Stochastic processes concepts

A stochastic process is a collection of random variables teJ, Two important cases are the discrete parameter process, when J cz = 0, 1, 2,. .. and the continuous parameter process, when T ci IR We shall mostly consider the second case, and, in addition, restrict ourselves to Markov processes (processes without after-effect). 

Systems under investigation are characterised at a fixed time, r, by a finitedimensional vector random variable ft), 2(0 a/(0- The state of the system can be characterised (not completely) by the absolute density function x(0 - g(x, t) where t) is the value of (r). The temporal evolution of the system is given by a partial differential equation for g. Very often we have to be satisfied with deriving and solving differential equations for the first two moments only. Since we are interested in Markov processes, we have to know the conditional probability density function / f(y, s x, t) gives the probability density that a system will be at jc at a time r, if at the time s it was at y. The connection between the absolute g( ) and conditional probability function/( ) is given by 

To study the diflTerent classes of processes the state-space and the character of the motion have to be specified  

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In this chapter stochastic processes are defined, together with a number of auxiliary concepts. The reason why they occur in physics is discussed. General properties of them are derived and some examples are treated in detail. 

The generalization of the concept of a characteristic function to stochastic processes is the characteristic functional (In a different connection this idea was used in section II.3.) Let Y(t) be a given random process. Introduce an arbitrary auxiliary test function k(t). Then the characteristic or moment generating functional is defined as the following functional of k(t ... 

It has been remarked in III.4 that by imposing a condition on the sample functions of a stochastic process one defines a subensemble. This concept of... 

As a final note, it has to be stressed out that Eqs. (4.49) and (4.50) and Eqs. (4.52) and (4.53) hold for an arbitrary stochastic process. These evolution equations cannot give any information about whether or not the process is Markovian.135 The master equation concept has been used to analyze some examples of multistate relaxation processes.139... 

The individual tolerance concept has some unrealistic properties (Kooijman 1996 Newman and McCloskey 2000). Most importantly, if there is a distribution in sensitivities, this would imply that the survivors from an experiment are the less sensitive individuals. Experiments with sequential exposure show that this prediction fails (at least as the dominant mechanism) (Newman and McCloskey 2000 Zhao and Newman 2007). There is sufficient reason to conclude that the individual threshold model is not sufficient to explain the observed dose-response relationships, and that mortality is a stochastic process at the level of the individual... 

Chapter 4 is devoted to the description of stochastic mathematical modelling and the methods used to solve these models such as analytical, asymptotic or numerical methods. The evolution of processes is then analyzed by using different concepts, theories and methods. The concept of Markov chains or of complete connected chains, probability balance, the similarity between the Fokker-Plank-Kolmogorov equation and the property transport equation, and the stochastic differential equation systems are presented as the basic elements of stochastic process modelling. Mathematical models of the application of continuous and discrete polystochastic processes to chemical engineering processes are discussed. They include liquid and gas flow in a column with a mobile packed bed, mechanical stirring of a liquid in a tank, solid motion in a liquid fluidized bed, species movement and transfer in a porous media. Deep bed filtration and heat exchanger dynamics are also analyzed. 

Destabilization of a recursive state in the present model occurs through the decrease of the population of the minority molecules in the core hypercycle. As this molecular species is taken over by parasitic molecules, the switching starts to occur. In this sense, the process in the switching is not random, but is restricted to specific routes within the phase space of chemical composition, as in the chaotic itinerancy. It is interesting to study the present switching over the recursive state by generalizing the concept of chaotic itinerancy [46], to include stochastic process. 

Some concepts from the general theory of stochastic processes... 

The merit of the pulse model is that it is capable of yielding quite different forms of the scalar interaction from the same physical concept. Undoubtedly other stochastic processes that can be explained on a... 

It is important to consider whether or not a steady-state distribution exists in a PBN, of which, treatment of these concepts will not be discussed here. These can be found in many standard textbooks on stochastic processes. If the chain has a steady-state distribution, we can investigate the following question In the long mn, what is the probability that the chain is in state i regardless of the initial state That is, with currently observed data of a patient such as gene expression microarray, we can predict the probability of the patient s risk of a disease in the future. 

