Evaluate an Integral of Squared Brownian Motion with Itô's Lemma

Summary

The document asks how to evaluate the stochastic integral of the differential of squared standard Brownian motion. The response first recognizes the integrand as a process, setting the squared Brownian motion equal to a new process. Integrating its differential from the initial time to the endpoint then reduces to the process's endpoint value minus its initial value.

To make that expression explicit, the response recommends applying Itô's lemma to derive the dynamics of the squared process and solving the resulting stochastic differential equation. For standard Brownian motion, this gives the familiar decomposition into a stochastic integral and a time term. The post provides a compact problem-solving route rather than a detailed derivation, and it does not discuss extensions to other stochastic processes or conditions beyond the stated Brownian setup.

Key ideas

• Treat the square of Brownian motion as a process in its own right.
• The integral of a process differential over an interval equals its endpoint increment.
• Use Itô's lemma to derive the dynamics of squared Brownian motion.
• The resulting stochastic differential equation makes the integral's terms explicit.

Tags

# Let $W_t$ denote a standard Brownian motion. Evaluate this integral


# Let $W_t$ denote a standard Brownian motion. Evaluate this integral












$$ \int_{0}^{t}d(W_{u}^2) $$

How can I deal with this kind of problem? If there is no function given to apply Itô's formula.

## Answer by d_797 (score 3)

https://quant.stackexchange.com/a/59467

Write $X_t = W^2_t$, then you are trying to find

$\int_0^t dX_t = X_t - X_0 $.

Now use Ito's Lemma to find the dynamics of $W^2_t$ and try to solve the SDE.

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