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.
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.
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 …
This paper gives a rigorous interpretation of a Feynman path integral on a Riemannian manifold M with non-positive sectional curvature. A L2 Riemannian metric GP is given on the space of piecewise geodesic paths HP(M) adapted to the partition P of [0,1], whence a finite-dimensional approximation of Wiener …
Formalizes quantum path integrals using groupoids and differential forms.
problem Formalizing Feynman's path integral in quantum mechanics.
method Shifted focus to pair groupoid, using van Est map and piecewise linear structures.
result Developed a coordinate-free approach to integration of differential forms.
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…
Reduces path integrals for interacting systems using dependent coordinates.
problem Reducing path integrals for systems with symmetry.
method Reduction procedure based on Wiener-type path integral, optimal nonlinear filtering, and projection of mean curvature vector field.
result Shows non-invariance of the measure in the path integral under reduction and generates the Jacobian.
Paper develops a finite dimensional approximation scheme for Riemannian manifolds.
problem Integration on Riemannian manifolds.
method New finite dimensional approximation scheme motivated by categorical colimit.
result Establishes a generalization for L1-functionals on Riemannian manifolds. Functional integrals explain quantum mechanics and field theory.
problem Explaining quantum mechanics and field theory using functional integrals.
method Describes Feynman's path integral approach to quantum mechanics and field theory.
result Equivalence of path integral formalism to classical mechanics and quantum mechanics.
Formulae prove integration by parts for foliated Wiener measure.
problem Integration by parts formula for foliated Wiener measure.
method Proved integration by parts formula for horizontal Wiener measure on totally geodesic Riemannian foliations.
result Horizontal Wiener measure has quasi-invariance under certain flows.
Geometrically represents path integral reduction Jacobian for interacting systems.
problem Quantizing a model mechanical system with dependent coordinates.
method Geometric representation using scalar curvature and Christoffel symbols in a nonholonomic basis.
result Found a geometric representation for the path integral reduction Jacobian.
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…
This work explores functional expansions to handle path dependence in various fields.
problem Path dependence and infinite-dimensional problems in non-Markovian systems.
method Generalizes Wiener series and functional Taylor expansion to handle static and dynamic functionals.
result Elegant separation of functionals from future trajectories in dynamic cases.
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…
Geometrically represents the Jacobian for a mechanical system with symmetry.
problem Path integral reduction for a mechanical system with symmetry.
method Geometric representation using scalar curvature and adapted coordinates.
result Obtained geometric representation of the Jacobian.
Counterexample shows Ito integrand needn't be locally square integrable.
problem Ito integrand's square integrability condition is not always met.
method Provided a counterexample to Ito's Lemma's integrability condition.
result 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.
Paper proves existence and uniqueness of stochastic integral.
problem Existence and uniqueness of stochastic integral with Wiener process.
method Characterizes the Ito integral through two properties: simple process calculation and convergence of squared integrands.
result Existence and uniqueness theorem for stochastic integral.
Model predicts cash accumulation for assets with unknown prices.
problem Cash accumulation for assets with unpredictable future prices.
method Discretized Wiener Process matched using ordinary integrals.
result Model efficiently predicts cash accumulation for various asset scenarios.
Proves existence of solutions to stochastic heat equations on manifolds.
problem Existence of solutions to stochastic heat equations on Riemannian manifolds.
method Proved existence using Dirichlet forms and Wiener measure.
result Established log-Sobolev inequality for the Dirichlet form in the path space.
New algorithm prices Bermudan options using Wiener chaos expansion for non-Markovian processes.
problem Pricing Bermudan options with non-Markovian payoff processes.
method Modified Longstaff Schwartz algorithm with Wiener chaos expansion for non-Markovian settings.
result Embarrassingly parallel algorithm for efficient computation.
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…
New methods for Z-transform inversion and Wiener-Hopf factorization.
problem Efficient numerical inversion of Z-transforms and factorization of functions. method Sinh-deformations of contours, variable changes, and simplified trapezoid rule.
result High precision and speed in evaluating moments and constructing filters.
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 models asset prices using Wiener chaos expansions for efficient calibration to implied volatility surfaces.
problem Calibrating to implied volatility surfaces using flexible martingale models.
method Constructing an over-parameterized martingale model based on Wiener chaos expansions and conditional expectations.
result The method enables fast calibration to implied volatility surfaces and demonstrates flexibility through numerical experiments.
Derives an explicit formula for optimal portfolios in financial markets.
problem Optimal investment problem in complete financial markets driven by Wiener process.
method Functional Itô calculus approach, relying only on integrability condition.
result Derives an explicit formula for the optimal portfolio process.
