Stochastic Differential Equations: An Introduction

Stochastic Differential Equations and Ito Calculus

When Deterministic Models Are Not Enough

Many real-world systems are subject to random disturbances that are fundamentally impossible to ignore. The Brownian motion of a molecule in a liquid is determined by random collisions with neighboring molecules. The price of a stock on the exchange undergoes random fluctuations under the influence of news. A nerve impulse contains thermal noise. An ecological system is subject to random climate fluctuations.

Stochastic differential equations (SDEs) include randomness explicitly in the dynamics of the system. This is not a “model imprecision”—it is a recognition of the fundamental role of noise.

Brownian Motion as a Building Block

The Wiener process W(t) (Brownian motion) is a stochastic process that satisfies:

1. W(0) = 0,
2. W(t) is continuous in t (almost surely),
3. The increments W(t) − W(s) ~ N(0, t − s) for t > s,
4. Increments on non-overlapping intervals are independent.

Key property: W(t) is nowhere differentiable (almost surely). Its “derivative” dW/dt—“white noise”—does not exist in the usual sense. However, the stochastic integral ∫ f(t) dW(t) can be defined as the limit of sums using a special (Ito-iterative) procedure.

Scaling: For small dt, the increment dW ~ N(0, dt) ~ √dt · N(0,1). That is, |dW| ~ √dt—this is √dt times greater than an “ordinary” increment dt. This means that Ito processes behave differently than ordinary functions.

The Ito Stochastic Equation

The SDE in Ito form: dX(t) = a(X(t), t) dt + b(X(t), t) dW(t).

The first term a is the deterministic drift, describing the “average” movement. The second term b is the diffusion (random) term, describing fluctuations.

The solution X(t) is a stochastic process that satisfies the integral equation:

X(t) = X(t₀) + ∫{t₀}^t a(X(s), s) ds + ∫{t₀}^t b(X(s), s) dW(s).

The second integral is the Ito stochastic integral: a limit of sums evaluated at the left end of the subinterval.

Ito’s Formula: An Analogue of the Chain Rule for Differentiation

For ordinary functions: dF = F'(x) dx. But for stochastic processes this is not valid due to the property dW ~ √dt.

Ito’s Lemma (Ito’s formula): If X(t) satisfies an SDE with drift a and diffusion b, then for a smooth function F(x, t):

dF = (∂F/∂t + a ∂F/∂x + (b²/2) ∂²F/∂x²) dt + b ∂F/∂x dW.

The key difference from the deterministic case is the Ito correction (b²/2)(∂²F/∂x²): an additional “drift” term that arises due to the quadratic nature of the noise.

Heuristic justification: (dW)² ≈ dt (the mean square increment!), therefore the Taylor expansion must be taken up to second order: dF ≈ ∂_t F dt + ∂_x F dX + (1/2) ∂²_x F (dX)² = ∂_t F dt + ∂_x F (a dt + b dW) + (1/2)∂²_x F b² dt.

Geometric Brownian Motion and the Black–Scholes Formula

Stock price model:

dS = μS dt + σS dW.

Here, μ is the expected rate of return, σ is the volatility (a measure of randomness). This is geometric Brownian motion: the logarithm of the price follows ordinary Brownian motion.

Applying Ito’s formula for F = ln S:

dF = (μ − σ²/2) dt + σ dW.

Integrate: ln S(t) − ln S(0) = (μ − σ²/2)t + σW(t).

Solution: S(t) = S(0) · exp((μ − σ²/2)t + σW(t)).

The correction −σ²/2 compared to the naive formula e^{μt} is a direct consequence of the Ito correction.

From this model, Black and Scholes in 1973 derived the famous option pricing formula:

C = S₀N(d₁) − Ke^{-rT}N(d₂),

where d₁ = (ln(S₀/K) + (r + σ²/2)T) / (σ√T), d₂ = d₁ − σ√T.

N is the normal distribution function, K is the option strike price, r is the risk-free rate, T is the time to expiration. This formula revolutionized financial engineering—its authors received the Nobel Prize in Economics in 1997. Ito’s formula is its foundation.

The Fokker–Planck Equation

Instead of an individual trajectory X(t), it is often the probability distribution p(x, t) that is of interest. It evolves according to the Fokker–Planck equation:

∂p/∂t = −∂(ap)/∂x + (1/2) ∂²(b²p)/∂x².

This is a parabolic-type partial differential equation (an analogue of the heat conduction equation). From it, we find the probability of the system being found in state x at time t.

Stationary distribution: When ∂p/∂t = 0, we find the equilibrium distribution. For linear drift (a harmonic oscillator with random noise) the stationary distribution is normal (Gaussian)—the basis of thermodynamic equilibrium.

Question for reflection: The Black–Scholes formula assumes that the volatility σ is constant. Real markets display variable volatility (the volatility smile). How can the model be generalized to take this into account? What class of SDEs describes stochastic volatility?

Numerical Methods for SDEs: Euler–Maruyama and Milstein

The main numerical method for SDEs dX = a(X,t)dt + b(X,t)dW is the Euler–Maruyama method: Xₙ₊₁ = Xₙ + a(Xₙ,tₙ)Δt + b(Xₙ,tₙ)ΔWₙ, where ΔWₙ ~ N(0, Δt)—normally distributed increments. The order of strong convergence is 0.5 (half as good as the deterministic Euler method). The Milstein method adds the correction term bb'(ΔWₙ² − Δt)/2 and achieves order 1.0. For the Heston model (stochastic volatility): the price S and volatility v are described by a system of two coupled SDEs with two correlated Brownian motion processes. This system does not have an analytic solution for most parameter values.