2.3 Brownian motion and diffusion processes

Definition of Brownian motion

Brownian motion provides a mathematical framework for modeling continuous random movement over time. In actuarial work, it's the foundation for modeling asset prices, interest rates, and other quantities that evolve unpredictably. Nearly every stochastic model you'll encounter in financial mathematics builds on it.

Mathematical formulation

Brownian motion is a continuous-time stochastic process $\{B(t), t \geq 0\}$ defined by these properties:

• Starts at zero: $B(0) = 0$
• Normal increments: For any $0 \leq s < t$, the increment $B(t) - B(s)$ is normally distributed with mean zero and variance $t - s$. In notation: $B(t) - B(s) \sim N(0, t-s)$
• Independent increments: For any collection of non-overlapping time intervals, the increments are independent of each other

At any time $t$, the distribution of $B(t)$ is $N(0, t)$. This means the uncertainty (measured by variance) grows linearly with time, while the standard deviation grows proportionally to $\sqrt{t}$.

Physical interpretation

Brownian motion is named after the botanist Robert Brown, who in 1827 observed the erratic, jittery motion of pollen grains suspended in water. The grains weren't alive; they were being bombarded by water molecules undergoing thermal agitation. Each collision imparts a tiny random force, and the cumulative effect produces the irregular paths we call Brownian motion.

This physical picture carries over to the mathematical model: many small, independent random shocks accumulate into a continuous but highly irregular trajectory.

Key properties

Several properties make Brownian motion especially useful (and mathematically distinctive):

• Continuous sample paths: The trajectory $B(t)$ is a continuous function of time with probability 1. There are no jumps.
• Nowhere differentiable: Despite being continuous, the paths are so jagged that they have no derivative at any point (almost surely). You can't define a "velocity" for Brownian motion in the classical sense.
• Martingale property: $\mathbb{E}[B(t) \mid B(s)] = B(s)$ for $s < t$. The best forecast of a future value, given the present, is just the current value.
• Self-similarity: Rescaling time by a factor $c$ and space by $\sqrt{c}$ produces a process with the same statistical properties. Formally, $\{B(ct)/\sqrt{c}\}$ has the same distribution as $\{B(t)\}$.

The nowhere-differentiable property is what forces us to use stochastic calculus rather than ordinary calculus when working with these processes.

Wiener process

The Wiener process is the rigorous mathematical formalization of Brownian motion, named after Norbert Wiener, who constructed it on a solid measure-theoretic foundation. In practice, "Wiener process" and "Brownian motion" are used interchangeably in most actuarial and financial contexts.

Standard Wiener process

The standard Wiener process $\{W(t), t \geq 0\}$ satisfies:

• $W(0) = 0$
• Independent increments: $W(t) - W(s)$ is independent of $\{W(u) : 0 \leq u \leq s\}$
• Normal increments: $W(t) - W(s) \sim N(0, t-s)$
• Continuous sample paths

A useful result for calculations: the covariance structure is $\text{Cov}(W(s), W(t)) = \min(s, t)$. This follows directly from the independent increments property and is frequently used when computing expectations involving products of Wiener process values.

Generalized Wiener process

The generalized Wiener process adds a deterministic drift to the standard Wiener process:

$X(t) = \mu t + \sigma W(t)$

• $\mu$ is the drift, controlling the average rate of change per unit time
• $\sigma$ is the diffusion coefficient, scaling the random fluctuations
• $W(t)$ is a standard Wiener process

At time $t$, $X(t) \sim N(\mu t, \sigma^2 t)$. This is the simplest extension that lets you model a process with both a trend and randomness. For example, if you set $\mu = 0.05$ and $\sigma = 0.2$, the process drifts upward on average while fluctuating randomly around that trend.

Wiener process vs Brownian motion

Strictly speaking, "Brownian motion" refers to the physical phenomenon (particles jostled by fluid molecules), while the "Wiener process" is the mathematical model capturing its essential features. In actuarial mathematics and finance, the distinction rarely matters, and you'll see both terms used for the same object. Just know that when someone writes $W(t)$, they mean the mathematical construct.

