Definition 2.1(The Partial Distribution). Let S be a non-negative stochastic variable, and it follows the distribution of density

(1)

then S is said to have a Partial Distribution, and denotes SεP( µ, σ²). The partial distribution is a kind of truncated normal distribution.

Definition 2.2(The Partial Process).

If stochastic variable S is related to time, i.e. , we have S(t)εP( µ(t), σ²(t)), then the {S(t), tε[0,∞)} is called a partial process.

In general, the stock price varies with time, therefore we have

Assumption 2.1. Let µ(t) be the cost price of stock at the time t, and σ²(t) be the variance of cost price at the time t. If the market prices of stock satisfy the basic assumptions in 2.1, thus suppose that S(t), the market price variable, follows the partial distribution at time t, and denotes S(t)εP( µ(t), σ²(t)).

S(t)εP( µ(t), σ²(t)) can be a stock or the market price of the stock. From [16], we have the following theorem 2.1 and theorem 2.2:

Theorem 2.1. Let S, the market price variable of a stock, follow the partial distribution P( µ, σ²), thus

(a) The expected value E(S) of S, means the average price on market exchange, is as follows

(2)

where

is the average trading profit.

(b) The variance, D(S), of the market price variable S, which means the risk of the market price, is as follows

(3)

Theorem 2.2. For any xε[0,∞], the following equations are correct approximately:

where,

Essentially, the partial distribution describes stock prices in its distribution construction.

Prof. Feng Dai, Prof. Zifu Qin

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