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The closed-form value of the Itô integral of a standard Brownian motion with respect to itself is $\frac{1}{2}W(T)^2 - \frac{1}{2}T$. This result is derived by applying Itô's lemma to $W^2$, integrating the resulting differential identity over the interval $[0, T]$, and using the initial condition $W(0)=0$.

Conditions: $W(t)$ is a standard Brownian motion.; The integral is evaluated from $0$ to $T$.; The integral is interpreted in the Itô sense.

The closed-form value of the Itô integral of a standard Brownian motion with respect to itself is $\frac{1}{2}W(T)^2 - \frac{1}{2}T$. This result is derived by applying Itô's lemma to $W^2$, integrating the resulting differential identity over the interval $[0, T]$, and using the initial condition $W(0)=0$.

Conditions: $W(t)$ is a standard Brownian motion.; The integral is evaluated from $0$ to $T$.; The integral is interpreted in the Itô sense.