Related papers: Phynance
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…
The goal of these notes is to provide an introduction to rough partial differential equations. For this purpose, we will present the theory of rough paths to the extend as it is required. Applications to stochastic partial differential…
Lecture notes for a master-level mathematics course on martingales and stochastic calculus, held at the University of Orl\'eans, France. With corrected exercises. Contents: Discrete-time martingales, stopping times, convergence theorems.…
These notes are an extended version of the course "Introduction to rough paths theory" given at the XXV Brazilian School of Probability in Campinas in August 2022. Their aim is to give a consise overview to Lyon's theory of rough paths with…
Recent releases of open-source research codes and solvers for numerically solving partial differential equations in Python present a great opportunity for educators to integrate these codes into the classroom in a variety of ways. The ease…
This course, intended for undergraduates familiar with elementary calculus and linear algebra, introduces the extension of differential calculus to functions on more general vector spaces, such as functions that take as input a matrix and…
In this paper we derive a efficient Monte Carlo approximation for the price of path-dependent derivatives under the multiscale stochastic volatility models of Fouque \textit{et al}. Using the formulation of this pricing problem under the…
The Feynman path integral has revolutionized modern approaches to quantum physics. Although the path integral formalism has proven very successful and spawned several approximation schemes, the direct evaluation of real-time path integrals…
It is shown how the pre-exponential factor of the Feynman propagator for a large class of potentials can be computed using contour integrals. This is of direct relevance in the context of tunnelling processes in quantum theories. The…
It seems that in the current age, computers, computation, and data have an increasingly important role to play in scientific research and discovery. This is reflected in part by the rise of machine learning and artificial intelligence,…
This is a review about financial dependencies which merges efforts in econophysics and financial economics during the last few years. We focus on the most relevant contributions to the analysis of asset markets' dependencies, especially…
Stochastic differential equation (SDE) models are the foundation for pricing and hedging financial derivatives. The drift and volatility functions in SDE models are typically chosen to be algebraic functions with a small number (less than…
These lecture notes for the course APM 351 at the University of Toronto are aimed at mathematicians and physicists alike. It is not meant as an introductory course to PDEs, but rather gives an overview of how to view and solve differential…
In this paper, we present a novel Feynman-Kac formula and investigate learning-based methods for approximating general nonlinear time-dependent Schr\"odinger equations which may be high-dimensional. Our formulation integrates both the…
The theory of fractional calculus in the complex plane was not built with a specific application in mind. The main obstacle to application was the difficulty with obtaining analytic continuations of fractional derivatives and integrals. It…
A kink-based expression for the canonical partition function is developed using Feynman's path integral formulation of quantum mechanics and a discrete basis set. The approach is exact for a complete set of states. The method is tested on…
Invited talk given at the ``International Workshop on `Symmetry Methods in Physics' in memory of Ya.\ A.\ Smorodinsky, 5--10 July 1993, Dubna, Russia; to appear in the proceedings. In this contribution I present further results on steps…
This document consists of lecture notes for a graduate course, which focuses on the relations between Information Theory and Statistical Physics. The course is aimed at EE graduate students in the area of Communications and Information…
We shall study backward stochastic differential equations and we will present a new approach for the existence of the solution. This type of equation appears very often in the valuation of financial derivatives in complete markets.…
Finite differences have been widely used in mathematical theory as well as in scientific and engineering computations. These concepts are constantly mentioned in calculus. Most frequently-used difference formulas provide excellent…