methods achieve a good balance between accuracy and computational efficiency.
This paper models short rates with jumps using PDEs.
problem Capturing jumps and spikes in interest rates.
method PDE approach for pricing interest rate derivatives.
result Established Feynman-Kač representation and derived solutions.
New methods solve complex PDEs with mixed boundary conditions.
problem Solving inhomogeneous Robin type boundary value problems for linear PDEs.
method Odd and even Hilbert transforms.
result Non-standard solutions to various PDEs in finance, stochastic analysis, etc.
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…
Optimizes portfolios with constraints and stochastic factors, deriving explicit solutions.
problem Optimizing expected utility in an incomplete market with stochastic factors and convex constraints.
method Fundamental duality results and HJB PDE, derived condition for exponential affine solutions.
result Explicit expressions for optimal allocations and Riccati ODE solutions in specific markets.
New integration method improves BSDE-based PDE solvers.
problem Discretization bias in standard BSDE-based solvers.
method Proposed Stratonovich-based BSDE formulation with stochastic Heun integration.
result Eliminates bias issues and outperforms EM-based variants.
High-dimensional partial differential equations (PDE) appear in a number of models from the financial industry, such as in derivative pricing models, credit valuation adjustment (CVA) models, or portfolio optimization models. The PDEs in such applications are high-dimensional as the dimension corresponds to the number …
New formulation tackles arbitrage in volatile markets using eigenvalue bounds.
problem Arbitrage opportunities in volatile markets beyond a certain time horizon.
method Formulated as a stochastic optimal control problem, solved via PDE.
result Characterized arbitrage time horizon through PDE solution.
Study rough volatility models using path-dependent PDEs and fractional Brownian motions.
problem Modeling and analyzing rough volatility in financial markets.
method Showed conditional expectations are unique classical solutions to path-dependent PDEs derived from functional Itô formula. Leverage these to study weak rates of convergence for discretized stochastic integrals.
result Obtained optimal weak error rates for approximating log-stock prices in rough volatility models.
Quantum machine learning solves high-dimensional PDEs with lower variance and improved accuracy.
problem Approximating solutions to high-dimensional parabolic PDEs.
method Pure Variational Quantum Circuit (VQC) for BSDE approximation, using temporal discretization and Monte Carlo simulation.
result VQC achieves lower variance and improved accuracy in most cases, particularly in highly nonlinear regimes.
Meta-learning base distributions for efficient PDE solutions.
problem Efficiently solving parametric parabolic PDEs across different scenarios.
method Meta-learning base distributions to compute PDE solutions.
result Improves generalization to new parameter regimes.
This work bridges stochastic interpolants to infinite-dimensional Hilbert spaces.
problem Limited flexibility in generating arbitrary distributions for function-valued data.
method Establishes a rigorous framework for stochastic interpolants in infinite-dimensional Hilbert spaces.
result Achieves state-of-the-art results in conditional generation for complex PDE-based benchmarks.
Paper tackles DOCTR-L with SciPhy RL, solving neural PDEs from data.
problem High-dimensional optimal control with stochastic policies.
method Soft HJB equation, Neural PDEs, Physics-Informed Neural Networks.
result Reduces DOCTR-L to solving neural PDEs from data.
Long maturity options or a wide class of hybrid products are evaluated using a local volatility type modelling for the asset price S(t) with a stochastic interest rate r(t). The calibration of the local volatility function is usually time-consuming because of the multi-dimensional nature of the problem. In this paper, …
New algorithm optimizes nonlinear SDEs online with convergence guarantees.
problem Optimizing nonlinear stochastic differential equations (SDEs) is computationally challenging.
method Forward propagation algorithm that solves an SDE derived using forward differentiation.
result Convergence theorem for nonlinear dissipative SDEs with bounds on stochastic fluctuations.
RS-PINN uses randomized smoothing to speed up high-dimensional PDE simulations without sacrificing accuracy.
problem High computational cost and bias in PINNs for high-dimensional PDEs.
method Introduces Gaussian noise for stochastic smoothing of PINNs, enabling Monte Carlo derivative approximation.
result Proposes bias correction techniques and a hybrid method to optimize the bias-variance trade-off.
