Particle Filter

Each particle represents a hypothesis about the hidden market state. Particles are drawn from a proposal distribution and assigned importance weights proportional to how well they explain the observed data. Weights are normalized to sum to one, creating a discrete approximation of the posterior distribution. This mechanism allows the filter to handle non-Gaussian noise and multimodal state densities that are common during market regime transitions.

To combat particle degeneracy—where most particles carry negligible weight—the SIR algorithm periodically resamples the particle set. Particles with high weights are replicated, while low-weight particles are discarded. This focuses computational resources on the most probable regions of the state space. Common resampling schemes include multinomial, systematic, and residual resampling, each offering different trade-offs between variance and computational cost.

Unlike the Kalman filter, which assumes linear dynamics and Gaussian noise, particle filters make no parametric assumptions. They can estimate latent states in models with:

• Stochastic volatility with fat-tailed distributions
• Regime-switching jump diffusions
• Threshold autoregression with abrupt structural breaks This makes them ideal for inferring hidden market regimes from observed returns that exhibit skewness and excess kurtosis.

The Auxiliary Particle Filter improves sampling efficiency by incorporating the latest observation before propagating particles. It computes first-stage weights based on a point estimate of each particle's predictive likelihood, then resamples before the propagation step. This produces a more evenly weighted particle set and reduces the variance of estimates, particularly valuable when observations are highly informative about the latent state, such as during volatility clustering events.

Particle filters provide an unbiased estimate of the marginal likelihood of the observed data, which is critical for Bayesian model comparison. The log-likelihood can be computed recursively as the sum of log predictive densities, enabling formal comparison between competing regime-switching specifications—such as a two-state versus three-state Markov switching model—using Bayes factors or information criteria.

When the state-space model contains a conditionally linear Gaussian substructure, the Rao-Blackwellized particle filter marginalizes out the linear components analytically using a Kalman filter and only applies particle methods to the nonlinear components. This variance reduction technique dramatically improves estimation accuracy for models like regime-switching stochastic volatility, where volatility is nonlinear but the mean equation is linear conditional on the regime.

The canonical particle filter algorithm that combines importance sampling with a resampling step to prevent weight degeneracy. The algorithm propagates particles through the state transition density, updates weights based on the observation likelihood, and then resamples particles proportionally to their weights. This resampling step eliminates particles with negligible weights and replicates high-weight particles, focusing computational resources on regions of high posterior probability. SIR is particularly effective in regime-switching models where the observation density is highly multimodal.

A Monte Carlo technique for estimating properties of a target distribution by drawing samples from a proposal distribution and weighting them by the ratio of target to proposal densities. In particle filtering, importance sampling enables recursive updating of particle weights as new observations arrive. The choice of proposal distribution critically affects filter efficiency: the optimal proposal minimizes weight variance by conditioning on both the previous state and the current observation, though it is often intractable for nonlinear models.

Methods for redrawing particles from the current weighted empirical distribution to combat sample impoverishment and weight degeneracy. Common approaches include:

• Multinomial resampling: Draws N particles independently from the categorical distribution defined by normalized weights
• Systematic resampling: Uses a single uniform random number with evenly spaced offsets for lower variance
• Residual resampling: Combines deterministic replication of high-weight particles with stochastic resampling of residuals
• Stratified resampling: Divides the unit interval into strata for more uniform particle selection Resampling is typically triggered when the effective sample size falls below a threshold.

A mathematical framework that separates the latent state process from the observation process, forming the generative model that particle filters operate upon. The state equation describes how the hidden regime evolves over time, while the observation equation maps the latent state to measurable market data. In quantitative finance, state-space models capture unobservable quantities like stochastic volatility, market regimes, or time-varying parameters. Particle filters excel when these relationships are nonlinear or the noise distributions are non-Gaussian.

The optimal recursive estimator for linear Gaussian state-space models, providing the analytical baseline against which particle filters are compared. The Kalman filter propagates mean and covariance estimates through linear dynamics and updates them via closed-form Bayesian equations. While computationally efficient, it fails in regime-switching contexts where dynamics are nonlinear or innovations exhibit fat tails. Particle filters generalize the Kalman approach by representing the entire posterior distribution nonparametrically through weighted samples rather than parametric sufficient statistics.

A class of algorithms for sampling from complex probability distributions by constructing a Markov chain whose stationary distribution equals the target posterior. While MCMC methods like Gibbs sampling and Metropolis-Hastings are powerful for batch estimation of regime-switching models, particle filters provide a sequential alternative that updates estimates online as new market data arrives. Hybrid approaches such as particle MCMC combine both paradigms, using particle filters as proposals within MCMC samplers for joint parameter and state estimation.