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Construction of the paths of Brownian motions on star graphs I

Vadim Kostrykin, Jürgen Potthoff, Robert Schrader
2012 Communications on Stochastic Analysis  
In the present article and its follow-up article [23] pathwise constructions of Brownian motions which satisfy all possible boundary conditions at the vertex of star graphs are given.  ...  In the opinion of the authors this leads to much simpler computations of the transition kernels than those in [17, 18, 20] for a Brownian motion on ℝ + .  ...  The Brownian motions constructed here on star graphs are then the basic building blocks of Brownian motions on general, finite metric graphs in the article [24] of the present authors.  ... 
doi:10.31390/cosa.6.2.04 fatcat:fwrwwo7ojbg6bjdsd7lzxizg4q

An Efficient Algorithm to Simulate a Brownian Motion Over Irregular Domains

S. Zein, A. Lejay, M. Deaconu
2010 Communications in Computational Physics  
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  ...  In 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.  ...  So far, the Brownian motion is characterized by the solution of some PDE. Conversely, the solution of a PDE can be computed using the Brownian motion.  ... 
doi:10.4208/cicp.240209.031209a fatcat:mcy7gujib5c7jodm4knma3hety

Stratified sampling and quasi-Monte Carlo simulation of Lévy processes

G. Leobacher
2006 Monte Carlo Methods and Applications  
We provide a method for the generation of paths of Lévy processes which has many of the benefits that the Brownian bridge construction has for Brownian motion.  ...  As a numerical example we consider the problem of pricing an asian option in the so-called hyperbolic market model.  ...  on a Brownian motion.  ... 
doi:10.1515/156939606778705155 fatcat:otpclcm2abfl3amsjifpenfvy4

Construction of the paths of Brownian motions on star graphs II

Vadim Kostrykin, Jürgen Potthoff, Robert Schrader
2012 Communications on Stochastic Analysis  
In this article and its predecessor [9] , pathwise constructions of Brownian motions which satisfy all possible boundary conditions at the vertex of star graphs are given.  ...  In article I we constructed the Walsh process on the single vertex graph from a standard Brownian motion on the real line.  ...  Hence, altogether we have shown that is a Brownian motion on in the sense of definition 1.1. Now we want to compute the generator of , and first we argue that is not a trap for .  ... 
doi:10.31390/cosa.6.2.05 fatcat:5bcspnxzkjeyhet3t3hsfvqgdq

Sticky Brownian motion as the strong limit of a sequence of random walks

Madjid Amir
1991 Stochastic Processes and their Applications  
We provide here a constructive definition of the sticky Brownian motion as we show that it is the almost sure uniform limit of path functions of a time changed random walk.  ...  The transition distribution of this process is also derived. F" P,) P,-a.s. W*(t) has speed 0304.4149/91/$03.50 @ 1991-Elsevier Science Publishers B.V. (North-Holland)  ...  In most texts, it is referred to the sticky Brownian motion, a Brownian motion 'slowly' reflecting at zero or with a sticky boundary at zero. Here we construct a sticky Brownian motion on (-00, +a).  ... 
doi:10.1016/0304-4149(91)90080-v fatcat:6eucef3zufgytjlkq6s2urbfai

Page 1948 of Mathematical Reviews Vol. , Issue 99c [page]

1991 Mathematical Reviews  
The authors take Brownian motion on the Sierpinski gasket and construct a superprocess over this spatial motion.  ...  The proofs use the scaling of the transition density of the Brownian motion on the gasket.  ... 

A multiscale guide to Brownian motion

Denis S Grebenkov, Dmitry Belyaev, Peter W Jones
2015 Journal of Physics A: Mathematical and Theoretical  
We revise the Levy's construction of Brownian motion as a simple though still rigorous approach to operate with various Gaussian processes.  ...  Such a wavelet representation gives a closed formula mapping of the unit interval onto the functional space of Brownian paths.  ...  BROWNIAN MOTION In this section, we derive a wavelet representation of Brownian motion in a simple explicit way, allowing one to gain an intuitive feeling of this constructive approach. A.  ... 
doi:10.1088/1751-8113/49/4/043001 fatcat:zollpg3nebelhpcf2djhsawvxi

Page 3771 of Mathematical Reviews Vol. , Issue 98F [page]

1998 Mathematical Reviews  
The construction relies on a clever coupling argument which constructs a sequence of Moran models on the same sample space as the “look down” particles such that the first m components share the same empirical  ...  Summary: “Let X(r) (¢ € R”) be a fractional Brownian motion of index a in R¢.  ... 

