Solves infinite horizon portfolio problem with path-dependent labor income.
problem Infinite horizon portfolio choice with path-dependent labor income.
method Solves an infinite dimensional stochastic optimal control problem using explicit solutions to the HJB equation.
result Explicit solutions to the optimal controls in feedback form are found.
A 3D space of hyperbolic manifolds is connected but not path-connected.
problem Proving connectivity and non-path-connectedness of framed hyperbolic 3-manifolds.
method Two proofs using density theorems for Kleinian groups, constructing dense sets of framings, and discussing paths.
result The space of framed infinite volume hyperbolic 3-manifolds is not path-connected.
Inf-FS selects features by graph paths, ranking them for infinite feature sets.
problem Feature selection in large datasets with relevance and redundancy.
method Graph-based feature selection with infinite paths, evaluating feature subsets using matrix power series and Markov chains.
result Inf-FS outperforms other methods in various feature selection scenarios.
Study shows infinitely many path components for positive Ricci curvature metrics on certain spin manifolds.
problem Understanding the space of metrics with positive Ricci curvature on spin manifolds.
method Analyzing spin manifolds of dimension 4k-1 for k ≥ 2, focusing on metrics with a specific property.
result The space of metrics has infinitely many path components.
The study examines moduli spaces of metrics with positive Ricci or non-negative sectional curvature on sphere bundles.
problem Classifying and understanding moduli spaces of metrics with specific curvature properties on sphere bundles.
method Analyzing total spaces of S7-bundles over S8 and quotients of Milnor and Shimada spheres. result The moduli space of metrics has infinitely many path components.
Adjoint sampler targets infinite-dimensional function spaces for efficient sampling.
problem Limited theory and algorithms for sampling infinite-dimensional function spaces.
method Adjoint Sampler for infinite-dimensional function spaces based on stochastic maximum principle.
result FAS achieves superior performance in synthetic and real systems.
Study shows infinite families of manifolds with nonnegative curvature.
problem Finding nonnegatively curved metrics on manifolds.
method Exhibited infinite families of manifolds with specific properties.
result Moduli space of nonnegatively curved metrics has infinitely many components.
3D hyperbolic spaces have endless simple paths.
problem Finding simple paths in complex 3D spaces.
method Analyzing geodesics in hyperbolic 3-manifolds.
result Cusped hyperbolic 3-manifolds have infinitely many simple closed geodesics.
Let M be any n dimensional smooth manifold and PM be the space of all smooth paths, then we showed that PM is a smooth manifold modelled over a complete normable space. We discussed many geometric structure on Path spaces and its relation to ambient space.
Study shows infinitely many metrics with nonnegative sectional or positive Ricci curvature on specific 5D quotients.
problem Finding metrics with specific curvature properties on Brieskorn quotients.
method Analyzing moduli spaces of metrics with nonnegative sectional or positive Ricci curvature.
result Moduli spaces have infinitely many path components for both nonnegative sectional and positive Ricci curvature.
Novel signature approach for pricing and hedging path-dependent options with market frictions.
problem Pricing and hedging path-dependent options with market frictions.
method Signature approach, mean-quadratic variation criterion, non-standard infinite-dimensional Riccati equations, time-augmented signature, non-Markovian stochastic control problem.
result Effective hedging strategies in frictional markets with low-truncated signature approximations.
This paper considers possible price paths of a financial security in an idealized market. Its main result is that the variation index of typical price paths is at most 2, in this sense, typical price paths are not rougher than typical paths of Brownian motion. We do not make any stochastic assumptions and only assume t…
We extend the notion of unicorn paths between two arcs introduced by Hensel, Przytycki and Webb to the case where we replace one arc with a geodesic asymptotic to a lamination. Using these paths, we give new proofs of the results of Klarreich and Schleimer identifying the Gromov boundaries of the curve graph and the ar…
This paper proposes new get-rich-quick schemes that involve trading in a financial security with a non-degenerate price path. For simplicity the interest rate is assumed zero. If the price path is assumed continuous, the trader can become infinitely rich immediately after it becomes non-constant (if it ever does). If i…
Using symplectic techniques and spectral analysis of smooth paths of self-adjoint operators, we characterize the set of conjugate instants along a geodesic in an infinite dimensional Riemannian Hilbert manifold.
