Let's start with a regular BM with a known drift rate denoted "m", and known variance parameter value, denoted "s":

dX = m dt + s dB

_{t}_{}
In other words,

X

_{t}– X_{0 }= m*t + s*B_{t}
Here, "... + B

_{t}" has a non-trivial meaning. It is not same as adding two numbers or adding two variables, but rather a signal-noise formula.. . It describes a Process, with a non-random, deterministic part and a random part whose variance at time t is equal to s^{2}t
Next, we construct or encounter a random process related but not derived from this BM:

dG/G = m dt + s dB

_{t}
It turns out this process can be exactly described by

G = G

_{0}exp[ (m- ½ s^{2})t + s B_{t}]
Again, the simple-looking "... + s B

_{t}" expression has a non-trivial meaning. It describes a Process, whose log value has a deterministic component, and a random component whose variance is s^{2}t.
Note in the formula above (m- ½ s

^{2}) isn't the drift of GBM process, because left hand size is "dG / G" rather than dG itself.
In contrast, (m- ½ s

^{2}) is a drift rate in the "log" process, i.e. the log of the GBM. This log process is a BM. We can think of the original drift eroded by (½ s^{2})^{2 }]Y_t) dt

^{2}) dt, not "...L_t dt"

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