Brownian bridge

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Two independent Brownian bridges with time horizon 1. Double the standard deviation (ellipse) is indicated in gray as the marginal confidence interval

A Brownian bridge is a special stochastic process that arises from the Wiener process (also called Brownian movement ). In contrast to this, however, it has a finite time horizon with a deterministic (i.e. not random) end value, which is normally the same as the start value. The Brownian bridge is used to model random developments in data, the value of which is known at two points in time.

definition

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

Brownian bridge of length . The only difference is in the fact that it requires is that at the time again returns to zero. So the probability distribution of at any point in time is given by the conditional probability

.

In particular, of course . Hence the name of the process: A bridge is built between 0 and , where you have "solid ground under your feet" again.

properties

Some fundamental properties of the Wiener process are retained in the transition to Brown's bridge, but others are lost:

  • The Brownsche Brücke almost certainly has steady , nowhere differentiable paths everywhere .
  • The expectation function of the Brownian bridge is constant .
  • The covariance function is .

In particular we have for the variance : .

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

simulation

In principle, the same possibilities are available for simulating a Brownian bridge as with the Wiener process, because a Brownian bridge can be used to gain a time horizon from a Wiener process . So you can simply simulate a Brownian movement up to the point in time and then convert it into a Brownian bridge with the above transformation.

But there are other possibilities: If the Brownian movement is generated by means of a dyadic decomposition (confusingly, this method is often also referred to as a Brownian bridge ) or spectral decomposition , then you can simply omit the first step that determines the end point , and then you get automatically a Brownian bridge. In the case of the spectral decomposition, the representation would be

are, with independent standard normally distributed .

Generalizations

  • As an alternative to the definition above, which guarantees, it is also possible for any by
Define a bridge that ends at any predetermined level (figuratively speaking, the bridge becomes a ramp ). The corresponding transformation is then .
  • In addition, the original Brownian movement can also be given any volatility (see: generalized Wiener process ). : The formulas for expected value and covariance are then
respectively
.
Interestingly enough, c has no influence on the expected value and c has no influence on the covariance. A possible drift in the Brownian movement would not affect the distribution of the process at all.

literature