Bridge to Stochastic Differential Equations¶

Having established in § Why Deterministic Models Fail that deterministic models cannot reproduce observed return behaviour, we now construct the continuous-time model consistent with the discrete-time dynamics. This section is purely constructive: the empirical motivation is not repeated, and the only question is what continuous-time object is compatible with the discrete return scaling.

The construction has three spans: (1) the discrete-time model, (2) the scaling limit that arises as the time step shrinks, and (3) the SDE notation that encodes that limit. The goal is to explain structure, not machinery---the rigorous definitions of the Ito integral, Ito's lemma, and SDE theory follow in Sections 2.1--2.3.

Constructing the Continuous-Time Limit¶

Step 1 --- The Discrete-Time Model¶

Fix a time step \(\Delta t > 0\). Suppose log-prices evolve via

Equivalently, in multiplicative form:

This model has three desirable properties:

• Positivity: \(S_n > 0\) for all \(n\) whenever \(S_0 > 0\).
• Correct variance scaling: \(\operatorname{Var}(\log S_{n+1} - \log S_n) = \sigma^2\Delta t\).
• Multiplicative structure: Volatility is proportional to price level.

Step 2 --- Refining the Time Step¶

Let \(t = n\,\Delta t\) and write the change over one step as

For small \(\Delta t\) the first-order Taylor expansion \(e^x - 1 \approx x\) gives the heuristic approximation

Rearranging:

As \(\Delta t \to 0\), the term \(Z/\sqrt{\Delta t}\) diverges---the limit is not a classical derivative but a distributional object. This is precisely why ordinary calculus fails and a new framework (Ito calculus, Section 2.1) is needed. The correct approach takes a weak limit of the entire rescaled random walk.

Step 3 --- Donsker's Theorem (Functional CLT)¶

Recall (see § Donsker's Theorem): the rescaled partial-sum process \(W^{(\Delta t)}(t) = \sqrt{\Delta t}\sum_{i=1}^{\lfloor t/\Delta t\rfloor} Z_i\), built from i.i.d. mean-zero unit-variance increments, converges in distribution to standard Brownian motion \(W\) on \([0,T]\) as \(\Delta t\to 0\).

Two consequences are essential for the bridge:

1. Universality. The limit is Brownian motion regardless of the distribution of \(Z_i\), provided mean is zero and variance is one. This justifies using Brownian motion as the canonical noise for any asset whose log-returns have finite variance.
2. The \(\sqrt{\Delta t}\) scaling is decisive. Scaling by \(\Delta t\) would collapse the walk to zero; no rescaling would cause divergence. Only \(\sqrt{\Delta t}\) produces a non-degenerate limit.

Brownian motion is therefore not an assumption but the unique scaling limit of discrete noise — exactly the object the failure of deterministic dynamics demanded.

Step 4 --- The SDE Limit¶

With Donsker's theorem in hand, the discrete model

converges in distribution, as \(\Delta t \to 0\), to

This is Geometric Brownian Motion (GBM), the foundation of the Black--Scholes model. Thus the SDE is not postulated — it is the only continuous-time object compatible with discrete return scaling.

Recall (see § Itô Calculus Applications): the explicit solution is \(S_t = S_0 \exp[(\mu - \sigma^2/2)t + \sigma W_t]\), with the \(-\sigma^2/2\) drift correction arising from \((dW_t)^2 = dt\).

Recall (see § Itô Integral Construction): \(dW_t\) is shorthand — the SDE always denotes the integral equation \(S_t = S_0 + \int_0^t \mu S_s\,ds + \int_0^t \sigma S_s\,dW_s\) with the stochastic integral defined via Itô's construction.

Emergence of the General SDE¶

The GBM derivation above used constant drift \(\mu\) and constant proportional volatility \(\sigma\). Replacing these with state- and time-dependent coefficients \(\mu(X_t,t)\) and \(\sigma(X_t,t)\) yields the general Itô SDE \(dX_t = \mu(X_t, t)\,dt + \sigma(X_t, t)\,dW_t\). Recall (see § Stochastic Differential Equations): the formal definition of this object, its drift/diffusion decomposition, and the catalogue of canonical models (OU, CIR, …) live in the SDE chapter.

The Construction in One Diagram¶

Discrete random walk ──────────────────► Brownian motion W_t │ (Donsker, √Δt scaling) │ │ (formal limit) │ (rigorous foundation) ▼ ▼ dS_t = μ S_t dt + σ S_t dW_t ◄──── Itô calculus │ │ (extend drift/diffusion functions) ▼ General SDEs → Quantitative finance models

What Comes Next¶

The bridge is conceptually complete. The rigorous machinery — Itô integration, Itô's lemma, and the theory of SDEs (existence, uniqueness, canonical models) — is developed in the chapters that follow: § Itô Integral, § Itô's Lemma, and § Stochastic Differential Equations.

Summary¶

The passage from discrete to continuous time is governed by one observation: the \(\sqrt{\Delta t}\) scaling of random shocks is the only scaling consistent with a non-degenerate continuous-time limit. Donsker's theorem makes this precise---the rescaled random walk converges to Brownian motion universally across all finite-variance innovations. The Ito SDE

is therefore not a choice but the unique continuous-time object compatible with discrete return scaling. Making this construction rigorous is the goal of the next three sections. \(\square\)

Exercises¶

Exercise 1. Starting from the discrete model \(\log S_{n+1} - \log S_n = \mu\,\Delta t + \sigma\sqrt{\Delta t}\cdot Z_n\) with \(Z_n \sim \mathcal{N}(0,1)\) i.i.d., derive the mean and variance of \(\log S_N\) after \(N\) steps. Show that \(\operatorname{Var}(\log S_N) = \sigma^2 T\) where \(T = N\Delta t\), independent of the step size.

Exercise 2. Donsker's theorem requires only i.i.d. increments with mean zero and variance one. Suppose \(Z_i\) follows a Rademacher distribution (\(P(Z_i = +1) = P(Z_i = -1) = 1/2\)) instead of a Gaussian. Does Donsker's theorem still apply? What is the limiting process? Explain why this universality matters for financial modeling.

Exercise 3. The heuristic first-order expansion gives \(S(t+\Delta t) - S(t) \approx \mu\,S(t)\,\Delta t + \sigma\,S(t)\,\sqrt{\Delta t}\cdot Z\). The second-order expansion includes an additional term \(\frac{1}{2}\sigma^2 S(t)\,\Delta t\). Show that this term is \(O(\Delta t)\) and therefore the same order as the drift. Explain how this leads to the \(-\sigma^2/2\) correction in the GBM solution.

Exercise 4. For each of the following models, identify the drift function \(\mu(x,t)\) and diffusion function \(\sigma(x,t)\), and state whether the diffusion is additive or multiplicative:

(a) Geometric Brownian Motion: \(dS_t = \mu S_t\,dt + \sigma S_t\,dW_t\)

(b) Ornstein--Uhlenbeck: \(dX_t = \kappa(\theta - X_t)\,dt + \sigma\,dW_t\)

(c) Cox--Ingersoll--Ross: \(dr_t = \kappa(\theta - r_t)\,dt + \xi\sqrt{r_t}\,dW_t\)

Exercise 5. Explain why the \(\sqrt{\Delta t}\) scaling in Donsker's theorem is the unique scaling that produces a non-degenerate limit. What happens if increments are scaled by \(\Delta t\) instead? By \(1\) (no rescaling)? Connect this to the non-differentiability of Brownian paths.