Doob's Martingale Inequality

	Doob's Martingale Inequality In mathematics, Doob's martingale inequality, also known as Kolmogorov’s submartingale inequality is a result in the study of stochastic processes. It gives a bound on the probability that a submartingale exceeds any given value over a given interval of time. As the name suggests, the result is usually given in the case that the process is a martingale, but the result is also valid for submartingales. The inequality is due to the American mathematician Joseph L. Doob. Statement of the inequality The setting of Doob's inequality is a submartingale relative to a filtration of the underlying probability space. The probability measure on the sample space of the martingale will be denoted by . The corresponding expected value of a random variable , as defined by Lebesgue integration, will be denoted by . Informally, Doob's inequality states that the expected value of the process at some final time controls the probability that a sample path will reach above any particular value befor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Layer Cake Representation In mathematics, the layer cake representation of a non- negative, real-valued measurable function f defined on a measure space (\Omega,\mathcal,\mu) is the formula :f(x) = \int_0^\infty 1_ (x) \, \mathrmt, for all x \in \Omega, where 1_E denotes the indicator function of a subset E\subseteq \Omega and L(f,t) denotes the super-level set :L(f, t) = \. The layer cake representation follows easily from observing that : 1_(x) = 1_(t) and then using the formula :f(x) = \int_0^ \,\mathrmt. The layer cake representation takes its name from the representation of the value f(x) as the sum of contributions from the "layers" L(f,t): "layers"/values t below f(x) contribute to the integral, while values t above f(x) do not. It is a generalization of Cavalieri's principle and is also known under this name. An important consequence of the layer cake representation is the identity \int_\Omega f(x) \, \mathrm\mu(x) = \int_0^ \mu(\)\,\mathrmt, which follows from it by applying the Fubini-Ton ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Probabilistic Inequalities Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty."Kendall's Advanced Theory of Statistics, Volume 1: Distribution Theory", Alan Stuart and Keith Ord, 6th Ed, (2009), .William Feller, ''An Introduction to Probability Theory and Its Applications'', (Vol 1), 3rd Ed, (1968), Wiley, . The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]