s<t<t+u Suppose, meet the following conditions for a stochastic process{Wt} to be called a Wiener process.
(i): W0=0
(ii): (Wt+u−Wt)⊥Ws
(iii): (Wt+u−Wt)∼N(0,u)
(iv): Sample paths of Wt are almost surely continuous.
The Wiener process has the following properties:
[1]: Wt∼N(0,t)
[2]: E(Wt)=0
[3]: Var(Wt)=t
[4]: cov(Wt,Ws)=21(∣t∣+∣s∣−∣t−s∣)=min{t,s}
Consider a very short infinitesimal interval [t,t+dt] of the Wiener process{Wt}t≥0. Although it is not an analytically rigorous assumption, let dt>0 be (dt)1/2>0, and small enough to be considered as (dt)k=0 for any k=2,3,⋯. In algebraic terms, under this assumption, we treat α+βdt as an infinitesimal.
Now, suppose dWt:=Wt+dt−Wt, and we want to consider the multiplications between dt and dWt.
Part 1. (dt)2=0
Of course, dt>0 holds, but let’s assume dt is small enough such that (dt)2=0.
Part 2. dtdWt=0
Since Wt is assumed to follow a Wiener process, it adheres to a normal distribution as dWt∼N(0,dt2).
The expectation of dtdWt brings the constant dt outside,
E(dtdWt)=dtE(dWt)=dt⋅0=0
Similarly, the variance of dtdWt also removes the squared dt,
Var(dtdWt)=(dt)2Var(dWt)=0⋅Var(dWt)=0
Thus, since dtdWt has a variance of 0 and an expectation of 0, it must be
dtdWt=dWtdt=0
Part 3. (dWt)2=dt
From Var(dWt)=dt, computing the expectation of dWt⋅dWt,
dt===Var(dWt)E((dWt)2)−[E(dWt)]2E((dWt)2)−02
Thus, it is E((dWt)2)=dt.
Assuming dWt∼N(0,dt2), the variable dWt follows a normal distribution with zero mean, and the variance of (dWt)2 follows from the expectation of the squared random variable E(X2n)=(2n−1)!!σ2n,
Var((dWt)2)======E([(dWt)2]2)−[E((dWt)2)]2E((dWt)2⋅2)−[dt]2(2⋅2−1)dt2⋅2−dt23dt2−dt22dt20
Therefore, (dWt)2 should carry an expected value of dt followed by
(dWt)2=dt