using dB_t vs B_t in a formula

label – stoch

(dW is a common synonym of dB)


Whenever I see dB, I used to see it as a differential of “B”. But it’s undefined – no such thing as a differential for a Brownian motion!


Actually any dB equation is an integral equation involving a “stochastic integral” which has a non-trivial definition. Greg Lawler spent a lot of time explaining the meaning of a stochastic integral.


I seldom see B itself, rather than dB, used in a formula describing a random process. The one big exception is the exact formula describing a GBM process.


<!–[if gte msEquation 12]>St=S0exptm-s22+Bt s<![endif]–>

Greg confirmed that the regular BM can also be described in B rather than dB:

given <!–[if gte msEquation 12]>dXt=m dt+s dBt<![endif]–>

<!–[if gte msEquation 12]>Xt=X0+m t+s Bt<![endif]–>

This signal-noise formula (see other post) basically says random walk position X at time t is a non-random, predictable position with an chance element superimposed. The chance element is a random variable ~ N(mean 0, variance s2t).


This is actually the most precise description of the random process X(t).


We can also see this as a generalized Brownian motion, expressed using a Standard Brownian motion. Specifically, it adds drift rate and variance parameter to the SBM.


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