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Stochastic Control
Stochastic control or stochastic optimal control is a sub field of control theory that deals with the existence of uncertainty either in observations or in the noise that drives the evolution of the system. The system designer assumes, in a Bayesian probability-driven fashion, that random noise with known probability distribution affects the evolution and observation of the state variables. Stochastic control aims to design the time path of the controlled variables that performs the desired control task with minimum cost, somehow defined, despite the presence of this noise. The context may be either discrete time or continuous time. Certainty equivalence An extremely well-studied formulation in stochastic control is that of linear quadratic Gaussian control. Here the model is linear, the objective function is the expected value of a quadratic form, and the disturbances are purely additive. A basic result for discrete-time centralized systems with only additive uncertainty is the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optimal Control
Optimal control theory is a branch of control theory that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and operations research. For example, the dynamical system might be a spacecraft with controls corresponding to rocket thrusters, and the objective might be to reach the Moon with minimum fuel expenditure. Or the dynamical system could be a nation's economy, with the objective to minimize unemployment; the controls in this case could be fiscal and monetary policy. A dynamical system may also be introduced to embed operations research problems within the framework of optimal control theory. Optimal control is an extension of the calculus of variations, and is a mathematical optimization method for deriving control policies. The method is largely due to the work of Lev Pontryagin and Richard Bellman in the 1950s, after contributions to calculus of v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asset Allocation
Asset allocation is the implementation of an investment strategy that attempts to balance risk versus reward by adjusting the percentage of each asset in an investment portfolio according to the investor's risk tolerance, goals and investment time frame. The focus is on the characteristics of the overall portfolio. Such a strategy contrasts with an approach that focuses on individual assets. Description Many financial experts argue that asset allocation is an important factor in determining returns for an investment portfolio. Asset allocation is based on the principle that different assets perform differently in different market and economic conditions. A fundamental justification for asset allocation is the notion that different asset classes offer returns that are not perfectly correlated, hence diversification reduces the overall risk in terms of the variability of returns for a given level of expected return. Asset diversification has been described as "the only f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Scheduling
Stochastic scheduling concerns scheduling problems involving random attributes, such as random processing times, random due dates, random weights, and stochastic machine breakdowns. Major applications arise in manufacturing systems, computer systems, communication systems, logistics and transportation, and machine learning, among others. Introduction The objective of the stochastic scheduling problems can be regular objectives such as minimizing the total flowtime, the makespan In operations research Operations research () (U.S. Air Force Specialty Code: Operations Analysis), often shortened to the initialism OR, is a branch of applied mathematics that deals with the development and application of analytical methods t ..., or the total tardiness cost of missing the due dates; or can be irregular objectives such as minimizing both earliness and tardiness costs of completing the jobs, or the total cost of scheduling tasks under likely arrival of a disastrous event such as a seve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Control Theory
Control theory is a field of control engineering and applied mathematics that deals with the control system, control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a desired state, while minimizing any ''delay'', ''overshoot'', or ''steady-state error'' and ensuring a level of control Stability theory, stability; often with the aim to achieve a degree of Optimal control, optimality. To do this, a controller with the requisite corrective behavior is required. This controller monitors the controlled process variable (PV), and compares it with the reference or Setpoint (control system), set point (SP). The difference between actual and desired value of the process variable, called the ''error'' signal, or SP-PV error, is applied as feedback to generate a control action to bring the controlled process variable to the same value as the set point. Other aspects ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. Examples include the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule. Stochastic processes have applications in many disciplines such as biology, chemistry, ecology Ecology () is the natural science of the relationships among living organisms and their Natural environment, environment. Ecology considers organisms at the individual, population, community (ecology), community, ecosystem, and biosphere lev ..., neuroscience, physics, image processing, signal processing, stochastic control, control theory, information theory, computer scien ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Backward Stochastic Differential Equation
A backward stochastic differential equation (BSDE) is a stochastic differential equation with a terminal condition in which the solution is required to be adapted with respect to an underlying filtration. BSDEs naturally arise in various applications such as stochastic control, mathematical finance, and nonlinear Feynman-Kac formula. Background Backward stochastic differential equations were introduced by Jean-Michel Bismut in 1973 in the linear case and by Étienne Pardoux and Shige Peng in 1990 in the nonlinear case. Mathematical framework Fix a terminal time T>0 and a probability space (\Omega,\mathcal,\mathbb). Let (B_t)_ be a Brownian motion with natural filtration (\mathcal_t)_. A backward stochastic differential equation is an integral equation of the type where f: ,Ttimes\mathbb\times\mathbb\to\mathbb is called the generator of the BSDE, the terminal condition \xi is an \mathcal_T-measurable random variable, and the solution (Y_t,Z_t)_ consists of stochastic proce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Itô's Lemma
