arXiv research

Neural networks solve SPDEs using Wiener chaos expansion.

The paper models asset prices using Wiener chaos expansions for efficient calibration to implied volatility surfaces.

This paper proposes a new method to optimize portfolio allocation with transaction costs using Wiener chaos expansion.

Study on spin random fields using chaos decomposition for cosmic microwave background modeling.

New method distinguishes data noise from GP uncertainty.

In this work, we propose a new policy iteration algorithm for pricing Bermudan options when the payoff process cannot be written as a function of a lifted Markov process. Our approach is based on a modification of the well-known Longstaff Schwartz algorithm, in which we basically replace the standard least square regre…

New method estimates SDE parameters efficiently using WCE and SGD.

In this we paper we recast the Cox--Ingersoll--Ross model of interest rates into the chaotic representation recently introduced by Hughston and Rafailidis. Beginning with the ``squared Gaussian representation'' of the CIR model, we find a simple expression for the fundamental random variable X. By use of techniques fro…

Study reveals three limiting regimes for neural network functionals.

The Wiener chaos approach to interest rate modelling arises from the observation that the pricing kernel admits a representation in terms of the conditional variance of a square-integrable random variable, which in turn admits a chaos expansion. When the expansion coefficients factorise into multiple copies of a single…

In the setting proposed by Hughston & Rafailidis (2005) we consider general interest rate models in the case of a Brownian market information filtration $(\mathcal{F}_t)_{t\geq0}$. Let $X$ be a square-integrable $\mathcal{F}_\infty$-measurable random variable, and assume the non-degeneracy condition that for all $t<\in…

Model reduction methods aim to describe complex dynamic phenomena using only relevant dynamical variables, decreasing computational cost, and potentially highlighting key dynamical mechanisms. In the absence of special dynamical features such as scale separation or symmetries, the time evolution of these variables typi…

This work explores functional expansions to handle path dependence in various fields.

New chaos formula simplifies variance calculation for Gaussian nodal volumes.

In this work, we propose an algorithm to price American options by directly solving the dual minimization problem introduced by Rogers. Our approach relies on approximating the set of uniformly square integrable martingales by a finite dimensional Wiener chaos expansion. Then, we use a sample average approximation tech…

Gaussian equivalence fails for simple polynomial embeddings in quadratic scaling RF models.

The paper proves a convergence theorem for Wiener measures on holonomy groups.

NeuralChaos efficiently approximates complex stochastic processes.

The study reveals chaos in geometric objects embedded in higher dimensions.

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…

We prove several versions of Driver's integration by parts formula for the horizontal Wiener measure on a totally geodesic Riemannian foliation and prove that the horizontal Wiener measure has a quasi-invariance property with respect to flows generated by suitable tangent processes.

The resilience of low-degree Rademacher chaos is studied, providing probabilistic lower bounds.

The paper discusses the main ideas of the chaos theory and presents mainly the importance of the nonlinearities in the mathematical models. Chaos and order are apparently two opposite terms. The fact that in chaos can be found a certain precise symmetry (Feigenbaum numbers) is even more surprising. As an illustration o…

New methods for $Z$-transform inversion and Wiener-Hopf factorization.

This paper gives a rigorous interpretation of a Feynman path integral on a Riemannian manifold M with non-positive sectional curvature. A $L^2$ Riemannian metric $G_P$ is given on the space of piecewise geodesic paths $H_P(M)$ adapted to the partition $P$ of $[0,1]$, whence a finite-dimensional approximation of Wiener …

The paper examines Wiener process for LID estimation methods.

Study on variance of Laplace eigenfunctions on manifolds.

In this paper we calibrate chaotic models for interest rates to market data using a polynomial-exponential parametrization for the chaos coefficients. We identify a subclass of one-variable models that allow us to introduce complexity from higher order chaos in a controlled way while retaining considerable analytic tra…

New insights into neural network training efficiency.

Cubature on Wiener space [Lyons, T.; Victoir, N.; Proc. R. Soc. Lond. A 8 January 2004 vol. 460 no. 2041 169-198] provides a powerful alternative to Monte Carlo simulation for the integration of certain functionals on Wiener space. More specifically, and in the language of mathematical finance, cubature allows for fast…

In an abstract Wiener space setting, we constract a rigorous mathematical model of the one-loop approximation of the perturbative Chern-Simons integral, and derive its explicit asymptotic expansion for stochastic Wilson lines.

Enhances polynomial chaos models with uncertainty intervals.

Study shows uniform-time chaos propagation in mean field Langevin dynamics.

Constructs non-asymptotic confidence regions for unknown functions in RKHS.

The paper introduces invariants to describe period-doubling routes to chaos in dynamical systems.

Uniform-in-time analysis for Stein Variational Gradient Descent across various metrics.

Researchers use quantum chaos and RMT to analyze turbulence, revealing unique scaling laws.

Neural networks can model chaos efficiently by becoming geometrically chaotic.

The paper explores Wiener-Granger causality and its computational enhancements.

The Financial Chaos Index models stock market volatility across three regimes based on mutual price fluctuations.

In this letter, as a proof of concept, we propose a deep learning-based approach to attack the chaos-based image encryption algorithm in \cite{guan2005chaos}. The proposed method first projects the chaos-based encrypted images into the low-dimensional feature space, where essential information of plain images has been …

New linear denoiser outperforms standard Wiener filter in noisy data.

A new neural network model uses polynomial chaos theory to improve neural signal processing.

Basis adaptation in Homogeneous Chaos spaces rely on a suitable rotation of the underlying Gaussian germ. Several rotations have been proposed in the literature resulting in adaptations with different convergence properties. In this paper we present a new adaptation mechanism that builds on compressive sensing algorith…

In this paper we study the price dynamics in a simple model of financial markets with heterogeneous agents. We concentrate on how increases in the total number of active traders influences fluctuations of asset prices. We find that a curious route to chaos is observed when the total number of [active traders] increases…

Wiener-Granger causality is a widely used framework of causal analysis for temporally resolved events. We introduce a new measure of Wiener-Granger causality based on kernelization of partial canonical correlation analysis with specific advantages in the context of large high-dimensional data. The introduced measure is…

Counterexample shows Ito integrand needn't be locally square integrable.

Chaos and nonlinear economic dynamics are addressed for a quantum coupled map lattice model of an artificial economy, with quantized supply and demand equilibrium conditions. The measure theoretic properties and the patterns that emerge in both the economic business volume dynamics' diagrams as well as in the quantum m…