A fundamental concept in DP is that of a state, denoted by s. The set S of all possible states is called the state space. The decision problem is often describe as a controlled stochastic process that... 

Models that seek to value options or describe a yield curve also describe the dynamics of asset price changes. The same process is said to apply to changes in share prices, bond prices, interest rates and exchange rates. The process by which prices and interest rates evolve over time is known as a stochastic process, and this is a fundamental concept in finance theory. Essentially, a stochastic process is a time series of random variables. Generally, the random variables in a stochastic process are related in a non-random manner, and so therefore we can capture them in a probability density function. A good introduction is given in Neftci (1996), and following his approach we very briefly summarise the main features here. 

A key element of the description of a stochastic process is a specification of the level of informatimi oti the behaviour of prices that is available to an observer at each point in time. As with the martingale property, a calculation of the expected future values of a price process requires information on current prices. Generally, fmancial valuation models require data on both the current and the historical security prices, but investors are only able to deal on the basis of current known information and do not have access to future information. In a stochastic model, this concept is captured via the process known as filtration. 

In Chapter 2, we introduced the concept of stochastic processes. Most but not all interest-rate models are essentially descriptions of the short-rate models in terms of stochastic process. Financial literature has tended to categorise models into one of up to six different types, but for our purposes we can generalise them into two types. Thus, we introduce some of the main models, according to their categorisation as equilibrium or arbitrage-free models. This chapter looks at the earlier models, including the first ever term structure model presented by Vasicek (1977). The next chapter considers what have been termed whole yield curve models, or the Heath-Jarrow-Morton family, while Chapter 5 reviews considerations in fitting the yield curve. 

To model the randomized motion of the Brownian particle, the concept of a random walk is typically used. A random walk is an example of a stochastic process, a collection of random variables parameterized by either a discrete or continuous index parameter [269, 314]. A random walk is a discrete stochastic process in which the state X at a given instant (defined by the index n) is related to the state X i, at step n — 1 by an offset that may be viewed as a random variable. That is, we have... 

The numerical analysis of stochastic differential equations is traditionally based on the concepts of weak and strong accuracy. Let a numerical method be given for solving an SDE in the form of a discrete stochastic process X +i = 0(X , h) for n = 0, l,...,v — 1, where vh = x is, fixed. We denote the stochastic solution of the SDE by X(t). In the case of strong accuracy, our measure of the global error is the quantity [200]... 

The treatment in this chapter is concept-oriented applications are not discussed, and the use of mathematics has been minimized, although some understanding of Fourier transforms, stochastic processes, estimation theory, and linear algebra is occasionally assumed. The Bibliography... 

This concept is depicted in Figure 1, with reference to the well known R — S (Resistance - Stress) (Melchers 1999) or load-capacity interference model, within a rehahility physics framework, where the system prohahihty o f failure Pf is evaluated by comparing the distributions of the two quantities, (Burgazzi 2003, Apostolakis et al. 2005). The curves 1 and 2 in Fig. lb represent respectively the system failure prohahihty Pf in case of time variant and time invariant stochastic process. 

Detachment of Adherent Particles as a Stochastic Process. Using concepts developed previously (see Section 3, p. 18) as to the detachment of particles as a stochastic process, we will now examine the removal of adherent particles by an air stream. 

The qualitative theory describes the long-term behaviour of stochastic systems without solving the equations. Recurrence, stationarity and ergodic properties are the most important concepts which characterise the stochastic process and/or the state-space. 

The stochastic version of a memory-free deterministic process is a Markov process — more precisely, a first-order Markov process. It is interesting to remark that in the theory of stochastic processes the concept of history-dependent processes, had been adopted by the time the theory was established (i.e. in the mid-thirties). 

The cybernetic description of systems of different types is characterized by concepts and definitions like feedback, delay time, stochastic processes, and stability. These aspects will, for the time being, be demonstrated on the control loop as an example of a simple cybernetic system, but one that contains all typical properties. Thereby the importance of the probability calculus and communication in cybernetics can be clearly explained. This is then followed by a general representation of cybernetic systems. 

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