Stochastic calculus

Because Brownian motion paths are nowhere differentiable, ordinary calculus breaks down. Stochastic calculus extends differentiation and integration to handle processes driven by Brownian motion. It's the mathematical engine behind derivative pricing, hedging, and interest rate modeling.

Stochastic integrals

The Itô integral is the standard stochastic integral used in finance. It integrates a stochastic process with respect to a Wiener process:

$\int_0^t X(s) \, dW(s)$

Three properties you need to know:

• Linearity: $\int_0^t (aX(s) + bY(s)) \, dW(s) = a\int_0^t X(s) \, dW(s) + b\int_0^t Y(s) \, dW(s)$
• Zero mean: $\mathbb{E}\left[\int_0^t X(s) \, dW(s)\right] = 0$
• Itô isometry: $\mathbb{E}\left[\left(\int_0^t X(s) \, dW(s)\right)^2\right] = \mathbb{E}\left[\int_0^t X(s)^2 \, ds\right]$

The Itô isometry is particularly useful because it converts a stochastic integral into an ordinary (Lebesgue) integral when computing second moments.

Itô's lemma

Itô's lemma is the stochastic version of the chain rule. If you have a smooth function $f(t, x)$ and a process $X(t)$ satisfying the SDE

$dX(t) = \mu(t, X(t)) \, dt + \sigma(t, X(t)) \, dW(t)$

then:

$df(t, X(t)) = \left(\frac{\partial f}{\partial t} + \mu\frac{\partial f}{\partial x} + \frac{1}{2}\sigma^2\frac{\partial^2 f}{\partial x^2}\right)dt + \sigma\frac{\partial f}{\partial x} \, dW(t)$

The critical difference from the ordinary chain rule is the extra term $\frac{1}{2}\sigma^2 \frac{\partial^2 f}{\partial x^2}$. This arises because $(dW)^2 = dt$ (in the mean-square sense), so second-order terms don't vanish as they do in classical calculus.

Example: To derive the solution of geometric Brownian motion, apply Itô's lemma to $f(t, S) = \ln S$ with $dS = \mu S \, dt + \sigma S \, dW$. You get:

$d(\ln S) = \left(\mu - \frac{1}{2}\sigma^2\right)dt + \sigma \, dW$

This is how you show that $\ln S(t)$ is normally distributed, which is central to the Black-Scholes framework.

Stratonovich integral vs Itô integral

There are two conventions for defining stochastic integrals, differing in where the integrand is evaluated within each time step:

• Itô integral: Evaluates the integrand at the left endpoint of each subinterval:

$\int_0^t X(s) \, dW(s) = \lim_{n \to \infty} \sum_{i=1}^n X(t_{i-1})(W(t_i) - W(t_{i-1}))$

• Stratonovich integral: Evaluates at the midpoint:

$\int_0^t X(s) \circ dW(s) = \lim_{n \to \infty} \sum_{i=1}^n X\!\left(\frac{t_{i-1}+t_i}{2}\right)(W(t_i) - W(t_{i-1}))$

The Itô integral is standard in finance because it respects the "no peeking into the future" requirement (the integrand depends only on information available up to time $t_{i-1}$). The Stratonovich integral obeys the ordinary chain rule, which makes it natural in physics. In actuarial and financial work, you'll almost always use Itô.

Stochastic differential equations (SDEs)

An SDE incorporates randomness directly into a differential equation. The general form is:

$dX(t) = \mu(t, X(t)) \, dt + \sigma(t, X(t)) \, dW(t)$

Here $\mu(t, X(t))$ is the drift (deterministic tendency) and $\sigma(t, X(t))$ is the diffusion (random component). The solution $X(t)$ is a stochastic process, not a single function.

Definition and properties

• SDEs are interpreted through stochastic integrals (typically Itô).
• Existence and uniqueness: Under Lipschitz and linear growth conditions on $\mu$ and $\sigma$, a unique strong solution exists. These are the standard regularity conditions you'll see in proofs.
• Markov property: The solution to an SDE is a Markov process, meaning the future depends on the past only through the current state. This is what makes SDEs tractable for pricing and risk calculations.

