Generating Financial Time Series by Matching Random Convolutional Features

Generating realistic financial time series is challenging as training data is often limited to a single historical path. With such scarce data, overfitting is hard to avoid, especially under adversarial training where a trained discriminator can memorize the training samples. To mitigate this, recent approaches train generators to minimize the discrepancy between untrained feature representations of real and generated time series. In these works, the feature maps are based on path signatures, which can fail to capture relevant time series properties at tractable truncation depths. In this work, we instead train generators by matching random convolutional features of real and generated time series. Existing random convolutional feature maps, such as Rocket and Hydra, have been shown to provide informative representations of real-world time series, but cannot supervise generative models because they are non-differentiable. We introduce SOCK (SOft Competing Kernels), a fully differentiable random convolutional feature map, suited to train generative time series models. We show that generators trained by matching random SOCK features consistently outperform signature and diffusion baselines across a wide range of small-sample financial datasets. We further demonstrate SOCK's expressiveness on two-sample hypothesis testing and time series classification tasks, where SOCK matches or outperforms existing unsupervised feature maps.

Random Neural Network Expressivity for Non-Linear Partial Differential Equations

Neural networks with randomly generated hidden weights (RaNNs) have been extensively studied, both as a standalone learning method and as an initialization for fully trainable deep learning methods. In this work, we study RaNN expressivity for learning solutions to non-linear partial differential equations (PDEs). Despite their widespread use in practical applications, a rigorous theoretical understanding of the approximation properties of RaNNs in this context remains limited. Here, we derive error bounds for RaNN approximations to time-dependent Sobolev functions and obtain a dimension-free approximation rate $\frac{1}{2}$ for sufficiently regular functions. We apply our results to two important classes of non-linear PDEs: Porous Medium Equations and Compressible Navier-Stokes Equations, showing that RaNNs are capable of efficiently approximating solutions to these complex, non-linear PDEs. Our theoretical analysis is supported by numerical experiments, showing that the obtained convergence rates extend beyond the considered setting.

Error analysis for the deep Kolmogorov method

The deep Kolmogorov method is a simple and popular deep learning based method for approximating solutions of partial differential equations (PDEs) of the Kolmogorov type. In this work we provide an error analysis for the deep Kolmogorov method for heat PDEs. Specifically, we reveal convergence with convergence rates for the overall mean square distance between the exact solution of the heat PDE and the realization function of the approximating deep neural network (DNN) associated with a stochastic optimization algorithm in terms of the size of the architecture (the depth/number of hidden layers and the width of the hidden layers) of the approximating DNN, in terms of the number of random sample points used in the loss function (the number of input-output data pairs used in the loss function), and in terms of the size of the optimization error made by the employed stochastic optimization method.

Distributional Adversarial Attacks and Training in Deep Hedging

In this paper, we study the robustness of classical deep hedging strategies under distributional shifts by leveraging the concept of adversarial attacks. We first demonstrate that standard deep hedging models are highly vulnerable to small perturbations in the input distribution, resulting in significant performance degradation. Motivated by this, we propose an adversarial training framework tailored to increase the robustness of deep hedging strategies. Our approach extends pointwise adversarial attacks to the distributional setting and introduces a computationally tractable reformulation of the adversarial optimization problem over a Wasserstein ball. This enables the efficient training of hedging strategies that are resilient to distributional perturbations. Through extensive numerical experiments, we show that adversarially trained deep hedging strategies consistently outperform their classical counterparts in terms of out-of-sample performance and resilience to model misspecification. Our findings establish a practical and effective framework for robust deep hedging under realistic market uncertainties.

Random Feature Representation Boosting

We introduce Random Feature Representation Boosting (RFRBoost), a novel method for constructing deep residual random feature neural networks (RFNNs) using boosting theory. RFRBoost uses random features at each layer to learn the functional gradient of the network representation, enhancing performance while preserving the convex optimization benefits of RFNNs. In the case of MSE loss, we obtain closed-form solutions to greedy layer-wise boosting with random features. For general loss functions, we show that fitting random feature residual blocks reduces to solving a quadratically constrained least squares problem. We demonstrate, through numerical experiments on 91 tabular datasets for regression and classification, that RFRBoost significantly outperforms traditional RFNNs and end-to-end trained MLP ResNets, while offering substantial computational advantages and theoretical guarantees stemming from boosting theory.

