Chaining a conditional likelihood function

Question

Before I proceed, allow me to apologise in advance for potential abuse of notations. Let $\{x_t:\Omega \to \mathbb{R}\}_{t=1,\cdots,n}$ be a stochastic process, defined on a probability space $(\Omega,\mathcal{F},P)$ and $\{x_t,\mathcal{F}_t\}_{t=1,\cdots,n}$ an adapted stochastic sequence, where $\mathcal{F}_t$ is a $\sigma$-field in $\Omega$, such that $\mathcal{F}_s\subseteq \mathcal{F}_t$ for $s<t$ and $\sigma(x_1,\cdots,x_t)\subset \mathcal{F}_t$. Let $S_t$ for $t=1,\cdots,n$ be a binary variable, such that \begin{equation} P[S_t=1\mid \mathcal{F}_{\infty }]=p_t \end{equation} where $\sigma(x_1,\cdots,x_n)\subset\mathcal{F}_\infty$. The likelihood function for $S_1,\cdots,S_n$ conditioned on the entire information set can be expressed as \begin{equation} \mathcal{L}(S_{1},\cdots,S_n\mid\mathcal{F}_{\infty})=\prod_{t=1}^nP(S_t=s_t\mid\mathcal{F}_{\infty}), \end{equation} where $s_t\in\{0,1\}$. However, suppose I change the assumption to, \begin{equation} P[S_t=1\mid \mathcal{F}_{t-1 }]=p_t \end{equation} and I do not wish any longer to condition the probabilities on the entire information set, rather on filtration up to time $t$ at each iteration, such that the likelihood function is instead expressed as (just an instance, as this maybe incorrect): \begin{equation} \prod_{t=1}^nP(S_t=s_t\mid\mathcal{F}_{t-1}), \end{equation} Would such representation of the likelihood function and iterative enlargement of the filtration be permissible? Perhaps using chaining approach? If so, I would be grateful if someone elaborates on this and show how notationally the likelihood function $\mathcal{L}$ would then be denoted