We are tasked with finding the stochastic differential equation (SDE) for the function

$G = \log \left( e^{r(T - t)} S \right),$

where $S$ is likely the stochastic process, often representing the price of an asset in financial mathematics. The equation we need to find is of the form:

$dG = a \, dt + b \, dB,$

where $dB$ represents a Wiener process (Brownian motion), and $a$ and $b$ are the drift and diffusion terms, respectively.

Step 1: Simplifying $G$

We start by simplifying the expression for $G$:

$G = \log \left( e^{r(T - t)} S \right) = \log\left(e^{r(T - t)}\right) + \log(S).$

Using the logarithmic properties, this simplifies to:

$G = r(T - t) + \log(S).$

Step 2: Applying Itô's Lemma

Next, we use Itô's Lemma to differentiate $G$. For a function $G(S, t) = r(T - t) + \log(S)$, Itô's Lemma gives:

$dG = \frac{\partial G}{\partial t} \, dt + \frac{\partial G}{\partial S} \, dS + \frac{1}{2} \frac{\partial^2 G}{\partial S^2} \, (dS)^2.$

• $\frac{\partial G}{\partial t} = -r$,
• $\frac{\partial G}{\partial S} = \frac{1}{S}$,
• $\frac{\partial^2 G}{\partial S^2} = -\frac{1}{S^2}$.

Now, we assume that $S$ follows the stochastic process:

$dS = \mu S \, dt + \sigma S \, dB,$

where $\mu$ is the drift and $\sigma$ is the volatility.

Step 3: Substituting into Itô's Lemma

We now substitute these into Itô's Lemma. First, compute $dG$:

$dG = \left( -r \right) dt + \frac{1}{S} \left( \mu S \, dt + \sigma S \, dB \right) + \frac{1}{2} \left( -\frac{1}{S^2} \right) (\sigma S)^2 \, dt.$

Simplifying:

$dG = -r \, dt + \mu \, dt + \sigma \, dB - \frac{1}{2} \sigma^2 \, dt.$

Thus, the final expression for $dG$ is:

$dG = (\mu - r - \frac{1}{2} \sigma^2) dt + \sigma \, dB.$

Conclusion

• The drift term $a$ is: $\mu - r - \frac{1}{2} \sigma^2$,
• The diffusion term $b$ is: $\sigma$.

Therefore, the stochastic differential equation for $G$ is:

$dG = \left( \mu - r - \frac{1}{2} \sigma^2 \right) dt + \sigma \, dB.$

Let me know if you need more details or have questions. Here are five follow-up questions you might consider:

1. How is Itô's Lemma applied in financial mathematics?
2. Can the volatility $\sigma$ change over time in a stochastic model?
3. What is the interpretation of the drift term $\mu - r - \frac{1}{2} \sigma^2$?
4. How would this model change if we assume the process follows a different SDE?
5. What are the practical applications of this SDE in option pricing?

Tip: When working with stochastic processes, Itô’s Lemma is a crucial tool to transform functions of stochastic variables into differential forms!