The nature of stochastic processes

Let us start with an example. Consider a stretch of highway between two intersections, and let the variable of interest be the number of cars within this road segment at any given time, N(t). This number is obviously a random function of time whose properties can be deduced from observation and also from experience and intuition. First, this function takes positive integer values but this is of no significance we could redefine A - N — N) and the new variable will assume both positive and negative values. Second and more significantly, this function is characterized by several timescales  

The apparent lack of systematic component in the time series displayed here reflects only a relatively short-time behavior. For time exceeding another characteristic time, 72, typically ofthe order 1 h forthis problem, we observe what appears to be a systematic trend as seen in Fig. 7.2. 

Here sampling is made every 3 h over a period of 24 hours and the line connecting the results has been added to aid the eye. Alternatively, we may perform coarse graining in the spirit of Eq. (1.180) using time intervals of, say, Az = 72 = 1 h, which will lead to a smoother display. The systematic trend shows the high and low traffic volumes at different times of day. 

If we extend our observation to longer times we will see other trends that occur on longer timescales. In this example, we may distinguish between 

Note that in the example considered above, the systematic behavior and the stochastic fluctuations arise from different causes. Sometimes the random motion itself gives, in time, a systematic signal. If we put an ink drop at the center of a pool of water the systematic spread of the ink by diffusion is caused by random motions of the individual molecules, each characterized by zero mean displacement. 

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The power spectrum was defined here as a property of a given stochastic process. In the physics literature it is customary to consider a closely related fimction that focuses on the properties of the thennal environment that couples to the system of interest and affects the stochastic nature of its evolution. This is the spectral density that was discussed in Section 6.5.2. (see also Section 8.2.6). To see the connection between these functions we recall that in applications of the theory of stochastic processes to physical phenomena, the stochastic process x(Z) represents a physical observable A, say a coordinate or a momentum of some observed particle. Suppose that this observable can be expanded in harmonic nonnal modes uj as in Eq. (6,79)... 

Bl. Bailey, N. T. J., The Elements of Stochastic Processes, with Applications to the Natural Sciences. Wiley, New York, 1964. 

Both of the numerical approaches explained above have been successful in producing realistic behaviour for lamellar thickness and growth rate as a function of supercooling. The nature of rough surface growth prevents an analytical solution as many of the growth processes are taking place simultaneously, and any approach which is not stochastic, as the Monte Carlo in Sect. 4.2.1, necessarily involves approximations, as the rate equations detailed in Sect. 4.2.2. At the expense of... 

There are various ways to classify mathematical models (5). First, according to the nature of the process, they can be classified as deterministic or stochastic. The former refers to a process in which each variable or parameter acquires a certain specific value or sets of values according to the operating conditions. In the latter, an element of uncertainty enters we cannot specify a certain value to a variable, but only a most probable one. Transport-based models are deterministic residence time distribution models in well-stirred tanks are stochastic. 

The third term corresponds to recoil is associated with spontaneous photon emission in a direction different from (x), justifying the change into I i(x). Such photon state is not correlated to the initial interacting state. The reason is due to the stochastic nature of spontaneous processes. The quantum state amplitudes are C3 = C4 = 0 and C3< = 1. The change is physically irreversible. 

By employing a very strong external field, a gedankexperiment may be set up whereby the natural thermal motion of the molecules is put in competition with the aligning effect of the field. This method reveals some properties of the molecular liquid state which are otherwise hidden. In order to explain the observable effects of the applied fields, it is necessary to use equations of motion more generally valid than those of Benoit. These equations may be incorporated within the general structure of reduced model theory " (RMT) and illustrate the use of RMT in the context of liquid-state molecular dynamics. (Elsewhere in this volume RMT is applied to problems in other fields of physics where consideration of stochastic processes is necessary.) In this chapter modifications to the standard methods are described which enable the detailed study of field-on molecular dynamics. 

MCMC methods are essentially Monte Carlo numerical integration that is wrapped around a purpose built Markov chain. Both Markov chains and Monte Carlo integration may exist without reference to the other. A Markov chain is any chain where the current state of the chain is conditional on the immediate past state only—this is a so-called first-order Markov chain higher order chains are also possible. The chain refers to a sequence of realizations from a stochastic process. The nature of the Markov process is illustrated in the description of the MH algorithm (see Section 5.1.3.1). 

For SPC of highly autocorrelated measurements, since it is essential to decorrelate the data, MSSPC with dyadic downsampling is used. The nature of the wavelet filters and the downsampling can decorrelate a wide variety of stochastic processes. Fig. 6 depicts the ARL for an AR(1) process given by... 

Time series models and SDEs deal with the same sort of stochastic process. Both differ only in the domain of variables, which are either discrete or continuous. For instance, chemical processes are continuous by nature. In practice, however, the condition of a... 

To model uncertainty, knowledge about the nature of the underlying stochastic process is required. More precisely, assumptions have to be made on how this process is ruled, which outcomes are possible, and what their corresponding probabiUties are. If the set... 

Let us examine more closely the nature of the underlying stochastic process Sa yit) for the time-space fractional equation (3.214). The latter is the long-time large-scale limit of the generalized Master equation (3.43) under the conditions that the symmetric jumps have a heavy-tailed density (3.204) with infinite variance and the waiting time PDF (t) decreases like with the index 0 < y < 1 as... 

Finally, we observe that by calculating cr t ) and generating random increments Rn J (0, y(tn)) we have a discrete representation of the continuous time stochastic process Y(t), effectively an exact stroboscopic representation. Also it is natural to view the stochastic integral as the solution of the stochastic differential equation (SDE) with zero initial condition ... 

This chapter considers some of the techniques used to fit the model-derived term structure to the observed one. The Vasicek, Brennan-Schwartz, Cox-Ingersoll-Ross, and other models discussed in chapter 4 made various assumptions about the nature of the stochastic process that drives interest rates in defining the term structure. The zero-coupon curves derived by those models differ from those constructed from observed market rates or the spot rates implied by market yields. In general, market yield curves have more-variable shapes than those derived by term-structure models. The interest rate models described in chapter 4 must thus be calibrated to market yield curves. This is done in two ways either the model is calibrated to market instruments, such as money market products and interest rate swaps, which are used to construct a yield curve, or it is calibrated to a curve constructed from market-instrument rates. The latter approach may be implemented through a number of non-parametric methods. 

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