Why Itô’s Lemma Adds a Variance Correction to Geometric Brownian Motion
Summary
The document resolves a discrepancy between two derivations of geometric Brownian motion. For an Itô process with proportional drift and diffusion, taking the logarithm requires Itô’s lemma: the quadratic variation contributes a correction of negative one-half the variance rate to the drift of the log price. Integrating this corrected differential gives the familiar lognormal solution, with the correction in the exponent.
The tempting alternative treats the differential of the logarithm as simply the asset’s relative differential. That substitution omits the quadratic-variation term and is therefore invalid under Itô calculus. A second response clarifies that the uncorrected chain-rule derivation corresponds instead to a Stratonovich stochastic differential equation, where the ordinary chain rule applies. The key caveat is that stochastic differential equations depend on their interpretation; switching between Itô and Stratonovich forms changes drift terms. The exchange explains the conceptual distinction but does not discuss parameter estimation, empirical evidence, or applications beyond this derivation.
Key ideas
- For an Itô process, the differential of the logarithm includes a quadratic-variation correction.
- The log-price drift in geometric Brownian motion is reduced by one-half the variance rate.
- Ordinary calculus cannot be applied directly to Itô differentials as if they were deterministic differentials.
- The uncorrected derivation is consistent with a Stratonovich interpretation, which uses the ordinary chain rule.
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# A question on Ito
# A question on Ito
If we know the dynamics of $S$, then we can estimate the value of $S$ at a time point, $t$. Here, I have a question concerning how to solve for $S_t$ by Itô because I obtained different results by different approaches.
For a geometric Brownian motion: $$dS_t=S_t μ dt+S_t σdW_t,$$ $$\frac{dS_t}{S_t} =μ dt+σdW_t,$$ and, in fact we have, $$\frac{dS_t}{S_t} =d\ln(S_t).$$ If we make $Z=d\ln(S_t)$, then, $$dZ=\frac{\partial Z}{\partial t} dt+\frac{\partial Z}{\partial S_t} dS_t+ \frac{1}{2} \frac{\partial^2 Z}{\partial S_t^2} (dS_t)^2=(μ-\frac{1}{2} σ^2 )dt+σdW_t,$$
$$Z_t= Z_0+\left(μ- \frac{σ^2}{2} \right) \int_0^tds+σ\int_0^tdW_s,$$ $$\ln(S_t )=\ln(S_0 )+(μ-\frac{1}{2} σ^2 )dt+σW_t,$$ $$S_t=S_0 \cdot e^\left((μ- \frac{1}{2} σ^2 )dt+σW_t \right).$$
However, if I use another approach, then I get the different result. Since we have $\frac{dS_t}{S_t} =d\ln(S_t)$ then, $$d\ln(S_t )=μdt+σdW_t$$ and \begin{align} \ln(S_t)&=\ln(S_0)+μ\int_0^tds+σ\int_0^t dW_s \\ &=\ln(S_0)+μt + σW_t, \end{align} $$S_t=S_0 \cdot e^{(μt+σW_t)}$$
I think both approaches are correct. But why are the results distinct?
## Answer by SRKX (score 9, accepted)
https://quant.stackexchange.com/a/8571
The part where you say that
$$\frac{dS_t}{S_t} = d\ln(S_t)$$
is wrong, because $S$ is a stochastic variable.
This is exactly what Itô tells you with his formula that you apply right do compute your $dZ$.
The difference comes from the quadratic variation of the process $S$ which you express as $(dS)^2$. If you don't add this term when the variable are stochastic, your derivation is wrong.
## Answer by horchler (score 1)
https://quant.stackexchange.com/a/8578
You're right, both approaches are correct in a way (but I think you have some messiness in how you written everything out as @SRKX pointed out) ... but under different formulations of stochastic calculus. Your second answer is the solution for the Stratonovich SDE:
$$\text{d}S_t = \mu S_t \text{d}t + \sigma S_t \circ \text{d}W_t,$$
Under the Stratonovich interpretation the generic calculus chain rule applies, so you don't need a form of Itô's formula/lemma, i.e., the chain rule for Itô stochastic calculus. These chain rules are used to remove state dependence (in your case, dependence on $S_t$) from the stochastic integrals that correspond to the SDEs, allowing them to be solved.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.