Diffusion Process Overview

Concept Definition

Let \((\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \ge 0}, \mathbb{P})\) be a filtered probability space satisfying the usual conditions (right-continuity of \((\mathcal{F}_t)\); \(\mathcal{F}_0\) contains all \(\mathbb{P}\)-null sets). Let

\[ W_t = (W_t^{1}, \dots, W_t^{m}) \]

be an \(m\)-dimensional Brownian motion adapted to \((\mathcal{F}_t)\).

Definition: Itô Diffusion

An \(\mathbb{R}^d\)-valued process \(X_t = (X_t^{1}, \dots, X_t^{d})\) is an Itô diffusion if it satisfies the stochastic differential equation

\[ \mathrm{d}X_t^{i} = b^{i}(t, X_t)\,\mathrm{d}t + \sigma^{i\alpha}(t, X_t)\,\mathrm{d}W_t^{\alpha}, \qquad i = 1, \dots, d,\quad \alpha = 1, \dots, m, \]

where:

• \(b^{i} : [0,\infty) \times \mathbb{R}^d \to \mathbb{R}\) is the drift field,
• \(\sigma^{i\alpha} : [0,\infty) \times \mathbb{R}^d \to \mathbb{R}\) is the diffusion matrix.

The Einstein summation convention is in force throughout (repeated Greek indices are summed from \(1\) to \(m\)).

The differential form is symbolic. The mathematically precise definition is the integral equation

\[ X_t^{i} = X_0^{i} + \int_0^t b^{i}(s, X_s)\,\mathrm{d}s + \int_0^t \sigma^{i\alpha}(s, X_s)\,\mathrm{d}W_s^{\alpha}. \]

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Explanation

Semimartingale Decomposition

Every Itô diffusion is a semimartingale: it decomposes as

\[ X_t^{i} = \underbrace{X_0^{i} + \int_0^t b^{i}(s, X_s)\,\mathrm{d}s}_{\text{finite variation (drift)}} + \underbrace{\int_0^t \sigma^{i\alpha}(s, X_s)\,\mathrm{d}W_s^{\alpha}}_{\text{local martingale (noise)}}. \]

Paths \(t \mapsto X_t(\omega)\) are continuous almost surely, since both integrals produce continuous processes.

Diffusion Matrix and Quadratic Covariation

Define the covariance matrix

\[ a^{ij}(t, x) := \sigma^{i\alpha}(t, x)\,\sigma^{j\alpha}(t, x). \]

This matrix is symmetric and non-negative definite. A defining identity for diffusions is

\[ \mathrm{d}\langle X^{i}, X^{j} \rangle_t = a^{ij}(t, X_t)\,\mathrm{d}t. \]

That is, quadratic covariation grows linearly in time at a rate determined by the current state.

Infinitesimal Generator

For \(f \in C^{1,2}([0,\infty)\times\mathbb{R}^d)\) (once differentiable in \(t\), twice in \(x\)), the infinitesimal generator acting on the spatial variables is

\[ (\mathcal{L}f)(t, x) = b^{i}(t, x)\,\frac{\partial f}{\partial x_i}(x) + \frac{1}{2}\,a^{ij}(t, x)\,\frac{\partial^2 f}{\partial x_i \partial x_j}(x). \]

Here \(\mathcal{L}\) acts on the spatial argument of \(f\); the coefficients \(b^i, a^{ij}\) carry the time dependence. For time-homogeneous coefficients one may restrict to \(f \in C^{2}(\mathbb{R}^d)\); the full Itô formula requires \(C^{1,2}\) regularity.

By Itô's formula,

\[ f(X_t) - f(X_0) - \int_0^t (\mathcal{L}f)(s, X_s)\,\mathrm{d}s = \int_0^t \frac{\partial f}{\partial x_i}(X_s)\,\sigma^{i\alpha}(s, X_s)\,\mathrm{d}W_s^{\alpha}, \]

which is a local martingale. This local martingale characterisation of diffusions is the foundation of the martingale problem formulation (see Martingale Problem — Stroock–Varadhan).

