Diffusion Process

Definition

A diffusion process is a continuous-time stochastic process satisfying the Itô stochastic differential equation (SDE): $dY_t = \mu(Y_t;\theta)\,dt + \sigma(Y_t;\theta)\,dW_t$ where $\mu:\mathbb{R}^d\times\Theta\to\mathbb{R}^d$ is the drift, $\sigma:\mathbb{R}^d\times\Theta\to\mathbb{R}^{d\times d}$ is the diffusion, and $W_t$ is a $d$ -dimensional Brownian motion. Under Lipschitz and linear-growth conditions a unique strong solution exists (Øksendal 1995). In finance, diffusions model interest rates, asset prices, stochastic volatility, and exchange rates.

Key Ideas

Discretization bias: Discrete observations require approximating the continuous-time transition density. The Euler scheme $\Delta Y_i = \mu(Y_i;\theta)\Delta t + \sigma(Y_i;\theta)\Delta W_i$ with $\Delta W_i \sim \mathcal{N}(0, I_d\Delta t)$ is Gaussian but biased when $\Delta t$ is large; naive maximum likelihood (ML) on discretized data can produce severely biased parameter estimates.
CEV (Constant Elasticity of Variance) model: $dY_t = (\theta_r + \kappa_r Y_t)dt + \sigma Y_t^\beta dW_t$ ; nests Vasicek ( $\beta=0$ ) and Cox-Ingersoll-Ross ( $\beta = \tfrac{1}{2}$ ); $\beta$ controls how diffusion scales with level. Used as a one-factor interest-rate model.
State variables: $Y_t = (X_t, Z_t)$ with $X_t$ observable and $Z_t$ latent (e.g., log-stochastic-volatility). Markov chain Monte Carlo (MCMC) augments $Z_t$ as additional missing data.
Exact transition density: Generally intractable in closed form for nonlinear $\mu$ and $\sigma$ ; Euler approximation with small $\Delta t$ replaces it with a tractable Gaussian.

How It Works

Data Augmentation MCMC (Eraker 2001)

The key insight: between each pair of observed values, introduce $m-1$ auxiliary latent data points at grid spacing $\Delta t = 1/m$ (chosen by the econometrician). The augmented $d\times n$ matrix $\widehat{Y}$ has density:

$\pi(\widehat{Y},\theta) \propto \prod_{i=1}^n p(\widehat{Y}_i \mid \widehat{Y}_{i-1};\theta)\,p(\theta)$

where each factor is Gaussian under the Euler approximation. As $m\to\infty$ the approximation error vanishes. The unobserved columns of $\widehat{Y}$ are integrated out by Gibbs sampling.

Brownian Bridge Conditional (Propositions 1–2):

Conditioning on both neighbors, the conditional density of $\widehat{Y}_i$ is approximately:

$\widehat{Y}_i \mid \widehat{Y}_{i-1}, \widehat{Y}_{i+1}, \theta \approx \mathcal{N}\!\left(\tfrac{1}{2}(\widehat{Y}_{i-1}+\widehat{Y}_{i+1}),\; \tfrac{1}{2}\sigma^2_{i-1}\Delta t\right)$

Proposition 1 (exact for constant $\mu$ , $\sigma$ ): $\widehat{Y}_i | \widehat{Y}_{i-1}, \widehat{Y}_{i+1}, \theta \sim \mathcal{N}(\tfrac{1}{2}(\widehat{Y}_{i-1}+\widehat{Y}_{i+1}), \tfrac{1}{2}\sigma^2\Delta t)$ .

Proposition 2: $\Delta t^{-1/2}[\widehat{Y}_i - \tfrac{1}{2}(\widehat{Y}_{i-1}+\widehat{Y}_{i+1})] \to \mathcal{N}(0, \tfrac{1}{2}\sigma^2_{i-1})$ as $\Delta t\to 0$ .

