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Using spectral decomposition techniques and singular perturbation theory, we develop a systematic method to approximate the prices of a variety of options in a fast mean-reverting stochastic volatility setting. Four examples are provided in…
Deep learning applies hierarchical layers of hidden variables to construct nonlinear high dimensional predictors. Our goal is to develop and train deep learning architectures for spatio-temporal modeling. Training a deep architecture is…
We consider the computation of model-free bounds for multi-asset options in a setting that combines dependence uncertainty with additional information on the dependence structure. More specifically, we consider the setting where the…
We present a non-intrusive gradient and a non-intrusive BFGS algorithm for parameter estimation problems in non-stationary elasticity. To avoid multiple (and potentially expensive) solutions of the underlying partial differential equation…
Experience has shown that trading in stock and cryptocurrency markets has the potential to be highly profitable. In this light, considerable effort has been recently devoted to investigate how to apply machine learning and deep learning to…
We propose a partial differential-integral equation (PDE) framework for deep neural networks (DNNs) and their associated learning problem by taking the continuum limits of both network width and depth. The proposed model captures the…
A new method to solve computationally challenging (random) parametric obstacle problems is developed and analyzed, where the parameters can influence the related partial differential equation (PDE) and determine the position and surface…
Deep learning for option pricing has emerged as a novel methodology for fast computations with applications in calibration and computation of Greeks. However, many of these approaches do not enforce any no-arbitrage conditions, and the…
The numerical solution of high dimensional partial differential equations (PDEs) is severely constrained by the curse of dimensionality (CoD), rendering classical grid--based methods impractical beyond a few dimensions. In recent years,…
In Hezaveh et al. 2017 we showed that deep learning can be used for model parameter estimation and trained convolutional neural networks to determine the parameters of strong gravitational lensing systems. Here we demonstrate a method for…
Variance reduction techniques are of crucial importance for the efficiency of Monte Carlo simulations in finance applications. We propose the use of neural SDEs, with control variates parameterized by neural networks, in order to learn…
Finding tight bounds on the optimal solution is a critical element of practical solution methods for discrete optimization problems. In the last decade, decision diagrams (DDs) have brought a new perspective on obtaining upper and lower…
We develop a probabilistic machine learning method, which formulates a class of stochastic neural networks by a stochastic optimal control problem. An efficient stochastic gradient descent algorithm is introduced under the stochastic…
Effective properties of materials with random heterogeneous structures are typically determined by homogenising the mechanical quantity of interest in a window of observation. The entire problem setting encompasses the solution of a local…
Deep learning models are increasingly deployed in safety-critical tasks where predictions must satisfy hard constraints, such as physical laws, fairness requirements, or safety limits. However, standard architectures lack built-in…
The method is proposed for the study of many-point boundary value problems for systems of nonlinear ODE, by reducing them to special equivalent integral equations, and allows us [in contrast with the known method [1]] to consider boundary…
We introduce a variational framework to learn the activation functions of deep neural networks. Our aim is to increase the capacity of the network while controlling an upper-bound of the actual Lipschitz constant of the input-output…
The neural network method of solving differential equations is used to approximate the electric potential and corresponding electric field in the slit-well microfluidic device. The device's geometry is non-convex, making this a challenging…
This paper addresses the problem of pricing involved financial derivatives by means of advanced of deep learning techniques. More precisely, we smartly combine several sophisticated neural network-based concepts like differential machine…
While time series momentum is a well-studied phenomenon in finance, common strategies require the explicit definition of both a trend estimator and a position sizing rule. In this paper, we introduce Deep Momentum Networks -- a hybrid…