Brownian bridge

A Brownian bridge is a special stochastic process, which emerges from the Wiener process ( Brownian motion also called ). But in contrast to this, it has a finite time horizon with a deterministic (not random ) final value, which is the starting value is normally the same. Brownian Bridge is used to model random developments in data whose value is, however, known at two points in time.

Definition

Be a standard Wiener process and a fixed chosen point in time. Then the process is called

Brownian bridge of length. The only difference is in the fact that it is conditioned that returns to the time to zero. Thus, the probability distribution of each time point is added to the conditional probability

In particular, of course, is true. Hence the name of the process: It is a bridge between 0 and beaten where you can then " solid ground under their feet" has.

Properties

Some fundamental properties of the Wiener process is preserved during the transition to the Brownian bridge, but others lost:

  • The Brownian bridge has almost certainly everywhere continuous, nowhere differentiable paths.
  • The expected value function of the Brownian bridge is constant.
  • The covariance function is.

In particular, therefore applies to the variance.

  • The Brownian bridge is a Markov process, but in contrast to Brownian motion neither Lévy process nor martingale.
  • The Brownian bridge is a Gaussian process, that is already determined by the above expected value and covariance function clearly.

Simulation

To simulate a Brownian bridge are one in principle, the same options are available as for the Wiener process, because of a Wiener process can be gained by a Brownian bridge with a time horizon. So, you can simulate a Brownian motion up to time and then convert them with the above transformation to a Brownian bridge.

However, there are other possibilities: If the Brownian motion ( is, confusingly, this method often also referred to as Brownian bridge ) using a dyadic decomposition or spectral decomposition generated, so you can there simply omit the first step, which determines the endpoint, and then obtained automatically a Brownian bridge. In the case of the spectral representation would therefore

Generalizations

  • Alternatively to the above definition that is guaranteed, it is also possible for any of
  • In addition, you can see the original Brownian motion with any volatility provided (see: generalized Wiener process ). The formulas for the expected value and covariance can then be written
  • Stochastic process
Almost surely Martingale (probability theory)
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