DF process and DF structure

Definition 2.3(DF process). If { ξ(t),tε[0,∞)} is a stochastic process, and

ξ (t)εP(µ(t), σ²(t)t)

then { ξ (t),tε[0,∞)} is called a DF process.

Definition 2.4. Let a and b be non-negative constants, If a>0,b=0, we define:

Definition 2.5(DF structure). Let X be the value of an asset related to stock S(t)εP( µ(t), σ²(t)), if and T>t, X_s(t,T)εP(X, D[S(t)](T-t)), then we call X_s(t,T) the DF stochastic structure of X on S(t). X_s(t,T) is called a DF structure of X for short.

When t=T, X_s(t,T)=X. So the actual meaning of the DF structure, X_s(t,T), is a stochastic value which is equal to that of an cash asset X in the future time T under-taking no discount of the interest rate.

Although the stock S(t) has certain connections with DF structure X_s(t,T) in variance, their stochastic movements may have no inevitable relation, so we have

Assumption 2.2. Let X_s(t,T) be the DF structure of X on S(t), thus X_s(t,T) and S(t) are independent of each other.

Prof. Feng Dai, Prof. Zifu Qin