Applying Itô’s Lemma to Powers of Geometric Brownian Motion

Summary

The document asks how to obtain an analytical expression for a process defined as a power of geometric Brownian motion. It states the geometric Brownian motion model, presents an application of Itô’s lemma for the powered process, and rewrites the dynamics in integral form. The author then attempts to evaluate the ordinary time integral using an endpoint expression and asks how to handle the stochastic integral.

The material is useful as a prompt about stochastic calculus and integral representations, but it does not include a response or a completed solution. In particular, the proposed evaluation of the time integral by multiplying process values by time is not generally valid, and the drift correction shown in the stated dynamics appears to omit a factor of volatility squared. The document provides no worked derivation, assumptions beyond the model statement, or numerical evidence. Readers would need to derive the integrals carefully rather than rely on the attempted endpoint calculation.

Key ideas

• A power of geometric Brownian motion can be analyzed by applying Itô’s lemma.
• The powered process can be written in integral form with both time and stochastic integrals.
• The proposed endpoint calculation for the time integral is not generally valid.
• The stated drift correction appears to require volatility squared.

Tags

# Analytical expression for SDE


# Analytical expression for SDE












I'm trying to find an analytical expression for the following. Suppose $X$ is a geometric Brownian motion, such that: $dX_{t} = \mu X_{t} dt + \sigma X_{t} dW_{t}$. Suppose furthermore, that the process $Y$ is defined by $Y_{t} = X_{t}^{n}$. I have found the dynamics of $Y$ which becomes:

$dY_{t} = X^{n}_{t}\left(n\mu + \frac{1}{2}n(n-1)\sigma\right)dt + nX^{n}_{t}\sigma dW_{t}$.

In integral form we have,

$Y_{t'} = Y_{t} + \left(n\mu + \frac{1}{2}n(n-1)\sigma\right)\int^{t'}_{t}X^{n}_{s}ds + n\sigma\int^{t'}_{t}X^{n}_{s} dW_{s}$,

assuming that $t' > t$. My problem is how to evaluate the two integrals. For the first one I thought about doing the following:

$\int^{t'}_{t}X^{n}_{s}ds = \left[X_{s}s\right]^{t'}_{t} = X_{t'}t' - X_{t}t = (X_{t'}-X_{t})(t'-t)$.

The stochastic integral is causing me some trouble. I would appreciate any kind of help to evaluate both the integrals.

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