Ornstein-Uhlenbeck stochastic process

The outlay of this chapter is as follows In Section 9.2, we introduce the system/ spin-bath model and derive the operator Langevin equation for the particle. This is followed by a discussion on stochastic dynamics in the presence of c-number noise, highlighting the role of the spectral density function in the high- and low-temperature regimes. A scheme for the generation of spin-bath noise as a superposition of several Ornstein-Uhlenbeck noise processes and its implementation in numerical simulation of the quantum Langevin equation are described in Section 9.3. Two examples have been worked out in Section 9.4 to illustrate the basic theoretical issues. This chapter is concluded in Section 9.5. 

The study of a Brownian particle suspended in a fluid lead also to the introduction of the exponentially correlated Ornstein-Uhlenbeck process [48], the only Markovian Gaussian non-white stochastic process [19, 22]. We present here the Langevin approach to this problem, hence we analyze the forces that act on a single Brownian particle. We suppose the particle having a mass m equal to unity, and we assume the force due to the hits with thermal activated molecules of the fluid to be a stochastic variable. Moreover, due to the viscosity of the fluid, a friction force proportional to the velocity of the particle has to be considered. All this yields the following equation... 

For multiplicative noise the determination of these moments requires a more detailed consideration of the stochastic integral since white noise is too irregular for Riemann integrals to be applied. Application of Stieltjes integration yields a dependence of the moments on how the limit to white noise is taken. If t) is the limit of the Ornstein-Uhlenbeck -process with r —> 0 (Stratonovich sense) the coefficients read [50]... 

It can be shown that ARCH and GARCH models are able to approximate stochastic differential processes if the latter fulfil certain properties. Albeit the goodness of fit is limited, both types of methods are related and can be converted into each other. Moreover, simple stochastic processes show quite simple auto-correlation structures similar to basic ARMA models. For instance, the Ornstein-Uhlenbeck process can be seen as the continuous equivalent of the AR(1) process. In other words, an Ornstein-Uhlenbeck process measured in discrete intervals can be interpreted/modeUed as an AR(1) process (see also (2.23), (2.60), and (2.61)). ... 

We saw before that we could think of the simple SDE initial value problem (6.8) as having a solution defined by a certain random process. In the same way we would like to obtain, for the Ornstein-Uhlenbeck SDE (6.19), an explicit stochastic process which is in some sense equivalent to solving the SDE. Multiply both sides of (6.19) by the integrating factor exp(yf), and observe that... 

It is difficult to see how their could be other solutions t, but we know from the Nos6-Hoover case that this must be possible. We improve the robustness of the method by introducing Ornstein-Uhlenbeck type stochastic processes in the two auxiliary variables and rj, yielding the system... 

The subscripts rr in (4.5) indicate partial derivatives—the derivative with respect to one variable of a function involving several variables. The terms dr and drY are dependent on the stochastic process that is selected for the short rate r. If this is the Ornstein-Uhlenbeck process represented in (4.4), the dynamics of P can be expressed as (4.6), which gives these dynamics in terms of the drift and volatility of the short rate. 

We take it to be the Ornstein-Uhlenbeck process given by the stochastic differential equation... 

The assumption that f(t) is purely random may actually be a highly restrictive and unnecessary limitation. In fact, many physical processes cannot be described by Equation 1.4 simply because they exhibit memory effects. Thus, we must introduce an extended Ornstein-Uhlenbeck process, deflned as the solution of the most general linear stochastic equation with additive noise, which we write as... 

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