New algorithm solves complex equations using deep learning.
problem High-dimensional nonlinear PDEs and BSDEs.
method Iterated time discretization, deep neural networks, stochastic gradient descent.
result Increased accuracy and reduced complexity compared to existing methods.
Hybrid LSMC-PDE method for Bermudan options under GDMR model.
problem Pricing Bermudan options under the GDMR model.
method Adapted Hybrid LSMC-PDE framework, combining Monte Carlo and PDE methods.
result Hybrid approach yields more accurate and lower error estimates than plain LSMC.
Scalable solver reduces PDE uncertainty with active learning.
problem High computational cost in solving PDEs.
method Stochastic dual descent and clustering-based active learning.
result Solver scales to large number of collocation points.
New method uses tensor trains for efficient PDE approximation.
problem High-dimensional PDEs and the curse of dimensionality.
method Tensor trains and backward stochastic differential equations for parabolic PDEs.
result Achieves a favorable trade-off between accuracy and computational efficiency.
Paper introduces a new method to solve complex PDEs efficiently.
problem Solving high-dimensional semilinear PDEs and BSDEs.
method Decomposes PDEs into linear and nonlinear parts, uses Deep BSDE solver with control variate method.
result Errors of the new method are much smaller than those of the original Deep BSDE solver.
New learning scheme solves high-dimensional semi-linear PDEs using sparse grids and Picard approximations.
problem Solving high-dimensional semi-linear parabolic PDEs.
method Probabilistic learning scheme based on Picard iteration with SGD, employing sparse grid approximation.
result Convergence proof and polynomial complexity in ε−1 for high-dimensional PDEs. Quantum algorithm for multi-asset option pricing under different volatility models.
problem Efficiently pricing multi-asset options under various volatility models using quantum computing.
method Developed an end-to-end quantum PDE framework for European option pricing, solving PDEs after discretization on spatial grids.
result Quantum framework provides polynomial improvement in resource usage compared to classical methods.
Study analyzes derivative-free loss method for solving PDEs and fluid problems.
problem Solving elliptic PDEs and fluid problems using neural networks.
method Derivative-free loss method with Feynman-Kac formulation and stochastic walkers.
result Training loss bias scales with time interval and spatial gradient, inversely with walker size.
New method solves SLV models faster using Lie algebra.
problem Local stochastic volatility models.
method Wei-Norman factorization method and Lie algebraic techniques.
result Reduces time-dependent SLV models to autonomous PDEs.
Extends XVA valuation under stochastic volatility, characterizing value processes via mild solutions.
problem Valuation of contingent claims in presence of default, collateral, and funding under stochastic volatility.
method Characterizes pre-default value processes via mild solutions to parabolic semilinear PDEs under stochastic volatility.
result Characterizes pre-default value processes via mild solutions to parabolic semilinear PDEs under stochastic volatility, providing sufficient conditions for existence and uniqueness.
Generative model improved using Liouville PDE-based sliced-Wasserstein flow.
problem Improving generative models for fair regression.
method Transformed sliced-Wasserstein flow into Liouville PDE-based formalism, handling density estimation with normalizing flows of neural ODE.
result Outperforms in convergence and fairness with reduced variance.
New sampling method uses stochastic interpolants and FBSDEs.
problem Sampling from high-dimensional distributions with unnormalized densities.
method Stochastic interpolants and FBSDEs to define and solve diffusion process.
result Effective sampling from challenging distributions.
Machine learning for scientific applications faces the challenge of limited data. We propose a framework that leverages a priori known physics to reduce overfitting when training on relatively small datasets. A deep neural network is embedded in a partial differential equation (PDE) that expresses the known physics and…
Investor optimizes worst-case portfolio in uncertain markets.
problem Optimizing investment in markets with potential crashes.
method Enhanced martingale approach via BSDEs and PDEs.
result Characterized indifference optimal strategies for various models.