Resolving Brownian Motion Notation in a Geometric Brownian Motion Solution

Summary

The document addresses an apparent loss of randomness in a video explanation of solving the geometric Brownian motion stochastic differential equation. The question arises when a function initially written in terms of time and Brownian motion appears later with a different argument, making the result look deterministic.

The accepted explanation identifies the notation change as the source of confusion: the solution should retain Brownian motion evaluated at time, alongside the time-dependent drift term. It gives the standard exponential form for the process, with the volatility multiplying the Brownian path and a drift correction involving volatility squared. The key lesson is to distinguish the process being solved from the Brownian input used to express its pathwise solution. The excerpt concerns one derivation and notation issue; it does not develop Ito’s lemma or discuss assumptions and applications of the model.

Key ideas

• A geometric Brownian motion model includes both drift and a Brownian random component.
• Applying Ito’s lemma leads to an exponential solution whose value depends on Brownian motion at time t.
• The apparent determinism comes from confusing the notation for the process with the Brownian input.
• The document resolves a notation issue in one example without developing the model’s broader assumptions.

Tags

# Not clear on an SDE solution example on YouTube


# Not clear on an SDE solution example on YouTube












This video, from about 6 to 12 minutes: https://youtu.be/qdbkvD4N-us

I feel like I’m following him ok, but then at the end his f(t,B(t)) has become an f(t,x) and there is no B(t) in his result, so it appears to be suddenly deterministic? If he has made a mistake, I’d appreciate a correction, or if someone can explain to me how I’m supposed to understand the randomness one would expect at some X of t

## Answer by Magic is in the chain (score 0, accepted)

https://quant.stackexchange.com/a/46891

So for everyone's benefit, this is an MIT OCW video 21.Stochastic Differential Equation, and the professor is explaining the solution of the GBM SDE:

$dX(t)=\mu X(t)dt+\sigma X(t)dB(t)$

He guesses a solution of the form:

$X(t)=f\left(t,B(t)\right)$

And then applies the Ito's lemma etc, and then write the solution as:

$f(t,X)=X_0 e^{\sigma X +\left(\mu-\frac{1}{2}\sigma^2 \right)t}$

So seems like X is used in two different senses, and this is the context to the question. The solution should indeed be:

$X(t)=f\left(t, B(t)\right)=X_0 e^{\sigma B(t) +\left(\mu-\frac{1}{2}\sigma^2 \right)t}$

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.