Sequential importance resampling for Feynman–Kac path targets

Use the finite-horizon target and proposal setting of Feynman–Kac path targets and normalizers. For admissible inputs require ${\mu }_0\ll {\nu }_0$ and, for each $k\ge 1$, $M_k(x,\cdot )\ll R_k(x,\cdot )$ for almost every $x$ under the last-coordinate marginal of the proposal path law $Q_{k-1}$. Select the joint measurable versions ${\rho }_0=d{\mu }_0\mathrm{/}d{\nu }_0$ and $r_k(x,\cdot )=d{M}_k(x,\cdot )\mathrm{/}d{R}_k(x,\cdot )$ supplied by Feynman–Kac path Radon–Nikodym weight. Fix $N\ge 1$ and select either multinomial or residual resampling at each time. The algorithm returns a weighted empirical path measure and a normalizer estimate.

1. Draw $X_0^1,\dots ,X_0^N$ independently from ${\nu }_0$. Set $a_0^i=G_0(X_0^i){\rho }_0(X_0^i)$ and ${\widehat{Z}}_0=N^{-1}{\sum }_{i=1}^N{a}_0^i$. If ${\sum }_i{a}_0^i>0$, put ${\bar{w}}_0^i=a_0^i\mathrm{/}{\sum }_j{a}_0^j$; otherwise put ${\bar{w}}_0^i=1\mathrm{/}N$. Each initial path is $X_{0:0}^i=X_0^i$.
2. For $k=1,\dots ,t$, do the following:
   1. Apply Multinomial and residual offspring resampling to the previous normalized weights $({\bar{w}}_{k-1}^1,\dots ,{\bar{w}}_{k-1}^N)$, obtaining ancestor indices $A_k^1,\dots ,A_k^N$.
   2. Given the previous paths and these indices, draw $(X_k^1,\dots ,X_k^N)$ with joint conditional law ${⨂}_{i=1}^N{R}_k(X_{k-1}^{A_k^i},\cdot )$ and set $X_{0:k}^i=(X_{0:k-1}^{A_k^i},X_k^i)$.
   3. Set $a_k^i=G_k(X_k^i)r_k(X_{k-1}^{A_k^i},X_k^i)$ and ${\widehat{Z}}_k={\widehat{Z}}_{k-1}(N^{-1}{\sum }_{i=1}^N{a}_k^i)$. If ${\sum }_i{a}_k^i>0$, put ${\bar{w}}_k^i=a_k^i\mathrm{/}{\sum }_j{a}_k^j$; otherwise put ${\bar{w}}_k^i=1\mathrm{/}N$.
3. Return ${\widehat{Z}}_t$ and the weighted empirical path probability measure ${\widehat{\pi }}_t^N={\sum }_{i=1}^N{\bar{w}}_t^i{\delta }_{X_{0:t}^i}$. Its last-coordinate marginal is the corresponding empirical filtering approximation.

The all-zero fallback only defines a probability measure for reporting and future sampling. Once a zero average weight occurs, $\widehat{Z}$ remains zero. The procedure makes no finite-particle accuracy, consistency, or long-time stability claim.

The input convention is that $t\in \{0,\dots ,T\}$ is the requested horizon, ${\nu }_0$ is a proposal probability on ${\mathsf{X}}_0$, $R_k$ are proposal probability kernels for $1\le k\le t$, and $N\in \mathbb{N}$ with $N\ge 1$. Thus the opening support condition ranges over $1\le k\le t$, and a resampling scheme is selected for each such $k$. Choose the finite-valued joint measurable versions ${\rho }_0$ and $r_k$ constructed in the proof of Feynman–Kac path Radon–Nikodym weight.

Resampling can add immediate sampling variance while reducing the chance that low-weight particles are carried forward. A trigger based only on current weight imbalance is a computational rule, not a certificate of target coverage or convergence.