We present two different approaches to stochastic integration in frictionless model free financial mathematics. The first one is in the spirit of Itô's integral and based on a certain topology which is induced by the outer measure corresponding to the minimal superhedging price. The second one is based on the controlle…
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48 results for “path integral control”
Adaptive algorithm learns latent dynamical systems from sequential data.
problem Learning low-dimensional latent dynamical systems from high-dimensional sequential data.
method Combines amortized inference with path integral control to approximate inference.
result Proposed method leads to tighter lower bounds in sequential data learning.
Paper explores rough path theory for frictionless markets, linking NCFL to unbiased rough integrators.
problem Tackles the limits of rough path theory in frictionless markets.
method Investigates the capacity of rough path theory to support No Free Lunch markets.
result Establishes a 'Rough Kreps-Yan' theorem linking NCFL to unbiased rough integrators.
New method for sampling from multivariate distributions using optimal control and quantum mechanics.
problem Sampling from continuous multivariate probability distributions efficiently and accurately.
method Harmonic Path Integral Diffusion (H-PID) framework, formulated as a Stochastic Optimal Control problem.
result Efficient sampling algorithms without neural networks, revealing dynamic phase transitions.
We present an embedding of stochastic optimal control problems, of the so called path integral form, into reproducing kernel Hilbert spaces. Using consistent, sample based estimates of the embedding leads to a model free, non-parametric approach for calculation of an approximate solution to the control problem. This fo…
New algorithms sample from complex path measures using neural networks.
problem Sampling from posterior path measures under a general prior process.
method Combines controlled equilibrium dynamics and optimization in infinite-dimensional probability space.
result The algorithms can be integrated with neural networks for learning target trajectory ensembles.
We derive a closed-form solution for the price of an average price as well as an average strike geometric Asian option, by making use of the path integral formulation. Our results are compared to a numerical Monte Carlo simulation. We also develop a pricing formula for an Asian option with a barrier on a control proces…
New BdryMatérn GP model for reliable boundary integration on irregular domains.
problem Incorporating boundary information in Gaussian process models for complex phenomena.
method Proposes a novel BdryMatérn GP framework with a new covariance kernel derived via path integral and stochastic PDE.
result Sample paths from the BdryMatérn GP satisfy desired boundaries with smoothness control on derivatives.
Develops a new model-free approach to portfolio theory using rough paths.
problem Handles more general portfolios without probabilistic assumptions.
method Rough path theory for stochastic portfolio theory (SPT).
result Asymptotic growth rates of various portfolios match.
New method improves feature selection by integrating stability paths.
problem Improving feature selection with tighter false positive control.
method Integrating stability paths to strengthen theoretical bounds on E(FP).
result Significantly more true positives with same E(FP) control.
HiDe learns hierarchical control for complex tasks by separating planning and control.
problem Solving long horizon control tasks with generalization to unseen scenarios.
method Functional decomposition of state-action spaces, RL-based planner, modular transfer of policy layers.
result Generalizes across unseen test environments and scales to longer horizons.
MF-PID uses interacting samples to efficiently transport probability mass.
problem Efficiently transporting probability mass in generative models.
method Introducing Mean-Field Path-Integral Diffusion (MF-PID) where samples become interacting agents.
result MF-PID achieves 19-24% reductions in control energy for demand-response control of energy systems.
We develop a technique based on Malliavin-Bismut calculus ideas, for asymptotic expansion of dual control problems arising in connection with exponential indifference valuation of claims, and with minimisation of relative entropy, in incomplete markets. The problems involve optimisation of a functional of Brownian path…
Plan2Vec learns image representations without labels, improving control tasks.
problem Learning image representations without labeled data.
method Constructs a weighted graph using near-neighbor distances and extrapolates to global embedding.
result Plan2Vec achieves accurate long-term value estimates in control tasks with reduced computational and memory costs.
Itô maps for any-step SDEsstat.ML
Itô maps provide a method for any-step SDE integration.
problem Stochastic dynamics
method Itô map formulation
result Empirical results on synthetic and image-generation benchmarks
New control theory for self-path-dependent problems solves unique constraints.
problem Optimal control with self-path-dependent constraints in stochastic systems.
method Introduces new HJB equations for variational inequalities with historical maximum controls.
result Value functions are viscosity solutions to HJB equations under Lipschitz conditions.
Optimal trading strategy with predictor and costs, derived equations and shape.
problem Optimal trading strategy in presence of price predictor, costs, and risk control.
method Path-integral method to derive equations for band edges, solved explicitly for Ornstein-Uhlenbeck predictor.
result Explicit equations and shape of the optimal band strategy derived and analyzed.
