Stochastic Processes: Brownian Motion, Diffusion, and Hitting Times

Brownian motion is the continuous-time reference model for accumulated small random shocks. It connects random walks, diffusion equations, barrier events, statistical tests, and stochastic integrals. Its paths are continuous yet so irregular that ordinary differential calculus no longer applies unchanged.

This lesson follows Stochastic Processes: Martingales and Markov Chains and Probability Theory for Finance. It expands the supplied lecture outline; numbered reference fragments without a bibliography are omitted. Numerical examples are hypothetical, and empirical return-fit claims are distinguished from mathematical identities.

Learning goals and notation

Define standard and volatility-scaled Brownian motion consistently.
Connect its Gaussian transition density to diffusion and random-walk limits.
Calculate maximum and first-passage probabilities using reflection.
Distinguish reflection, absorption, bridges, and random-time sampling.
Solve continuous gambler’s ruin with and without drift.
Explain quadratic variation and the conditions for a trading integral to be a martingale.

Throughout, Wₜ is standard Brownian motion, with Var(Wₜ)=t. σWₜ has variance σ²t. The diffusion with initial state x and drift μ is Xₜ=x+μt+σWₜ, where σ>0. Time, barriers, drift, and volatility must use compatible units. We reserve Φ for the standard normal CDF.

1. A short historical route

Milestone	Contribution
Robert Brown, 1827	Observed irregular microscopic motion while examining particles associated with pollen in water
Louis Bachelier, 1900	Used a Brownian-type model in his theory of financial speculation, including French market contracts
Albert Einstein, 1905	Connected particle displacement and molecular motion through diffusion theory
Norbert Wiener, 1923	Established a rigorous measure-theoretic construction on continuous paths

These are milestones rather than an exhaustive history. The mathematical Wiener process is an idealized limit, not a literal description of molecular motion at every physical time scale. University accounts discuss Brown and Bachelier’s historical context and Wiener’s path-space construction.

2. Definition, moments, and information

Standard Brownian motion {Wₜ:t≥0} satisfies:

W₀=0 almost surely.
Its sample paths are continuous almost surely.
Wₜ−Wₛ∼N(0,t−s) for 0≤s<t.
Increments over disjoint time intervals are independent.

The increment law depends only on interval length, so increments are stationary. The level process is not covariance stationary, because its variance grows with time.

It follows that E[Wₜ]=0, Var(Wₜ)=t, and Cov(Wₛ,Wₜ)=min(s,t). For s≤t, write Wₜ=Wₛ+(Wₜ−Wₛ); independence makes the covariance with the new increment zero, leaving Var(Wₛ)=s.

Conditional on Wₛ=x, the future value Wₜ has distribution N(x,t−s). This is the Markov property. Under a Brownian filtration ℱₜ, future increments are independent of ℱₛ, so E[Wₜ | ℱₛ]=Wₛ: W is also a martingale. Enlarging the information to reveal future values can destroy that property.

Example: for Bₜ=2Wₜ, Var(B₀.₂₅)=4×0.25=1, and Cov(B₀.₂₅,B₁)=1. Calling B “standard” while also assigning an arbitrary σ² would mix two conventions.

3. Diffusion and the heat equation

For Xₜ=x+σ(Wₜ−Wₛ) after starting at x at time s, the transition density for t>s is:

p(y,t | x,s)=[1/√(2πσ²(t−s))] exp[−(y−x)²/(2σ²(t−s))].

Differentiating this Gaussian kernel gives:

∂p/∂t=(σ²/2)∂²p/∂y².

This is the constant-coefficient, zero-drift diffusion equation: a special case of the Fokker–Planck or Kolmogorov forward equation, mathematically equivalent to a heat equation with diffusivity σ²/2. General Fokker–Planck equations can also have drift and state-dependent diffusion terms; the names do not make every such PDE identical to this simplest form.

The initial condition is a unit point mass at x as t decreases to s. On the whole real line, the Gaussian density integrates to one. Integrating the PDE preserves mass when the boundary flux vanishes appropriately. Initial and boundary conditions matter: an absorbing boundary can remove mass from the surviving interior density, with the remaining probability accounted for at the absorbing state if it is included.

With drift μ, the forward equation becomes ∂p/∂t=−μ∂p/∂y+(σ²/2)∂²p/∂y². This describes the evolution of a transition density. Later, a related differential operator will describe a hitting probability as a function of its starting point.

4. From random walks to a continuous process

Let ξ₁,ξ₂,… be i.i.d., with mean zero and variance one, and define Sₖ=Σᵢ≤kξᵢ. The scaled step process is:

W⁽ⁿ⁾(t)=S_floor(nt)/√n.

