Simulating and Analyzing Brownian Motion: The Continuous Evolution of Stochastic Processes

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Introduction: The Evolution of Stochastic Processes

In our previous studies, we explored:

  • Random Walks: discrete states, discrete time.
  • Poisson Processes: discrete states, continuous time.

Now we complete the picture with Brownian Motion (also called the Wiener Process): continuous states, continuous time. The connection between these processes is deep: Brownian motion emerges as the limiting case of a random walk when we take smaller and smaller steps at faster and faster rates, with the step sizes following a normal distribution. This is guaranteed by the Central Limit Theorem, just as the Poisson distribution emerged as the limit of binomial distributions.

Brownian motion represents the natural evolution from discrete random walks to continuous stochastic processes. While random walks operate on discrete steps and Poisson processes count discrete events over continuous time, Brownian motion introduces both continuous state space and continuous time.

Simulation Approach

Time Discretization and Continuous Approximation

The challenge in simulating Brownian motion is that, theoretically, the process evolves continuously with infinitely many random movements. Our practical approach discretizes time while preserving the essential continuous nature:

  1. Partition the time interval: Divide [0, T] into n subintervals of equal length Δt = T/n
  2. Normal increments: At each time step, the process moves by a random amount drawn from a normal distribution: Yt = Y(t-1) + σ√Δt · Z where Z ~ N(0,1) is a standard normal random variable, so that the increment has distribution N(0, σ²Δt).
  3. Scaling property: The Δt factor is crucial – it ensures that as we refine our time grid (n → ∞, Δt → 0), the process converges to true Brownian motion

Generating Normal Random Variables: The Box-Muller Transform

Unlike the Poisson process simulation where we used Bernoulli trials, Brownian motion requires generating values from a normal distribution. Since most programming languages provide only uniform random numbers U ~ Uniform(0,1), we need a transformation method.

The Box-Muller transform is an elegant algorithm that converts pairs of uniform random variables into pairs of independent standard normal variables:

Given U₁, U₂ ~ Uniform(0,1), we obtain:

  • Z₁ = √(-2 ln U₁) × cos(2π U₂)
  • Z₂ = √(-2 ln U₁) × sin(2π U₂)

where Z₁, Z₂ ~ N(0,1) are independent standard normal variables.

To generate N(0, σ²), we multiply by σ: X = σ · Z. This gives a random variable with mean 0 and variance σ².

Basic Cases and Their Interpretation

Case 1: Low Volatility (σ = 0.5) – Smooth, Stable Paths

When volatility is low (σ = 0.5), Brownian motion shows smooth, gentle fluctuations. The paths wander slowly and stay relatively close to their starting point. The variance grows linearly as 0.25t, meaning the process spreads out slowly over time. This resembles stable markets or large particles in viscous fluids. Unlike Poisson processes that jump between states, Brownian motion never stops moving – it just moves very little when volatility is low.

Case 2: Medium Volatility (σ = 2) – Balanced Random Motion

When volatility is medium (σ = 2), Brownian motion moves like a “drunkard’s walk.” The path moves randomly—sometimes going far from the start, sometimes coming back near zero. The movement is always smooth and continuous, with no sudden jumps. Key features:
Normal distribution: At any moment, the position follows a bell curve
No memory: Each step is independent of previous steps
Spreads out over time: The typical distance from start grows with the square root of time

Case 3: High Volatility (σ = 5) – Extreme Fluctuations

When volatility is high (σ = 5), Brownian motion becomes very wild and unpredictable. The path swings dramatically—it can move far from the starting point very quickly. Even though the movement is chaotic, it never jumps; the path remains continuous.

Theoretical Foundation

Properties of Brownian Motion

Brownian motion B(t) is defined by four fundamental properties that distinguish it from other stochastic processes:

  1. B(0) = 0: The process starts at the origin (or any fixed point if we add a constant)
  2. Independent increments: For any times 0 ≤ t₁ < t₂ < t₃ < t₄, the increments B(t₂) – B(t₁) and B(t₄) – B(t₃) are independent random variables. Past movements do not influence future movements.
  3. Stationary increments: The distribution of B(t+s) – B(s) depends only on t, not on s. The process is time-homogeneous, its statistical behavior doesn’t change over time. It means that the increments do not depend on the starting point, but only on the length of the time interval. NOTE: B(t+s) - B(s) represents the increment of the Brownian motion process over the time interval of length t starting at time s. In other words, it is: the change in the process between time s and time s+t calculated as (Final position) - (Initial position).
  4. Normal distribution: For any t > 0, the increment B(t) – B(s) follows a normal distribution: B(t) – B(s) ~ N(0, σ²(t-s)) [the change over any time interval is normally distributed]. This is the signature property that distinguishes Brownian motion from other continuous processes.
  5. Continuous paths: With probability 1, the function t ↦ B(t) is continuous. The process never jumps, unlike Poisson processes.