This paper analyzes shallow ReLU networks in L^p and Sobolev spaces, focusing on approximation and generalization.
problem Approximation and generalization of shallow ReLU networks in L^p and Sobolev spaces.
method Spherical harmonic analysis and embeddings into spectral Barron spaces for L^p spaces, path-norm control for Sobolev spaces.
result Minimax-optimal rates for nonparametric regression with shallow ReLU networks under path-norm control.
Computes Wilson Loop observable using Einstein-Hilbert and Chern-Simons path integrals.
problem Computing Wilson Loop observable using path integrals.
method Uses Einstein-Hilbert action and axial-gauge fixing to express as Chern-Simons integrals, then computes observable from link diagram.
result Wilson Loop observable can be computed from a hyperlink's link diagram, invariant under equivalence relation.
Survey on rigorous construction of supersymmetric path integral.
problem Rigorous construction of supersymmetric path integral.
method Construction based on joint work with collaborators.
result Rigorous construction of supersymmetric path integral.
Optimizes portfolios in fractional and rough Heston models.
problem Optimizing portfolios in models with fractional and rough volatility.
method Fractional Heston model, power utility functions, stochastic control, Marchaud fractional derivative.
result Explicit solutions and Laplace transform of integrated volatility.
A Rigorous Construction of the Supersymmetric Path Integral Associated to a Compact Spin Manifoldmath.DG
Constructs a rigorous path integral for supersymmetric spin manifolds.
problem Defining a rigorous path integral for N=1/2 supersymmetry.
method Using differential forms and iterated integrals on loop spaces of compact spin manifolds.
result Provides a rigorous background for Atiyah-Singer index theorem proofs.
Unified approach to stochastic control, filtering, and stopping using rough paths.
problem Addressing gaps in classical problems of stochastic control, filtering, and stopping.
method Combining rough path theory with controlled rough paths to provide a pathwise deterministic framework.
result Established rigorous connection between candidate solutions and Hamilton-Jacobi-Bellman equation.
Survey on heat kernels and path integrals.
problem Approximating Wiener measure on compact manifolds.
method Review of recent results on approximating Wiener measure.
result Approximation of Wiener measure by measures on spaces of piece-wise geodesics.
New control methods improve dynamic measure transport paths.
problem Improving paths for dynamic measure transport.
method Connecting mean-field games to optimization problems for learning paths, advocating for smoothness of velocities.
result Our method recovers more efficient and smooth transport models compared to untilted paths.
In a rigorous construction of the path integral for supersymmetric quantum mechanics on a Riemann manifold, based on Bär and Pfäffle's use of piecewise geodesic paths, the kernel of the time evolution operator is the heat kernel for the Laplacian on forms. The path integral is approximated by the integral of a form on …
Model for stock prices using non-Gaussian path integral.
problem Fit stock price dynamics with a small number of parameters.
method Generalized Ilinski's path integral model with a different action.
result Provides excellent fits for stock prices and indices.
Completes proof of index theorem using rigorous path integrals for supersymmetric quantum mechanics.
problem Analytic difficulties in generalizing Feynman's path integral to non-quadratic potentials.
method Develops rigorous path integrals for a class of Lagrangians including spinors on Riemannian manifolds.
result Steepest-descent approximation to path integral for twisted supersymmetric quantum mechanics is provably correct.
PAN improves graph neural networks using path integrals.
problem Efficiency and performance of graph neural networks.
method PAN uses path integrals to generalize graph Laplacian, incorporating all paths between nodes.
result PAN achieves state-of-the-art performance on benchmark tasks.
New path integrals for elasticity derived from differential complex theory.
problem Deriving path integrals for elasticity equations.
method Using Bernstein-Gelfand-Gelfand (BGG) construction and properties of the de Rham complex, derived path integral operators for elasticity.
result Path integral operators for elasticity satisfying and .
Path Integral and Asian Optionsq-fin.PR
In this paper we analytically study the problem of pricing an arithmetically averaged Asian option in the path integral formalism. By a trick about the Dirac delta function, the measure of the path integral is defined by an effective action functional whose potential term is an exponential function. This path integral …
Paper bridges ResNet and Feynman path integral for mathematical understanding.
problem Gradient vanishing issue in ResNet.
method Proves equivalence between ResNet and Feynman path integral, using partial differential equations.
result ResNet's advantage in gradient vanishing issue is mathematically explained.
