Graduate Texts in Mathematics Brownian Motion, Martingales, and Stochastic Calculus (original) (raw)

2020

This paper serves as a rigorous introduction to probability theory and stochastic calculus. It will first introduce basic facts about probability over uncountable spaces, and subsequently transition to a discussion of brownian motion and stochastic calculus.

A review on Brownian Motion and Stochastic Calculus

2022

Stochastic Calculus has found a wide range of applications in analyzing the evolution of many natural, but complex systems. In this article, we discuss Brownian motion and Stochastic Calculus. In Chapter 2, we have listed preliminary notions about Stochastic Processes. In Chapter 3, we focus on the definition and properties of Brownian motion. In Chapter 4, we first discuss the construction of Ito integral and then look at the change of variables formula of Stochastic Calculus, the Ito formula. We finish with a discussion on existence and uniqueness problem for solutions of stochastic differential equations.

Random Walk, Brownian Motion, and Martingales

Graduate Texts in Mathematics, 2021

Graduate Texts in Mathematics bridge the gap between passive study and creative understanding, offering graduate-level introductions to advanced topics in mathematics. The volumes are carefully written as teaching aids and highlight characteristic features of the theory. Although these books are frequently used as textbooks in graduate courses, they are also suitable for individual study.

Introduction to the theory of stochastic processes and Brownian motion problems

Graduate course notes (2005). Includes both translational and rotational Brownian motion

An introduction to the theory of stochastic processes based on several sources. The presentation follows the books of van Kampen and Wio. The introduction is essentially that of Gardiner's book, whereas the treatment of the Langevin equation and the methods for solving Fokker-Planck equations are based on the book of Risken.

General Stochastic Calculus and Applications

5.2 A t-dependence plot of the various constructed processes. Solid line represents the constructed process. Dotted line represents the final value of one differential equation within the time step being used as initial condition for the next one. .

An Outline of Stochastic Calculus

UNITEXT for Physics

For simplicity again, in this chapter we will only consider processes with just one component. We already remarked in the Sect. 6.3 that a white noise is a singular process whose main properties can be traced back to the non differentiability of some processes. As a first example we have shown indeed that the Poisson impulse process (6.63) and its associated compensated version (6.65) are white noises entailed by the formal derivation respectively of a simple Poisson process N (t) and of its compensated variant N (t). In the same vein we have shown then in the Example 6.22 that also the formal derivative of the Wiener process W (t)-not differentiable according to the Proposition 6.18-meets the conditions (6.69) to be a white noise, and in the Appendix H we also hinted that the role of the fluctuating force B(t) in the Langevin equation (6.78) for the Brownian motion is actually played by such a white noiseẆ (t). We can now give a mathematically more cogent justification for this identification in the framework of the Markovian diffusions. The Langevin equation (6.78) is a particular case of the more general equatioṅ X (t) = a(X (t), t) + b(X (t), t) Z (t) (8.1) where a(x, t) and b(x, t) are given functions and Z (t) is a process with E [Z (t)] = 0 and uncorrelated with X (t). From a formal integration of (8.1) we find X (t) = X (t 0) + t t 0 a(X (s), s) ds + t t 0 b(X (s), s)Z (s) ds so that, being X (t) assembled as a combination of Z (s) values with t 0 < s < t, to secure the non correlation of X (t) and Z (t) we should intuitively require also the non correlation of Z (s) and Z (t) for every pair s = t. Since moreover Z (t) is presumed

Notes on Brownian motion and related phenomena

Eprint Arxiv Physics 9903033, 1999

In this article we explore the phenomena of nonequilibrium stochastic process starting from the phenomenological Brownian motion. The essential points are described in terms of Einstein's theory of Brownian motion and then the theory extended to Langevin and Fokker-Planck formalism. Then the theory is applied to barrier crossing dynamics, popularly known as Kramers' theory of activated rate processes. The various regimes are discussed extensively and Smoluchowski equation is derived as a special case. Then we discuss some of the aspects of Master equation and two of its applications.

Essentials of Stochastic Processes

Between the first undergraduate course in probability and the first graduate course that uses measure theory, there are a number of courses that teach Stochastic Processes to students with many different interests and with varying degrees of mathematical sophistication. To allow readers (and instructors) to choose their own level of detail, many of the proofs begin with a nonrigorous answer to the question "Why is this true?" followed by a Proof that fills in the missing details. As it is possible to drive a car without knowing about the working of the internal combustion engine, it is also possible to apply the theory of Markov chains without knowing the details of the proofs. It is my personal philosophy that probability theory was developed to solve problems, so most of our effort will be spent on analyzing examples. Readers who want to master the subject will have to do more than a few of the twenty dozen carefully chosen exercises.

The Brownian motion

2019

This open access textbook is the first to provide Business and Economics Ph.D. students with a precise and intuitive introduction to the formal backgrounds of modern financial theory. It explains Brownian motion, random processes, measures, and Lebesgue integrals intuitively, but without sacrificing the necessary mathematical formalism, making them accessible for readers with little or no previous knowledge of the field. It also includes mathematical definitions and the hidden stories behind the terms discussing why the theories are presented in specific ways.

Introduction to Stochastic Analysis and Malliavin Calculus

2014

for any real, bounded and Borel function ϕ. We show its connection with the heat equation both in the real line and in the half-line. Then we consider Dirichlet, Neuman and Ventzell problems in the half-line. The chapter ends with a final section on some remarkable properties of the set of zeros of the Brownian motion.

Exercises in Stochastic Calculus

This, somewhat unusual collection of problems in Measure-Theoretic Probability and Stochastic Analysis, should have been more appropriately called: "The Problems I Like". The problems borrowed from the existing literature have solutions different than in the original sources, and for those full references and occasional historical remarks are provided. There are also original problems: