M.4.3 The Itô Integral & the Itô–Doeblin Lemma

Basic Prerequisites: 02 · Brownian Motion & Martingales.

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1. Intuition & Practical Objective

Ordinary "${\int }_0^T\Delta dW$" does not exist as a Riemann–Stieltjes integral because $W$ has infinite first-order variation. The Itô integral replaces it with a martingale-building machine: it is defined so that its output is a martingale with mean $0$, and its variance is exactly the object the smile/vol modeler controls - the integrated squared integrand. This is the mathematical object behind “delta-hedging gains,” and the Itô–Doeblin lemma is its chain rule.

Practical objective: by the end of this page you can (1) define $\int \Delta dW$ rigorously and state its three invariant properties (martingale, isometry, quadratic variation); (2) apply Itô–Doeblin to a GBM stock to derive $dS=\mu Sdt+\sigma SdW$ and see where the $\frac{1}{2}{\sigma }^2$ comes from; (3) prove ${\int }_0^{T}WdW=\frac{1}{2}W(T{)}^2-\frac{1}{2}T$, the canonical worked example.

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2. Mathematical Ground Truth & Derivations

2.1 Construction (Shreve II §4.2–4.3; Björk Ch 4)

1. Simple integrands $\Delta (t)={\Delta }_j$ (constant on $[t_j,t_{j+1})$): define $I(t)={\sum }_j{\Delta }_j(W(t_{j+1})-W(t_j))$ - the forward increment. It is a martingale, has mean $0$, and its quadratic variation is ${\sum }_j{\Delta }_j^2\Delta t$.
2. General integrands with $\mathbb{E}{\int }_0^T{\Delta }^2(t)dt<\infty$: approximate $\Delta$ by simple integrands and pass to the $L^2$ limit. The Itô isometry (Thm 4.2.2) makes this well-defined:

$$\mathbb{E}\left[\right({\int }_0^t\Delta dW{)}^2]=\mathbb{E}[{\int }_0^t{\Delta }^2(u)du].$$

Three invariants (Shreve II Thms 4.2.1–4.2.3):

• Martingale: $I(t)={\int }_0^t\Delta dW$ is a martingale, $\mathbb{E}[I(t)\mid \mathcal{F}(s)]=I(s)$, $\mathbb{E}[I(t)]=0$.
• Isometry: $\mathbb{E}[I(t{)}^2]={\int }_0^t\mathbb{E}[{\Delta }^2]du$.
• Quadratic variation: $[I,I](t)={\int }_0^t{\Delta }^{2}du$, i.e. $dI\cdot dI={\Delta }^2(t)dt$.

2.2 Itô–Doeblin formula (Shreve II Thm 4.4.1 & 4.4.6; Björk Thm 4.10)

For $f\in C^{1,2}$, integral form

$$f(T,W(T))=f(0,W(0))+{\int }_0^T{f}_{t}dt+{\int }_0^T{f}_{x}dW+\frac{1}{2}{\int }_0^T{f}_{xx}dt.$$

For an Itô process $dX(t)=\Theta (t)dt+\Delta (t)dW(t)$ (Shreve II Thm 4.4.6):

$$\boxed{df(t,X)=f_{t}dt+f_{x}dX+\frac{1}{2}{f}_{xx}(dX{)}^2,(dX{)}^2={\Delta }^2(t)dt.}$$

Why the $\frac{1}{2}$ term? Taylor-expand to second order; terms with $(dt{)}^2,dtdW\to 0$, but $(dW{)}^2=dt$ survives, contributing $\frac{1}{2}{f}_{xx}{\Delta }^{2}dt$. Itô product rule (Cor 4.6.3): $d(XY)=XdY+YdX+dXdY$.

2.3 Application: GBM (Shreve I §15.3; Ex 4.4.8)

The solution of $dS=\mu Sdt+\sigma SdW$ is

$$S(t)=S(0)\exp \{\sigma W(t)+(\mu -\frac{1}{2}{\sigma }^2\left)t\right\}.$$

Check: apply Itô–Doeblin to $f(t,x)=S(0)e^{\sigma x+(\mu -\frac{1}{2}{\sigma }^2)t}$; the $-\frac{1}{2}{\sigma }^2$ in the exponent cancels the $\frac{1}{2}{\sigma }^2$ from $f_{xx}$, so the total $dt$-coefficient is exactly $\mu S$. Without the correction you would get $\mu +\frac{1}{2}{\sigma }^2$, a pure QV artifact. With $\mu =0$, $S(t)=S(0)+{\int }_0^t\sigma SdW$ is a martingale.

2.4 The canonical integral (Shreve I p-167; Shreve II §4.2)

$${\int }_0^{T}W(u)dW(u)=\frac{1}{2}W(T{)}^2-\frac{1}{2}T.$$

The $-\frac{1}{2}T$ is the Itô correction: naive $\frac{1}{2}W(T{)}^2$ comes from treating $dW$ as $C^1$. This example is the audit test for any stochastic-calculus implementation.

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3. Computational Implementation - verify isometry & Itô–Doeblin numerically

Three stdlib checks: (a) the Itô integral with $\Delta \equiv 1$ equals $W(T)$, mean $\to 0$ and variance $\to T$ (isometry); (b) the discrete Itô sum reproduces $\frac{1}{2}W(T{)}^2-\frac{1}{2}T$ path-by-path; (c) Euler–Maruyama approximates GBM.

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4. Failure Modes & First-Principles Breakdowns

1. Applying the classical chain rule. ${\int }_0^{T}WdW=\frac{1}{2}W(T{)}^2$ is wrong; the correct Itô value subtracts $\frac{1}{2}T$. Any result that doesn’t subtract the quadratic-variation term is an $O(T)$ error.
2. Using backward vs forward increments. The Itô integral uses forward/left-endpoint increments $\Delta (t_j)(W(t_{j+1})-W(t_j))$, which makes it a martingale. Using the backward increment would yield the Stratonovich integral (no martingale property, different answers). Is your $\Delta$ predictable (adapted) - if it looks into the future, the integral is ill-defined and arbitrage appears.
3. Ignoring integrability $\mathbb{E}\int {\Delta }^{2}du<\infty$. Without it the $L^2$ extension and isometry fail; in extreme cases the integral isn’t even a martingale (local-martingale). This is the technical gate for Girsanov’s integrability condition in 05.
4. Reading $d{X}^2={\Delta }^{2}dt$ as optional. It is derived from QV, not assumed; skipping $f_{xx}$ (e.g. pretending the function is affine in $W$ on a curved payoff) drops the single term that drives all hedging P&L.

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5. References

• Shreve, Stochastic Calculus for Finance II
• Shreve, Stochastic Calculus for Finance I
• Björk, Arbitrage Theory in Continuous Time

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6. Connected Graph Bridges

• Back: 02 · BM & Martingales
• Forward: 04 · SDEs & Simulation · Index Hub
• Application: BSM: The PDE & Derivation