Brownian motion

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  • The mathematical theory now known as Malliavin calculus was first introduced by Paul Malliavin in [157] as an infinite-dimensional integration by parts technique. The purpose of this calculus was to prove the results about the smoothness of densities of solutions of stochastic differential equations driven by Brownian motion. For several years this was the only known application. Therefore, since this theory was considered quite complicated by many, Malliavin calculus remained a relatively unknown theory also among mathematicians for some time.

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  • Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí sinh học đề tài : Necessary and sufficient condition for the smoothness of intersection local time of subfractional Brownian motions

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  • Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : Necessary and sufficient condition for the smoothness of intersection local time of subfractional Brownian motions Guangjun Shen

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  • the years that have passed since the pioneering work of Kakutani, Kac, and Doob, it has been shown that Brownian motion can be used to prove many results in classical analysis, primarily concerning the behavior of harmonic and analytic functions and the solutions of certain partial differential equations. In spite of the many pages that have been written on this subject, the results in this area are not widely known, primarily because they appear in articles that are scattered throughout the literature and are written in a style appropriate for technical journals.

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  • Regular fractals and self-similarity, random fractals, regular and fractional brownian motion, iterative feedback processes and chaos, chaotic oscillations,... is the main content of the book "Fractals and Chaos". Invite you to consult the text book for more documents serving the academic needs and research.

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  • Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Generalized Cauchy identities, trees and multidimensional Brownian motions. Part I: bijective proof of generalized Cauchy identities ´ Piotr Sniady...

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  • This comprehensive guide to stochastic processes gives a complete overview of the theory and addresses the most important applications. Pitched at a level accessible to beginning graduate students and researchers from applied disciplines, it is both a course book and a rich resource for individual readers. Subjects covered include Brownian motion, stochastic calculus, stochastic differential equations, Markov processes, weak convergence of processes, and semigroup theory.

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  • Let T (x, ε) denote the first hitting time of the disc of radius ε centered at x for Brownian motion on the two dimensional torus T2 . We prove that supx∈T2 T (x, ε)/| log ε|2 → 2/π as ε → 0. The same applies to Brownian motion on any smooth, compact connected, two-dimensional, Riemannian manifold with unit area and no boundary. As a consequence, we prove a conjecture, due to Aldous (1989), that the number of steps it takes a simple random walk to cover all points of the lattice torus Z2 is asymptotic to 4n2 (log...

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  • This volume contains a collection of articles dedicated to the 70th anniversary of Albert Shiryaev. The majority of contributions are written by his former students, co-authors, colleagues and admirers strongly influenced by Albert’s scientific tastes as well as by his charisma. We believe that the papers of this Festschrift reflect modern trends in stochastic calculus and mathematical finance and open new perspectives of further development in these fascinating fields which attract new and new researchers.

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  • Asset allocation investigates the optimal division of a portfolio among different asset classes. Standard theory involves the optimal mix of risky stocks, bonds, and cash together with various subdivisions of these asset classes. Underlying this is the insight that diversification allows for achieving a balance between risk and return: by using different types of investment, losses may be limited and returns are made less volatile without losing too much potential gain.

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  • This book is an extension of “Probability for Finance” to multi-period financial models, either in the discrete or continuous-time framework. It describes the most important stochastic processes used in finance in a pedagogical way, especially Markov chains, Brownian motion and martingales. It also shows how mathematical tools like filtrations, Itô’s lemma or Girsanov theorem should be understood in the framework of financial models. It also provides many illustrations coming from the financial literature....

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  • The objective of this book is to introduce recent advances and state-of-the-art applications of Monte Carlo Simulation (MCS) in various fields. MCS is a class of statistical methods for performance analysis and decision making based on taking random samples from underly‐ ing systems or problems to draw inferences or estimations. Let us make an analogy by using the structure of an umbrella to define and exemplify the position of this book within the fields of science and engineering. Imagine that one can place MCS at the centerpoint of an umbrella and define the...

