Wiener Process - Related Processes

Related Processes

The stochastic process defined by

is called a Wiener process with drift μ and infinitesimal variance σ2. These processes exhaust continuous Lévy processes.

Two random processes on the time interval appear, roughly speaking, when conditioning the Wiener process to vanish on both ends of . With no further conditioning, the process takes both positive and negative values on and is called Brownian bridge. Conditioned also to stay positive on (0, 1), the process is called Brownian excursion. In both cases a rigorous treatment involves a limiting procedure, since the formula P(A|B) = P(A ∩ B)/P(B) does not apply when P(B) = 0.

A geometric Brownian motion can be written

It is a stochastic process which is used to model processes that can never take on negative values, such as the value of stocks.

The stochastic process

is distributed like the Ornstein–Uhlenbeck process.

The time of hitting a single point x > 0 by the Wiener process is a random variable with the Lévy distribution. The family of these random variables (indexed by all positive numbers x) is a left-continuous modification of a Lévy process. The right-continuous modification of this process is given by times of first exit from closed intervals .

The local time L_t(0) treated as a random function of t is a random process distributed like the process

The local time L_t(x) treated as a random function of x (while t is constant) is a random process described by Ray–Knight theorems in terms of Bessel processes.