Markov Property

Under standard Lipschitz and linear-growth conditions on \(b\) and \(\sigma\), the SDE has a unique strong solution and \(X_t\) is a strong Markov process:

\[ \mathbb{E}[f(X_t) \mid \mathcal{F}_s] = \mathbb{E}[f(X_t) \mid X_s], \qquad s \le t. \]

The extension to stopping times is the strong Markov property (see Strong Markov Property).

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Diagram / Example

Special Cases

Pure Brownian motion (\(b = 0\), \(\sigma = I\)):

\[ \mathrm{d}X_t^{i} = \mathrm{d}W_t^{i}. \]

Drifted Brownian motion:

\[ \mathrm{d}X_t^{i} = b^{i}(X_t)\,\mathrm{d}t + \mathrm{d}W_t^{i}. \]

Gradient diffusion (Langevin equation):

\[ \mathrm{d}X_t = -\nabla V(X_t)\,\mathrm{d}t + \sqrt{2}\,\mathrm{d}W_t, \]

which has invariant density \(\pi(x) \propto e^{-V(x)}\) whenever \(\int e^{-V} < \infty\). This is the prototypical example for invariant measures; see Invariant Measures and Stationarity.

Structure at a Glance

\[ \boxed{ \text{Diffusion} = \text{state-dependent drift} + \text{state-dependent Brownian noise} } \]

Component	Symbol	Role
Drift	\(b^i(t,x)\)	Deterministic push
Diffusion matrix	\(\sigma^{i\alpha}(t,x)\)	Amplitude of noise
Covariance matrix	\(a^{ij} = \sigma^{i\alpha}\sigma^{j\alpha}\)	Quadratic variation rate
Generator	\(\mathcal{L}\)	Infinitesimal mean evolution

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Proof / Derivation

The SDE integral equation is well-defined under standard conditions. We verify the semimartingale decomposition and the quadratic covariation identity.

Quadratic covariation. By bilinearity of quadratic covariation and \(\langle W^{\alpha}, W^{\beta} \rangle_t = \delta^{\alpha\beta} t\):

\[ \langle X^i, X^j \rangle_t = \left\langle \int_0^{\cdot} \sigma^{i\alpha}(s, X_s)\,\mathrm{d}W_s^{\alpha},\; \int_0^{\cdot} \sigma^{j\beta}(s, X_s)\,\mathrm{d}W_s^{\beta} \right\rangle_t = \int_0^t \sigma^{i\alpha}(s, X_s)\,\sigma^{j\beta}(s, X_s)\,\mathrm{d}\langle W^\alpha, W^\beta \rangle_s = \int_0^t \sigma^{i\alpha}(s, X_s)\,\sigma^{j\alpha}(s, X_s)\,\mathrm{d}s = \int_0^t a^{ij}(s, X_s)\,\mathrm{d}s. \]

(The Itô isometry gives \(\mathbb{E}[\langle M \rangle_t] = \mathbb{E}[\int_0^t h_s^2\,\mathrm{d}s]\) for an \(L^2\) martingale \(M = \int h\,\mathrm{d}W\); it is the bilinearity identity above that yields the pathwise quadratic covariation.)

Itô's formula. For \(f \in C^{1,2}([0,\infty) \times \mathbb{R}^d)\), the chain rule for semimartingales gives:

\[ \mathrm{d}f(t, X_t) = \frac{\partial f}{\partial t}\,\mathrm{d}t + \frac{\partial f}{\partial x_i}\,\mathrm{d}X_t^i + \frac{1}{2}\,\frac{\partial^2 f}{\partial x_i \partial x_j}\,\mathrm{d}\langle X^i, X^j \rangle_t. \]

