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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!

Find a stochastic differential equation for the function G = log(e^(r(T - t)) S).

Explore how to derive the stochastic differential equation for G = log(e^(r(T - t)) S) using Itô's Lemma. This problem applies key concepts in stochastic processes, particularly in financial mathematics. Suitable for students at the undergraduate level.

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