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  • To introduce the ]orward-backward stochastic differential equations (FBS- DEs, for short), let us begin with some examples. Unless otherwise speci- fled, throughout the book, we let (~, •, {Ft)t_0, P) be a complete filtered probability space on which is defined a d-dimensional standard Brownian motion W(t), such that {5~t }t_0 is the natural filtration of W(t), augmented by...

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  • SLEκ is a random growth process based on Loewner’s equation with driving parameter a one-dimensional Brownian motion running with speed κ. This process is intimately connected with scaling limits of percolation clusters and with the outer boundary of Brownian motion, and is conjectured to correspond to scaling limits of several other discrete processes in two dimensions. The present paper attempts a first systematic study of SLE. It is proved that for all κ = 8 the SLE trace is a path; for κ ∈ [0, 4] it is a simple path; for κ ∈ (4, 8) it is...

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  • The “Mathematical Finance Conference in Honor of the 60th Birthday of Dilip B. Madan” was held at the Norbert Wiener Center of the University of Maryland, College Park, from September 29 – October 1, 2006, and this volume is a Festschrift in honor of Dilip that includes articles from most of the conference’s speakers. Among his former students contributing to this volume are Ju-Yi Yen as one of the co-editors, along with Ali Hirsa and Xing Jin as co-authors of three of the articles.

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  • a a For a 0, let W1 (t) and W2 (t) be the a-neighbourhoods of two independent standard Brownian motions in Rd starting at 0 and observed until time t. We prove that, for d ≥ 3 and c 0, t→∞ t(d−2)/d lim 1 κ a a log P |W1 (ct) ∩ W2 (ct)| ≥ t = −Id a (c) κ and derive a variational representation for the rate constant Id a (c). Here, κa is the Newtonian capacity of the ball with radius a. We show that the optimal a a strategy to realise the above large deviation is for...

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  • Contents 9 Sample path properties of local times 9.1 Bounded discontinuities 9.2 A necessary condition for unboundedness 9.3 Sufficient conditions for continuity 9.4 Continuity and boundedness of local times 9.5 Moduli of continuity 9.6 Stable mixtures 9.7 Local times for certain Markov chains 9.8 Rate of growth of unbounded local times 9.9 Notes and references p-variation 10.1 Quadratic variation of Brownian motion 10.2 p-variation of Gaussian processes 10.3 Additional variational results for Gaussian processes 10.4 p-variation of local times 10.

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  • MULTIDIMENSIONAL KOLMOGOROV-PETROVSKY TEST FOR THE BOUNDARY REGULARITY AND IRREGULARITY OF SOLUTIONS TO THE HEAT EQUATION UGUR G. ABDULLA Received 25 August 2004 Dedicated to I. G. Petrovsky This paper establishes necessary and sufficient condition for the regularity of a characteristic top boundary point of an arbitrary open subset of RN+1 (N ≥ 2) for the diffusion (or heat) equation. The result implies asymptotic probability law for the standard Ndimensional Brownian motion. 1. Introduction and main result Consider the domain Ωδ = (x,t) ∈ RN+1 : |x| 0, N ≥ 2, x = (x1 ,...

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  • There have been some attempts to explain theoretically the behavior of the stock return volatility. Veronesi (1999) constructs a model with regime shifts in the endowments in which investors will- ingness to hedge against their own uncertainty on the true regime generates overreaction to good news in bad times and volatility clustering. In contrast to that paper, I assume that the exogenous state variables are not subject to regimes, neither do they exhibit mean-reversion.

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  • While the demand for options can arise from many sources, our focus on jumps stems from fundamental considerations regarding the nature of price pro- cesses in an arbitrage-free economy. Recently, Madan [19] has argued that all arbitrage-free continuous time price processes must be both semi-martingales and time-changed Brownian motion. Furthermore, it is argued that if the time change is not locally deterministic, then the resulting price process is discontinuous.

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