Applying Itô’s Lemma to a Product with a Stochastic Integral

Summary

The document presents an Itô calculus exercise involving a process formed from the square of one Brownian motion and an exponential whose exponent combines another Brownian motion with a stochastic integral. The stated challenge is differentiating the integral with respect to its integrator and its random integrand. The response recommends breaking the expression into intermediate processes instead of applying Itô’s formula to the full expression at once.

In particular, it names the exponent as a separate process and writes the original quantity as a product involving its exponential. This decomposition makes it possible to find the differential of each component and then apply the product and chain rules while accounting for quadratic variation. However, the document stops at the suggested setup: it does not work through the stochastic integral’s differential or give the final differential. It is therefore a concise problem-solving hint, not a complete derivation or a trading strategy.

Key ideas

• The exercise asks for the differential of a process built from Brownian motion and a stochastic integral.
• The response advises defining the exponent as a separate intermediate process.
• The decomposition prepares the expression for the chain and product rules in Itô calculus.
• The document does not provide the full calculation or final differential.

Tags

# Trouble With Applying Ito's Lemma


# Trouble With Applying Ito's Lemma












I am having trouble applying Ito's Formula to the following:

Let $Z_t = W_{1t}^2 e^{W_{1t}+ \int_0^t W_{3s}dW_{2s}}$. Find $dZ_t$. $W_1,W_2,W_3$ are independent Brownian motions.

I know the formula but I am having trouble differentiating the integral with respect to $W_2$ and $W_3$.

## Answer by masi (score 1)

https://quant.stackexchange.com/a/60154

The problem, I think, is that you are trying to do it in one step.

But, if you write, for example $Z_t = W_{1t}^2 e^{Y_t}$, where $Y_t = W_{1t} + \int_0^t W_{3s}dW_{2s}$, you should be able to see how to do it.

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.