M.4.1 Stochastic Calculus from Zero

Basic Prerequisites: Elementary probability only (a random variable, expectation, the normal distribution) - no measure theory required at this entry point; the measure-theoretic machinery is developed from 02 onward.

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1. Intuition & Practical Objective

The entire subject is one sentence: Brownian motion $(W(t))$ accumulates squared increments at rate 1 - that is, $(dW{)}^2=dt$ - so the “calculus” that describes prices is not the calculus you learned in school. Classical $df=f^{\prime }(x)dx$ works because $d{x}^2$ is negligible. For $W$, $d{W}^2$ is not negligible; instead, over any interval, $\sum (\Delta W{)}^2\to$ (elapsed time) deterministically. This is the single mechanism behind every surprise in quantitative finance (the $\frac{1}{2}{\sigma }^2$ in geometric-Brownian drift, the martingale measure, the Itô correction).

Build it in three “aha”s, from the discrete to the continuous:

1. A random walk is a martingale in disguise. Flip fair coins: $M_k={\sum }_{j=1}^k{X}_j$, $X_j=\pm 1$. It has zero drift, so $\mathbb{E}[M_{k+1}\mid {\mathcal{F}}_k]=M_k$ - a fair game. Scale steps down ($\pm 1/\sqrt{n}$) and speed up ($n$ steps per unit time): $M_k/\sqrt{k}$ looks Gaussian (CLT), and as $n\to \infty$ the interpolated process converges to Brownian motion (Shreve I §13.4; a functional CLT / Donsker).

2. The size of the step determines the smoothness. Because the scaled step is $\pm 1/\sqrt{n}$ (not $\pm 1/n$), squared steps are $1/n$ and don’t vanish when summed over $n$ steps: $\sum$ of (square) $\approx n\cdot (1/n)=1$ per unit time. That is exactly why BM has unbounded first-order variation but finite quadratic variation $T$. A $C^1$ function (step $\propto 1/n$) would have $(dt{)}^2\to 0$; a BM path moves too fast.

3. ”$\int$ with a $dW$” is a different beast. You cannot interpret ${\int }_0^{T}WdW$ as a Riemann–Stieltjes integral, because classical calculus yields $\frac{1}{2}W(T{)}^2$, but the true (Itô) value is $\frac{1}{2}W(T{)}^2-\frac{1}{2}T$. The subtraction is the quadratic-variation correction. The beginner must unlearn the classical answer before learning the Itô one.

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2. Mathematical Ground Truth & Derivations

From random walk to BM (Shreve I §13.3–13.5). Let $M_k$ be the symmetric random walk and define the scaled process

$$W^{(n)}(t)=\frac{1}{\sqrt{n}}{M}_{nt}.$$

The CLT (via MGFs, Shreve I Thm 3.39) gives $M_{nt}/\sqrt{nt}\stackrel{d}{\to }N(0,1)$; by independence of increments the whole interpolated trajectory converges weakly to BM. BM is defined (Shreve II Def 3.3.1) by

$$W(0)=0,W\text{ continuous},W(t)-W(s)\sim N(0,t-s)\text{independent increments}.$$

Key structural facts: covariance $\mathbb{E}[W(s)W(t)]=\min (s,t)$ (Shreve I §13.6), and $W$ is both a martingale and a Markov process (Shreve II Thm 3.3.4, §3.5).

Quadratic variation - the heart. For any partition $\Pi$ of $[0,T]$,

$$[W,W](T)=\underset{\Vert \Pi \Vert \to 0}{\lim }\sum _j(W(t_{j+1})-W(t_j){)}^2=T.$$

Why: each squared increment has mean $t_{j+1}-t_j$ and variance $2(t_{j+1}-t_j{)}^2$; by a law-of-large-numbers the sum collapses to $\sum (t_{j+1}-t_j)=T$ (Shreve II Thm 3.4.3). The informal bookkeeping is then

$$(dW{)}^2=dt,dWdt=0,dtdt=0\text{(Shreve II eqs 3.4.10–13).}$$

Consequence for the drift of a log-price. For geometric Brownian motion $S(t)=S(0)e^{\sigma W(t)+\left(\mu -\frac{1}{2}{\sigma }^2\right)t}$ (Shreve I Thm 15.3), applying Itô’s lemma recovers $dS=\mu Sdt+\sigma SdW$. The $\frac{1}{2}{\sigma }^2$ is forced: exponentiate the differential form and the $(dW{)}^2=dt$ rule generates the correction backwards - the answer is consistent only if the exponent carries $-\frac{1}{2}{\sigma }^2$. Missing it would make $\mathbb{E}[\log (S_T/S_0)]$ grow with $\frac{1}{2}{\sigma }^2$, a pure artifact of nonzero quadratic variation, not a real drift.

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3. Computational Implementation - see the quadratic variation

Simulate BM and watch $\sum (\Delta W{)}^2\to T$ regardless of path; then check that a pushed ($\sigma dW$) version has quadratic variation ${\sigma }^{2}T$ - the direct, quantitative meaning of "$(dW{)}^2=dt$". Stdlib only.

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4. Failure Modes & First-Principles Breakdowns

1. Treating $(dW{)}^2$ as negligible. This is the single most common and most damaging error. Ordinary calculus drops all second-order terms; for $W$ the second-order term is the leading stochastic one. Symptom: your Itô expansions disagree with simulation.
2. Confusing quadratic variation with total variation. BM has infinite first-order variation (${\int }_0^T|dW|=\infty$), but finite quadratic variation. The “infinite” part is why you cannot treat $dW$ as a classical $C^1$ differential, and why the Itô integral needs its own construction (Shreve II §4.2–4.3).
3. Random-walk intuition with wrong scaling. A symmetric walk with step $\pm 1$ (not $\pm 1/\sqrt{n}$) has quadratic variation $\to \infty$ and does not converge to BM. The correct scaling that produces finite quadratic variation is the essence of the construction.
4. Believing $W$ is differentiable. BM is nowhere differentiable; it has no Riemann–Stieltjes derivative. Any “chain rule” for $W$ must go through Itô–Doeblin, not the classical chain rule - see 03 · Itô Integral & Doeblin.

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5. References

• Shreve, Stochastic Calculus for Finance I
• Shreve, Stochastic Calculus for Finance II
• Glasserman, Monte Carlo Methods in Financial Engineering

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6. Connected Graph Bridges

• Base: Probability & Measure Theory
• Continue: 02 · Brownian Motion & Martingales · Index Hub