We study a quantum system in a Riemannian manifold M on which a Lie group G acts isometrically. The path integral on M is decomposed into a family of path integrals on a quotient space Q=M/G and the reduced path integrals are completely classified by irreducible unitary representations of G. It is not necessary to assu…
Researchers calculate the Ray-Singer Torsion for bundles.
problem Few explicit evaluations of path integrals in higher dimensions.
method Algebraic choice of gauge leading to factorization of path integral.
result Explicit calculation of Ray-Singer Torsion for bundles.
Complexity Measures for Neural Networks with General Activation Functions Using Path-based Normscs.LG
Complexity measures for neural nets with general activations using path-based norms.
problem Control complexity of neural networks with arbitrary activation functions.
method Approximate general activations with ReLU networks and derive path-based norms for complexity control.
result Preliminary analyses of function spaces and regularized estimators.
New algorithm preserves transport maps for better diffusion model training.
problem Training diffusion models with task-specific optimality structures.
method Generalized Schrödinger Bridge Matching (GSBM), inspired by conditional stochastic optimal control.
result GSBM better preserves transport maps, enabling stable convergence and improved scalability.
Computation of first-order Greeks for barrier options using chain rules for Wiener path integralsq-fin.MF
New method calculates barrier option Greeks using Wiener path integrals.
problem Computing first-order Greeks for barrier options efficiently.
method Developed chain rules for Wiener path integrals.
result Effectiveness demonstrated through numerical examples.
Path integral method calculates PDBS option prices with time-dependent parameters.
problem Pricing proportional double-barrier step options with time-dependent interest rates and volatilities.
method Path integral method applied to a quantum mechanical analogy of barrier options.
result Derivation of pricing kernel for PDBS options with time-dependent parameters.
Derives path-integrals for superstrings on curved backgrounds using string geometry theory.
problem Calculating path-integrals for superstrings on curved backgrounds.
method Derives path-integrals from string geometry theory by considering fluctuations around string backgrounds.
result Derives path-integrals for perturbative superstrings on all string backgrounds.
The paper proves signatures of non-geometric rough paths can approximate functionals uniformly.
problem Approximating functionals of non-geometric rough paths.
method Extending rough paths with time and quadratic variation terms, proving uniform approximation.
result Linear functionals of extended signatures uniformly approximate continuous functionals.
In the framework of path integral the evolution operator kernel for the Merton-Garman Hamiltonian is constructed. Based on this kernel option formula is obtained, which generalizes the well-known Black-Scholes result. Possible approximation numerical schemes for path integral calculations are proposed.
A new formula connects supersymmetric path integrals to Chern-Simons theory.
problem Constructing a rigorous path integral for supersymmetric theories on spin manifolds.
method Using Chen differential forms and non-commutative geometry, a Chern-Simons transgression formula is derived.
result The supersymmetric path integral induces a differential topological invariant.
Proposes Geodesic Integrated Gradients (GIG) for more accurate feature attributions in deep networks.
problem Flawed attributions using straight paths from Integrated Gradients (IG).
method Introduces a model-induced Riemannian metric and computes attributions along geodesics.
result GIG produces more faithful attributions than IG on benchmarks.
A Path Integral Approach for Time-Dependent Hamiltonians with Applications to Derivatives Pricingq-fin.CP
Improved path integral method for financial derivatives pricing.
problem Analytical intractability of financial derivative pricing models.
method Generalized semi-classical path integral approach to time-dependent Hamiltonians.
result Accuracy and computational efficiency of the path integral approach for derivatives pricing.
The paper analyzes how grid cells perform path integration and learns hexagon grid patterns.
problem Understanding how grid cells perform path integration calculations.
method Theoretical analysis of a general representation model of path integration by grid cells, identifying group representation and isotropic scaling conditions.
result The learned model of hexagon grid patterns is capable of accurate long distance path integration.
The paper associates path measures to surface curvature.
problem Detecting Gaussian curvature using path lengths.
method Define path measures and calculate integrals over paths.
result Integral equals zero for surfaces with zero Gaussian curvature.
Derives path integrals for perturbative strings on various backgrounds.
problem Calculating path integrals for strings on curved backgrounds.
method Derives path integrals from string geometry theory by considering fluctuations around string backgrounds.
result Derives path integrals of all order perturbative strings on various backgrounds.
Path Integral Method for Proportional Step and Proportional Double-Barrier Step Option Pricingq-fin.PR
Path integral method calculates barrier option prices.
problem Barrier option pricing in finance.
method Path integral method applied to trapezoid and square potential barriers.
result Analytical expressions for option pricing derived.