Fast Deep Hedging with Second-Order Optimization

Hedging exotic options in presence of market frictions is an important risk management task. Deep hedging can solve such hedging problems by training neural network policies in realistic simulated markets. Training these neural networks may be delicate and suffer from slow convergence, particularly for options with long maturities and complex sensitivities to market parameters. To address this, we propose a second-order optimization scheme for deep hedging. We leverage pathwise differentiability to construct a curvature matrix, which we approximate as block-diagonal and Kronecker-factored to efficiently precondition gradients. We evaluate our method on a challenging and practically important problem: hedging a cliquet option on a stock with stochastic volatility by trading in the spot and vanilla options. We find that our second-order scheme can optimize the policy in 1/4 of the number of steps that standard adaptive moment-based optimization takes.

An Overview on Machine Learning Methods for Partial Differential Equations: from Physics Informed Neural Networks to Deep Operator Learning

The approximation of solutions of partial differential equations (PDEs) with numerical algorithms is a central topic in applied mathematics. For many decades, various types of methods for this purpose have been developed and extensively studied. One class of methods which has received a lot of attention in recent years are machine learning-based methods, which typically involve the training of artificial neural networks (ANNs) by means of stochastic gradient descent type optimization methods. While approximation methods for PDEs using ANNs have first been proposed in the 1990s they have only gained wide popularity in the last decade with the rise of deep learning. This article aims to provide an introduction to some of these methods and the mathematical theory on which they are based. We discuss methods such as physics-informed neural networks (PINNs) and deep BSDE methods and consider several operator learning approaches.

Variance Norms for Kernelized Anomaly Detection

We present a unified theory for Mahalanobis-type anomaly detection on Banach spaces, using ideas from Cameron-Martin theory applied to non-Gaussian measures. This approach leads to a basis-free, data-driven notion of anomaly distance through the so-called variance norm of a probability measure, which can be consistently estimated using empirical measures. Our framework generalizes the classical $\mathbb{R}^d$, functional $(L^2[0,1])^d$, and kernelized settings, including the general case of non-injective covariance operator. We prove that the variance norm depends solely on the inner product in a given Hilbert space, and hence that the kernelized Mahalanobis distance can naturally be recovered by working on reproducing kernel Hilbert spaces. Using the variance norm, we introduce the notion of a kernelized nearest-neighbour Mahalanobis distance for semi-supervised anomaly detection. In an empirical study on 12 real-world datasets, we demonstrate that the kernelized nearest-neighbour Mahalanobis distance outperforms the traditional kernelized Mahalanobis distance for multivariate time series anomaly detection, using state-of-the-art time series kernels such as the signature, global alignment, and Volterra reservoir kernels. Moreover, we provide an initial theoretical justification of nearest-neighbour Mahalanobis distances by developing concentration inequalities in the finite-dimensional Gaussian case.

Universal randomised signatures for generative time series modelling

Randomised signature has been proposed as a flexible and easily implementable alternative to the well-established path signature. In this article, we employ randomised signature to introduce a generative model for financial time series data in the spirit of reservoir computing. Specifically, we propose a novel Wasserstein-type distance based on discrete-time randomised signatures. This metric on the space of probability measures captures the distance between (conditional) distributions. Its use is justified by our novel universal approximation results for randomised signatures on the space of continuous functions taking the underlying path as an input. We then use our metric as the loss function in a non-adversarial generator model for synthetic time series data based on a reservoir neural stochastic differential equation. We compare the results of our model to benchmarks from the existing literature.

Universal Approximation Theorem and error bounds for quantum neural networks and quantum reservoirs

Universal approximation theorems are the foundations of classical neural networks, providing theoretical guarantees that the latter are able to approximate maps of interest. Recent results have shown that this can also be achieved in a quantum setting, whereby classical functions can be approximated by parameterised quantum circuits. We provide here precise error bounds for specific classes of functions and extend these results to the interesting new setup of randomised quantum circuits, mimicking classical reservoir neural networks. Our results show in particular that a quantum neural network with $\mathcal{O}(\varepsilon^{-2})$ weights and $\mathcal{O} (\lceil \log_2(\varepsilon^{-1}) \rceil)$ qubits suffices to achieve accuracy $\varepsilon>0$ when approximating functions with integrable Fourier transform.