One of the Stochastic problems (HW3Q5.2) is revealing (Midterm2015Q6.4 also). We are given
dX = m(X,t) dt + s(X,t) dBt
where m() and s() can be very complicated functions. Now look at this unusual process definition, without Xt :
Appling Ito’s, we notice this function, denoted f(), is a function of t, not a function of Xt, so df/dx = 0. We get
dY = Yt Xt3 dt
So, There’s no dB term so the process Y has a drift only but no variance. However, the drift rate depends on X, which does have a dB component! How do you square the circle? Here are the keys:
Note we are talking about the variance of the Increment over a time interval delta_t
Key — there’s a filtration up to time t. At time t, the value of X and Y are already revealed and not random any more.
Key — variance of the increment is always proportional to delta_t, and the linear factor is the quasi-constant “variance parameter”. Just like instantaneous volatility, this variance parameter is assumed to be slow-changing.
(Ditto for the drift rate..)
In this case, the variance parameter is 0. The increment over the next interval has only a drift element, without a random element.
Therefore, the revealed, realized values of X and Y determine the drift rate over the Next interval of delta_t
