\begin{algorithm}[h]
\caption{Particle Filter}
\label{alg:pf}
\begin{enumerate}
\item{Input: data $y_{1:T}$, level $l\in\mathbb{N}_0$, particle number $N\in\mathbb{N}$ and parameter $\theta\in\Theta$.}
\item{Initialize: For $i\in\{1,\dots,N\}$, independently generate $\overline{W}_{\Delta_l:1}^i$ from $\mathcal{N}(0,\Delta_l)$. Set $t=1$, {$\hat{p}^N(y_{1:0})=1$ for convention} and go to step 3.}
\item{Iterate: For $i\in\{1,\dots,N\}$ compute
$$
u_t^i = \frac{\kappa_{t,l}(\overline{w}_{\Delta_l:t}^i)}{\sum_{j=1}^N\kappa_{t,l}(\overline{w}_{\Delta_l:t}^j)}.
$$
Set $\hat{p}^N(y_{1:t})={\hat{p}^N(y_{1:t-1})}\tfrac{1}{N}\sum_{i=1}^N\kappa_{t,l}(\overline{w}_{\Delta_l:t}^i)$.
Then sample $\overline{w}_{\Delta_l:t}^{1:N}$ with replacement from  $\overline{w}_{\Delta_l:t}^{1:N}$ using probabilities
$u_t^{1:N}$. For $i\in\{1,\dots,N\}$, independently generate $\overline{W}_{t+\Delta_l:t+1}^i$ from $\mathcal{N}(0,\Delta_l)$.
Set $t=t+1$ and if $t=T$ go to step 4, otherwise restart step 3.}
\item{Grand Selection: For $i\in\{1,\dots,N\}$ compute
$
u_T^i = \frac{\kappa_{T,l}(\overline{w}_{\Delta_l:T}^i)}{\sum_{j=1}^N\kappa_{T,l}(\overline{w}_{\Delta_l:T}^j)}.
$
Set $\hat{p}^N(y_{1:T})=\hat{p}^N(y_{1:T-1})\tfrac{1}{N}\sum_{i=1}^N\kappa_{t,l}(\overline{w}_{\Delta_l:T}^i)$.
Sample one $\overline{w}_{\Delta_l:T}$ from $\overline{w}_{\Delta_l:T}^{1:N}$ using $u_T^{1:N}$ and go to step 5.}
\item{Output: trajectory $\overline{w}_{\Delta_l:T}$ and normalizing constant estimate $\hat{p}^N(y_{1:T})$.}
\end{enumerate}
\end{algorithm}

\begin{algorithm}
[h]
\caption{Particle Filter}
\begin{enumerate}
\item{Input: data $y_{1:T}$, level $l\in\mathbb{N}_0$, particle number $N\in\mathbb{N}$ and parameter $\theta\in\Theta$.}
\item{Initialize: For $i\in\{1,\dots,N\}$, independently generate $\overline{W}_{\Delta_l:1}^i$ from $\mathcal{N}(0,\Delta_l)$. Set $t=1$, {$\hat{p}^N(y_{1:0})=1$ for convention} and go to step 3.}
\item{Iterate: For $i\in\{1,\dots,N\}$ compute
$$
u_t^i = \frac{\kappa_{t,l}(\overline{w}_{\Delta_l:t}^i)}{\sum_{j=1}^N\kappa_{t,l}(\overline{w}_{\Delta_l:t}^j)}.
$$
Set $\hat{p}^N(y_{1:t})={\hat{p}^N(y_{1:t-1})}\tfrac{1}{N}\sum_{i=1}^N\kappa_{t,l}(\overline{w}_{\Delta_l:t}^i)$.
Then sample $\overline{w}_{\Delta_l:t}^{1:N}$ with replacement from $\overline{w}_{\Delta_l:t}^{1:N}$ using probabilities
$u_t^{1:N}$. For $i\in\{1,\dots,N\}$, independently generate $\overline{W}_{t+\Delta_l:t+1}^i$ from $\mathcal{N}(0,\Delta_l)$.
Set $t=t+1$ and if $t=T$ go to step 4, otherwise restart step 3.}
\item{Grand Selection: For $i\in\{1,\dots,N\}$ compute
$
u_T^i = \frac{\kappa_{T,l}(\overline{w}_{\Delta_l:T}^i)}{\sum_{j=1}^N\kappa_{T,l}(\overline{w}_{\Delta_l:T}^j)}.
$
Set $\hat{p}^N(y_{1:T})=\hat{p}^N(y_{1:T-1})\tfrac{1}{N}\sum_{i=1}^N\kappa_{t,l}(\overline{w}_{\Delta_l:T}^i)$.
Sample one $\overline{w}_{\Delta_l:T}$ from $\overline{w}_{\Delta_l:T}^{1:N}$ using $u_T^{1:N}$ and go to step 5.}
\item{Output: trajectory $\overline{w}_{\Delta_l:T}$ and normalizing constant estimate $\hat{p}^N(y_{1:T})$.}
\end{enumerate}
\end{algorithm}
{
    "title": "bayesian parameter inference for partially observed stochastic volterra   equations",
    "id": "2310.03114",
    "abstract": "in this article we consider bayesian parameter inference for a type of partially observed stochastic volterra equation (sve). sves are found in many areas such as physics and mathematical finance. in the latter field they can be used to represent long memory in unobserved volatility processes. in many cases of practical interest, sves must be time-discretized and then parameter inference is based upon the posterior associated to this time-discretized process. based upon recent studies on time-discretization of sves (e.g. richard et al. 2021), we use euler-maruyama methods for the afore-mentioned discretization. we then show how multilevel markov chain monte carlo (mcmc) methods (jasra et al. 2018) can be applied in this context. in the examples we study, we give a proof that shows that the cost to achieve a mean square error (mse) of $\\mathcal{o}(\\epsilon^2)$, $\\epsilon>0$, is {$\\mathcal{o}(\\epsilon^{-\\tfrac{4}{2h+1}})$, where $h$ is the hurst parameter. if one uses a single level mcmc method then the cost is $\\mathcal{o}(\\epsilon^{-\\tfrac{2(2h+3)}{2h+1}})$} to achieve the same mse. we illustrate these results in the context of state-space and stochastic volatility models, with the latter applied to real data.",
    "categories": "stat.co stat.me",
    "doi": "",
    "created": "2023-10-04",
    "updated": "2024-02-19",
    "authors": [
        "ajay jasra",
        "hamza ruzayqat",
        "amin wu"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2310.03114"
}
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