Donsker’s invariance principle says this process converges in distribution as a random function to standard Brownian motion on a fixed finite horizon. The step version is naturally treated in a suitable space of right-continuous paths with the Skorokhod topology; linearly interpolated walks give a continuous-path formulation. This is stronger than a central limit theorem at a single date because it controls the joint path behavior.

The limit is the same for fair ±1 steps or other i.i.d. zero-mean, unit-variance steps. For variance v, divide by √(nv) to obtain a standard limit, or retain the appropriate volatility scale. Dependent increments or infinite-variance tails require different theorems and can have different limits. Universality holds under hypotheses, not for every market process.

Scaling and simulation

Brownian scaling gives W_ct distributed as √c Wₜ as processes. Over a grid interval Δt, simulate an increment as √Δt Z, where Z∼N(0,1), or σ√Δt Z for scaled motion. Using Δt Z would make the accumulated variance vanish as the grid is refined.

Grid simulation gives correct Gaussian values at the sampled dates, but it can miss a barrier crossed between dates. Continuous monitoring and discrete monitoring are different contracts or events, even when the sample grid looks fine.

5. Reflection principle and the running maximum

For standard W, define Mₜ=max₀≤u≤t Wᵤ and the first hit of a>0:

τ_a=inf{u≥0:Wᵤ=a}.

After the stopping time τ_a, replace the path by 2a−Wᵤ, leaving it unchanged beforehand. Brownian symmetry and the strong Markov property imply that the reflected process has the same law as the original one. This reflection argument does not extend unchanged to arbitrary drifted processes.

Paths that hit a and end below it are paired with paths ending above a. Therefore, for t>0:

P(Mₜ≥a)=2P(Wₜ≥a)=2[1−Φ(a/√t)].

For σW, replace √t with σ√t. At positive barriers, strict and non-strict inequalities give the same probability because the distributions have no atom there. See MIT’s Brownian motion lecture for the reflection argument.

Worked barrier probability

For standard W and horizon t=1, a barrier a=2 has terminal exceedance probability about 2.275%, but the probability of reaching it at any point before time 1 is about 4.550%. Finishing below a barrier does not mean it was never crossed.

This difference matters for barrier options, stop rules, and credit triggers. Applying the formula to an actual asset requires the right process, scale, and barrier definition. A positive stock modeled by geometric Brownian motion needs a log transformation and drift treatment rather than direct use of a zero-drift price formula.

6. First hitting times: finite almost surely, infinite mean

Since {τ_a≤t}={Mₜ≥a}, differentiate the CDF to obtain the standard-Brownian first-passage density:

f_τ(t)=a/[√(2π)t³ᐟ²] exp[−a²/(2t)], for t>0.

For σW reaching a, it becomes a/[σ√(2π)t³ᐟ²] exp[−a²/(2σ²t)]. This is a Lévy first-passage law. It is unimodal, with mode a²/(3σ²), but its right tail is very long.

As t tends to infinity, the hitting CDF tends to one, so P(τ_a<∞)=1 for zero drift. Yet E[τ_a]=∞. For example, a=2 and σ=1 give a density mode at 4/3, not a finite average waiting time. “Eventually with probability one” and “finite expected time” are different statements.

This also explains an optional-stopping pitfall. W starts at zero, but W_τ_a=a. One cannot conclude E[W_τ_a]=0 merely because τ_a is almost surely finite. The stopped family lacks the conditions needed to exchange the unbounded-time limit and expectation; the discrete-time lecture gives the corresponding random-walk example.

7. Reflection, absorption, and bridges

Reflected Brownian motion

Starting at zero, Rₜ=abs(Wₜ) has reflected Brownian motion law on the nonnegative half-line. At a fixed time it has a half-normal distribution:

E[Rₜ]=√(2t/π),

Var(Rₜ)=(1−2/π)t.

For σabs(W), multiply the mean by σ and the variance by σ². Reflection prevents negative states, but R is not a zero-drift martingale: the boundary creates an upward push represented more formally through local time. Folding a sampled path this way is a particular construction, not a claim that a reflected process has unchanged independent Gaussian increments.

Absorbed Brownian motion

Start at x>0, define Xₜ=x+σWₜ, and let τ₀=inf{t≥0:Xₜ=0}. The absorbed process is Aₜ=X_min(t,τ₀). It remains zero after its first hit. Starting an absorbed process at zero would make it identically zero, so the positive initial state is essential.

At later times its distribution has an atom at zero plus a continuous surviving part on positive states. Absorption can be a stylized default mechanism, but a corporate model also needs a meaningful asset/liability barrier, recovery, and economic dynamics. The mathematical boundary alone does not establish an empirical bankruptcy model.

