Dupire's functional Itô calculus provides an alternative approach to the classical Malliavin calculus for the computation of sensitivities, also called Greeks, of path-dependent derivatives prices. In this paper, we introduce a measure of path-dependence of functionals within the functional Itô calculus framework. Name…
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48 results for “path computation”
The paper introduces surface signatures for irregular surfaces and rough surfaces.
problem Characterizing and integrating highly irregular paths and surfaces.
method Introducing surface signatures and proving extension theorems.
result Surface signatures are universal for surface holonomy and rough surfaces.
Linear time dynamic programming for the exact path of optimal models selected from a finite setcs.LG
New algorithm speeds up path computation for optimal models.
problem Finding the exact path of optimal models from a finite set.
method Dynamic programming approach for linear time computation.
result Dynamic programming achieves linear time for breakpoints computation.
The regularization path of the Lasso can be shown to be piecewise linear, making it possible to "follow" and explicitly compute the entire path. We analyze in this paper this popular strategy, and prove that its worst case complexity is exponential in the number of variables. We then oppose this pessimistic result to a…
We introduce a new feature map for barcodes that arise in persistent homology computation. The main idea is to first realize each barcode as a path in a convenient vector space, and to then compute its path signature which takes values in the tensor algebra of that vector space. The composition of these two operations …
This work develops a generic framework, called the bag-of-paths (BoP), for link and network data analysis. The central idea is to assign a probability distribution on the set of all paths in a network. More precisely, a Gibbs-Boltzmann distribution is defined over a bag of paths in a network, that is, on a representati…
Computation of first-order Greeks for barrier options using chain rules for Wiener path integralsq-fin.MF
This paper presents a new methodology to compute first-order Greeks for barrier options under the framework of path-dependent payoff functions with European, Lookback, or Asian type and with time-dependent trigger levels. In particular, we develop chain rules for Wiener path integrals between two curves that arise in t…
The paper proposes using path signatures for better inference in time series data.
problem Simulation models with time series data often lack tractable likelihood functions.
method Approximate Bayesian Computation with path signatures to handle sequential data.
result Theoretical guarantees on the resultant posteriors for Bayesian parameter inference.
Computes sections of a submersion and applies to evasion path problem.
problem Evasion path problem for mobile sensor networks.
method Computation of sections from fiber homotopy groups and time-varying homology/cohomology.
result Necessary and sufficient conditions for evasion paths and lower bounds.
This work derives closed-form expressions computing the expectation of co-presence and of number of co-occurrences of nodes on paths sampled from a network according to general path weights (a bag of paths). The underlying idea is that two nodes are considered as similar when they often appear together on (preferably s…
Path-independent equilibrium models improve network performance on harder problems.
problem Improving network performance on harder problem instances.
method Investigated path-independent equilibrium models and their impact on network performance.
result Path independence correlates with better performance on harder problem instances.
In this paper I develop a new computational method for pricing path dependent options. Using the path integral representation of the option price, I show that in general it is possible to perform analytically a partial averaging over the underlying risk-neutral diffusion process. This result greatly eases the computati…
A hyperlink is a finite set of non-intersecting simple closed curves in . We compute the Wilson Loop observable using a path integral with an Einstein-Hilbert action. Using axial-gauge fixing, we can write this path integral as the limit of a sequence of Chern-Simons integrals, studied e…
Branched Signature Modelmath.NA
Paper introduces branched signature model for efficient computation and data-driven applications.
problem Efficient computation and data-driven modeling of branched rough paths.
method Develops a universal approximation theorem and constructs an extension map to realize branched signatures.
result Explicit construction of branched signatures via an extension map for efficient computation.
A new path gradient estimator speeds up normalizing flows without sacrificing accuracy.
problem High computational cost and limited scalability of path gradient estimators for normalizing flows.
method Proposed a fast path gradient estimator that improves computational efficiency and scalability.
result The new estimator achieves superior performance and reduced variance across various applications.
Path signatures adapted for Lie groups improve action recognition in computer vision.
problem Improving action recognition in computer vision with geometric constraints.
method Lifting path signatures to Lie groups and proving universality and characteristic property.
result Path signatures on Lie groups provide comparable performance to shallow learning approaches in action recognition.