Examples of SDEs

Three SDEs appear constantly in actuarial and financial applications:

Geometric Brownian motion (GBM): $dS(t) = \mu S(t) \, dt + \sigma S(t) \, dW(t)$

Used to model stock prices. The key feature is that both drift and diffusion are proportional to $S(t)$, so the process stays positive. The solution is $S(t) = S(0) \exp\!\left[\left(\mu - \frac{1}{2}\sigma^2\right)t + \sigma W(t)\right]$.

Ornstein-Uhlenbeck (OU) process: $dX(t) = \theta(\mu - X(t)) \, dt + \sigma \, dW(t)$

Models mean-reverting behavior. When $X(t)$ is above $\mu$, the drift pulls it down; when below, it pulls it up. Used for interest rates and commodity prices.

Cox-Ingersoll-Ross (CIR) model: $dr(t) = \kappa(\theta - r(t)) \, dt + \sigma\sqrt{r(t)} \, dW(t)$

Also mean-reverting, but the diffusion term $\sigma\sqrt{r(t)}$ ensures the process stays non-negative (provided $2\kappa\theta \geq \sigma^2$, the Feller condition). Widely used for interest rate modeling.

Solution methods for SDEs

Analytical solutions exist for some important cases:

• GBM has a closed-form solution (shown above)
• The OU process has an explicit solution involving an integral of the Wiener process
• Linear SDEs with constant coefficients are generally solvable in closed form

Numerical methods are needed for most other SDEs:

1. Euler-Maruyama method: The simplest scheme. Discretize time into steps of size $\Delta t$ and approximate: $X_{n+1} = X_n + \mu(t_n, X_n)\Delta t + \sigma(t_n, X_n)\Delta W_n$ where $\Delta W_n \sim N(0, \Delta t)$. This has strong convergence order 0.5.

2. Milstein method: Adds a correction term involving $\frac{\partial \sigma}{\partial x}$, improving strong convergence to order 1.0: $X_{n+1} = X_n + \mu \Delta t + \sigma \Delta W_n + \frac{1}{2}\sigma \frac{\partial \sigma}{\partial x}[(\Delta W_n)^2 - \Delta t]$

3. Higher-order methods: Stochastic Runge-Kutta and other schemes offer better accuracy at higher computational cost.

The Euler-Maruyama method is the default starting point. Use Milstein when you need better accuracy without much extra implementation effort.

Diffusion processes

Diffusion processes are continuous-time Markov processes with continuous sample paths. Every SDE of the form above defines a diffusion process (under appropriate regularity conditions). They're the broadest class of continuous stochastic models you'll work with.

Definition and properties

A diffusion process $\{X(t), t \geq 0\}$ is characterized by:

• Continuous sample paths: No jumps or discontinuities
• Markov property: $P(X(t) \in A \mid X(s), s \leq u) = P(X(t) \in A \mid X(u))$ for $u < t$
• Infinitesimal generator: The local behavior is captured by the drift $\mu(x,t)$ and diffusion coefficient $\sigma(x,t)$, which together define the infinitesimal generator:

$\mathcal{L} = \mu(x,t)\frac{\partial}{\partial x} + \frac{1}{2}\sigma^2(x,t)\frac{\partial^2}{\partial x^2}$

The drift and diffusion coefficients fully specify the dynamics of the process. All the PDEs that follow (Fokker-Planck, Kolmogorov) are expressed in terms of these two functions.

Fokker-Planck equation

The Fokker-Planck equation (also called the forward Kolmogorov equation) governs how the probability density of a diffusion process evolves over time:

$\frac{\partial p(x, t)}{\partial t} = -\frac{\partial}{\partial x}[\mu(x, t) \, p(x, t)] + \frac{1}{2}\frac{\partial^2}{\partial x^2}[\sigma^2(x, t) \, p(x, t)]$

Here $p(x, t)$ is the probability density of finding the process at state $x$ at time $t$. Think of it this way: the first term on the right describes how drift transports probability, and the second term describes how diffusion spreads it out.