The Parameter σ: Volatility

Mathematical Meaning

The parameter σ quantifies the volatility or diffusion coefficient of the process. It controls how rapidly the process spreads out over time:

  • Variance at time t: Var[B(t)] = σ²t – the variance grows linearly with time
  • Standard deviation: SD[B(t)] = σ√t – the standard deviation grows as the square root of time (slower than variance)
  • Quadratic variation: Over interval [0,T], the “accumulated squared changes” converge to σ²T as Δt → 0

Practical Interpretation

The value of σ determines the observable behavior:

  • Low volatility (σ = 0.1-1): Stable systems with small random perturbations. The process stays close to its starting point for extended periods.
  • Medium volatility (σ = 1-5): Balanced random motion, ideal for studying theoretical properties. Clear wandering behavior without extreme outliers.
  • High volatility (σ > 5): Highly unstable systems dominated by randomness. Large deviations occur rapidly, making long-term predictions unreliable.

Connection to Random Walk

Brownian motion is the continuous limit of a scaled random walk. Consider a random walk where at each time step Δt:

  • Move up by +√Δt with probability 1/2
  • Move down by -√Δt with probability 1/2

As Δt → 0 and the number of steps n → ∞ (keeping T = nΔt fixed), this random walk converges to Brownian motion with σ = 1. This is a consequence of the Donsker Invariance Principle (also called the Functional Central Limit Theorem), which extends the classical CLT to stochastic processes.

The √Δt scaling is critical: it ensures that the variance accumulates at the correct rate σ²t.

The Euler-Maruyama Method for Brownian Motion Simulation

Stochastic Differential Equations Framework

While we previously described Brownian motion through discrete increments, the formal mathematical framework uses Stochastic Differential Equations (SDEs). A general SDE has the form:

dX(t) = μ(X,t) dt + σ(X,t) dW(t)

Where:

  • μ(X,t) is the drift coefficient (deterministic component)
  • σ(X,t) is the diffusion coefficient (stochastic component)
  • dW(t) represents the increment of a Wiener process

For standard Brownian motion with volatility parameter σ, the SDE simplifies to:

dB(t) = σ dW(t)

This means μ = 0 (no drift) and the diffusion coefficient is constant. The process is purely driven by random fluctuations.

Numerical Solution: The Euler-Maruyama Scheme

Since SDEs cannot generally be solved analytically, we use numerical methods. The Euler-Maruyama method is the stochastic analog of the classical Euler method for ordinary differential equations.

Discretization Formula

Given the SDE dX(t) = μ dt + σ dW(t), the Euler-Maruyama discretization over a time step Δt is:

X(t + Δt) = X(t) + μ Δt + σ ΔW

and the Euler-Maruyama discretization over a time step Δt for a Brownion Motion is

X(t + Δt) = X(t) + σ √Δt · Z

Where:

  • Z ~ N(0,1) is a standard normal random variable
  • √Δt · Z represents the Brownian increment ΔW, which has distribution N(0, Δt)

This is precisely the simulation approach we described earlier, now formalized within the SDE framework.

Algorithm Steps

  1. Initialize: Set X₀ = starting value, choose time horizon T and number of steps n
  2. Compute step size: Δt = T/n
  3. Iterate: For each time step i = 0, 1, …, n-1:
    • Generate Z ~ N(0,1) using Box-Muller transform
    • Compute X(i+1) = X(i) + μ Δt + σ √Δt · Z
  4. Return: The path {X₀, X₁, …, Xₙ}

For standard Brownian motion with volatility σ:

  • Set μ = 0
  • The formula becomes: B(t + Δt) = B(t) + σ √Δt · Z

Implementation

Verification of Theoretical Properties

We can verify that our Euler-Maruyama simulation produces the correct statistical properties. Following images show the code and the output:

Expected output should show close agreement between theoretical and empirical values.

Generalization to Arbitrary SDEs

The Euler-Maruyama method easily extends to more complex stochastic processes. We can create a general simulator that accepts arbitrary drift and diffusion functions:

Conclusion

Brownian motion completes the natural progression from discrete to continuous processes in probability theory. Through simulation and analysis, we’ve explored this fundamental process that evolves continuously in both time and space.

Our simulation approach demonstrates a practical method for generating Brownian paths: by taking small normal-distributed steps at each time interval, scaled by σ√Δt, we create paths whose variance grows linearly with time while maintaining continuous movement. This bridges computational practicality with mathematical theory.

The three volatility cases clearly show how σ controls the process behavior:

  • Low σ (0.5): Gentle, stable fluctuations
  • Medium σ (2): Balanced random motion
  • High σ (5): Wild, unpredictable swings

Across all cases, Brownian motion maintains its defining properties: independent increments, stationary increments, normal distributions, and continuous paths. The connection to random walks through the scaling limit shows how simple discrete steps can converge to complex continuous motion.

Sources and Further Reading

Brownian Motion

Box-Muller