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In the formula $\frac{1}{2}W(T)^2 - \frac{1}{2}T$, the term $\frac{1}{2}W(T)^2$ depends on the terminal position of the path, while the term $-\frac{1}{2}T$ is a correction that comes from accumulated randomness over time.

Conditions: The formula is the result of evaluating the Itô integral $\int_0^T W\,dW$.; $W(T)$ is the value of the Brownian motion at time $T$.; $T$ is the elapsed time.

In the formula $\frac{1}{2}W(T)^2 - \frac{1}{2}T$, the term $\frac{1}{2}W(T)^2$ depends on the terminal position of the path, while the term $-\frac{1}{2}T$ is a correction that comes from accumulated randomness over time.

Conditions: The formula is the result of evaluating the Itô integral $\int_0^T W\,dW$.; $W(T)$ is the value of the Brownian motion at time $T$.; $T$ is the elapsed time.