Ornstein–Uhlenbeck process

The Ornstein - Uhlenbeck process ( often abbreviated OU ) process is a special stochastic process which, after the two Dutch physicists George Uhlenbeck ( 1900-1988 ) and Leonard Ornstein ( 1880-1941 ) is named. He is in addition to the geometric Brownian motion one of the simplest and most important defined by a stochastic differential equation processes.

Definition and parameters

Let and be constants. A stochastic process is called Ornstein - Uhlenbeck process with initial value, equilibrium level, stiffness and diffusion if it solves the following stochastic initial value problem:

With a standard Wiener process.

The parameters are easily interpreted and thus use in the modeling of a stochastic time series simply as " screws ":

  • Is the equilibrium level of the process (English: mean reversion level ). Above that value, the drift term is negative and the drift will tend to "pull" the process down. X is smaller, the drift is positive and the process is drawn in anticipation upwards.
  • ( engl: mean reversion speed or mean reversion rate) indicates how strong the above-described " attraction " of. For small values ​​of this effect disappears for large values ​​, X is very stiff to develop.
  • Indicates how strong is the influence of ( ie of chance ) on the process. For X is simply converge exponentially against, with strong diffusion this convergence is disturbed randomly.

The difference to the well -equipped with the mean- reversion mechanism root diffusion process or the geometric Brownian motion essentially consists, is that the OU process of diffusion term constant, that is independent of X. This means that the process can take OU in contrast to the other two negative values.

Solution of the differential equation

In contrast to the root diffusion process, the above differential equation is explicitly solvable, though not ( as in the geometric Brownian motion ) is integrally represented without: one turns on the two-dimensional function on the one hand, the lemma of Ito, on the other hand, the usual chain rule of differential calculus to, we obtain

The above integrated up identity of 0 to t ( wherein ) yields the solution

Properties

  • The above solution looks at you that it is the Ornstein - Uhlenbeck process to a Gaussian process: The integrand is deterministic, ie the value of the Ito integral is always normally distributed.
  • As a Gaussian process of the OU process is uniquely determined by its expected value and covariance in its distribution. These arise as
  • Since both the expected value and variance converge, there exists a stationary distribution for the Markov process X: it involves a normal distribution with mean and variance. In contrast to the Wiener process of the Ornstein - Uhlenbeck process is thus (weakly ) stationary. We then say that the process is an " invariant measure " has: Then for each t

So the process has in no asymptote.

  • The OU process is, as well as the root diffusion process an affine process.
  • In some ways, the OU process is more complicated than the Wiener process. For large time scales, however, the Wiener process can serve as an approximation of the OU process. It is in the sense of convergence in distribution

Lévy processes

If the defining differential equation driven by a Brownian motion not, but by a Lévy process, the result is also a (non - Gaussian ) Ornstein - Uhlenbeck process.

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