How does the video connect Itô integrals to finance?

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.

The Itô integral is defined as the limit of a sum where time is divided into small intervals. In each interval, the value of the stochastic process at the beginning of the interval is multiplied by the increment of the process over that interval.

Conditions: The integrator is a martingale, typically Brownian motion.; The integrand must be adapted to the filtration of the integrator.

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.

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.