Substituting \(\mathrm{d}X_t^i = b^i\,\mathrm{d}t + \sigma^{i\alpha}\,\mathrm{d}W_t^\alpha\) and \(\mathrm{d}\langle X^i,X^j\rangle_t = a^{ij}\,\mathrm{d}t\), and collecting all \(\mathrm{d}t\) terms:

\[ \mathrm{d}f(t, X_t) = \underbrace{\left(\frac{\partial f}{\partial t} + b^i\frac{\partial f}{\partial x_i} + \frac{1}{2}a^{ij}\frac{\partial^2 f}{\partial x_i\partial x_j}\right)}_{= \,\partial_t f \,+\, \mathcal{L}f}\mathrm{d}t + \frac{\partial f}{\partial x_i}\,\sigma^{i\alpha}\,\mathrm{d}W_t^{\alpha} = \left(\frac{\partial f}{\partial t} + \mathcal{L}f\right)\mathrm{d}t + \frac{\partial f}{\partial x_i}\,\sigma^{i\alpha}\,\mathrm{d}W_t^{\alpha}. \]

The stochastic integral term is a local martingale; the remaining term is finite variation. \(\square\)

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What to Remember

• A diffusion is a continuous-path Markov semimartingale whose quadratic covariation is \(\mathrm{d}\langle X^i, X^j\rangle_t = a^{ij}(t, X_t)\,\mathrm{d}t\).
• The drift \(b\) and diffusion matrix \(\sigma\) fully specify the dynamics; \(a = \sigma\sigma^\top\) is the effective covariance.
• The generator \(\mathcal{L}\) encodes the infinitesimal mean behaviour and connects SDEs to PDEs (Kolmogorov equations).
• The SDE defines the process pathwise; the martingale problem defines it in law via \(\mathcal{L}\) alone.

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Exercises

Exercise 1. Let \(X_t\) be a one-dimensional Itô diffusion with drift \(b(x) = -\alpha x\) and diffusion coefficient \(\sigma(x) = \beta\), where \(\alpha, \beta > 0\) are constants. Write down the covariance matrix \(a(x)\) and the infinitesimal generator \(\mathcal{L}\). Apply \(\mathcal{L}\) to the test function \(f(x) = x^2\) and interpret the result.

Solution to Exercise 1

The diffusion coefficient is \(\sigma(x) = \beta\), so the covariance matrix is

\[ a(x) = \sigma(x)^2 = \beta^2 \]

The infinitesimal generator for a one-dimensional diffusion with drift \(b(x) = -\alpha x\) and covariance \(a(x) = \beta^2\) is

\[ \mathcal{L}f(x) = b(x)\,f'(x) + \frac{1}{2}\,a(x)\,f''(x) = -\alpha x\,f'(x) + \frac{\beta^2}{2}\,f''(x) \]

Applying \(\mathcal{L}\) to \(f(x) = x^2\): we have \(f'(x) = 2x\) and \(f''(x) = 2\), so

\[ \mathcal{L}(x^2) = -\alpha x \cdot 2x + \frac{\beta^2}{2} \cdot 2 = -2\alpha x^2 + \beta^2 \]

Interpretation. By Itô's formula, \(f(X_t) - f(X_0) - \int_0^t \mathcal{L}f(X_s)\,\mathrm{d}s\) is a local martingale, so

\[ \frac{\mathrm{d}}{\mathrm{d}t}\mathbb{E}[X_t^2] = \mathbb{E}[\mathcal{L}(X_t^2)] = -2\alpha\,\mathbb{E}[X_t^2] + \beta^2 \]

The term \(-2\alpha\,\mathbb{E}[X_t^2]\) represents mean reversion pulling the second moment toward zero, while \(\beta^2\) represents the constant injection of variance by the noise. At equilibrium, \(\mathbb{E}[X_t^2] = \beta^2/(2\alpha)\).

