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The extra $dt$ term in the Itô expansion of $W^2$ arises from the randomness of the Brownian motion path. This term distinguishes Itô calculus from ordinary deterministic calculus, where the chain rule would only produce a $2W\,dW$ term.

Conditions: $W$ is a standard Brownian motion.; Itô's lemma is being applied to $W^2$.

The extra $dt$ term in the Itô expansion of $W^2$ arises from the randomness of the Brownian motion path. This term distinguishes Itô calculus from ordinary deterministic calculus, where the chain rule would only produce a $2W\,dW$ term.

Conditions: $W$ is a standard Brownian motion.; Itô's lemma is being applied to $W^2$.

Classical integration sums smooth and predictable quantities, like accumulating steady bond interest, and is visually represented by the area under a smooth curve. Stochastic integration sums highly irregular, random movements, such as wild stock price jumps, and is visually depicted as summing areas under a jagged, erratic path.

Conditions: Classical integration applies to deterministic, smooth functions.; Stochastic integration applies to processes with non-smooth, random paths.

Classical integration sums smooth and predictable quantities, like accumulating steady bond interest, and is visually represented by the area under a smooth curve. Stochastic integration sums highly irregular, random movements, such as wild stock price jumps, and is visually depicted as summing areas under a jagged, erratic path.

Conditions: Classical integration applies to deterministic, smooth functions.; Stochastic integration applies to processes with non-smooth, random paths.

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.