Chapter 3: Stochastic Differential Equations¶

This chapter develops stochastic differential equations as the natural continuous-time limit of random systems and builds the analytical framework needed to study them. The entire chapter is driven by the single identity arising from quadratic variation:

\[(dW_t)^2 = dt\]

From this one fact the Itô integral, Itô's formula, and the theory of diffusions all follow.

The structure follows a single pipeline:

\[\text{data} \;\longrightarrow\; \text{randomness} \;\longrightarrow\; \text{Itô calculus} \;\longrightarrow\; \text{SDEs} \;\longrightarrow\; \text{structure}\]

Key Concepts¶

Empirical Motivation¶

Financial returns exhibit statistical properties — heavy tails, volatility clustering, and absence of autocorrelation — that deterministic models cannot reproduce.

This forces the introduction of randomness and motivates stochastic modeling. The required integration theory is developed in the next section.

Itô Integration¶

Classical integration fails for Brownian motion because its paths have unbounded first variation but non-vanishing quadratic variation:

\[(dW_t)^2 = dt\]

The Itô integral is defined as an \(L^2\)-limit of adapted Riemann sums evaluated at left endpoints and is characterized by the Itô isometry:

\[\mathbb{E}\!\left[\left(\int_0^t H_s\, dW_s\right)^{\!2}\right] = \mathbb{E}\!\left[\int_0^t H_s^2\, ds\right]\]

This defines the fundamental stochastic integral and produces continuous martingales. The same quadratic variation identity is what forces a correction term in the chain rule below.

Itô's Formula¶

Itô's formula is the stochastic chain rule — the fundamental theorem of stochastic calculus. For \(f \in C^{1,2}\) and an Itô process \(X_t\):

\[df = \frac{\partial f}{\partial t}\, dt + \frac{\partial f}{\partial x}\, dX_t + \frac{1}{2}\frac{\partial^2 f}{\partial x^2}\, (dX_t)^2\]

The extra second-order term arises from quadratic variation. This is the central mechanism that transforms randomness into drift and makes SDEs solvable in closed form when a transformation is available.

Stochastic Differential Equations¶

An SDE

\[dX_t = b(t, X_t)\, dt + \sigma(t, X_t)\, dW_t\]

defines a stochastic process via its drift \(b\) and diffusion coefficient \(\sigma\). Solving an SDE means constructing a process satisfying the corresponding integral equation.

Analytical solutions are obtained via Itô transformations; numerical schemes approximate paths when closed forms are unavailable. The question of when a solution exists at all is addressed next.

Existence and Uniqueness¶

Under Lipschitz and linear growth conditions on \(b\) and \(\sigma\), an SDE admits a unique strong solution. The solution is constructed via Picard iteration, paralleling the classical theory for ODEs.

This establishes when SDE models are mathematically well-defined, setting the stage for studying their structure.

Diffusion Processes¶

Solutions of SDEs with Markov coefficients form diffusion processes — continuous-path Markov processes completely determined by \(b\) and \(\sigma\). They can be characterized equivalently by:

SDEs (pathwise description),
infinitesimal generators (analytic description),
martingale problems (law-based description).

These three perspectives describe the same object through different lenses and are connected by the generator introduced below.

Infinitesimal Generator and Dynkin's Formula¶

The generator

\[\mathcal{L}f = b \cdot \nabla f + \tfrac{1}{2}\, a : \nabla^2 f, \qquad a = \sigma\sigma^{\top}\]

captures the instantaneous rate of change of \(\mathbb{E}^x[f(X_t)]\). The fundamental identity

\[f(X_t) = f(X_0) + \int_0^t \mathcal{L}f(X_s)\, ds + M_t\]

decomposes the dynamics of \(f(X_t)\) into a predictable drift and a martingale fluctuation \(M_t\). Taking expectations yields Dynkin's formula:

\[\mathbb{E}^x[f(X_t)] = f(x) + \mathbb{E}^x\!\left[\int_0^t \mathcal{L}f(X_s)\, ds\right]\]

which governs the evolution of expectations and is the bridge from stochastic models to pricing PDEs.

Conceptual Flow¶

flowchart LR
  A[Empirical Data]
  --> B[Randomness Required]
  --> C[Itô Integral]
  --> D[Itô Formula]
  --> E[SDE]
  --> F[Existence & Uniqueness]
  --> G[Diffusion Structure]
  --> H[Generator & Dynkin]

Each idea appears exactly once, in the section that owns it; the sections below develop the details.