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Exercise 2. Consider the two-dimensional diffusion \((X_t^1, X_t^2)\) satisfying

\[ \mathrm{d}X_t^1 = X_t^2\,\mathrm{d}t + \mathrm{d}W_t^1, \qquad \mathrm{d}X_t^2 = -X_t^1\,\mathrm{d}t + \mathrm{d}W_t^2. \]

Compute the covariance matrix \(a^{ij}\) and the quadratic covariation \(\langle X^1, X^2 \rangle_t\). Is \(a\) degenerate or non-degenerate?

Solution to Exercise 2

The drift is \(b(x) = (x^2, -x^1)^\top\) and the diffusion matrix is \(\sigma = I_{2 \times 2}\) (the identity). Therefore

\[ a^{ij} = \sigma^{i\alpha}\sigma^{j\alpha} = \delta^{ij} \]

so

\[ a = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]

The quadratic covariation is

\[ \langle X^1, X^2 \rangle_t = \int_0^t a^{12}(X_s)\,\mathrm{d}s = \int_0^t 0\,\mathrm{d}s = 0 \]

The matrix \(a = I\) is strictly positive definite (all eigenvalues equal \(1\)), so it is non-degenerate. The two components of the diffusion are driven by independent Brownian motions, despite the coupling in the drift.

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Exercise 3. Verify the semimartingale decomposition for the geometric Brownian motion \(\mathrm{d}S_t = \mu S_t\,\mathrm{d}t + \sigma S_t\,\mathrm{d}W_t\). Identify the drift field \(b(s)\), the diffusion coefficient \(\sigma(s)\), and the covariance matrix \(a(s)\). Write down the infinitesimal generator \(\mathcal{L}\) and compute \(\mathcal{L}f\) for \(f(s) = \log s\).

Solution to Exercise 3

For geometric Brownian motion \(\mathrm{d}S_t = \mu S_t\,\mathrm{d}t + \sigma S_t\,\mathrm{d}W_t\), the coefficients are:

• Drift field: \(b(s) = \mu s\)
• Diffusion coefficient: \(\sigma(s) = \sigma s\) (using \(\sigma\) for both the constant and the function, with the understanding that the diffusion coefficient function is \(s \mapsto \sigma s\))
• Covariance matrix: \(a(s) = (\sigma s)^2 = \sigma^2 s^2\)

The semimartingale decomposition is

\[ S_t = S_0 + \underbrace{\int_0^t \mu S_s\,\mathrm{d}s}_{\text{finite variation}} + \underbrace{\int_0^t \sigma S_s\,\mathrm{d}W_s}_{\text{local martingale}} \]

The infinitesimal generator is

\[ \mathcal{L}f(s) = \mu s\,f'(s) + \frac{1}{2}\sigma^2 s^2\,f''(s) \]

For \(f(s) = \log s\): we have \(f'(s) = 1/s\) and \(f''(s) = -1/s^2\), so

\[ \mathcal{L}(\log s) = \mu s \cdot \frac{1}{s} + \frac{1}{2}\sigma^2 s^2 \cdot \left(-\frac{1}{s^2}\right) = \mu - \frac{\sigma^2}{2} \]

This is a constant, confirming that \(\log S_t - (\mu - \sigma^2/2)t\) is a local martingale — consistent with the well-known result \(\log S_t = \log S_0 + (\mu - \sigma^2/2)t + \sigma W_t\).

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Exercise 4. Let \(X_t\) be an Itô diffusion in \(\mathbb{R}^d\) with generator \(\mathcal{L}\). Using Itô's formula, show that for \(f \in C^2(\mathbb{R}^d)\), the process

\[ M_t^f := f(X_t) - f(X_0) - \int_0^t (\mathcal{L}f)(X_s)\,\mathrm{d}s \]

is a local martingale. Under what additional condition on \(f\) and \(\sigma\) does \(M_t^f\) become a true martingale?

