(1) It is a Markov process. Therefore, the current value of the process is all the information needed for future prediction.
⑵ Wiener process has independent increment. The probability distribution of a process changing in any time interval has nothing to do with its probability of changing in any other time interval.
⑶ Its change in any finite time obeys normal distribution, and its variance increases linearly with the length of time interval.
Given the second moment process {w (t), t >；; =0}, if it satisfies
1. has independent increments.
2. For any t>s & gt=0, increment
W (t)-w (s) ~ n (0, σ 2 (t-s)), and s >；; 0
⒊W(0)=0
This process is called Wiener process.
Wiener process is a mathematical model of Brownian motion. Brown, a British botanist, observed the tiny particles floating on the surface of a calm liquid under a microscope and found that they were constantly moving in disorder. This phenomenon was later called Brownian motion. W(t) means from time t=0 to time t>0 (the ordinate can also be discussed), and let W(0)=0. According to the theory put forward by Einstein 1905, this movement of particles is the result of a large number of random and independent molecular collisions. Therefore, the displacement of a particle in the time period (s, t) can be regarded as the algebraic sum of many tiny displacements. Then w (t)-w (.
The distribution of Wiener process increment is only related to the time difference, so it is a homogeneous independent increment process. It is also a normal process. Its distribution is completely determined by its mean function and autocorrelation function. Wiener process is not only a mathematical model of Brownian motion, but also the thermal noise of electronic components at constant temperature can be attributed to Wiener process.
In the BS model of the futures pricing model, both the futures price and the underlying asset price are affected by the same uncertain factor, and both of them follow the same Wiener process.