In 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 ar ..., Itô's lemma or Itô's formula (also called the Itô–Döblin formula) is an identity used in Itô calculus to find the differential of a time-dependent function of a stochastic process. It serves as the stochastic calculus counterpart of the chain rule. It can be heuristically derived by forming the Taylor series expansion of the function up to its second derivatives and retaining terms up to first order in the time increment and second order in the Wiener process increment. The Lemma (mathematics), lemma is widely employed in mathematical finance, and its best known application is in the derivation of the Black–Scholes equation for option values. This result was discovered by Japanese mathematician Kiyoshi It ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
2008 Financial Crisis
The 2008 financial crisis, also known as the global financial crisis (GFC), was a major worldwide financial crisis centered in the United States. The causes of the 2008 crisis included excessive speculation on housing values by both homeowners and financial institutions that led to the 2000s United States housing bubble, exacerbated by predatory lending for subprime mortgages and deficiencies in regulation. Cash out refinancings had fueled an increase in consumption that could no longer be sustained when home prices declined. The first phase of the crisis was the subprime mortgage crisis, which began in early 2007, as mortgage-backed securities (MBS) tied to U.S. real estate, and a vast web of Derivative (finance), derivatives linked to those MBS, collapsed in value. A liquidity crisis spread to global institutions by mid-2007 and climaxed with the bankruptcy of Lehman Brothers in September 2008, which triggered a stock market crash and bank runs in several countries. The crisis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jerome Stein
Jerome (; ; ; – 30 September 420), also known as Jerome of Stridon, was an early Christian priest, confessor, theologian, translator, and historian; he is commonly known as Saint Jerome. He is best known for his translation of the Bible into Latin (the translation that became known as the Vulgate) and his commentaries on the whole Bible. Jerome attempted to create a translation of the Old Testament based on a Hebrew version, rather than the Septuagint, as prior Latin Bible translations had done. His list of writings is extensive. In addition to his biblical works, he wrote polemical and historical essays, always from a theologian's perspective. Jerome was known for his teachings on Christian moral life, especially those in cosmopolitan centers such as Rome. He often focused on women's lives and identified how a woman devoted to Jesus should live her life. This focus stemmed from his close patron relationships with several prominent female ascetics who were members of aff ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Halil Mete Soner
Halil Mete Soner is a Turkish American mathematician born in Ankara and is the Normal John Sollenberger Professor at Princeton University. Soner's research interests are nonlinear partial differential equations; asymptotic analysis of Ginzburg-Landau type systems, viscosity solutions, and mathematical finance. Currently he is working on mean field games and control and related nonlinear partial differential equations on Wasserstein spaces. Education After graduating from the Ankara Science High School, he started his university education at the Middle East Technical University in Ankara, later transferred to Boğaziçi University, Istanbul in 1977. He received a B.Sc. in mathematics and another in electrical engineering simultaneously in 1981, both in first-rank. Soner then attended Brown University in Providence, RI, U.S. on a research fellowship, where he obtained his M.Sc. (1983) and Ph.D. (1986) in applied mathematics. Career In 1985, Soner was research associate at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Raymond Rishel
Raymond is a male given name of Germanic origin. It was borrowed into English from French (older French spellings were Reimund and Raimund, whereas the modern English and French spellings are identical). It originated as the Germanic ᚱᚨᚷᛁᚾᛗᚢᚾᛞ (''Raginmund'') or ᚱᛖᚷᛁᚾᛗᚢᚾᛞ (''Reginmund''). ''Ragin'' ( Gothic) and ''regin'' (Old German) meant "counsel". The Old High German ''mund'' originally meant "hand", but came to mean "protection". This etymology suggests that the name originated in the Early Middle Ages, possibly from Latin. Alternatively, the name can also be derived from Germanic Hraidmund, the first element being ''Hraid'', possibly meaning "fame" (compare ''Hrod'', found in names such as Robert, Roderick, Rudolph, Roland, Rodney and Roger) and ''mund'' meaning "protector". Despite the German and French origins of the English name, some of its early uses in English documents appear in Latinized form. As a surname, its first recorded ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wendell Fleming
Wendell Helms Fleming (March 7, 1928 – February 18, 2023) was an American mathematician, specializing in geometrical analysis and stochastic differential equations. Fleming received his PhD in 1951 under Laurence Chisholm Young at the University of Wisconsin–Madison with a thesis entitled ''Boundary and related notions for generalized parametric surfaces''. Fleming was a professor at Brown University, where he retired in 2009 as professor emeritus. Fleming was with Herbert Federer a pioneer of geometric measure theory. Later in his career, he worked on stochastic processes, stochastic differential equations and their applications in control theory. In 1976–1977 he was a Guggenheim Fellow. In 1982 he gave a plenary address (''Optimal control of Markov Processes'') at the ICM in Warsaw. Awards and honors In 1987 he received with Federer the Leroy P. Steele Prize of the American Mathematical Society. In 1994 he won the Reid Prize from the Society for Industrial and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