Solution to Exercise 4

By Itô's formula applied to \(f \in C^2(\mathbb{R}^d)\) (taking \(f\) time-independent):

\[ f(X_t) = f(X_0) + \int_0^t \frac{\partial f}{\partial x_i}(X_s)\,\mathrm{d}X_s^i + \frac{1}{2}\int_0^t \frac{\partial^2 f}{\partial x_i \partial x_j}(X_s)\,\mathrm{d}\langle X^i, X^j \rangle_s \]

Substituting \(\mathrm{d}X_s^i = b^i(X_s)\,\mathrm{d}s + \sigma^{i\alpha}(X_s)\,\mathrm{d}W_s^\alpha\) and \(\mathrm{d}\langle X^i, X^j\rangle_s = a^{ij}(X_s)\,\mathrm{d}s\):

\[ f(X_t) - f(X_0) = \int_0^t \left(b^i \partial_i f + \frac{1}{2}a^{ij}\partial_i\partial_j f\right)(X_s)\,\mathrm{d}s + \int_0^t \partial_i f(X_s)\,\sigma^{i\alpha}(X_s)\,\mathrm{d}W_s^\alpha \]

The first integral is \(\int_0^t (\mathcal{L}f)(X_s)\,\mathrm{d}s\), so

\[ M_t^f = f(X_t) - f(X_0) - \int_0^t (\mathcal{L}f)(X_s)\,\mathrm{d}s = \int_0^t \partial_i f(X_s)\,\sigma^{i\alpha}(X_s)\,\mathrm{d}W_s^\alpha \]

This is a stochastic integral with respect to Brownian motion, hence a local martingale.

\(M_t^f\) is a true martingale if the integrand is square-integrable:

\[ \mathbb{E}\!\int_0^t \sum_{i,\alpha} |\partial_i f(X_s)|^2\,|\sigma^{i\alpha}(X_s)|^2\,\mathrm{d}s < \infty \]

A sufficient condition is that \(\nabla f\) is bounded (e.g. \(f \in C_c^2(\mathbb{R}^d)\)) and \(\sigma\) satisfies the linear growth condition \(|\sigma(x)| \le C(1 + |x|)\), combined with finite moments \(\mathbb{E}[\sup_{s \le t}|X_s|^2] < \infty\).

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Exercise 5. For the gradient diffusion \(\mathrm{d}X_t = -\nabla V(X_t)\,\mathrm{d}t + \sqrt{2}\,\mathrm{d}W_t\) with \(V(x) = \frac{1}{2}k|x|^2\) in \(\mathbb{R}^d\) (where \(k > 0\)), identify the drift, diffusion matrix, and generator. Verify that \(\pi(x) \propto e^{-V(x)}\) is a Gaussian density and compute its mean and covariance.

Solution to Exercise 5

For the gradient diffusion \(\mathrm{d}X_t = -\nabla V(X_t)\,\mathrm{d}t + \sqrt{2}\,\mathrm{d}W_t\) with \(V(x) = \frac{1}{2}k|x|^2\):

• Drift: \(b^i(x) = -\partial_i V(x) = -kx^i\)
• Diffusion matrix: \(\sigma^{i\alpha} = \sqrt{2}\,\delta^{i\alpha}\), so \(a^{ij} = 2\delta^{ij}\)
• Generator:

\[ \mathcal{L}f(x) = -kx^i\,\partial_i f(x) + \frac{1}{2}\cdot 2\delta^{ij}\,\partial_i\partial_j f(x) = -kx \cdot \nabla f(x) + \Delta f(x) \]

The invariant density is \(\pi(x) \propto e^{-V(x)} = e^{-k|x|^2/2}\). Normalizing:

\[ \pi(x) = \left(\frac{k}{2\pi}\right)^{d/2} \exp\!\left(-\frac{k}{2}|x|^2\right) \]

This is a \(d\)-dimensional Gaussian with:

• Mean: \(\mathbb{E}[X] = 0\)
• Covariance: \(\mathrm{Cov}(X^i, X^j) = \frac{1}{k}\,\delta^{ij}\), i.e. \(\Sigma = \frac{1}{k}I_d\)

This is verified by matching the exponent: \(-\frac{k}{2}|x|^2 = -\frac{1}{2}x^\top(kI)x\), which is the exponent of \(\mathcal{N}(0, k^{-1}I)\).

