相关论文: Universal approximation on non-geometric rough pat…

We establish a universal approximation theorem for signatures of rough paths that are not necessarily weakly geometric. By extending the path with time and its rough path bracket terms, we prove that linear functionals of the signature of…

We establish universal approximation theorems for infinite-dimensional geometric rough paths, i.e., we show that continuous functions on the space of infinite-dimensional weakly geometric H\"older continuous rough paths can be approximated…

We establish $L^p$-type universal approximation theorems for general and non-anticipative functionals on suitable rough path spaces, showing that linear functionals acting on signatures of time-extended rough paths are dense with respect to…

The universal approximation theorem asserts that a single hidden layer neural network approximates continuous functions with any desired precision on compact sets. As an existential result, the universal approximation theorem supports the…

We introduce so-called functional input neural networks defined on a possibly infinite dimensional weighted space with values also in a possibly infinite dimensional output space. To this end, we use an additive family to map the input…

We introduce signature payoffs, a family of path-dependent derivatives that are given in terms of the signature of the price path of the underlying asset. We show that these derivatives are dense in the space of continuous payoffs, a result…

We establish global universal approximation theorems on spaces of piecewise linear paths, stating that linear functionals of the corresponding signatures are dense with respect to $L^p$- and weighted norms, under an integrability condition…

In this work we produce a framework for constructing universal function approximators on graph isomorphism classes. We prove how this framework comes with a collection of theoretically desirable properties and enables novel analysis. We…

We prove an extension to the classical continuity theorem in rough paths. We show that two $p$-rough paths are close in all levels of iterated integrals provided the first $\lfl p \rfl$ terms are close in a uniform sense. Applications…

In this paper, we introduce the branched signature model, motivated by the branched rough path framework of [Gubinelli, Journal of Differential Equations, 248(4), 2010], which generalizes the classical geometric rough path. We establish a…

It is a classical result in rational approximation theory that certain non-smooth or singular functions, such as $|x|$ and $x^{1/p}$, can be efficiently approximated using rational functions with root-exponential convergence in terms of…

We describe the pricing and hedging of financial options without the use of probability using rough paths. By encoding the volatility of assets in an enhancement of the price trajectory, we give a pathwise presentation of the replication of…

A computational technique borrowed from the physical sciences is introduced to obtain accurate closed-form approximations for the transition probability of arbitrary diffusion processes. Within the path integral framework the same technique…

In the spirit of Arrow-Debreu, we introduce a family of financial derivatives that act as primitive securities in that exotic derivatives can be approximated by their linear combinations. We call these financial derivatives signature…

The non-linear sewing lemma constructs flows of rough differential equations from a braod class of approximations called almost flows. We consider a class of almost flows that could be approximated by solutions of ordinary differential…

Traditional nonparametric estimation methods often lead to a slow convergence rate in large dimensions and require unrealistically enormous sizes of datasets for reliable conclusions. We develop an approach based on partial derivatives,…

A novel generative machine learning approach for the simulation of sequences of financial price data with drawdowns quantifiably close to empirical data is introduced. Applications such as pricing drawdown insurance options or developing…

The derivative expansion approach to the calculation of the interaction between two surfaces, is a generalization of the proximity force approximation, a technique of widespread use in different areas of physics. The derivative expansion…

The universal approximation theorem states that a neural network with one hidden layer can approximate continuous functions on compact sets with any desired precision. This theorem supports using neural networks for various applications,…

In this paper we consider a family of algorithms for approximate implicitization of rational parametric curves and surfaces. The main approximation tool in all of the approaches is the singular value decomposition, and they are therefore…