Weiner process

Wiener processes are continuous but not differentiable. To generate a particular time history for a set of L random variables, we use some set of M statistically independent Wiener processes W t),..., Wuit), which will herafter be labeled by indices m, = 1M. The number of Weiner processes M used to generate random processes for L coordinates need not always equal L. 

To construct a sequence of ODEs whose solutions converge to that of a corresponding Stratonovich SDE driven by a known set of Weiner processes W f),..., WM f), the random functions /m(f) may be taken as the time derivatives f ( ) = dWm (f)fdt of a sequence of differentiable functions Wm t) that approach the specified Weiner processes Wm t) in the limit e 0, so that... 

Many interest rate models assume that the movement of interest rates over time follows a Weiner process. 

The Weiner process is usually denoted with W although Z and z are also used. For a Weiner process W(i),i > 0 it can be shown that after an infinitesimal... 

A Weiner process is not differentiable but a generalised Weiner process termed an Ito process is differentiable and is described in the form... 

In Merton s (1971) model, the interest-rate process is assumed to be a generalised Weiner process, described by Equation (3.31) ... 

The markets assume that the state variables evolve through a geometric Brownian motion, or Weiner process. It is therefore possible to model their evolution using a stochastic differential equation. The market also assumes that the cash flow stream of assets such as bonds and equities is a function of the state variables. 

Changes or jumps in yield that follow a Weiner process are scaled by the volatility of the stochastic process that drives interest rates, which is denoted by <7. The stochastic process for change in yields is expressed by (4.2). 

The behavior of underlying asset prices follows a geometric Brownian motion, or Weiner process, with a variance rate proportional to the square root of the price. This is stated formally in (8.11). 

In a strict mathematical sense, Brownian motion can be classified as a Weiner process. 

A Weiner process is a class of stochastic processes which satisfies the following conditions [8] ... 

This definition shows that Brownian motion is closely linked to the Gaussian/normal distribution. The formalism of Weiner processes opens stochastic processes to rigorous mathematical analysis and has enabled the use of Weiner processes in the field of stochastic differential equations. Stochastic differential equations are analogs of classical differential equations where... 

Therefore dW t) is a combined three-dimensional vector of three correlated Brownian or Weiner processes, with a correlation of p. The volatility of each bond and the index is therefore also a three-element vector. 

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