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Exercise 6. Consider a one-dimensional diffusion with \(b(x) = 0\) and \(\sigma(x) = \sqrt{1 + x^2}\). Compute the covariance function \(a(x)\) and write down the generator \(\mathcal{L}\). Does this diffusion satisfy the standard Lipschitz condition? Does the linear growth condition hold?

Solution to Exercise 6

With \(b(x) = 0\) and \(\sigma(x) = \sqrt{1 + x^2}\), the covariance function is

\[ a(x) = \sigma(x)^2 = 1 + x^2 \]

The generator is

\[ \mathcal{L}f(x) = \frac{1}{2}(1 + x^2)\,f''(x) \]

Lipschitz condition. We need \(|\sigma(x) - \sigma(y)| \le L|x - y|\) for some constant \(L\). Compute

\[ \sigma'(x) = \frac{x}{\sqrt{1 + x^2}} \]

Since \(|\sigma'(x)| = |x|/\sqrt{1+x^2} < 1\) for all \(x\), by the mean value theorem \(|\sigma(x) - \sigma(y)| \le |x - y|\). So the Lipschitz condition holds with \(L = 1\).

Linear growth condition. We need \(|\sigma(x)| \le C(1 + |x|)\) for some constant \(C\). Since \(\sqrt{1 + x^2} \le 1 + |x|\) (squaring both sides: \(1 + x^2 \le 1 + 2|x| + x^2\), which is true), the linear growth condition holds with \(C = 1\).

Therefore, by the standard existence and uniqueness theorem, the SDE \(\mathrm{d}X_t = \sqrt{1+X_t^2}\,\mathrm{d}W_t\) has a unique strong solution.

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Exercise 7. Let \(X_t\) be a \(d\)-dimensional Itô diffusion with constant drift \(b \in \mathbb{R}^d\) and constant diffusion matrix \(\sigma \in \mathbb{R}^{d \times m}\). Show that \(X_t\) is a Gaussian process and compute \(\mathbb{E}[X_t]\) and \(\mathrm{Cov}(X_s, X_t)\) for \(s \le t\) in terms of \(b\), \(a = \sigma\sigma^\top\), and the initial condition \(X_0 = x_0\).

Solution to Exercise 7

With constant \(b\) and \(\sigma\), the SDE \(\mathrm{d}X_t = b\,\mathrm{d}t + \sigma\,\mathrm{d}W_t\) has the explicit solution

\[ X_t = x_0 + bt + \sigma W_t \]

Since \(W_t\) is a Gaussian process (every finite collection \((W_{t_1}, \ldots, W_{t_k})\) is jointly Gaussian), and \(X_t\) is an affine transformation of \(W_t\), \(X_t\) is also a Gaussian process.

Mean:

\[ \mathbb{E}[X_t] = x_0 + bt \]

Covariance: For \(s \le t\), using \(a = \sigma\sigma^\top\):

\[ \mathrm{Cov}(X_s^i, X_t^j) = \mathrm{Cov}\!\left(\sigma^{i\alpha}W_s^\alpha,\, \sigma^{j\beta}W_t^\beta\right) = \sigma^{i\alpha}\sigma^{j\beta}\,\mathrm{Cov}(W_s^\alpha, W_t^\beta) \]

Since \(\mathrm{Cov}(W_s^\alpha, W_t^\beta) = \min(s,t)\,\delta^{\alpha\beta} = s\,\delta^{\alpha\beta}\) for \(s \le t\):

\[ \mathrm{Cov}(X_s^i, X_t^j) = \sigma^{i\alpha}\sigma^{j\alpha}\,s = a^{ij}\,s \]

In matrix form:

\[ \mathrm{Cov}(X_s, X_t) = a\,\min(s, t) = \sigma\sigma^\top \min(s, t) \]