scholarly journals Monte-Carlo methods for the pricing of American options: a semilinear BSDE point of view

2019 ◽  
Vol 65 ◽  
pp. 294-308x ◽  
Author(s):  
Bruno Bouchard ◽  
Ki Wai Chau ◽  
Arij Manai ◽  
Ahmed Sid-Ali
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Methods ◽  
Numerical Experiments ◽  
Branching Processes ◽  
Type Equation ◽  
Reaction Diffusion ◽  
Payoff Function ◽  
Point Of View ◽  
Option Prices ◽  
Diffusion Type

We extend the viscosity solution characterization proved in [5] for call/put American option prices to the case of a general payoff function in a multi-dimensional setting: the price satisfies a semilinear reaction/diffusion type equation. Based on this, we propose two new numerical schemes inspired by the branching processes based algorithm of [8]. Our numerical experiments show that approximating the discontinuous driver of the associated reaction/diffusion PDE by local polynomials is not efficient, while a simple randomization procedure provides very good results.

scholarly journals Comparison of Direct and Adjoint k and α-Eigenfunctions

2021 ◽  
Vol 2 (2) ◽  
pp. 132-151
Author(s):  
Vito Vitali ◽  
Florent Chevallier ◽  
Alexis Jinaphanh ◽  
Andrea Zoia ◽  
Patrick Blaise
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Methods ◽  
Energy Transport ◽  
Distinct Point ◽  
Point Of View ◽  
Critical Systems ◽  
Small Deviations ◽  
Power Iteration ◽  
Continuous Energy ◽  
Fission Probability

Modal expansions based on k-eigenvalues and α-eigenvalues are commonly used in order to investigate the reactor behaviour, each with a distinct point of view: the former is related to fission generations, whereas the latter is related to time. Well-known Monte Carlo methods exist to compute the direct k or α fundamental eigenmodes, based on variants of the power iteration. The possibility of computing adjoint eigenfunctions in continuous-energy transport has been recently implemented and tested in the development version of TRIPOLI-4®, using a modified version of the Iterated Fission Probability (IFP) method for the adjoint α calculation. In this work we present a preliminary comparison of direct and adjoint k and α eigenmodes by Monte Carlo methods, for small deviations from criticality. When the reactor is exactly critical, i.e., for k0 = 1 or equivalently α0 = 0, the fundamental modes of both eigenfunction bases coincide, as expected on physical grounds. However, for non-critical systems the fundamental k and α eigenmodes show significant discrepancies.

A stochastic simulation for solving scalar reaction–diffusion equations

1990 ◽  
Vol 22 (01) ◽  
pp. 88-100
Author(s):  
B. Chauvin ◽  
Rouault
Keyword(s):  
Monte Carlo ◽  
Limit Theorem ◽  
Branching Processes ◽  
Martingale Problem ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Convergence Problem ◽  
One Dimensional ◽  
Spaces Of Measures

A recent Monte Carlo method for solving one-dimensional reaction–diffusion equations is considered here as a convergence problem for a sequence of spatial branching processes with interaction. The martingale problem is studied and a limit theorem is proved by embedding spaces of measures in Sobolev spaces.

On intersection volumes of confidence hyper-ellipsoids and two geometric Monte Carlo methods

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nima Rabiei ◽  
Elias G. Saleeby
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Methods ◽  
Numerical Experiments ◽  
High Probability ◽  
Statistical Decision ◽  
Time Series Clustering ◽  
Sample Mean ◽  
Wide Range ◽  
Geometric Type ◽  
Statistical Decision Making

Abstract The intersection or the overlap region of two n-dimensional ellipsoids plays an important role in statistical decision making in a number of applications. For instance, the intersection volume of two n-dimensional ellipsoids has been employed to define dissimilarity measures in time series clustering (see [M. Bakoben, T. Bellotti and N. M. Adams, Improving clustering performance by incorporating uncertainty, Pattern Recognit. Lett. 77 2016, 28–34]). Formulas for the intersection volumes of two n-dimensional ellipsoids are not known. In this article, we first derive exact formulas to determine the intersection volume of two hyper-ellipsoids satisfying a certain condition. Then we adapt and extend two geometric type Monte Carlo methods that in principle allow us to compute the intersection volume of any two generalized convex hyper-ellipsoids. Using the exact formulas, we evaluate the performance of the two Monte Carlo methods. Our numerical experiments show that sufficiently accurate estimates can be obtained for a reasonably wide range of n, and that the sample-mean method is more efficient. Finally, we develop an elementary fast Monte Carlo method to determine, with high probability, if two n-ellipsoids are separated or overlap.

A stochastic simulation for solving scalar reaction–diffusion equations

1990 ◽  
Vol 22 (1) ◽  
pp. 88-100 ◽  
Author(s):  
B. Chauvin ◽  
Rouault
Keyword(s):  
Monte Carlo ◽  
Limit Theorem ◽  
Branching Processes ◽  
Martingale Problem ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Convergence Problem ◽  
One Dimensional ◽  
Spaces Of Measures

A recent Monte Carlo method for solving one-dimensional reaction–diffusion equations is considered here as a convergence problem for a sequence of spatial branching processes with interaction. The martingale problem is studied and a limit theorem is proved by embedding spaces of measures in Sobolev spaces.

scholarly journals Computational Biophysical Modeling of the Radiation Bystander Effect in Irradiated Cells

Radiation ◽  
2021 ◽  
Vol 2 (1) ◽  
pp. 33-51
Author(s):  
Paweł Wysocki ◽  
Krzysztof W. Fornalski
Keyword(s):  
Monte Carlo ◽  
Probability Distribution ◽  
Ionizing Radiation ◽  
Monte Carlo Methods ◽  
Bystander Effect ◽  
Direct Interaction ◽  
Point Of View ◽  
Biophysical Modeling ◽  
New Approach ◽  
Probability Tree

It is well known that ionizing radiation can cause damages to cells that interact with it directly. However, many studies have shown that damages also occur in cells that have not experienced direct interaction. This is due to the so-called bystander effect, which is observed when the irradiated cell sends signals that can damage neighboring cells. Due to the complexity of this effect, it is not easy to strictly describe it biophysically, and thus it is also difficult to simulate. This article reviews various approaches to modeling and simulating the bystander effect from the point of view of radiation biophysics. In particular, the last model presented within this article is part of a larger project of modeling the response of a group of cells to ionizing radiation using Monte Carlo methods. The new approach presented here is based on the probability tree, the Poisson distribution of signals and the saturated dose-related probability distribution of the bystander effect’s appearance, which makes the model very broad and universal.

scholarly journals Monte-Carlo algorithms for a forward Feynman–Kac-type representation for semilinear nonconservative partial differential equations

2018 ◽  
Vol 24 (1) ◽  
pp. 55-70 ◽  
Author(s):  
Anthony Le Cavil ◽  
Nadia Oudjane ◽  
Francesco Russo
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Monte Carlo ◽  
Numerical Experiments ◽  
Type Equation ◽  
Parabolic Partial Differential Equation ◽  
Partial Differential ◽  
Type Representation ◽  
Monte Carlo Algorithms ◽  
Semilinear Parabolic

Abstract The paper is devoted to the construction of a probabilistic particle algorithm. This is related to a nonlinear forward Feynman–Kac-type equation, which represents the solution of a nonconservative semilinear parabolic partial differential equation (PDE). Illustrations of the efficiency of the algorithm are provided by numerical experiments.

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