Sub-Riemannian geometry connects bike paths to mathematical curves.
problem Understanding bike paths and their mathematical properties.
method Relating sub-Riemannian geometry to bicycle motion and curve shapes.
result Geodesics in sub-Riemannian geometry correspond to specific bike paths.
Project infinite time series graphs to finite marginal models using number theory.
problem Handling infinite time series graphs for causal inference.
method Projection method using number theory to find common ancestors in infinite graphs.
result Developed algorithm to project infinite graphs to finite marginal models.
A new method detects anomalies in multivariate streams without unit dependence.
problem Detect anomalies in multivariate streams without unit dependence.
method Proposes SigMahaKNN combining variance norm and path signature.
result SigMahaKNN detects anomalies better than existing methods.
We show that the moduli space of metrics of nonnegative sectional curvature on every homotopy RP5 has infinitely many path components. We also show that in each dimension 4k+1 there are at least 22k homotopy RP4k+1s of pairwise distinct oriented diffeomorphism type for which the…
We establish a local function version of a classical result claiming that a bivector field on a manifold M is Poisson if and only if cotangent paths form a coisotropic set of the infinite dimensional symplectic manifold of paths valued in T∗M. Our purpose here is to prove this result without using the Banach manif…
Local gluing connects flow lines in finite time intervals.
problem Connecting flow lines in finite time intervals.
method Functional analytic approach to define local gluing map.
result Explicit construction of local gluing map in Euclidean case; intricate construction in non-Euclidean case.
Study geodesic paths on flat surfaces, comparing length and singularity counts.
problem Comparing geometric length and singularity counts on geodesic paths.
method Apply counting limit laws to infinite graphs and then to flat surfaces.
result Statistical comparison of geometric length and singularity counts on geodesic paths.
We present an explicit realization of abelian extensions of infinite dimensional Lie groups using abelian extensions of path groups, by generalizing Mickelsson's approach to loop groups and the approach of Losev-Moore-Nekrasov-Shatashvili to current groups. We apply our method to coupled cocycles on current Lie algebra…
We consider idealized financial markets in which price paths of the traded securities are cadlag functions, imposing mild restrictions on the allowed size of jumps. We prove the existence of quadratic variation for typical price paths, where the qualification "typical" means that there is a trading strategy that risks …
Study geodesics on graphs with random lengths, proving bi-infinite paths exist.
problem Existence of bi-infinite geodesic paths on graphs with random edge lengths.
method Sublinear Morse geodesics and first passage percolation analysis.
result Proves the existence of bi-infinite geodesic paths in graphs with specific properties.
Develops methods for estimating constrained function-valued parameters in infinite-dimensional models.
problem Estimating function-valued parameters with structural constraints in complex models.
method Characterizes constrained solutions as minimizers of penalized population risk, using a Lagrange-type formulation and path through unconstrained space.
result Proposes estimators that achieve optimal risk and constraint satisfaction, applicable across various statistical learning approaches.
The paper calculates the growth rates of billiard languages in hyperbolic polygons.
problem Computing the exponential growth rates of billiard languages in polygons.
method New methods relating to minimal tiling paths.
result Explicit computation of exponential growth rates for q even, and bounds for q odd. We develop a Malliavin calculus on the horizontal path space of a totally geodesic Riemannian foliation. As a first application, under suitable assumptions, we prove a log-Sobolev inequality for a natural one-parameter family of infinite-dimensional Ornstein-Uhlenbeck type operators. As a second application, we obtain …
New method for integrating singular foliations via paths.
problem Characterize holonomy and fundamental groupoids of singular foliations.
method Quotient of an infinite dimensional space of paths, extending classical construction for regular foliations.
result Characterization of holonomy and fundamental groupoids of singular foliations.
We aim to generalize the results of Cai and Nitta (2007) by allowing both the utility and production function to depend on time. We also consider an additional intertemporal optimality criterion. We clarify the conditions under which the limit of the solutions for the finite horizon problems is optimal among all attain…
For a principal bundle P→M equipped with a connection Aˉ, we study an infinite dimensional bundle PAˉdecP over the space of paths on M, with the points of PAˉdecP being horizontal paths on P decorated with elements of a second structure group. We co…
The paper solves optimal control problems for stochastic delay equations.
problem Optimal control of stochastic delay differential equations.
method Rewriting the problem in an infinite-dimensional Hilbert space, using dynamic programming and viscosity solutions.
result Characterizes the value function as the unique viscosity solution of the Hamilton-Jacobi-Bellman equation.