Brownian bridge

On [0,T], define:

βₜ=Wₜ−(t/T)W_T.

It is pinned at β₀=β_T=0, with E[βₜ]=0, Var(βₜ)=t(1−t/T), and Cov(βₛ,βₜ)=min(s,t)−st/T. For T=1 these become t(1−t) and min(s,t)−st.

At t=0.5 for a unit bridge, variance is 0.25 and standard deviation is 0.5. Unlike Brownian motion, the bridge has correlated increments and is pulled toward its endpoint. It is Gaussian and Markov in its own natural information structure, but is not a martingale. The displayed construction uses future W_T, so it is not adapted to the original W filtration before T.

Its law corresponds to Brownian motion conditioned to finish at zero, understood through Gaussian conditional distributions rather than conditioning by an ordinary positive-probability event. For i.i.d. observations with continuous known CDF F, the empirical process √n(Fₙ(x)−F(x)) converges to β(F(x)) on the unit-bridge scale under the standard functional limit theorem. Uniform data give β(u) directly. Estimating F’s parameters or allowing dependent observations changes standard goodness-of-fit reference laws. Berkeley’s empirical-process notes explain the bridge limit.

8. Random time and Laplace marginals

Let E be an independent exponential time with rate λ>0. Conditional on E=t, W_E is N(0,t). Its characteristic function is:

E[exp(iuW_E)]=E[exp(−u²E/2)]=λ/(λ+u²/2).

This is a centered Laplace distribution with scale b=1/√(2λ) and density exp(−abs(x)/b)/(2b). For σW_E, b=σ/√(2λ), and variance is σ²/λ. At λ=2 and σ=1, b=0.5 and variance is 0.5.

This exact identity requires the independence and distributional assumptions. To form a time-evolving process, the clock must be specified consistently. Sampling W at cumulative independent exponential waiting times gives independent Laplace increments in the discrete observation index. Independently drawing unrelated clock values at every date does not guarantee an increasing time change.

A continuous-time subordinator is an increasing process with stationary independent increments. A gamma clock gives a Brownian time-change model with Laplace marginals at a suitable unit shape, but increments at arbitrary clock lengths are not all the identical Laplace law. Such time changes can also produce jumps, unlike ordinary Brownian paths.

Laplace tails decay exponentially and are heavier than Gaussian tails, but they are not power-law heavy tails in the usual technical sense. The lecture’s claim of a substantially better S&P 500 fit is an empirical hypothesis requiring its dataset, horizon, estimation, and comparison metrics. A better marginal fit alone would not capture volatility clustering, dependence, or option-pricing consistency.

9. Drifted Brownian motion and its local moments

For Xₜ=x+μt+σWₜ, a small step h has increment ΔX=μh+σΔW, where ΔW∼N(0,h). Therefore:

E[ΔX]=μh; Var(ΔX)=σ²h; E[(ΔX)²]=σ²h+μ²h².

For powers c>2, E[abs(ΔX)ᶜ]=O(hᶜᐟ²)=o(h) as h decreases to zero, for fixed μ and σ. Absolute moments avoid the ambiguity of noninteger powers of negative increments. Higher-order expectations become negligible relative to h, but individual path increments are not literally deterministic square roots of time.

The operator ℒf=μf′+(σ²/2)f″ is the generator of this diffusion. A second-order term survives because the increment’s second moment is order h rather than order h².

10. Continuous gambler’s ruin

Start X at x∈(a,b), and stop when it first reaches a or b. Let u(x)=Pₓ(X_τ=b). With constant μ, σ>0, the Markov property and a short-step expansion imply that inside the interval:

μu′(x)+(σ²/2)u″(x)=0, with u(a)=0, u(b)=1.

This is a backward equation in the starting point x. It differs in purpose from the forward equation evolving a density in time.

For zero drift:

u(x)=(x−a)/(b−a).

For μ≠0:

u(x)=[1−exp(−2μ(x−a)/σ²)]/[1−exp(−2μ(b−a)/σ²)].

This is equivalent to the exponential ratio in the supplied outline. Its μ→0 limit gives the linear zero-drift result. Numerically, near-zero μ requires care with subtracting nearly equal exponentials.

Worked exit probabilities and waiting time

Take a=0, b=10, and x=4. Without drift, the upper-hit probability is 40%. With μ=0.1 and σ=1 in compatible units:

u(4)=[1−exp(−0.8)]/[1−exp(−2)]≈63.69%.