You use this equation when you know where the process starts and want to find the distribution at a later time.

Kolmogorov equations

The Kolmogorov equations come in a pair:

Forward equation (Fokker-Planck): Describes the evolution of the probability density forward in time, as shown above.

Backward equation: Describes how expected values of future quantities depend on the current state:

$\frac{\partial u(x, t)}{\partial t} + \mu(x, t)\frac{\partial u(x, t)}{\partial x} + \frac{1}{2}\sigma^2(x, t)\frac{\partial^2 u(x, t)}{\partial x^2} = 0$

where $u(x, t) = \mathbb{E}[f(X(T)) \mid X(t) = x]$.

Note the sign convention: the backward equation has a $+$ in front of the time derivative (or equivalently, the equation equals zero with all terms on one side). You use the backward equation when you want to compute the expected payoff of some function at a future time $T$, starting from state $x$ at time $t$. This is exactly the setup for pricing derivatives.

Ornstein-Uhlenbeck process

The Ornstein-Uhlenbeck (OU) process is the prototypical mean-reverting diffusion:

$dX(t) = \theta(\mu - X(t)) \, dt + \sigma \, dW(t)$

• $\theta > 0$: mean-reversion speed (larger $\theta$ means faster reversion)
• $\mu$: long-term mean level
• $\sigma > 0$: volatility of the random shocks

Key properties:

• Mean-reversion: The drift $\theta(\mu - X(t))$ always points toward $\mu$. If $X(t) > \mu$, the drift is negative; if $X(t) < \mu$, positive.
• Stationary distribution: The process has a Gaussian stationary distribution with mean $\mu$ and variance $\frac{\sigma^2}{2\theta}$. Higher mean-reversion speed $\theta$ produces a tighter stationary distribution.
• Autocorrelation: $\text{Corr}(X(t), X(t+h)) = e^{-\theta h}$, decaying exponentially. The "half-life" of the autocorrelation is $\frac{\ln 2}{\theta}$.

The OU process is used in the Vasicek interest rate model, for modeling commodity prices that revert to a production cost level, and for the velocity of particles in a fluid (its original physical application).

Applications in finance

Brownian motion and diffusion processes underpin most of quantitative finance. The two models below are among the most important for actuarial exams and practice.

Black-Scholes model

The Black-Scholes model assumes the underlying asset price follows geometric Brownian motion:

$dS(t) = \mu S(t) \, dt + \sigma S(t) \, dW(t)$

where $S(t)$ is the asset price, $\mu$ is the expected return, and $\sigma$ is the volatility.

Under risk-neutral pricing (replacing $\mu$ with the risk-free rate $r$), the Black-Scholes formula for a European call option with strike $K$ and maturity $T$ is:

$C = S(0)\Phi(d_1) - Ke^{-rT}\Phi(d_2)$

where $d_1 = \frac{\ln(S(0)/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}$ and $d_2 = d_1 - \sigma\sqrt{T}$.

The derivation uses Itô's lemma applied to a portfolio of the option and the underlying asset, leading to a PDE (the Black-Scholes PDE) that can be solved in closed form. Key assumptions include constant volatility, no transaction costs, and continuous trading.

Vasicek model

The Vasicek model describes the short-term interest rate as an Ornstein-Uhlenbeck process:

$dr(t) = \kappa(\theta - r(t)) \, dt + \sigma \, dW(t)$

• $\kappa > 0$: speed of mean reversion
• $\theta$: long-term mean interest rate
• $\sigma$: interest rate volatility

The model produces analytical formulas for zero-coupon bond prices and European bond options, which makes it very tractable. The bond price has the form $P(t, T) = A(t,T) \exp(-B(t,T) r(t))$ where $A$ and $B$ are deterministic functions of the model parameters and the time to maturity.

One limitation: because the OU process is Gaussian, the Vasicek model allows negative interest rates. While this was once considered a drawback, negative rates have been observed in practice (e.g., European government bonds in the 2010s). For applications where non-negativity is required, the CIR model (which uses $\sigma\sqrt{r(t)}$ instead of $\sigma$) is the standard alternative.