Hedging with Monte Carlo Simulation [chapter]

Jaksa Cvitanić, Levon Goukasian, Fernando Zapatero
2002 Applied Optimization  
Monte Carlo simulation provides a simple procedure to price securities numerically; however, it does not immediately yield the weights necessary to construct a replicating portfolio.  ...  Compared to the alternative procedure, the method we suggest has the advantage that the computational cost is independent of the number of dimensions (unlike the traditional procedure where the cost increases  ...  In this Section we describe the implementation of the RVM in a setting with more than one Brownian motion process.  ... 
doi:10.1007/978-1-4757-3613-7_18 fatcat:52iwp5znfbesnozrw6mggnxoyi

Effective Langevin equations for constrained stochastic processes [article]

Satya N. Majumdar, Henri Orland
2015 arXiv   pre-print
The method is illustrated on the generation of Brownian bridges, Brownian meanders, Brownian excursions and constrained Ornstein-Uehlenbeck processes.  ...  In addition, we show how to solve this equation in the case of a general force acting on the particle. As an example, we show how to generate constrained path joining the two minima of a double-well.  ...  ACKNOWLEDGMENTS We thank the hospitality of the Galileo Galilei Institute (Florence) during the workshop "Advances in Nonequilibrium Statistical Mechanics: large deviations and long-range correlations,  ... 
arXiv:1503.02639v2 fatcat:rmojmqjn35gnpiwwpjpv5enwku

Fractional Brownian fields, duality, and martingales [chapter]

Vladimir Dobrić, Francisco M. Ojeda
2006 High Dimensional Probability  
The Gaussian field, due to inherited "duality", reveals a new way of constructing martingales associated with the odd and even part of a fractional Brownian motion and therefore of the fractional Brownian  ...  In this paper the whole family of fractional Brownian motions is constructed as a single Gaussian field indexed by time and the Hurst index simultaneously.  ...  That property leads to a construction of martingales associated to fractional Brownian motions.  ... 
doi:10.1214/074921706000000770 fatcat:e4qwzlxqfvea7ofkxqrlvpcy2u

Monte Carlo methods for fissured porous media: a gridless approach *

Antoine Lejay
2004 Monte Carlo Methods and Applications  
In this article, we present two Monte Carlo methods to solve some problems related to the Darcy law in geophysics. Both methods do not require any discretization and are exact methods.  ...  Now, locally around a vertex, a Brownian motion (X, P x ) on a graph is a generalization of the Skew Brownian motion: on each edge, the process behaves like a one-dimensional Brownian motion until it hit  ...  For any α ∈ (0, 1), it is possible to construct a stochastic process called a Skew Brownian motion such that: X behaves like a Brownian motion on R * + and R * − (that is the infinitesimal generator is  ... 
doi:10.1515/mcma.2004.10.3-4.385 fatcat:3btb24kmgjekhfid72xrtb7wzm

Page 6907 of Mathematical Reviews Vol. , Issue 96k [page]

1996 Mathematical Reviews  
The key point to T. Lindstrem’s construction of Brownian motion on nested fractals [Mem. Amer. Math.  ...  The paper contains a construction of a pair of planar Brownian motions starting from the origin and conditioned not to intersect until they hit the unit circle.  ... 

On White Noise Space and Levy's Brownian Motion on the Circle [article]

Chunfeng Huang, Ao Li
2020 arXiv   pre-print
In this article, we show that the Brownian motion on the circle constructed in Levy (1959) is a regular Euclidean Brownian motion on the half-circle with its own mirror image on the other half-circle,  ...  This raises the question of what the white noise is on the circle. We then formally define the white noise space and its associated Brownian bridge.  ...  In Euclidean spaces, a white noise, sometimes, can be thought of as the generalized derivative of a Brownian motion.  ... 
arXiv:1911.03374v2 fatcat:navb5rai4fa35hyj6t4nuskfiy

Page 5191 of Mathematical Reviews Vol. , Issue 86k [page]

1986 Mathematical Reviews  
To avoid a certain dependency on dimension which appears in Pinsky’s con- struction, the author uses the central hitting measure of the surface of the unit ball with respect to the Brownian motion on the  ...  In this expository article the author recapitulates his construc- tion of Brownian motion on a smooth manifold as a weak limit of “Isotropic transport processes”.  ... 
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