What is Ito integral?

What is Ito integral?

Approach: Define as limit of Riemann sums with left-point evaluation.

∫_0^t X_s dW_s defined as limit in probability of ∑ X_{t_i} (W_{t_{i+1}} - W_{t_i}) where partition mesh → 0. Key: evaluator at left endpoint gives martingale.

Follow-up: Why not use right endpoint?

Key concepts: martingale, quadratic variation, stochastic integral.