Advanced Monte Carlo methods for barrier and related exotic options

In this work, we present advanced Monte Carlo techniques applied to the pricing of barrier options and other related exotic contracts. It covers in particular the Brownian bridge approaches, the barrier shifting techniques (BAST) and their extensions as well. We leverage the link between discrete and continuous monitoring to design efficient schemes, which can be applied to the Black-Scholes model but also to stochastic volatility or Merton's jump models. This is supported by theoretical results and numerical experiments.

Sharp two-sided heat kernel estimates of twisted tubes and applications

We prove on-diagonal bounds for the heat kernel of the Dirichlet Laplacian $-\Delta^D_\Omega$ in locally twisted three-dimensional tubes $\Omega$. In particular, we show that for any fixed $x$ the heat kernel decays for large times as $\mathrm{e}^{-E_1t}\, t^{-3/2}$, where $E_1$ is the fundamental eigenvalue of the Dirichlet Laplacian on the cross section of the tube. This shows that any, suitably regular, local twisting speeds up the decay of the heat kernel with respect to the case of straight (untwisted) tubes. Moreover, the above large time decay is valid for a wide class of subcritical operators defined on a straight tube. We also discuss some applications of this result, such as Sobolev inequalities and spectral estimates for Schr\"odinger operators $-\Delta^D_\Omega-V$.Comment: To appear in Arch. Rat. Mech. Ana

An Efficient Algorithm to Simulate a Brownian Motion Over Irregular Domains

International audienceIn this paper, we present an algorithm to simulate a Brownian motion by coupling two numerical schemes: the Euler scheme with the random walk on the hyper-rectangles. This coupling algorithm has the advantage to be able to compute the exit time and the exit position of a Brownian motion from an irregular bounded domain (with corners at the boundary), and being of order one with respect to the time step of the Euler scheme. The efficiency of the algorithm is studied through some numerical examples by comparing the analytical solution with the Monte Carlo solution of some Poisson problems. The Monte Carlo solution of these PDEs requires simulating Brownian motions of different types (natural, reflected or drifted) over an irregular domain

Advanced Monte Carlo methods for barrier and related exotic options

International audienceIn this work, we present advanced Monte Carlo techniques applied to the pricing of barrier options and other related exotic contracts. It covers in particular the Brownian bridge approaches, the barrier shifting techniques (BAST) and their extensions as well. We leverage the link between discrete and continuous monitoring to design efficient schemes, which can be applied to the Black-Scholes model but also to stochastic volatility or Merton's jump models. This is supported by theoretical results and numerical experiments

Persistence and First-Passage Properties in Non-equilibrium Systems

In this review we discuss the persistence and the related first-passage properties in extended many-body nonequilibrium systems. Starting with simple systems with one or few degrees of freedom, such as random walk and random acceleration problems, we progressively discuss the persistence properties in systems with many degrees of freedom. These systems include spins models undergoing phase ordering dynamics, diffusion equation, fluctuating interfaces etc. Persistence properties are nontrivial in these systems as the effective underlying stochastic process is non-Markovian. Several exact and approximate methods have been developed to compute the persistence of such non-Markov processes over the last two decades, as reviewed in this article. We also discuss various generalisations of the local site persistence probability. Persistence in systems with quenched disorder is discussed briefly. Although the main emphasis of this review is on the theoretical developments on persistence, we briefly touch upon various experimental systems as well.Comment: Review article submitted to Advances in Physics: 149 pages, 21 Figure

Confocal microscopy

Chapter focusing on confocal microscopy. A confocal microscope is one in which the illumination is confined to a small volume in the specimen, the detection is confined to the same volume and the image is built up by scanning this volume over the specimen, either by moving the beam of light over the specimen or by displacing the specimen relative to a stationary beam. The chief advantage of this type of microscope is that it gives a greatly enhanced discrimination of depth relative to conventional microscopes. Commercial systems appeared in the 1980s and, despite their high cost, the world market for them is probably between 500 and 1000 instruments per annum, mainly because of their use in biomedical research in conjunction with fluorescent labelling methods. There are many books and review articles on this subject ( e.g. Pawley ( 2006) , Matsumoto( 2002), Wilson (1990) ). The purpose of this chapter is to provide an introduction to optical and engineering aspects that may be o f interest to biomedical users of confocal microscopy

Probabilistic approach to non-local equations

The work focuses on probabilistic representation of solutions of non-local equations, where the considered non-local operators are either Caputo-type derivatives or trasformations of the Laplace operator via Bernstein functions. The first chapter is devoted to the introduction of the main tools, that is to say Bernstein functions, subordinators and their inverse processes, Caputo-type derivatives and Bochner subordination. In the second chapter we focus on theoretical results concerning probabilistic representation of non-local equations involving Caputo-type derivatives. First we study a general theory for abstract Cauchy problems involving such kind of derivatives, focusing also on the eigenfunctions of such Caputo-type derivatives and Gronwall-type inequalities. Then we exploit the link between time-changed birth-death processes and abstract Cauchy problems in suitable Banach sequence spaces, via a spectral decomposition approach. Then we consider how the Fokker-Planck equation of a non-Markov Gaussian process changes after applying a time-change via the inverse of a subordinator. Finally, we consider exit times of time-changed processes, their asymptotic problems and the link between their survival probability and solutions of non-local (in time) parabolic PDEs. The third chapter is devoted to applications of the previously presented theoretical results. In particular we focus on queueing theory and computational neuroscience. Moreover, we also exploit some simulation properties to work with such processes. In the fourth and last chapter we focus on non-local operators in space with two exemplary problems. The first one concerns the integral representation of Bernstein functions of the Laplace operator. In particular we prove asymptotic properties of the singular kernel of such integral representations depending on asymptotic properties of the Levy measure of the considered Bernstein function. The second problem deals with spectral properties of a Marchaud-type operator on the sphere. In particular, we prove an identity involving the first eigenvalue of the aforementioned operator and moments of the length of random segments in the unit ball

Dust Transportation and Settling within the Mine Ventilation Network

Dust is ubiquitous in underground mine activities. Continuous inhalation of dust could lead to irreversible occupational diseases. Dust particles of size lower than 75.0 µm, also known as float coal dust, can trigger a coal dust explosion following a methane ignition. Ventilation air carries the float coal dust from the point of production to some distance before it’s deposited on the surfaces of underground coal mine. Sources of dust are widely studied, but study of dust transportation has been mainly based on experimental data and simplified models. An understanding of dust transportation in the mine airways is instrumental in the implementation of local dust control strategies. This thesis presents techniques for sampling float coal dust, computational fluid dynamics (CFD) analysis, and mathematical modeling to estimate average dust deposition in an underground coal mine. Dust samples were taken from roof, ribs, and floor at multiple areas along single air splits from longwall and room and pillar mines. Thermogravimetric analysis of these samples showed no conclusive trends in float coal dust deposition rate with location and origin of dust source within the mine network. CFD models were developed using the Lagrangian particle tracking approach to model dust transportation in reduced scale model of mine. Three dimensional CFD analysis showed random deposition pattern of particle on the mine model floor. A pseudo 2D model was generated to approximate the distance dust particles travel when released from a 7 ft. high coal seam. The models showed that lighter particles released in a high airflow field travel farthest. NIOSH developed MFIRE software was adopted to simulate dust transportation in a mine airway analogous to fume migration. The simulations from MFIRE can be calibrated using the dust sampling results to estimate dust transportation in the ventilation network