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Optimal Stopping
In mathematics, the theory of optimal stopping or early stopping : (For French translation, secover storyin the July issue of ''Pour la Science'' (2009).) is concerned with the problem of choosing a time to take a particular action, in order to Optimization (mathematics), maximise an expected reward or minimise an expected cost. Optimal stopping problems can be found in areas of statistics, economics, and mathematical finance (related to the pricing of American options). A key example of an optimal stopping problem is the secretary problem. Optimal stopping problems can often be written in the form of a Bellman equation, and are therefore often solved using dynamic programming. Definition Discrete time case Stopping rule problems are associated with two objects: # A sequence of random variables X_1, X_2, \ldots, whose joint distribution is something assumed to be known # A sequence of 'reward' functions (y_i)_ which depend on the observed values of the random variables in 1: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a σ-algebra that satisfies Measure (mathematics), measure properties such as ''countable additivity''. The difference between a probability measure and the more general notion of measure (which includes concepts like area or volume) is that a probability measure must assign value 1 to the entire space. Intuitively, the additivity property says that the probability assigned to the union of two disjoint (mutually exclusive) events by the measure should be the sum of the probabilities of the events; for example, the value assigned to the outcome "1 or 2" in a throw of a dice should be the sum of the values assigned to the outcomes "1" and "2". Probability measures have applications in diverse fields, from physics to finance and biology. Definition The requirements for a set function \mu to be a probability measure on a σ-algebra are that: * \mu must return results in the unit interval ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variational Inequality
In mathematics, a variational inequality is an inequality involving a functional, which has to be solved for all possible values of a given variable, belonging usually to a convex set. The mathematical theory of variational inequalities was initially developed to deal with equilibrium problems, precisely the Signorini problem: in that model problem, the functional involved was obtained as the first variation of the involved potential energy. Therefore, it has a variational origin, recalled by the name of the general abstract problem. The applicability of the theory has since been expanded to include problems from economics, finance, optimization and game theory. History The first problem involving a variational inequality was the Signorini problem, posed by Antonio Signorini in 1959 and solved by Gaetano Fichera in 1963, according to the references and : the first papers of the theory were and , . Later on, Guido Stampacchia proved his generalization to the Lax–Milg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinitesimal Generator (stochastic Processes)
In mathematics — specifically, in stochastic analysis — the infinitesimal generator of a Feller process (i.e. a continuous-time Markov process satisfying certain regularity conditions) is a Fourier multiplier operator that encodes a great deal of information about the process. The generator is used in evolution equations such as the Kolmogorov backward equation, which describes the evolution of statistics of the process; its ''L''2 Hermitian adjoint is used in evolution equations such as the Fokker–Planck equation, also known as Kolmogorov forward equation, which describes the evolution of the probability density functions of the process. The Kolmogorov forward equation in the notation is just \partial_t \rho = \mathcal A^* \rho, where \rho is the probability density function, and \mathcal A^* is the adjoint of the infinitesimal generator of the underlying stochastic process. The Klein–Kramers equation is a special case of that. Definition General case For ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Open Set
In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line. In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two points), an open set is a set that, with every point in it, contains all points of the metric space that are sufficiently near to (that is, all points whose distance to is less than some value depending on ). More generally, an open set is a member of a given Set (mathematics), collection of Subset, subsets of a given set, a collection that has the property of containing every union (set theory), union of its members, every finite intersection (set theory), intersection of its members, the empty set, and the whole set itself. A set in which such a collection is given is called a topological space, and the collection is called a topology (structure), topology. These conditions are very loose, and allow enormous flexibility in the choice ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poisson Random Measure
Let (E, \mathcal A, \mu) be some measure space with \sigma- finite measure \mu. The Poisson random measure with intensity measure \mu is a family of random variables \_ defined on some probability space (\Omega, \mathcal F, \mathrm) such that i) \forall A\in\mathcal,\quad N_A is a Poisson random variable with rate \mu(A). ii) If sets A_1,A_2,\ldots,A_n\in\mathcal don't intersect then the corresponding random variables from i) are mutually independent. iii) \forall\omega\in\Omega\;N_(\omega) is a measure on (E, \mathcal ) Existence If \mu\equiv 0 then N\equiv 0 satisfies the conditions i)–iii). Otherwise, in the case of finite measure \mu, given Z, a Poisson random variable with rate \mu(E), and X_, X_,\ldots, mutually independent random variables with distribution \frac, define N_(\omega) = \sum\limits_^ \delta_(\cdot) where \delta_(A) is a degenerate measure located in c. Then N will be a Poisson random measure. In the case \mu is not finite the measure N can be obtained ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brownian Motion
Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical sources. This motion pattern typically consists of Randomness, random fluctuations in a particle's position inside a fluid sub-domain, followed by a relocation to another sub-domain. Each relocation is followed by more fluctuations within the new closed volume. This pattern describes a fluid at thermal equilibrium, defined by a given temperature. Within such a fluid, there exists no preferential direction of flow (as in transport phenomena). More specifically, the fluid's overall Linear momentum, linear and Angular momentum, angular momenta remain null over time. The Kinetic energy, kinetic energies of the molecular Brownian motions, together with those of molecular rotations and vibrations, sum up to the caloric component of a fluid's in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Differential Equation
A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs have many applications throughout pure mathematics and are used to model various behaviours of stochastic models such as stock prices,Musiela, M., and Rutkowski, M. (2004), Martingale Methods in Financial Modelling, 2nd Edition, Springer Verlag, Berlin. random growth models or physical systems that are subjected to thermal fluctuations. SDEs have a random differential that is in the most basic case random white noise calculated as the distributional derivative of a Brownian motion or more generally a semimartingale. However, other types of random behaviour are possible, such as jump processes like Lévy processes or semimartingales with jumps. Stochastic differential equations are in general neither differential equations nor random differential equations. Random differential equation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lévy Process
In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is a stochastic process with independent, stationary increments: it represents the motion of a point whose successive displacements are random, in which displacements in pairwise disjoint time intervals are independent, and displacements in different time intervals of the same length have identical probability distributions. A Lévy process may thus be viewed as the continuous-time analog of a random walk. The most well known examples of Lévy processes are the Wiener process, often called the Brownian motion process, and the Poisson process. Further important examples include the Gamma process, the Pascal process, and the Meixner process. Aside from Brownian motion with drift, all other proper (that is, not deterministic) Lévy processes have discontinuous paths. All Lévy processes are additive processes. Mathematical definition A Lévy process is a stochastic process X=\ that satisfi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stefan Problem
In mathematics and its applications, particularly to phase transitions in matter, a Stefan problem is a particular kind of boundary value problem for a system of partial differential equations (PDE), in which the boundary between the phases can move with time. The classical Stefan problem aims to describe the evolution of the boundary between two phases of a material undergoing a phase change, for example the melting of a solid, such as ice to water. This is accomplished by solving heat equations in both regions, subject to given boundary and initial conditions. At the interface between the phases (in the classical problem) the temperature is set to the phase change temperature. To close the mathematical system a further equation, the Stefan condition, is required. This is an energy balance which defines the position of the moving interface. Note that this evolving boundary is an unknown (hyper-)surface; hence, Stefan problems are examples of free boundary problems. Analogou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Boundary Problem
In mathematics, a free boundary problem (FB problem) is a partial differential equation to be solved for both an unknown function u and an unknown domain \Omega. The segment \Gamma of the boundary of \Omega which is not known at the outset of the problem is the free boundary. FBs arise in various mathematical models encompassing applications that ranges from physical to economical, financial and biological phenomena, where there is an extra effect of the medium. This effect is in general a qualitative change of the medium and hence an appearance of a phase transition: ice to water, liquid to crystal, buying to selling (assets), active to inactive (biology), blue to red (coloring games), disorganized to organized (self-organizing criticality). An interesting aspect of such a criticality is the so-called sandpile dynamic (or Internal DLA). The most classical example is the melting of ice: Given a block of ice, one can solve the heat equation given appropriate initial and boundary c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Snell Envelope
The Snell envelope, used in stochastics and mathematical finance, is the smallest supermartingale dominating a stochastic process. The Snell envelope is named after James Laurie Snell. Definition Given a filtered probability space (\Omega,\mathcal,(\mathcal_t)_,\mathbb) and an absolutely continuous probability measure \mathbb \ll \mathbb then an adapted process U = (U_t)_ is the Snell envelope with respect to \mathbb of the process X = (X_t)_ if # U is a \mathbb-supermartingale # U dominates X, i.e. U_t \geq X_t \mathbb-almost surely for all times t \in ,T/math> # If V = (V_t)_ is a \mathbb-supermartingale which dominates X, then V dominates U. Construction Given a (discrete) filtered probability space (\Omega,\mathcal,(\mathcal_n)_^N,\mathbb) and an absolutely continuous probability measure \mathbb \ll \mathbb then the Snell envelope (U_n)_^N with respect to \mathbb of the process (X_n)_^N is given by the recursive scheme :U_N := X_N, :U_n := X_n \lor \mathbb^ _ \mid \ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
