P91 of Greg Lawler’s lecture notes states that the most basic, simple SDE

dXt = A_{t} dBt (1)

can be intuitively interpreted this way — X_{t} is like a process that at time t evolves like a BM with zero drift and variance A_{t}^{2}.

In order to make sense of it, let’s back track a bit. A regular BM with 0 drift and variance_parameter = 33 is a random walker. At any time like 64 days after the start (assuming days to be the unit of time), the walker still has 0 drift and variance_param=33. The position of this walker is a random variable ~ N(0, 64*33). However, If we look at the next interval from time 64 to 64.01, the BM’s increment is a different random variable ~ N(0, 0.01*33).

This is a process with constant variance parameter. In contrast, our Xt process has a … time-varying variance parameter! This random walker at time 64 is also a BM walker, with 0 drift, but variance_param= A_{t}^{2}. If we look at the interval from time 64 to 64.01, (due to slow-changing A_{t}), the BM’s increment is a random variable ~ N(0, 0.01A_{t}^{2}).

Actually, the LHS “dXt” represents that __signed__ increment. As such, it is a random variable ~ N(0, dt A_{t}^{2}).

Formula (1) is another signal-noise formula, but without a signal. It precisely describes the distribution of the next increment. This is as precise as possible.

Note BS-E is a PDE not a SDE, because BS-E has no dB or dW term.