The study finds infinitely many periodic orbits that can be used to modify Anosov flows.
problem Can surgeries on periodic orbits of Anosov flows produce equivalent flows?
method Analyzing suspension Anosov flows, the study identifies pairs of periodic orbits that can be used to modify the flow.
result For some suspension Anosov flows, there exist infinitely many pairs of periodic orbits that can be used to modify the flow.
New method uses LSTM and signature theory to solve complex financial PDEs.
problem Solving path-dependent PDEs for financial derivatives pricing.
method Combining LSTM networks and rough paths theory.
result Efficient algorithms for pricing and hedging path-dependent derivatives.
Study proves hyperfiniteness of mapping class group actions on surface graphs.
problem Hyperfiniteness of mapping class group actions on surface graphs.
method Infinite unicorn paths and Gromov boundaries of arc and curve graphs.
result Proves hyperfiniteness of orbit equivalence relations induced by mapping class group actions.
The infinite Viterbi alignment is the limiting maximum a-posteriori estimate of the unobserved path in a hidden Markov model as the length of the time horizon grows. For models on state-space Rd satisfying a new ``decay-convexity'' condition, we develop an approach to existence of the infinite Viterbi ali…
We generalize the classical Bochner formula for the heat flow on evolving manifolds (M,gt)t∈[0,T] to an infinite-dimensional Bochner formula for martingales on parabolic path space PM of space-time M=M×[0,T]. Our new Bochner formula and the inequalities that follow from it a…
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.
The relaxed maximum entropy problem is concerned with finding a probability distribution on a finite set that minimizes the relative entropy to a given prior distribution, while satisfying relaxed max-norm constraints with respect to a third observed multinomial distribution. We study the entire relaxation path for thi…
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.
Study growth rates of subgroups in groups with a constricting element.
problem Understanding growth rates of subgroups in groups with a constricting element.
method Examining the spectrum of relative and quotient exponential growth rates of quasi-convex subgroups.
result Determine when growth rates of subgroups are strictly smaller or coincide with the group's growth rate.
This paper shows how infinitely wide Tensor Networks converge to Gaussian Processes.
problem Understanding the relationship between Tensor Networks and Gaussian Processes.
method Analyzing the infinite-width limit of Tensor Networks and comparing them to Gaussian Processes.
result Infinitely wide Tensor Networks converge to Gaussian Processes, proving their equivalence.
The symmetries of paths in a manifold M are classified with respect to a given pointwise proper action of a Lie group G on M. Here, paths are embeddings of a compact interval into M. There are at least two types of symmetries: Firstly, paths that are parts of an integral curve of a fundamental vector field on $…
We investigate the short-time expansion of the heat kernel of a Laplace type operator on a compact Riemannian manifold and show that the lowest order term of this expansion is given by the Fredholm determinant of the Hessian of the energy functional on a space of finite energy paths. This is the asymptotic behavior to …
The paper introduces a new volatility model using Fourier techniques for pricing and hedging.
problem Pricing and hedging of financial derivatives with stochastic volatility.
method A Fourier-based approach to price and hedge European and path-dependent options in a stochastic volatility model.
result The model includes and extends popular volatility models like Stein-Stein, Bergomi, and Heston.
The paper proves a category of dg manifolds with finite positive amplitude.
problem Understanding the structure of dg manifolds with finite positive amplitude.
method Using path spaces and homotopy transfer theorem for curved L∞[1]-algebras. result Proves that dg manifolds of finite positive amplitude form a category of fibrant objects.
We consider the variational complex on infinite jet space and the complex of variational derivatives for Lagrangians of multidimensional paths and study relations between them. The discussion of the variational (bi)complex is set up in terms of a flat connection in the jet bundle. We extend it to supercase using a part…
Volterra signature provides a clear, interpretable feature for history-dependent systems.
problem Learning from non-Markovian time series with implicit memory mechanisms.
method Develops Volterra signature as a tensor algebra representation weighted by a temporal kernel, proving injectivity and universal approximation.
result Volterra signature leads to linear functionals and universal approximation, improving dynamic learning tasks.