Global invariant for path structures and differential equations defined on torus.
problem Global invariant for path structures and differential equations.
method Computed as a secondary invariant from a Cartan connection on a canonical bundle.
result Formula for global invariant of second order differential equations on torus.
Kernel for Lévy rough paths derived from PDE system.
problem Computing similarity measures for Lévy rough paths.
method Developed a PDE system for the expected signature of inhomogeneous Lévy processes.
result Gaussian martingales' expected signature kernel satisfies a Goursat PDE.
A multiobjective continuation method to compute the regularization path of deep neural networkscs.LG
Algorithm approximates regularization path for deep neural networks efficiently.
problem Computing the regularization path for high-dimensional deep neural networks.
method Multiobjective continuation method for non-smooth objectives.
result Approximation of the entire Pareto front for regularization path.
Path weighting sensitivitiesmath.PR
The paper calculates sensitivities for financial derivatives using path weighting methods.
problem Computing sensitivities for path-dependent financial derivatives with high variance and degeneracy issues.
method Proposes explicit path weighting formula, variance reduction adjustment, and covariance inflation technique.
result Effective methods to address high variance and degeneracy in sensitivities computation.
The recently developed bag-of-paths (BoP) framework consists in setting a Gibbs-Boltzmann distribution on all feasible paths of a graph. This probability distribution favors short paths over long ones, with a free parameter (the temperature ) controlling the entropic level of the distribution. This formalism enables…
A Machine Learning Framework for Computing the Most Probable Paths of Stochastic Dynamical Systemsmath.DS
Develops a machine learning framework for computing most probable paths in stochastic systems.
problem Computing the most probable paths in stochastic dynamical systems.
method Reformulates the boundary value problem of Hamiltonian systems and uses a neural network to solve the Euler-Lagrange equation for the Onsager-Machlup action functional.
result Demonstrates the efficacy and accuracy of the machine learning approach in computing most probable paths for stochastic systems with various types of noise.
Estimates path-valued data using signature metrics and local kernels.
problem Nonparametric regression and classification for path-valued data.
method Combines signature transform and local kernel regression.
result Establishes convergence bounds and demonstrates competitive accuracy.
Geometric approach clusters intersecting manifolds with high probability.
problem Clustering intersecting d-dimensional manifolds.
method Compute locality graph on d-simplices using dihedral angles, then compute LAPD to separate manifold components.
result The method separates manifold components with high probability under random sampling.
For a variety of regularized optimization problems in machine learning, algorithms computing the entire solution path have been developed recently. Most of these methods are quadratic programs that are parameterized by a single parameter, as for example the Support Vector Machine (SVM). Solution path algorithms do not …
Parallel American Monte Carloq-fin.CP
In this paper we introduce a new algorithm for American Monte Carlo that can be used either for American-style options, callable structured products or for computing counterparty credit risk (e.g. CVA or PFE computation). Leveraging least squares regressions, the main novel feature of our algorithm is that it can be fu…
New estimator for digital options using path splitting and MLMC.
problem Estimating digital options with stochastic differential equations.
method Repeated path splitting, Multilevel Monte Carlo (MLMC).
result Estimator complexity similar to MLMC for Lipschitz payoffs.
In this paper, we address the challenging problem of selecting tuning parameters for high-dimensional sparse regression. We propose a simple and computationally efficient method, called path thresholding (PaTh), that transforms any tuning parameter-dependent sparse regression algorithm into an asymptotically tuning-fre…
Solar: solution path averaging for fast and accurate variable selection in high-dimensional datastat.ML
Solar algorithm selects variables faster and more accurately in high-dimensional data.
problem Variable selection in high-dimensional data with high accuracy and stability.
method Subsample-ordered least-angle regression (solar) and its coordinate descent generalization (solar-cd) using norm solution path averaging.
result Solar selects variables with high accuracy and stability, reducing redundant variable selection.
Efficient algorithms for clustered Lasso and OSCAR reduce computational costs.
problem High dimensional regression with feature clustering.
method Efficient path algorithms for clustered Lasso and OSCAR, reducing computational costs.
result Proposed algorithms are more efficient than existing methods in numerical experiments.
