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Stochastic Calculus / Stochastic Integration and Itô's Formula

Stochastic Integration and Itô's Formula: Overview

Classical calculus builds integrals as sums of increments: ∫0tf(s) ds=lim⁡n∑jf(sj)Δsj\int_0^t f(s) \, ds = \lim_n \sum_j f(s_j) \Delta s_j. This works because ff is (usually) Lipschitz or at least bounded variation, so the sum converges as increments shrink.

Brownian motion has infinite total variation, so the naive sum ∑jHsjΔWsj\sum_j H_{s_j} \Delta W_{s_j} doesn't converge as a Stieltjes integral. Instead, the increments ΔWj\Delta W_j contribute randomness that cancels in L2L^2: the Itô isometry E[(∫Hs dWs)2]=E[∫Hs2 ds]\mathbb{E}[(\int H_s \, dW_s)^2] = \mathbb{E}[\int H_s^2 \, ds] shows the stochastic integral is defined as an L2L^2-limit over elementary (step-function) processes.
The central object built in this topic is the Itô integral ∫0tHs dWs\int_0^t H_s \, dW_s, and its key consequence is Itô's formula — the chain rule for stochastic processes, which differs from the classical chain rule by a second-order correction term coming from [W,W]t=t[W,W]_t = t. These tools are the engine behind Black-Scholes, Girsanov's theorem, and the Feynman-Kac representation.

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  3. Finance reading
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