Hybrid accept-reject Metropolis-Hastings (AR-MH): Proposal $q = \mathcal{N}(m_i, \tfrac{1}{2}[\sigma^2_{i-1}]^2\Delta t)$ with scale constant $C = p(m_i|\cdot)/q(m_i|\cdot)$ ensures supp $(q)\subseteq$ supp $(p)$ . Acceptance rates 0.90–0.98 for one-factor models.

Parameter Block

Given the augmented path $\widehat{Y}$ :

Linear drift parameters (e.g., $\theta_r, \kappa_r$ ): Linear regression on $y_i = (\widehat{Y}_i - \widehat{Y}_{i-1})/(\sigma\widehat{Y}_{i-1}^\beta\sqrt{\Delta t})$ with design $X_i = [\sqrt{\Delta t}\widehat{Y}_{i-1}^{-\beta},\, \sqrt{\Delta t}\widehat{Y}_{i-1}^{1-\beta}]$ → Normal posterior $(\theta_r, \kappa_r)|\widehat{Y} \sim \mathcal{N}(\hat\theta, \sigma^2(X'X)^{-1})$ .
Diffusion variance $\sigma^2$ : Inverse-Gamma posterior $\sigma^{-2}|\widehat{Y} \sim \text{IG}(n-2, \hat{s}^2)$ .
Nonlinear parameters (e.g., $\beta$ ): Log-concave posterior → Metropolis step with second-order Taylor-expanded Normal proposal $\mathcal{N}(\beta^{(h-1)} - f'(\beta^{(h-1)})/f''(\beta^{(h-1)}),\; -(f''(\beta^{(h-1)}))^{-1})$ .

CEV+SV (Stochastic Volatility) Reparameterization (Appendix D)

The log-volatility process $\{Z_t^{(g)}\}$ has high serial dependence; its running mean $\bar{Z}^{(g)}$ is nearly constant across iterations, making the conditional mean of $\kappa_z$ biased downward. Fix: at each Gibbs step subtract $k = \bar{Z}^{(g)}$ and set $Z_t^* \leftarrow Z_t^{(g)} - k$ , $\sigma_r^* \leftarrow \sigma_r^{(g)}\exp(\tfrac{1}{2}k)$ . This restores unbiased $\kappa_z$ conditionals without changing the stationary distribution.

Discretization Bias vs. Convergence Trade-off

$m$	Discretization bias	MCMC convergence	Strategy
1 ( $\Delta t=1$ )	High	Fast	Avoid
4 ( $\Delta t=1/4$ )	Moderate	Good	Starting point
8 ( $\Delta t=1/8$ )	Negligible	Slower	Production
$\infty$	Zero	Fails	Limit

Practical approach: start with large $\Delta t$ , decrease $m$ until posterior means stabilize. For CEV, convergence occurs at $m\approx 2$ – $4$ ; $\beta$ requires higher $m$ due to larger discretization bias.

Why It Matters

Provides a fully Bayesian, exact-in-principle framework for any Itô diffusion model — not restricted to linear drift or constant diffusion, and not requiring auxiliary model specification as in Efficient Method of Moments (EMM).
Handles latent state variables (stochastic volatility, unobserved factors) naturally within the same augmented-data MCMC framework.
Demonstrates that CEV is misspecified for short-term US interest rates: the SV component is necessary to capture both the leptokurtosis and the conditional heteroscedasticity of interest-rate changes.
The Brownian bridge AR-MH proposal becomes exact as $\Delta t \to 0$ — unlike Pedersen (1995) unconditional simulation, which conditions on less information and is less efficient.

Open Questions

As $\Delta t \to 0$ MCMC convergence slows (Dirac-collapsing conditionals); optimal $m$ balances bias and variance.
Column-by-column missing-data sampling is slower than block Independence Metropolis (Elerian-Chib-Shephard 1999); the trade-off between block size and acceptance rate is problem-specific.
Extension to jump-diffusions (compound Poisson jumps in returns or volatility) requires augmenting the jump times and sizes, handled in Eraker-Johannes-Polson (2003).
The reparameterization trick for highly persistent latent factors generalizes beyond the log-volatility case to any model where the running chain mean of a latent state induces bias in a mean-reversion parameter.