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High Quality Content by WIKIPEDIA articles! A stochastic differential equation is a differential equation in which one or more of the terms is a stochastic process, thus resulting in a solution which is itself a stochastic process. Typically, SDEs incorporate white noise which can be thought of as the derivative of Brownian motion, however, it should be mentioned that other types of random fluctuations are possible, such as jump processes.

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The LMM is an effective framework for the pricing of interest rate derivatives, not least because it models observable market quantities. There exist three main techniques for incorporating a volatility smile/skew in any modelling framework: allowing a local volatility function, stochastic volatility and jump dynamics. Here various ways to incorporate smile/skew are studied, loosely based on the above three approaches.Both the CEV and displaced-diffusion processes give rise to an implied volatility skew. The two processes produce closely matching prices for European call options over a variety of strikes and maturities. Here, this similarity in prices is analytically quantified using asymptotic expansion techniques.A regime shifting model may be viewed as a reduced form of a full stochastic volatility model. A two state, continuous time Markov Chain model, characterised by a time dependent volatility in each state is implemented. Finally, the Levy LIBOR model is considered as a generalisation of jump processes.

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Stock return predictability continues to attract an enormous amount of attention and yet the empirical evidence struggles to meet a general consensus. While a number of studies debate on the ability of economically meaningful variables such as dividend yield, term spread and consumption-wealth ratio to predict future stock returns, an important strand of the literature focuses on how to accurately incorporate the effect of stylized facts such as stochastic volatility and jumps on the data generating process of stock returns. However attempts have produced mixed results and mainly examined model specifications by using statistical measures. The economic advantage of using double-jump models remains largely unexplored. We find that, under both latent volatility and realized volatility measures, although jumps clearly affect the optimal weights, the pure diffusion model has better portfolio performance than jump-diffusion model, as stochastic volatility alone delivers the best portfolio performance. In addition, adding jumps in volatility yields more economic gains over the jump-diffusion model.

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High Quality Content by WIKIPEDIA articles! Stochastic optimization (SO) methods are optimization algorithms which incorporate probabilistic (random) elements, either in the problem data (the objective function, the constraints, etc.), or in the algorithm itself (through random parameter values, random choices, etc.), or in both . The concept contrasts with the deterministic optimization methods, where the values of the objective function are assumed to be exact, and the computation is completely determined by the values sampled so far.Partly-random input data arise in such areas as real-time estimation and control, simulation-based optimization where Monte Carlo simulations are run as estimates of an actual system, and problems where there is experimental (random) error in the measurements of the criterion. In such cases, knowledge that the function values are contaminated by random "noise" leads naturally to algorithms that use statistical inference tools to estimate the "true" values of the function and/or make statistically optimal decisions about the next steps.

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The present investigation is about the process of decision making in supply chain management using a real options analysis framework. Specifically, we address issues regarding the optimal inventory level to hedge against demand uncertainties, the timing for equipment capacity implementation under market product mix complexity, the timing for workforce capacity reinforcement aiming market service requirements, and the decisions between integration and outsourcing in an uncertainty environment. Discrete and continuous time methodologies were used to identify the optimal value and timing of the options to adopt, when the demand is stochastic. Additionally, the effect of market requirements, such as product mix complexity and service level, were also taken into consideration. The increasing uncertainty in demand has promoted supply chains agility and flexibility, limiting the use of many of the traditional management techniques, because of their inability to incorporate the effects of uncertainty. Flexibility is clearly a competitive advantage that companies should have and therefore must be quantified.

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This research aims to incorporate travel time constraint into the Stochastic User Equilibrium framework by means of a non-linear perceived travel time function. This modified model focuses primarily on discretionary travel behavior, and hence also allows the possibility of deferring travel decisions by incorporating an additional choice alternative, the Shop Less Frequently alternative. This model is compared to the traditional SUE model by means of simulated travel scenarios on a test network designed to reflect a range of practical planning situations. These simulations show when attractiveness levels are increased by the introduction of new shopping opportunities, the presence travel time constraints can lead to significantly smaller predicted travel volumes than the traditional SUE model. Travel to those shopping destinations with enhanced attractiveness can actually decrease for some origins. These findings suggest that when attempting to evaluate the impact of planning alternatives on future traffic patterns, it is vital to consider not only the cost of time itself but also the time tradeoffs between travel and other human activities.

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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Stochastic partial differential equations (SPDEs) are similar to ordinary stochastic differential equations. They are essentially partial differential equations that have additional random terms. They can be exceedingly difficult to solve. However, they have strong connections with quantum field theory and statistical mechanics. A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, thus resulting in a solution which is itself a stochastic process. SDE are used to model diverse phenomena such as fluctuating stock prices or physical system subject to thermal fluctuations. Typically, SDEs incorporate white noise which can be thought of as the derivative of Brownian motion (or the Wiener process), however, it should be mentioned that other types of random fluctuations are possible, such as jump processes.

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This book is aimed at offering some insight into the dynamics of evolutionary algorithms, which are primarily stochastic heuristic schemes. Since the comprehensive theoretical formulation encompassing the entire spectrum of evolutionary schemes is difficult, if not impossible, the present treatise may be viewed as an effort to make mathematical chracterization of the dynamics of some restricted class of evolutionary algorithms. Naturally, the endeavour can further be extended to incorporate even wider class of evolutionary algorithms. In that sense, this treatment may be viewed as the beginning of theorization of the dynamics of such class of probabilistic heuristics. As such, the authors intend to dedicate this book to this community to invite them in such efforts. However, the authors are also engaged in extending their present work further. An appreciation from the readership, even if critical, will be treated as of immense help in their future effort.

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The considerable influence of inherent uncertainties on structural behavior has led the engineering community to recognize the importance of a stochastic approach to structural problems. Issues related to uncertainty quantification and its influence on the reliability of the computational models are continuously gaining in significance. In particular, the problems of dynamic response analysis and reliability assessment of structures with uncertain system and excitation parameters have been the subject of continuous research over the last two decades as a result of the increasing availability of powerful computing resources and technology.This book is a follow up of a previous book with the same subject (ISBN 978-90-481-9986-0) and focuses on advanced computational methods and software tools which can highly assist in tackling complex problems in stochastic dynamic/seismic analysis and design of structures. The selected chapters are authored by some of the most active scholars in their respective areas and represent some of the most recent developments in this field.The book consists of 21 chapters which can be grouped into several thematic topics including dynamic analysis of stochastic systems, reliability-based design, structural control and health monitoring, model updating, system identification, wave propagation in random media, seismic fragility analysis and damage assessment.This edited book is primarily intended for researchers and post-graduate students who are familiar with the fundamentals and wish to study or to advance the state of the art on a particular topic in the field of computational stochastic structural dynamics. Nevertheless, practicing engineers could benefit as well from it as most code provisions tend to incorporate probabilistic concepts in the analysis and design of structures.

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