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The standard Black-Scholes theory of option pricing is extended to cope with underlying return fluctuations described by general probability distributions. A Langevin process and its related Fokker-Planck equation are devised to model the…
The accurate numerical simulation of high Reynolds number incompressible flows is a challenging topic in computational fluid dynamics. Classical inf-sup stable methods like the Taylor-Hood element or only $L^2$-conforming discontinuous…
We investigate scaling and efficiency of the deep neural network multigrid method (DNN-MG). DNN-MG is a novel neural network-based technique for the simulation of the Navier-Stokes equations that combines an adaptive geometric multigrid…
First-order methods such as stochastic gradient descent (SGD) are currently the standard algorithm for training deep neural networks. Second-order methods, despite their better convergence rate, are rarely used in practice due to the…
Deep learning techniques for improving fluid flow modelling have gained significant attention in recent years. Advanced deep learning techniques achieve great progress in rapidly predicting fluid flows without prior knowledge of the…
We propose a new, data-driven approach for efficient pricing of - fixed- and float-strike - discrete arithmetic Asian and Lookback options when the underlying process is driven by the Heston model dynamics. The method proposed in this…
Transformed Gaussian Processes (TGPs) are stochastic processes specified by transforming samples from the joint distribution from a prior process (typically a GP) using an invertible transformation; increasing the flexibility of the base…
This paper introduces a high order numerical framework for efficient and robust simulation of compressible flows. To address the inefficiencies of standard hybridized discontinuous Galerkin (HDG) methods in large scale settings, we develop…
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…
In this study, we consider the simulation of subsurface flow and solute transport processes in the stationary limit. In the convection-dominant case, the numerical solution of the transport problem may exhibit non-physical diffusion and…
Financial derivatives pricing aims to find the fair value of a financial contract on an underlying asset. Here we consider option pricing in the partial differential equations framework. The contemporary models lead to one-dimensional or…
Pricing of high-dimensional options is one of the most important problems in Mathematical Finance. The objective of this manuscript is to present an original self-contained treatment of the multidimensional pricing. During the past decades…
Deep Gaussian Processes (DGPs) are multi-layer, flexible extensions of Gaussian processes but their training remains challenging. Sparse approximations simplify the training but often require optimization over a large number of inducing…
Deep generative replay has emerged as a promising approach for continual learning in decision-making tasks. This approach addresses the problem of catastrophic forgetting by leveraging the generation of trajectories from previously…
The discontinuous Galerkin (DG) algorithm is a representative high order method in Computational Fluid Dynamics (CFD) area which possesses considerable mathematical advantages such as high resolution, low dissipation, and dispersion.…
Recognizing the importance of jump risk in option pricing, we propose a neural jump stochastic differential equation model in this paper, which integrates neural networks as parameter estimators in the conventional jump diffusion model. To…
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…
Deep Learning Recommendation Models (DLRMs) often rely on extensive manual feature engineering to improve accuracy and user experience, which increases system complexity and limits scalability of model performance with respect to…
This paper mainly discusses the American option's hedging strategies via binomialmodel and the basic idea of pricing and hedging American option. Although the essential scheme of hedging is almost the same as European option, small…
An implicit multiscale method with multiple macroscopic prediction for steady state solutions of gas flow in all flow regimes is presented. The method is based on the finite volume discrete velocity method (DVM) framework. At the cell…