arXiv research

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.

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

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.

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 …

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

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

Certain natural geometric approximation schemes are developed for Wiener measure on a compact Riemannian manifold. These approximations closely mimic the informal path integral formulas used in the physics literature for representing the heat semi-group on Riemannian manifolds. The path space is approximated by finite …

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…

Formalizes quantum path integrals using groupoids and differential forms.

Paper develops a finite dimensional approximation scheme for Riemannian manifolds.

We consider a standard optimal investment problem in a complete financial market driven by a Wiener process and derive an explicit formula for the optimal portfolio process in terms of the vertical derivative from functional It^o calculus. An advantage with this approach compared to the Malliavin calculus approach is t…

This paper presents a numerical model to solve the problem of cash accumulation strategies for products with an unknown future price, like assets. Stock prices are modeled by a discretized Wiener Process, and by the means of ordinary integrals this Wiener Process will be exactly matched at a preset terminal time. Three…

In this survey article, we review the relation between heat kernels and path integrals. In particular, we review recent results on the approximation of the Wiener measure on compact manifold by measures on (finite-dimensional) spaces of piece-wise geodesics.

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…

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

This paper provides an existence-and-uniqueness theorem characterizing the stochastic integral with respect to a Wiener process. The integral is represented as a mapping from the space of measurable and adapted pathwise locally integrable processes to the space of continuous adapted processes. It is characterized in te…

Develops trinomial models using cubature methods for financial derivative pricing.

Reduces path integrals for interacting systems using dependent coordinates.

Study on stochastic covariant derivatives in curved space-time.

Fast method developed for pricing barrier options and joint Lévy process distributions.

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

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…

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…

Neural networks solve SPDEs using Wiener chaos expansion.

These notes were inspired by the course ''Quantum Field Theory from a Functional Integral Point of View'' given at the University of Zurich in Spring 2017 by Santosh Kandel. We describe Feynman's path integral approach to quantum mechanics and quantum field theory from a functional integral point of view, where the mai…

We introduce a class of non-commutative, complex, infinite-dimensional Heisenberg like Lie groups based on an abstract Wiener space. The holomorphic functions which are also square integrable with respect to a heat kernel measure $μ$ on these groups are studied. In particular, we establish a unitary equivalence between…

The paper examines Wiener process for LID estimation methods.

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…

We show that the jumps correlation matrix of a multivariate Hawkes process is related to the Hawkes kernel matrix through a system of Wiener-Hopf integral equations. A Wiener-Hopf argument allows one to prove that this system (in which the kernel matrix is the unknown) possesses a unique causal solution and consequentl…

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

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

Modeling stock price fluctuations using Brownian motion and stochastic differential equations.

New linear denoiser outperforms standard Wiener filter in noisy data.

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…

New method distinguishes data noise from GP uncertainty.

A new error bound improves safety in Bayesian optimization.

Paper proposes a new model for better engine control.

Let $W_i=\{W_i(t_i), t_i\in \R_+\}, i=1,2,\ldots,d$ are independent Wiener processes. $W=\{W(\mathbf{t}),t\in \R_+^d\}$ be the additive Wiener field define as the sum of $W_i$. For any trend $f$ in $\kHC$ (the reproducing kernel Hilbert Space of $W$), we derive upper and lower bounds for the boundary non-crossing proba…

Geometrically represents path integral reduction Jacobian for interacting systems.

We prove a Paley-Wiener Theorem for a class of symmetric spaces of the compact type, in which all root multiplicities are even. This theorem characterizes functions of small support in terms of holomorphic extendability and exponential type of their (discrete) Fourier transforms. We also provide three independent new p…

The paper creates nonparametric confidence bands for band-limited functions.

We introduce a class of non-commutative Heisenberg like infinite dimensional Lie groups based on an abstract Wiener space. The Ricci curvature tensor for these groups is computed and shown to be bounded. Brownian motion and the corresponding heat kernel measures, $\{ν_t\}_{t>0},$ are also studied. We show that these he…

The problem of image restoration in cryo-EM entails correcting for the effects of the Contrast Transfer Function (CTF) and noise. Popular methods for image restoration include `phase flipping', which corrects only for the Fourier phases but not amplitudes, and Wiener filtering, which requires the spectral signal to noi…

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…

The paper ensures positivity of solutions to stochastic equations with positive initial data.

This paper considers the valuation of exotic path-dependent options in Lévy models, in particular options on the supremum and the infimum of the asset price process. Using the Wiener--Hopf factorization, we derive expressions for the analytically extended characteristic function of the supremum and the infimum of a Lév…

These notes represent a much expanded and updated version of the \textquotedblleft mini course\textquotedblright that the author gave at the ETH (Zürich) and the University of Zürich in February of 1995. The purpose of these notes is to first provide some basic background to Riemannian geometry and stochastic calculus …