Stochastic process Monte Carlo method

While static Monte Carlo methods generate a sequence of statistically independent configurations, dynamic MC methods are always based on some stochastic Markov process, where subsequent configurations X of the system are generated from the previous configuration X —X —X" — > with some transition probability IF(X —> X ). Since to a large extent the choice of the basic move X —X is arbitrary, various methods differ in the choice of the basic unit of motion . Also, the choice of transition probability IF(X — > X ) is not unique the only requirement is that the principle... 

They point out that at the heart of technical simulation there must be unreality otherwise, there would not be need for simulation. The essence of the subject linder study may be represented by a model of it that serves a certain purpose, e.g., the use of a wind tunnel to simulate conditions to which an aircraft may be subjected. One uses the Monte Carlo method to study an artificial stochastic model of a physical or mathematical process, e.g., evaluating a definite integral by probability methods (using random numbers) using the graph of the function as an aid. 

An HMM is essentially a Markov chain (—> Monte Carlo methods). Each state inside the Markov chain can produce a letter, and it does so by a stochastic Poisson process that chooses one of a finite number of letters in an alphabet. Each letter has a characteristic probability of being chosen. That probability depends on the state and on the letter. After the state produces a letter, the Markov chain moves to another state. This happens according to a transition probability that depends on the prior and succeeding state. 

The Monte Carlo method, however, is prone to model risk. If the stochastic process chosen for the underlying variable is unrealistic, so will be the estimate of VaR. This is why the choice of the underlying model is particularly important. The geometric Brownian motion model described above adequately describes the behavior of some financial variables, but certainly not that of short-term fixed-income securities. In the Brownian motion, shocks on prices are never reversed. This does not represent the price process for default-free bonds, which must converge to their face value at expiration. 

In order to fully account for finite surface mobilities and the heterogeneous surface structure, we have to employ a stochastic description of the surface processes. The kinetic Monte Carlo method enables the incorporation of structural details at an atomistic level. This method has been apphed successfully in the field of heterogeneous (electro-) catalysis [59 1,65] and is further discussed in the chapter by Phil Ross in this book. In the model, hexagonal grids represent catalyst particles. The active sites are randomly distributed on the grid. Adsorbates are considered to bind to on-top sites. The first reaction method was used [66,67]. 

In this paper, the results of a Monte Carlo method for the simulation of the stochastic time evolution of the micellization process are presented. The computational algorithm [1] used represents an optimization of a general procedure introduced by Gillespie some years ago [2]. It was applied to the case of surfactant reversible association according to the general mechanism reported in Fig. 1 that allows associations and dissociations among -mers of whatever aggregation number. 

M.H.C. Everdij and H.A.P. Blom (2003). Petri-nets and hybrid-state Markov processes in a power-hierarchy of dependability models. In Engel, Gueguen, Zaytoon (eds.), Analysis and design of hybrid systems, Elsevier, pp. 313-318 M.H.C. Everdij and H.A.P. Blom (2005), Piecewise deterministic Markov processes represented by dynamically coloured Petri nets. Stochastics Vol. 77, pp.1-29 P. Glasserman (2004), Monte Carlo methods in financial engineering. Springer. 

Dynamic Monte Carlo methods are based on stochastic Markov processes where subsequent configurations X are generated from the previous one (Xi X2 X3 ) with some transition probability W(Xi -> X2). To some extent, the choice of the basic move Xj X2 is arbitrary. Various methods, as shown in Fig. 4, just differ in the choice of the basic unit of motion. Furthermore, W is not uniquely defined We only require the principle of detailed balance with the equilibrium distribution Pg X),... 

Monte Carlo methods for the artificial realization of the system behavior can be divided into time-driven and event-driven Monte Carlo simulations. In the former approach, the time interval At is chosen, and the realization of events within this time interval is determined stochastically. Whereas in the latter, the time interval between two events is determined based on the rates of processes. In general, the coalescence rates in granulation processes can be extracted from the coalescence kernel models. The event-driven Monte Carlo can be further divided into constant volume methods... 

Monte Carlo methods can be classified as static, quasi-static or dynamic. Static methods are those that generate a sequence of statistically independent samples from the desired probability distribution tt. Quasi-static methods are those that generate a sequence of statistically independent batches of samples from the desired probability distribution ir the correlations within a batch are often difficult to describe. Dynamic methods are those that generate a sequence of correlated samples from some stochastic process (usually a Markov process) having the desired probability distribution TT as its unique equilibrium distribution. 

A particularly usefid way to measure the effects of uncertainty in the dynamic response of structural systems is the so-called first-excursion probability. This probability is widely used in stochastic structural dynamics and measures the chances that one or more structural responses exceed a prescribed threshold level within the duration of a dynamical excitation (Soong and Grigoriu 1993). First-excursion probability estimation is particularly challenging as characterization of uncertain loading usually comprises stochastic processes whose discrete representation can involve hundreds or even thousands of random variables. Similarly, the number of possible failure criteria involved can be extremely large as well, i.e., there can be several responses of interest that must be controlled at a large number of discrete time instants. Hence, several different techniques have been proposed in order to estimate first-excursion probabilities. Among these, methods based on simulation (such as the Monte Carlo method and its more advanced variants) have been shown to be the most appropriate approach to compute these probabilities (Schueller et al. 2004). 

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