Doob's Decomposition Theorem

This lecture covers Doob's decomposition theorem for submartingales, which states that a submartingale can be decomposed into two processes: a martingale and a non-decreasing predictable process. The proof involves defining a natural candidate process A and showing its predictability through induction. Additionally, the lecture discusses Brownian motion as a continuous-time process with specific properties, such as starting at 0 and having certain distributions. It also explores the connection between Brownian motion and classical random walks, emphasizing the quadratic variation of functions and their variations. The lecture concludes with insights into continuous martingales and Levy's theorem, highlighting the relationship between martingales and Brownian motion.

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related lectures (54)

Related concepts (48)

COM-417: Advanced probability and applications

In this course, various aspects of probability theory are considered. The first part is devoted to the main theorems in the field (law of large numbers, central limit theorem, concentration inequaliti

Quadratic variation

In mathematics, quadratic variation is used in the analysis of stochastic processes such as Brownian motion and other martingales. Quadratic variation is just one kind of variation of a process. Suppose that is a real-valued stochastic process defined on a probability space and with time index ranging over the non-negative real numbers. Its quadratic variation is the process, written as , defined as where ranges over partitions of the interval and the norm of the partition is the mesh.

Semimartingale

In probability theory, a real valued stochastic process X is called a semimartingale if it can be decomposed as the sum of a local martingale and a càdlàg adapted finite-variation process. Semimartingales are "good integrators", forming the largest class of processes with respect to which the Itô integral and the Stratonovich integral can be defined. The class of semimartingales is quite large (including, for example, all continuously differentiable processes, Brownian motion and Poisson processes).

Martingale (probability theory)

In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time, the conditional expectation of the next value in the sequence is equal to the present value, regardless of all prior values. Originally, martingale referred to a class of betting strategies that was popular in 18th-century France. The simplest of these strategies was designed for a game in which the gambler wins their stake if a coin comes up heads and loses it if the coin comes up tails.

Wiener process

In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is often also called Brownian motion due to its historical connection with the physical process of the same name originally observed by Scottish botanist Robert Brown.

Geometric Brownian motion

A geometric Brownian motion (GBM) (also known as exponential Brownian motion) is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion (also called a Wiener process) with drift. It is an important example of stochastic processes satisfying a stochastic differential equation (SDE); in particular, it is used in mathematical finance to model stock prices in the Black–Scholes model.

Numerical Analysis: Implicit Schemes MATH-351: Advanced numerical analysis

Covers implicit schemes in numerical analysis for solving partial differential equations.

Martingale Convergence Theorem COM-417: Advanced probability and applications

Covers the proof of the martingale convergence theorem and the convergence of the martingale sequence almost surely.

Implicit Schemes in Numerical Analysis MATH-351: Advanced numerical analysis

Explores implicit schemes in numerical analysis, emphasizing stability and convergence properties in solving differential equations.

Harmonic Forms and Riemann Surfaces MATH-680: Monstrous moonshine

Explores harmonic forms on Riemann surfaces, covering uniqueness of solutions and the Riemann bilinear identity.

Cohomology Real Projective Space MATH-506: Topology IV.b - cohomology rings

Covers cohomology in real projective spaces, focusing on associative properties and algebraic structures.