B1) BM with dB and drift = 0? the Standard BM, simple but not useless. A cornerstone building block of more complex stoch processes.
G1) GBM with dB and drift = 0? The simplest, purest GBM. The tilting function used in Girsanov theorem. See my blog “GBM + zero drift”
G2) GBM with drift without dB? a deterministic exponential growth path. Example – bank account. Used in BS and pricing.
B2) BM with drift without dB? a linear, non-random walker, not a real BM, useless in stoch.
B3) Now, how do we deal with a BM with both drift and dB? use G1 construct as “tilting function” to effect a change of measure. In a nutshell,
B3 –(G1)–> B1
G3) GBM with drift + dB? The most common stock price model.