With zero drift, expected exit time from this bounded interval is (x−a)(b−x)/σ². At σ=1 it is 24 time units. It is finite because either boundary ends the experiment, unlike the one-sided zero-drift hitting problem’s infinite mean.

11. Continuous but nowhere differentiable

Brownian paths are almost surely continuous everywhere and differentiable nowhere. The scale calculation:

(Wₜ₊ₕ−Wₜ)/h∼N(0,1/h)

shows why an ordinary finite instantaneous slope is implausible: its variance grows without bound as h shrinks. This is useful intuition, not by itself a complete proof of nowhere differentiability simultaneously at all times. The full path theorem requires stronger arguments.

The path also has infinite total variation on every nontrivial finite interval, almost surely. Its squared increments, however, have a remarkably stable accumulated limit.

12. Quadratic variation and the Itô correction

For a deterministic partition 0=t₀<…<tₙ=T, define:

Q_Π=Σⱼ(W_tⱼ−W_tⱼ₋₁)².

Gaussian independent increments give E[Q_Π]=T and Var(Q_Π)=2Σⱼ(Δtⱼ)²≤2T×mesh(Π). As the mesh tends to zero, Q_Π→T in mean square, hence in probability.

Along standard refining sequences such as dyadic partitions, convergence also holds almost surely. One should not assert a single almost-sure statement covering every arbitrary, potentially path-selected partition merely from the deterministic-partition calculation. The convergence mode and allowed partitions are part of the theorem.

For Xₜ=x+μt+σWₜ, quadratic variation over [0,T] is σ²T; the finite-variation drift contributes zero. This is often written d[W]ₜ=dt. The shorthand (dW)²=dt describes an accumulation limit, not a literal identity for every infinitesimal random move.

For smooth f, Itô’s formula reads:

df(Xₜ)=[μf′(Xₜ)+½σ²f″(Xₜ)]dt+σf′(Xₜ)dWₜ.

For f(x)=x² and X=W, it gives d(Wₜ²)=2WₜdWₜ+dt. Thus Wₜ²−t is a martingale, the continuous-time counterpart of the squared random walk minus accumulated variance. The second derivative term is exactly what an ordinary chain rule would miss.

13. Trading integrals: adaptedness is not enough

A filtration ℱₜ records available information. For a suitable predictable or progressively measurable integrand Y, the Itô integral Gₜ=∫₀ᵗYₛdWₛ models accumulated exposure to Brownian shocks.

A useful sufficient condition is E[∫₀ᵀYₛ²ds]<∞ for every finite T. Then G is a square-integrable martingale, E[Gₜ]=0, and the Itô isometry gives E[Gₜ²]=E[∫₀ᵗYₛ²ds]. Under weaker local conditions, one may obtain only a local martingale. Being adapted alone guarantees neither an integral’s existence nor a true martingale property. See the University of Chicago stochastic-integral introduction for construction and integrability issues.

For a constant exposure Y=3, Gₜ=3Wₜ and Var(G_T)=9T. For Y=W, ∫₀ᵀWₛdWₛ=(W_T²−T)/2, with zero mean. Future-aware choices such as using W_T to choose today’s position violate the usual non-anticipation requirement.

In finance, the underlying integrator is a modeled discounted price or gains process under an appropriate probability measure. Predictable trading, financing, costs, and admissibility must be specified before interpreting a zero expected stochastic integral as a no-arbitrage statement. Brownian mathematics alone does not say all real-world strategies have zero expected return.

14. Practice questions

What is Var(3W₂), and what is Cov(W₁,W₄)?
If the time step shrinks from 1 to 0.01, by what factor does the standard deviation of a Brownian increment shrink?
For standard W, how does P(M₁≥2) compare with P(W₁≥2)?
Does almost sure finiteness of a one-sided hitting time imply a finite expected hitting time?
What is the variance of a unit Brownian bridge at t=0.25?
For a zero-drift diffusion starting at 3 between barriers 0 and 12, what is the upper-hit probability? If σ=2, what is the expected exit time?
What is the quadratic variation of μt+2Wₜ over [0,3]?
Does an adapted trading exposure automatically make its stochastic integral a true martingale?

Answers

18 and 1, respectively.
By a factor of 0.1, since standard deviation scales as √Δt.
The maximum probability is twice the terminal exceedance probability: about 4.55% versus 2.275%.
No. A positive one-sided barrier is reached almost surely by zero-drift Brownian motion, but its mean hitting time is infinite.
0.25×0.75=0.1875.
3/12=25%; expected exit time 3×9/4=6.75 in the chosen time units.
4×3=12. The drift does not contribute.
No. Appropriate measurability and integrability conditions are also needed, and weaker conditions can yield only a local martingale.