Related papers: PDE formulation of some SABR/LIBOR market models a…
In this article, we show how the scaling symmetry of the SABR model can be utilized to efficiently price European options. For special kinds of payoffs, the complexity of the problem is reduced by one dimension. For more generic payoffs,…
In this article, we propose a new numerical approach to high-dimensional partial differential equations (PDEs) arising in the valuation of exotic derivative securities. The proposed method is extended from Reisinger and Wittum (2007) and…
We introduce a general, analytical framework to express and to approximate partial differential equations (PDEs) numerically on graphs and networks of surfaces---generalized by the term hypergraphs. To this end, we consider PDEs on…
Parabolic partial differential equations (PDEs) appear in many disciplines to model the evolution of various mathematical objects, such as probability flows, value functions in control theory, and derivative prices in finance. It is often…
Partial differential equations (PDEs) are typically used as models of physical processes but are also of great interest in PDE-based image processing. However, when it comes to their use in imaging, conventional numerical methods for…
We develop several deep learning algorithms for approximating families of parametric PDE solutions. The proposed algorithms approximate solutions together with their gradients, which in the context of mathematical finance means that the…
This paper includes an original self contained proof of well-posedness of an initial-boundary value problem involving a non-local parabolic PDE which naturally arises in the study of derivative pricing in a generalized market model. We call…
Using a combination of recurrent neural networks and signature methods from the rough paths theory we design efficient algorithms for solving parametric families of path dependent partial differential equations (PPDEs) that arise in pricing…
We propose algorithms for solving high-dimensional Partial Differential Equations (PDEs) that combine a probabilistic interpretation of PDEs, through Feynman-Kac representation, with sparse interpolation. Monte-Carlo methods and…
Sparsity plays a central role in recent developments in signal processing, linear algebra, statistics, optimization, and other fields. In these developments, sparsity is promoted through the addition of an $L^1$ norm (or related quantity)…
Gas transport and other complex real-world challenges often require solving and controlling partial differential equations (PDEs) defined on graph structures, which typically demand substantial memory and computational resources. The Random…
In financial engineering, prices of financial products are computed approximately many times each trading day with (slightly) different parameters in each calculation. In many financial models such prices can be approximated by means of…
The numerical approximation of partial differential equations (PDEs) poses formidable challenges in high dimensions since classical grid-based methods suffer from the so-called curse of dimensionality. Recent attempts rely on a combination…
In mathematical finance a popular approach for pricing options under some Levy model is to consider underlying that follows a Poisson jump diffusion process. As it is well known this results in a partial integro-differential equation (PIDE)…
In this short note, using our geometric method introduced in a previous paper \cite{phl} and initiated by \cite{ave}, we derive an asymptotic swaption implied volatility at the first-order for a general stochastic volatility Libor Market…
This paper includes a proof of well-posedness of an initial-boundary value problem involving a system of degenerate non-local parabolic PDE which naturally arises in the study of derivative pricing in a generalized market model. In a…
This paper introduces a no-arbitrage, Monte Carlo-free approach to pricing path-dependent interest rate derivatives. The Heath-Jarrow-Morton model gives arbitrage-free contingent claims prices but is infinite-dimensional, making traditional…
The work in this paper is four-fold. Firstly, we introduce an alternative approach to solve fractional ordinary differential equations as an expected value of a random time process. Using the latter, we present an interesting numerical…
We propose machine learning methods for solving fully nonlinear partial differential equations (PDEs) with convex Hamiltonian. Our algorithms are conducted in two steps. First the PDE is rewritten in its dual stochastic control…
Relying on the classical connection between Backward Stochastic Differential Equations (BSDEs) and non-linear parabolic partial differential equations (PDEs), we propose a new probabilistic learning scheme for solving high-dimensional…