In this note we apply the recently established Wiener-Hopf Monte Carlo (WHMC) simulation technique for Levy processes from Kuznetsov et al. [17] to path functionals, in particular first passage times, overshoots, undershoots and the last maximum before the passage time. Such functionals have many applications, for inst…
In the setting proposed by Hughston & Rafailidis (2005) we consider general interest rate models in the case of a Brownian market information filtration (Ft)t≥0. Let X be a square-integrable F∞-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.
problem Optimizing portfolio allocation with transaction costs in multi-period settings.
method Wiener chaos expansion approach to represent and solve the optimization problem.
result The proposed method finds an optimal strategy for portfolio allocation with transaction costs.
Develops trinomial models using cubature methods for financial derivative pricing.
problem Pricing financial derivatives in complex stochastic market models.
method Cubature methods applied to Wiener space for constructing trinomial models.
result Numerical solutions compare favorably with Black-Scholes model.
Study on stochastic covariant derivatives in curved space-time.
problem Analyzing covariant derivatives in curved space-time under stochastic processes.
method Using Itô-Wiener processes and stochastic calculus, including Besov spaces, Schrödinger operators, and white noise.
result Developed a framework for stochastic geodesics and white noise in fractoid spaces.
Fast method developed for pricing barrier options and joint Lévy process distributions.
problem Accurate pricing of barrier options and joint distributions in Lévy models.
method Dual space calculations, Wiener-Hopf factorization, sinh-deformations, Gaver-Wynn Rho acceleration.
result Achieves precision of 10−15 in seconds and 10−9−10−8 in fractions of a second. Develops interpretable model for latent stochastic systems from noisy data.
problem Learning interpretable models of latent stochastic dynamical systems from noisy data.
method Semi-parametric model using Gaussian process for drift, inference of latent paths with sparse variational description.
result Flexible nonparametric model of dynamics with interpretable portraits.
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…
The paper proves a convergence theorem for Wiener measures on holonomy groups.
problem Understanding convergence of Wiener measures on holonomy groups.
method Using stochastic parallel transports along convergent metric connections.
result Proves a convergence theorem for push-forward Wiener measures on holonomy groups.
In this paper we consider the problem of pricing a perpetual American put option in an exponential regime-switching Lévy model. For the case of the (dense) class of phase-type jumps and finitely many regimes we derive an explicit expression for the value function. The solution of the corresponding first passage problem…
This paper establishes a non-stochastic analogue of the celebrated result by Dubins and Schwarz about reduction of continuous martingales to Brownian motion via time change. We consider an idealized financial security with continuous price path, without making any stochastic assumptions. It is shown that typical price …
In this paper we prove the probabilistic continuous complexity conjecture. In continuous complexity theory, this states that the complexity of solving a continuous problem with probability approaching 1 converges (in this limit) to the complexity of solving the same problem in its worst case. We prove the conjecture ho…
New method infers population dynamics from snapshots using path space optimization.
problem Recover dynamics of a population from its temporal marginals.
method Grid-free algorithm using Schrödinger bridges coupled via noisy gradient descent in mean-field limit.
result Global convergence to min-entropy estimator with end-to-end theoretical guarantees.
Neural networks solve SPDEs using Wiener chaos expansion.
problem Solving stochastic partial differential equations (SPDEs) numerically.
method Using neural networks in the truncated Wiener chaos expansion.
result Approximation rates for learning SPDE solutions with noise.
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.
Theory of covariant Schrödinger semigroups on Riemannian manifolds developed.
problem Developing theory for Schrödinger semigroups on Riemannian manifolds.
method Sobolev spaces, heat kernels, differential operators, Wiener measure, Dynkin and Kato potentials.
result Properties and continuity of covariant Schrödinger semigroups established.
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.
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.
Paper develops a new framework for analyzing certainty equivalents and dynamic risk premia using Malliavin calculus and Wiener chaos analysis.
problem Limitations of Arrow-Pratt approximation for arbitrary sequences of vanishing risks.
method Develops a new framework based on Malliavin calculus and Wiener chaos analysis, combining Itô calculus, the Clark--Ocone representation, and the Wiener chaos decomposition.
result Establishes a unified framework linking expected utility theory, stochastic analysis, and Wiener chaos expansions, revealing higher-order certainty equivalents and dynamic risk premia.
We use GANs and signatures to approximate conditional laws in filtering and prediction of diffusion processes.
problem Approximating conditional laws for diffusion processes with noisy observations.
method Conditional GANs combined with signatures for approximation.
result Efficient approximation of conditional laws for diffusion processes.