Ornstein-Uhlenbeck processes

The Ornstein-Uhlenbeck process is stationary, Gaussian, and Markovian. Doob s theoremstates that it is essentially the only process with these three properties. Essentially means that one must allow for linear transformations of y and t, and that there is one other, although trivial, process with these properties, see (3.22) below. We sketch the proof. 

This is identical with the generating functional of the Ornstein-Uhlenbeck process. 

These are the so-called forward and backward Kolmogorov equations for the Ornstein-Uhlenbeck process. Their paramount importance will appear in VIII.4 under the more familiar name of Fokker-Planck equation. 

Exercise. Y(t) being the Ornstein-Uhlenbeck process define Z(t) = jo Y(t ) dt for t O. The process Z(t) is Gaussian but neither stationary nor Markovian. Show... 

Exercise. In the Ornstein-Uhlenbeck process rescale the variables y = ay, t = fit and show that in a suitably chosen limit of a and ft the P reduces to that of the Wiener process. 

If A < 0 the stationary solution (1.4) is Gaussian. In fact, in that case it is possible by shifting y and rescaling, to reduce (1.5) to (IV.3.20), so that one may conclude the stationary Markov process determined by the linear Fokker-Planck equation is the Ornstein-Uhlenbeck process. For Al 0 there is no stationary probability distribution. 

This is a linear Fokker-Planck equation. Apart from constants which can be scaled away, it is identical with the equation (IV.3.20) obeyed by the transition probability of the Ornstein-Uhlenbeck process. The stationary solution of (4.6) is the same as the Pl given in (IV.3.10). Thus, in equilibrium V(t) is the Ornstein-Uhlenbeck process. 

Thus our additional approximation for the neighborhood of rf leads to a linear Fokker-Planck equation of the same form as (4.6). The fluctuations in the stationary state are therefore again an Ornstein-Uhlenbeck process. It will be shown in X.4 that (5.6) is a consistent approximation.510... 

Exercise. The Ornstein-Uhlenbeck process (IV.3.10), (IV.3.11) satisfies the generalized Langevin equation with memory kernel ... 

This is the master equation for the Ornstein-Uhlenbeck process. This result is readily extended to more variables, but we emphasize that the equations must be linear. 

Exercise. The generating functional of the solution of (1.1) can be found explicitly with the aid of (1.4) and (3.2). Show in this way that V(t) is the Ornstein-Uhlenbeck process that is given by (VIII.4.6). 

In particular let us take for the stationary solution of (1.10) = s = 1. Then (1.11) reduces to a time-independent Fokker-Planck equation whose solution is the Ornstein-Uhlenbeck process. More directly one finds from (1.12b)... 

Exercise. Verify that the linear noise approximation always leads to an Ornstein-Uhlenbeck process for the fluctuations in a stable stationary state. 

Exercise. Far below threshold the nonlinear terms in (7.6) and (7.7) may be neglected. In this case E is an Ornstein-Uhlenbeck process and... 

Exercise. Again in (1.2) take a> = co0 + a but let be the Ornstein-Uhlenbeck process. Show that in proper units... 

Equation (58) is equivalent to the fractional Rayleigh equation [75, 77], and therefore we refer to Eq. (58) as the fractional Ornstein-Uhlenbeck process. For the sharp initial condition Wo(x) = <5(x — xo), the solution to this process is, according to Eq. (46), given by... 

At this point it has to be emphasized the links of the Langevin description with the diffusion processes. By comparing the transition density functions (4.121) and (4.130), it is clear that the Langevin equation (4.126) is equivalent to the Ornstein-Uhlenbeck process. Equation (4.130) satisfies the following one-dimensional Fokker-Planck... 

For a harmonic oscillator immersed in a Brownian medium, the displacement—in the high-friction approximation—is a position Ornstein-Uhlenbeck process. With U(x) = mwV/2, the solution of Eqs. (4.152) and (4.153) is... 

In the computational procedure, the Ornstein-Uhlenbeck process is applied to onedimensional random walks under a harmonic velocity field. The velocity field obtained by the Gaussian function eqn (4-C-5) is. 

As we know, if the random force had Gaussian statistics and was delta-correlated in time, we would have an Ornstein-Uhlenbeck process. The variance of the system response would increase linearly in time for early times and be constant at late times. However, when the random force is Levy-stable the second moment of the system response is infinite. 

The standard random walk problem in physics is the Ornstein-Uhlenbeck process, which is a model of the Brownian motion in a dissipative medium. We are now looking at the possibility to generalise this to the quantum mechanical dynamics. To this end we introduce the one-dimensional canonical variables [x, p = ih, where we retain the quantum constant for dimensional reasons. We assume that these co-ordinates are physical in the sense that the laboratory positions are given by x and the physical forces are supposed to act on the momentum p only. 

The flawed time evolution tells us that, starting from some proper quantum states, we may obtain improper ones. However, Eq. (35) is the correct evolution equation for the Ornstein-Uhlenbeck process. Thus, starting from allowed classical states, it will always produce allowed classical ones. This strange situation indicates that irreversible time evolution is more intricate in quantum theory than we may guess from the classical counterpart. 

This is the function known from the Ornstein-Uhlenbeck process for short times it goes like t2 but for long times it gives... 

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]... 

For colored noise sources the derivation of evolution equations for the probability densities is more difficult. In a Markovian embedding, i.e. if the Ornstein-Uhlenbeck process is defined via white noise (cf. chapter 1.3.2) and v t) is part of the phase space one again gets a Fokker-Planck equation for the density P x,y, Similarly, one finds in case of the telegraph... 

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)). ... 

See e.g. Haeeler (2007, p. 236) about the similarity between both processes and Uhlenbeck and Omstein (1930) or Vaeicek (1977) for details about the Ornstein-Uhlenbeck process. 

To give an example for the influence of the length of the recording interval, let y(t) denote the value of an Ornstein-Uhlenbeck process as introduced in (2.23) at time t. For this specific process the autocorrelation p for ( oo can be expressed as ... 

BDT, HW, and BK models extended the Ho-Lee model to match a term structure volatility curve (for example the cap prices) in addition to the term structure. The BK model is a generalization of the BDT model and it overcomes the problem of negative interest rates assuming that the short rate r is the exponential of an Ornstein-Uhlenbeck process having time-dependent coefficients. It is popular with practitioners because it fits the swaption volatility surface well. Nevertheless, it does not have closed formulae for bonds or options on bonds. 

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