M.4.2 Brownian Motion & Martingales

Basic Prerequisites: Probability & Measure Theory and 01 · From Zero.

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1. Intuition & Practical Objective

Brownian motion is the canonical fair game in continuous time: it moves, but has no drift, so its best predictor of the future given the present information is the present value. That single property - the martingale property $\mathbb{E}[W(t)\mid \mathcal{F}(s)]=W(s)$ - is the backbone of all no-arbitrage pricing, because “prices under the risk-neutral measure are martingales” is how modern pricing is stated.

The practical objective of this page: understand the three linked faces of BM - (1) Gaussian increments and the covariance $\min (s,t)$; (2) the martingale and Markov properties and their exponential cousin $e^{\sigma W-\frac{1}{2}{\sigma }^{2}t}$; (3) the fact that BM “runs fast,” i.e. quadratic variation $=t$ and infinite first-order variation. These give the toolkit (independent increments, stopping, first-passage) used constantly in barrier options, simulation, and Girsanov.

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2. Mathematical Ground Truth & Derivations

2.1 Definition & covariance (Shreve II Def 3.3.1; Shreve I §13.5–13.7)

A process $W(t),t\ge 0$ is BM if: $W(0)=0$; paths are continuous; increments $W(t_{i+1})-W(t_i)$ are independent and $N(0,t_{i+1}-t_i)$-distributed. Then $\mathrm{Cov}(W(s),W(t))=\min (s,t)$ and finite-dimensional laws are jointly normal with $C_{ij}=t_i\wedge t_j$. Equivalent characterizations (Shreve II Thm 3.3.2): independent normal increments ⇔ jointly-normal-with-that-covariance ⇔ the joint MGF form (3.3.5). Normal increments make “uncorrelated ⇒ independent” true here - a huge simplification that fails for general processes.

2.2 Martingale property (Shreve II Thm 3.3.4)

Using independence of the future increment $W(t)-W(s)$ from $\mathcal{F}(s)$:

$$\mathbb{E}[W(t)\mid \mathcal{F}(s)]=\mathbb{E}[W(s)+(W(t)-W(s))\mid \mathcal{F}(s)]=W(s)+0=W(s).$$

$W$ is also Markov: the only relevant information in $\mathcal{F}(s)$ is $W(s)$ itself, with transition density $p(\tau ,x,y)=\frac{1}{\sqrt{2\pi \tau }}{e}^{-\frac{(y-x{)}^2}{2\tau }},\mathbb{E}[f(W(t))\mid \mathcal{F}(s)]=\int f(y)p(t-s,W(s),y)dy.$ (Shreve II §3.5; via the Independence Lemma, Shreve II Lemma 2.3.4.)

2.3 The exponential martingale (Shreve II Thm 3.6.1; Shreve I Thm 9.41)

For constant $\sigma$,

$$Z(t)=\exp \{\sigma W(t)-\frac{1}{2}{\sigma }^{2}t\}⟹\mathbb{E}[Z(t)\mid \mathcal{F}(s)]=Z(s).$$

Proof. Condition, factor $Z(s)$, and use the increment MGF $\mathbb{E}{e}^{\sigma (W(t)-W(s))}=e^{\frac{1}{2}{\sigma }^2(t-s)}$, which exactly cancels the $\frac{1}{2}{\sigma }^2$ correction. This process is the seed of Girsanov’s change of measure (05 · Girsanov).

2.4 Quadratic variation & the first-passage time (Shreve II §3.4, §3.6)

$$[W,W](T)=\lim \sum _j(\Delta W_j{)}^2=T\Rightarrow (dW{)}^2=dt.$$

Let ${\tau }_m=\min \{t\ge 0:W(t)=m\}$ ($m>0$). Stopping the exponential martingale at ${\tau }_m$ and letting $t\to \infty$ via dominated convergence gives (Shreve II Thm 3.6.2):

$$\mathbb{E}\left[e^{-\alpha {\tau }_m}\right]=e^{-m\sqrt{2\alpha }}(\alpha >0),\mathbb{E}{\tau }_m=\infty .$$

So BM reaches every level with probability 1 but takes, on average, infinitely long - a striking, exactly-quantified tension. The reflection principle (Shreve II eq 3.7.6) turns barrier/threshold probabilities into tail probabilities:

$$\mathbb{P}\{M(t)\ge m,W(t)\le w\}=\mathbb{P}\{W(t)\ge 2m-w\},M(t)=\underset{0\le s\le t}{\max }W(s).$$

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3. Computational Implementation - verify martingale & exponential-martingale in one shot

Simulate many paths, condition on the state at a fixed time $s$, and check that the sample mean of the later value equals the conditioned state (martingale property); then average the exponential martingale and compare to 1. Stdlib only.

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4. Failure Modes & First-Principles Breakdowns

1. “Zero drift” ≠ “stays put.” A martingale can have enormous variance - BM, discounted asset prices, everything fair-but-volatile. The martingale property says only that the conditional mean of the future is the present, never that the path is flat. Treating “martingale” as “predictable at the level” misreads the definition.
2. Uncorrelated ≠ independent (outside Gaussian). Only for jointly normal processes does zero covariance imply independence (Shreve II §2.2, Example). Many intuitions that “work” for BM fail for general processes; the independence cutoff is the Gaussian assumption, not a general fact.
3. $\mathbb{E}{\tau }_m=\infty$ surprises everyone. First-passage is almost-sure but mean-infinite; naive simulation (finite horizon) massively understates hitting probabilities, and pricing barrier options from a short simulation is biased. This underlies the continuity-correction and Brownian-interpolation literature (Glasserman Ch 6).
4. Mismatched variance in simulation. Mis-scale the step variance: drawing $\Delta W\sim N(0,dt)$ from $N(0,\Delta t)$ over a different grid quietly violates $\mathrm{Var}=t-s$ and breaks every downstream QV/martingale check.

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5. References

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

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6. Connected Graph Bridges

• Back: 01 · From Zero
• Forward: 03 · Itô Integral & Doeblin · Index Hub
• Theory: Probability & Measure Theory · No-Arbitrage & Binomial Trees