Path Integral and Asset Pricingq-fin.MF
We give a pragmatic/pedagogical discussion of using Euclidean path integral in asset pricing. We then illustrate the path integral approach on short-rate models. By understanding the change of path integral measure in the Vasicek/Hull-White model, we can apply the same techniques to "less-tractable" models such as the …
This paper improves tail dependence analysis by introducing a path-based approach.
problem The classical tail dependence coefficient fails to capture non-exchangeable features of tail dependence.
method The paper introduces a path-based maximal tail dependence approach to capture the most pronounced feature of dependence over all possible paths.
result The paper proves the existence and provides an explicit characterization of the path-based maximal TDC, improving analytical and computational tractability.
Deep signature algorithm for pricing path-dependent options.
problem Pricing path-dependent options with complex payoff functions.
method Extended backward scheme for state-dependent FBSDEs with reflections, incorporating signature layer for path-dependent FBSDEs.
result Convergence analysis of the algorithm with explicit dependence on truncation order and neural network approximation errors.
Control Contraction Metrics (CCMs) provide a nonlinear controller design involving an offline search for a Riemannian metric and an online search for a shortest path between the current and desired trajectories. In this paper, we generalize CCMs to Finsler geometry, allowing the use of non-Riemannian metrics. We provid…
A toolkit for path-norms enhances neural network generalization bounds.
problem Establishing generalization bounds for modern neural networks.
method Introducing a comprehensive toolkit for path-norms in ReLU networks with various operations.
result Established generalization bounds for modern neural networks that are the most widely applicable and recover/beat the sharpest known bounds.
Paper tackles DR problem with scalable signature-based approach.
problem Memory and computation cost issues in DR solutions.
method Signature-based features, novel distance approximator.
result Proposes scalable DR solution with reduced estimation uncertainty.
We compute the integral homology of the space of paths in with endpoints in , and its algebra structure with respect to the Pontryagin-Chas-Sullivan product with -coefficients.
This paper describes a novel framework for computing geodesic paths in shape spaces of spherical surfaces under an elastic Riemannian metric. The novelty lies in defining this Riemannian metric directly on the quotient (shape) space, rather than inheriting it from pre-shape space, and using it to formulate a path energ…
The calculation of minimum energy paths for transitions such as atomic and/or spin re-arrangements is an important task in many contexts and can often be used to determine the mechanism and rate of transitions. An important challenge is to reduce the computational effort in such calculations, especially when ab initio …
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.
We use Karhunen-Loève expansion for efficient pricing of exotic derivatives.
problem Efficient pricing of path-dependent options.
method Karhunen-Loève expansion and Monte Carlo simulation.
result Fast and accurate computation of exotic derivatives pricing.
A new method uses string method to explore diffusion models.
problem Understanding the geometry of learned distributions in diffusion models.
method String method to compute continuous paths between samples.
result The string method identifies realistic morphing sequences and transition pathways.
Quantum Encoding and Analysis on Continuous Time Stochastic Process with Financial Applicationsquant-ph
Quantum computing speeds up analysis of financial stochastic processes.
problem Challenging simulation and analysis of continuous time stochastic processes.
method Established a quantum framework for efficient state preparation and information extraction.
result Extraction of path-dependent and history-sensitive information from stochastic processes efficiently.
CoMPNetX uses neural networks to efficiently solve constrained motion planning problems.
problem Finding collision-free paths on constraint manifolds efficiently.
method Neural generator and discriminator with neural gradients-based projection operator.
result CoMPNetX finds path solutions with high success rates and lower computation times.
Efficiently computes sparse signature coefficients using kernels.
problem Lack of efficient methods for sparse signature coefficients.
method Signature kernels and PDE-based methods.
result Sparse groups of signature coefficients can be isolated effectively.
New algorithm reduces MFGs with common noise complexity.
problem Prohibitive computational cost in solving MFGs with common noise.
method Signatured deep fictitious play based on rough path theory.
result Significantly reduced computational complexity and improved efficiency.
We generalize the classical Bochner formula for the heat flow on M to martingales on the path space PM, and develop a formalism to compute evolution equations for martingales on path space. We see that our Bochner formula on PM is related to two sided bounds on Ricci curvature in much the same manner that the classical…
In this paper, we recover sparse signals from their noisy linear measurements by solving nonlinear differential inclusions, which is based on the notion of inverse scale space (ISS) developed in applied mathematics. Our goal here is to bring this idea to address a challenging problem in statistics, \emph{i.e.} finding …