Particle weight update
w_t^(i) proportional to w_(t-1)^(i) p(y_t | x_t^(i))Particles that better explain the observation receive more posterior weight.
Research & Systems
Particle and Kalman Filtering in Finance
Filters infer hidden states (value, volatility, regimes) from noisy price or order-flow observations.
Mental model
Filtering turns noisy observations into evolving beliefs about hidden financial states.
Kalman filters solve linear-Gaussian state-space models exactly with recursive updates.
Particle filters generalise to non-linear or non-Gaussian cases by propagating weighted particles.
Explore the mechanism
When does a distribution of particles represent a hidden financial state better than a Gaussian mean and covariance?
A latent state (grey) generates noisy observations (red points). The Kalman filter (blue) and a particle filter (green) both try to recover that state in real time. Change noise levels and particle count to see how smoothing and tracking trade off.
Numbers
σ_state ≈ 20.0%, σ_obs ≈ 15.0%
Particles ≈ 40
Mean |error| Kalman ≈ 0.102
Mean |error| particle ≈ 0.140
Interpretation
With low observation noise, both filters can track the grey state closely. As σ_obs rises, the red dots jump more: the Kalman filter smooths these jumps using its model, while the particle filter uses a cloud of scenarios. Increasing the particle count reduces the green error line: the ensemble is a better approximation of the full Bayesian update.
In finance, the latent state might be trend, volatility, or “efficient price”, and observations are microstructure-noisy quotes. Linear–Gaussian structure favours the Kalman side; non-linear or heavy-tailed effects motivate particle approaches, at the cost of computation.
UNDERSTAND
What to notice
In finance, filters track latent volatility, trend, liquidity, or order-flow imbalance in real time.
The quantitative structure
Notation
x_t^(i)- particle i representing the hidden state
w_t^(i)- normalized particle weight
y_t- new observation
Effective sample size
ESS = 1 / sum_i (w_t^(i))^2A small ESS signals weight concentration and the need to consider resampling.
This model assumes
- The transition and observation likelihoods can be simulated or evaluated.
- The particle count is sufficient for the dimension and tail behavior of the posterior.
Where it appears
- estimating stochastic volatility
- tracking nonlinear latent factors
- inferring market regimes from noisy observations
Where the intuition breaks
What this view leaves out
- Particle filters can suffer weight degeneracy and high computational cost, while both particle and Kalman methods remain sensitive to model specification.
Common mistakes
“A particle filter is automatically more accurate than a Kalman filter.”
Not quite.Its flexibility helps with nonlinearity and non-Gaussianity, but finite particles and poor proposals can make it noisier or less stable.
Where this concept sits
Follow the strongest prerequisites, applications, extensions, and contrasts.
Connections for Particle and Kalman Filtering in Finance
Particle and Kalman Filtering in FinanceYOU ARE HERE
ExtendsKalman FilterThe combined filtering view extends linear-Gaussian Kalman filtering toward nonlinear or non-Gaussian state estimation with particle methods.
Extends · Kalman Filter
The combined filtering view extends linear-Gaussian Kalman filtering toward nonlinear or non-Gaussian state estimation with particle methods.
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